types of analogies in critical thinking

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Analogy and Analogical Reasoning

An analogy is a comparison between two objects, or systems of objects, that highlights respects in which they are thought to be similar. Analogical reasoning is any type of thinking that relies upon an analogy. An analogical argument is an explicit representation of a form of analogical reasoning that cites accepted similarities between two systems to support the conclusion that some further similarity exists. In general (but not always), such arguments belong in the category of ampliative reasoning, since their conclusions do not follow with certainty but are only supported with varying degrees of strength. However, the proper characterization of analogical arguments is subject to debate (see §2.2 ).

Analogical reasoning is fundamental to human thought and, arguably, to some nonhuman animals as well. Historically, analogical reasoning has played an important, but sometimes mysterious, role in a wide range of problem-solving contexts. The explicit use of analogical arguments, since antiquity, has been a distinctive feature of scientific, philosophical and legal reasoning. This article focuses primarily on the nature, evaluation and justification of analogical arguments. Related topics include metaphor , models in science , and precedent and analogy in legal reasoning .

1. Introduction: the many roles of analogy

2.1 examples, 2.2 characterization, 2.3 plausibility, 2.4 analogical inference rules, 3.1 commonsense guidelines, 3.2 aristotle’s theory, 3.3 material criteria: hesse’s theory, 3.4 formal criteria: the structure-mapping theory, 3.5 other theories, 3.6 practice-based approaches, 4.1 deductive justification, 4.2 inductive justification, 4.3 a priori justification, 4.4 pragmatic justification, 5.1 analogy and confirmation, 5.2 conceptual change and theory development, online manuscript, related entries.

Analogies are widely recognized as playing an important heuristic role, as aids to discovery. They have been employed, in a wide variety of settings and with considerable success, to generate insight and to formulate possible solutions to problems. According to Joseph Priestley, a pioneer in chemistry and electricity,

analogy is our best guide in all philosophical investigations; and all discoveries, which were not made by mere accident, have been made by the help of it. (1769/1966: 14)

Priestley may be over-stating the case, but there is no doubt that analogies have suggested fruitful lines of inquiry in many fields. Because of their heuristic value, analogies and analogical reasoning have been a particular focus of AI research. Hájek (2018) examines analogy as a heuristic tool in philosophy.

Example 1 . Hydrodynamic analogies exploit mathematical similarities between the equations governing ideal fluid flow and torsional problems. To predict stresses in a planned structure, one can construct a fluid model, i.e., a system of pipes through which water passes (Timoshenko and Goodier 1970). Within the limits of idealization, such analogies allow us to make demonstrative inferences, for example, from a measured quantity in the fluid model to the analogous value in the torsional problem. In practice, there are numerous complications (Sterrett 2006).

At the other extreme, an analogical argument may provide very weak support for its conclusion, establishing no more than minimal plausibility. Consider:

Example 2 . Thomas Reid’s (1785) argument for the existence of life on other planets (Stebbing 1933; Mill 1843/1930; Robinson 1930; Copi 1961). Reid notes a number of similarities between Earth and the other planets in our solar system: all orbit and are illuminated by the sun; several have moons; all revolve on an axis. In consequence, he concludes, it is “not unreasonable to think, that those planets may, like our earth, be the habitation of various orders of living creatures” (1785: 24).

Such modesty is not uncommon. Often the point of an analogical argument is just to persuade people to take an idea seriously. For instance:

Example 3 . Darwin takes himself to be using an analogy between artificial and natural selection to argue for the plausibility of the latter:

Why may I not invent the hypothesis of Natural Selection (which from the analogy of domestic productions, and from what we know of the struggle of existence and of the variability of organic beings, is, in some very slight degree, in itself probable) and try whether this hypothesis of Natural Selection does not explain (as I think it does) a large number of facts…. ( Letter to Henslow , May 1860 in Darwin 1903)

Here it appears, by Darwin’s own admission, that his analogy is employed to show that the hypothesis is probable to some “slight degree” and thus merits further investigation. Some, however, reject this characterization of Darwin’s reasoning (Richards 1997; Gildenhuys 2004).

Sometimes analogical reasoning is the only available form of justification for a hypothesis. The method of ethnographic analogy is used to interpret

the nonobservable behaviour of the ancient inhabitants of an archaeological site (or ancient culture) based on the similarity of their artifacts to those used by living peoples. (Hunter and Whitten 1976: 147)

For example:

Example 4 . Shelley (1999, 2003) describes how ethnographic analogy was used to determine the probable significance of odd markings on the necks of Moche clay pots found in the Peruvian Andes. Contemporary potters in Peru use these marks (called sígnales ) to indicate ownership; the marks enable them to reclaim their work when several potters share a kiln or storage facility. Analogical reasoning may be the only avenue of inference to the past in such cases, though this point is subject to dispute (Gould and Watson 1982; Wylie 1982, 1985). Analogical reasoning may have similar significance for cosmological phenomena that are inaccessible due to limits on observation (Dardashti et al. 2017). See §5.1 for further discussion.

As philosophers and historians such as Kuhn (1996) have repeatedly pointed out, there is not always a clear separation between the two roles that we have identified, discovery and justification. Indeed, the two functions are blended in what we might call the programmatic (or paradigmatic ) role of analogy: over a period of time, an analogy can shape the development of a program of research. For example:

Example 5 . An ‘acoustical analogy’ was employed for many years by certain nineteenth-century physicists investigating spectral lines. Discrete spectra were thought to be

completely analogous to the acoustical situation, with atoms (and/or molecules) serving as oscillators originating or absorbing the vibrations in the manner of resonant tuning forks. (Maier 1981: 51)

Guided by this analogy, physicists looked for groups of spectral lines that exhibited frequency patterns characteristic of a harmonic oscillator. This analogy served not only to underwrite the plausibility of conjectures, but also to guide and limit discovery by pointing scientists in certain directions.

More generally, analogies can play an important programmatic role by guiding conceptual development (see §5.2 ). In some cases, a programmatic analogy culminates in the theoretical unification of two different areas of inquiry.

Example 6 . Descartes’s (1637/1954) correlation between geometry and algebra provided methods for systematically handling geometrical problems that had long been recognized as analogous. A very different relationship between analogy and discovery exists when a programmatic analogy breaks down, as was the ultimate fate of the acoustical analogy. That atomic spectra have an entirely different explanation became clear with the advent of quantum theory. In this case, novel discoveries emerged against background expectations shaped by the guiding analogy. There is a third possibility: an unproductive or misleading programmatic analogy may simply become entrenched and self-perpetuating as it leads us to “construct… data that conform to it” (Stepan 1996: 133). Arguably, the danger of this third possibility provides strong motivation for developing a critical account of analogical reasoning and analogical arguments.

Analogical cognition , which embraces all cognitive processes involved in discovering, constructing and using analogies, is broader than analogical reasoning (Hofstadter 2001; Hofstadter and Sander 2013). Understanding these processes is an important objective of current cognitive science research, and an objective that generates many questions. How do humans identify analogies? Do nonhuman animals use analogies in ways similar to humans? How do analogies and metaphors influence concept formation?

This entry, however, concentrates specifically on analogical arguments. Specifically, it focuses on three central epistemological questions:

• What criteria should we use to evaluate analogical arguments?
• What philosophical justification can be provided for analogical inferences?
• How do analogical arguments fit into a broader inferential context (i.e., how do we combine them with other forms of inference), especially theoretical confirmation?

Following a preliminary discussion of the basic structure of analogical arguments, the entry reviews selected attempts to provide answers to these three questions. To find such answers would constitute an important first step towards understanding the nature of analogical reasoning. To isolate these questions, however, is to make the non-trivial assumption that there can be a theory of analogical arguments —an assumption which, as we shall see, is attacked in different ways by both philosophers and cognitive scientists.

2. Analogical arguments

Analogical arguments vary greatly in subject matter, strength and logical structure. In order to appreciate this variety, it is helpful to increase our stock of examples. First, a geometric example:

Example 7 (Rectangles and boxes). Suppose that you have established that of all rectangles with a fixed perimeter, the square has maximum area. By analogy, you conjecture that of all boxes with a fixed surface area, the cube has maximum volume.

Two examples from the history of science:

Example 8 (Morphine and meperidine). In 1934, the pharmacologist Schaumann was testing synthetic compounds for their anti-spasmodic effect. These drugs had a chemical structure similar to morphine. He observed that one of the compounds— meperidine , also known as Demerol —had a physical effect on mice that was previously observed only with morphine: it induced an S-shaped tail curvature. By analogy, he conjectured that the drug might also share morphine’s narcotic effects. Testing on rats, rabbits, dogs and eventually humans showed that meperidine, like morphine, was an effective pain-killer (Lembeck 1989: 11; Reynolds and Randall 1975: 273).

Example 9 (Priestley on electrostatic force). In 1769, Priestley suggested that the absence of electrical influence inside a hollow charged spherical shell was evidence that charges attract and repel with an inverse square force. He supported his hypothesis by appealing to the analogous situation of zero gravitational force inside a hollow shell of uniform density.

Finally, an example from legal reasoning:

Example 10 (Duty of reasonable care). In a much-cited case ( Donoghue v. Stevenson 1932 AC 562), the United Kingdom House of Lords found the manufacturer of a bottle of ginger beer liable for damages to a consumer who became ill as a result of a dead snail in the bottle. The court argued that the manufacturer had a duty to take “reasonable care” in creating a product that could foreseeably result in harm to the consumer in the absence of such care, and where the consumer had no possibility of intermediate examination. The principle articulated in this famous case was extended, by analogy, to allow recovery for harm against an engineering firm whose negligent repair work caused the collapse of a lift ( Haseldine v. CA Daw & Son Ltd. 1941 2 KB 343). By contrast, the principle was not applicable to a case where a workman was injured by a defective crane, since the workman had opportunity to examine the crane and was even aware of the defects ( Farr v. Butters Brothers & Co. 1932 2 KB 606).

What, if anything, do all of these examples have in common? We begin with a simple, quasi-formal characterization. Similar formulations are found in elementary critical thinking texts (e.g., Copi and Cohen 2005) and in the literature on argumentation theory (e.g., Govier 1999, Guarini 2004, Walton and Hyra 2018). An analogical argument has the following form:

• \(S\) is similar to \(T\) in certain (known) respects.
• \(S\) has some further feature \(Q\).
• Therefore, \(T\) also has the feature \(Q\), or some feature \(Q^*\) similar to \(Q\).

(1) and (2) are premises. (3) is the conclusion of the argument. The argument form is ampliative ; the conclusion is not guaranteed to follow from the premises.

\(S\) and \(T\) are referred to as the source domain and target domain , respectively. A domain is a set of objects, properties, relations and functions, together with a set of accepted statements about those objects, properties, relations and functions. More formally, a domain consists of a set of objects and an interpreted set of statements about them. The statements need not belong to a first-order language, but to keep things simple, any formalizations employed here will be first-order. We use unstarred symbols \((a, P, R, f)\) to refer to items in the source domain and starred symbols \((a^*, P^*, R^*, f^*)\) to refer to corresponding items in the target domain. In Example 9 , the source domain items pertain to gravitation; the target items pertain to electrostatic attraction.

Formally, an analogy between \(S\) and \(T\) is a one-to-one mapping between objects, properties, relations and functions in \(S\) and those in \(T\). Not all of the items in \(S\) and \(T\) need to be placed in correspondence. Commonly, the analogy only identifies correspondences between a select set of items. In practice, we specify an analogy simply by indicating the most significant similarities (and sometimes differences).

We can improve on this preliminary characterization of the argument from analogy by introducing the tabular representation found in Hesse (1966). We place corresponding objects, properties, relations and propositions side-by-side in a table of two columns, one for each domain. For instance, Reid’s argument ( Example 2 ) can be represented as follows (using \(\Rightarrow\) for the analogical inference):

Hesse introduced useful terminology based on this tabular representation. The horizontal relations in an analogy are the relations of similarity (and difference) in the mapping between domains, while the vertical relations are those between the objects, relations and properties within each domain. The correspondence (similarity) between earth’s having a moon and Mars’ having moons is a horizontal relation; the causal relation between having a moon and supporting life is a vertical relation within the source domain (with the possibility of a distinct such relation existing in the target as well).

In an earlier discussion of analogy, Keynes (1921) introduced some terminology that is also helpful.

Positive analogy . Let \(P\) stand for a list of accepted propositions \(P_1 , \ldots ,P_n\) about the source domain \(S\). Suppose that the corresponding propositions \(P^*_1 , \ldots ,P^*_n\), abbreviated as \(P^*\), are all accepted as holding for the target domain \(T\), so that \(P\) and \(P^*\) represent accepted (or known) similarities. Then we refer to \(P\) as the positive analogy .

Negative analogy . Let \(A\) stand for a list of propositions \(A_1 , \ldots ,A_r\) accepted as holding in \(S\), and \(B^*\) for a list \(B_1^*, \ldots ,B_s^*\) of propositions holding in \(T\). Suppose that the analogous propositions \(A^* = A_1^*, \ldots ,A_r^*\) fail to hold in \(T\), and similarly the propositions \(B = B_1 , \ldots ,B_s\) fail to hold in \(S\), so that \(A, {\sim}A^*\) and \({\sim}B, B^*\) represent accepted (or known) differences. Then we refer to \(A\) and \(B\) as the negative analogy .

Neutral analogy . The neutral analogy consists of accepted propositions about \(S\) for which it is not known whether an analogue holds in \(T\).

Finally we have:

Hypothetical analogy . The hypothetical analogy is simply the proposition \(Q\) in the neutral analogy that is the focus of our attention.

These concepts allow us to provide a characterization for an individual analogical argument that is somewhat richer than the original one.

An analogical argument may thus be summarized:

It is plausible that \(Q^*\) holds in the target, because of certain known (or accepted) similarities with the source domain, despite certain known (or accepted) differences.

In order for this characterization to be meaningful, we need to say something about the meaning of ‘plausibly.’ To ensure broad applicability over analogical arguments that vary greatly in strength, we interpret plausibility rather liberally as meaning ‘with some degree of support’. In general, judgments of plausibility are made after a claim has been formulated, but prior to rigorous testing or proof. The next sub-section provides further discussion.

Note that this characterization is incomplete in a number of ways. The manner in which we list similarities and differences, the nature of the correspondences between domains: these things are left unspecified. Nor does this characterization accommodate reasoning with multiple analogies (i.e., multiple source domains), which is ubiquitous in legal reasoning and common elsewhere. To characterize the argument form more fully, however, is not possible without either taking a step towards a substantive theory of analogical reasoning or restricting attention to certain classes of analogical arguments.

Arguments by analogy are extensively discussed within argumentation theory. There is considerable debate about whether they constitute a species of deductive inference (Govier 1999; Waller 2001; Guarini 2004; Kraus 2015). Argumentation theorists also make use of tools such as speech act theory (Bermejo-Luque 2012), argumentation schemes and dialogue types (Macagno et al. 2017; Walton and Hyra 2018) to distinguish different types of analogical argument.

Arguments by analogy are also discussed in the vast literature on scientific models and model-based reasoning, following the lead of Hesse (1966). Bailer-Jones (2002) draws a helpful distinction between analogies and models. While “many models have their roots in an analogy” (2002: 113) and analogy “can act as a catalyst to aid modeling,” Bailer-Jones observes that “the aim of modeling has nothing intrinsically to do with analogy.” In brief, models are tools for prediction and explanation, whereas analogical arguments aim at establishing plausibility. An analogy is evaluated in terms of source-target similarity, while a model is evaluated on how successfully it “provides access to a phenomenon in that it interprets the available empirical data about the phenomenon.” If we broaden our perspective beyond analogical arguments , however, the connection between models and analogies is restored. Nersessian (2009), for instance, stresses the role of analog models in concept-formation and other cognitive processes.

To say that a hypothesis is plausible is to convey that it has epistemic support: we have some reason to believe it, even prior to testing. An assertion of plausibility within the context of an inquiry typically has pragmatic connotations as well: to say that a hypothesis is plausible suggests that we have some reason to investigate it further. For example, a mathematician working on a proof regards a conjecture as plausible if it “has some chances of success” (Polya 1954 (v. 2): 148). On both points, there is ambiguity as to whether an assertion of plausibility is categorical or a matter of degree. These observations point to the existence of two distinct conceptions of plausibility, probabilistic and modal , either of which may reflect the intended conclusion of an analogical argument.

On the probabilistic conception, plausibility is naturally identified with rational credence (rational subjective degree of belief) and is typically represented as a probability. A classic expression may be found in Mill’s analysis of the argument from analogy in A System of Logic :

There can be no doubt that every resemblance [not known to be irrelevant] affords some degree of probability, beyond what would otherwise exist, in favour of the conclusion. (Mill 1843/1930: 333)

In the terminology introduced in §2.2, Mill’s idea is that each element of the positive analogy boosts the probability of the conclusion. Contemporary ‘structure-mapping’ theories ( §3.4 ) employ a restricted version: each structural similarity between two domains contributes to the overall measure of similarity, and hence to the strength of the analogical argument.

On the alternative modal conception, ‘it is plausible that \(p\)’ is not a matter of degree. The meaning, roughly speaking, is that there are sufficient initial grounds for taking \(p\) seriously, i.e., for further investigation (subject to feasibility and interest). Informally: \(p\) passes an initial screening procedure. There is no assertion of degree. Instead, ‘It is plausible that’ may be regarded as an epistemic modal operator that aims to capture a notion, prima facie plausibility, that is somewhat stronger than ordinary epistemic possibility. The intent is to single out \(p\) from an undifferentiated mass of ideas that remain bare epistemic possibilities. To illustrate: in 1769, Priestley’s argument ( Example 9 ), if successful, would establish the prima facie plausibility of an inverse square law for electrostatic attraction. The set of epistemic possibilities—hypotheses about electrostatic attraction compatible with knowledge of the day—was much larger. Individual analogical arguments in mathematics (such as Example 7 ) are almost invariably directed towards prima facie plausibility.

The modal conception figures importantly in some discussions of analogical reasoning. The physicist N. R. Campbell (1957) writes:

But in order that a theory may be valuable it must … display an analogy. The propositions of the hypothesis must be analogous to some known laws…. (1957: 129)

Commenting on the role of analogy in Fourier’s theory of heat conduction, Campbell writes:

Some analogy is essential to it; for it is only this analogy which distinguishes the theory from the multitude of others… which might also be proposed to explain the same laws. (1957: 142)

The interesting notion here is that of a “valuable” theory. We may not agree with Campbell that the existence of analogy is “essential” for a novel theory to be “valuable.” But consider the weaker thesis that an acceptable analogy is sufficient to establish that a theory is “valuable”, or (to qualify still further) that an acceptable analogy provides defeasible grounds for taking the theory seriously. (Possible defeaters might include internal inconsistency, inconsistency with accepted theory, or the existence of a (clearly superior) rival analogical argument.) The point is that Campbell, following the lead of 19 th century philosopher-scientists such as Herschel and Whewell, thinks that analogies can establish this sort of prima facie plausibility. Snyder (2006) provides a detailed discussion of the latter two thinkers and their ideas about the role of analogies in science.

In general, analogical arguments may be directed at establishing either sort of plausibility for their conclusions; they can have a probabilistic use or a modal use. Examples 7 through 9 are best interpreted as supporting modal conclusions. In those arguments, an analogy is used to show that a conjecture is worth taking seriously. To insist on putting the conclusion in probabilistic terms distracts attention from the point of the argument. The conclusion might be modeled (by a Bayesian) as having a certain probability value because it is deemed prima facie plausible, but not vice versa. Example 2 , perhaps, might be regarded as directed primarily towards a probabilistic conclusion.

There should be connections between the two conceptions. Indeed, we might think that the same analogical argument can establish both prima facie plausibility and a degree of probability for a hypothesis. But it is difficult to translate between epistemic modal concepts and probabilities (Cohen 1980; Douven and Williamson 2006; Huber 2009; Spohn 2009, 2012). We cannot simply take the probabilistic notion as the primitive one. It seems wise to keep the two conceptions of plausibility separate.

Schema (4) is a template that represents all analogical arguments, good and bad. It is not an inference rule. Despite the confidence with which particular analogical arguments are advanced, nobody has ever formulated an acceptable rule, or set of rules, for valid analogical inferences. There is not even a plausible candidate. This situation is in marked contrast not only with deductive reasoning, but also with elementary forms of inductive reasoning, such as induction by enumeration.

Of course, it is difficult to show that no successful analogical inference rule will ever be proposed. But consider the following candidate, formulated using the concepts of schema (4) and taking us only a short step beyond that basic characterization.

Rule (5) is modeled on the straight rule for enumerative induction and inspired by Mill’s view of analogical inference, as described in §2.3. We use the generic phrase ‘degree of support’ in place of probability, since other factors besides the analogical argument may influence our probability assignment for \(Q^*\).

It is pretty clear that (5) is a non-starter. The main problem is that the rule justifies too much. The only substantive requirement introduced by (5) is that there be a nonempty positive analogy. Plainly, there are analogical arguments that satisfy this condition but establish no prima facie plausibility and no measure of support for their conclusions.

Here is a simple illustration. Achinstein (1964: 328) observes that there is a formal analogy between swans and line segments if we take the relation ‘has the same color as’ to correspond to ‘is congruent with’. Both relations are reflexive, symmetric, and transitive. Yet it would be absurd to find positive support from this analogy for the idea that we are likely to find congruent lines clustered in groups of two or more, just because swans of the same color are commonly found in groups. The positive analogy is antecedently known to be irrelevant to the hypothetical analogy. In such a case, the analogical inference should be utterly rejected. Yet rule (5) would wrongly assign non-zero degree of support.

To generalize the difficulty: not every similarity increases the probability of the conclusion and not every difference decreases it. Some similarities and differences are known to be (or accepted as being) utterly irrelevant and should have no influence whatsoever on our probability judgments. To be viable, rule (5) would need to be supplemented with considerations of relevance , which depend upon the subject matter, historical context and logical details particular to each analogical argument. To search for a simple rule of analogical inference thus appears futile.

Carnap and his followers (Carnap 1980; Kuipers 1988; Niiniluoto 1988; Maher 2000; Romeijn 2006) have formulated principles of analogy for inductive logic, using Carnapian \(\lambda \gamma\) rules. Generally, this body of work relates to “analogy by similarity”, rather than the type of analogical reasoning discussed here. Romeijn (2006) maintains that there is a relation between Carnap’s concept of analogy and analogical prediction. His approach is a hybrid of Carnap-style inductive rules and a Bayesian model. Such an approach would need to be generalized to handle the kinds of arguments described in §2.1 . It remains unclear that the Carnapian approach can provide a general rule for analogical inference.

Norton (2010, and 2018—see Other Internet Resources) has argued that the project of formalizing inductive reasoning in terms of one or more simple formal schemata is doomed. His criticisms seem especially apt when applied to analogical reasoning. He writes:

If analogical reasoning is required to conform only to a simple formal schema, the restriction is too permissive. Inferences are authorized that clearly should not pass muster… The natural response has been to develop more elaborate formal templates… The familiar difficulty is that these embellished schema never seem to be quite embellished enough; there always seems to be some part of the analysis that must be handled intuitively without guidance from strict formal rules. (2018: 1)

Norton takes the point one step further, in keeping with his “material theory” of inductive inference. He argues that there is no universal logical principle that “powers” analogical inference “by asserting that things that share some properties must share others.” Rather, each analogical inference is warranted by some local constellation of facts about the target system that he terms “the fact of analogy”. These local facts are to be determined and investigated on a case by case basis.

To embrace a purely formal approach to analogy and to abjure formalization entirely are two extremes in a spectrum of strategies. There are intermediate positions. Most recent analyses (both philosophical and computational) have been directed towards elucidating criteria and procedures, rather than formal rules, for reasoning by analogy. So long as these are not intended to provide a universal ‘logic’ of analogy, there is room for such criteria even if one accepts Norton’s basic point. The next section discusses some of these criteria and procedures.

3. Criteria for evaluating analogical arguments

Logicians and philosophers of science have identified ‘textbook-style’ general guidelines for evaluating analogical arguments (Mill 1843/1930; Keynes 1921; Robinson 1930; Stebbing 1933; Copi and Cohen 2005; Moore and Parker 1998; Woods, Irvine, and Walton 2004). Here are some of the most important ones:

These principles can be helpful, but are frequently too vague to provide much insight. How do we count similarities and differences in applying (G1) and (G2)? Why are the structural and causal analogies mentioned in (G5) and (G6) especially important, and which structural and causal features merit attention? More generally, in connection with the all-important (G7): how do we determine which similarities and differences are relevant to the conclusion? Furthermore, what are we to say about similarities and differences that have been omitted from an analogical argument but might still be relevant?

An additional problem is that the criteria can pull in different directions. To illustrate, consider Reid’s argument for life on other planets ( Example 2 ). Stebbing (1933) finds Reid’s argument “suggestive” and “not unplausible” because the conclusion is weak (G4), while Mill (1843/1930) appears to reject the argument on account of our vast ignorance of properties that might be relevant (G3).

There is a further problem that relates to the distinction just made (in §2.3 ) between two kinds of plausibility. Each of the above criteria apart from (G7) is expressed in terms of the strength of the argument, i.e., the degree of support for the conclusion. The criteria thus appear to presuppose the probabilistic interpretation of plausibility. The problem is that a great many analogical arguments aim to establish prima facie plausibility rather than any degree of probability. Most of the guidelines are not directly applicable to such arguments.

Aristotle sets the stage for all later theories of analogical reasoning. In his theoretical reflections on analogy and in his most judicious examples, we find a sober account that lays the foundation both for the commonsense guidelines noted above and for more sophisticated analyses.

Although Aristotle employs the term analogy ( analogia ) and discusses analogical predication , he never talks about analogical reasoning or analogical arguments per se . He does, however, identify two argument forms, the argument from example ( paradeigma ) and the argument from likeness ( homoiotes ), both closely related to what would we now recognize as an analogical argument.

The argument from example ( paradeigma ) is described in the Rhetoric and the Prior Analytics :

Enthymemes based upon example are those which proceed from one or more similar cases, arrive at a general proposition, and then argue deductively to a particular inference. ( Rhetoric 1402b15) Let \(A\) be evil, \(B\) making war against neighbours, \(C\) Athenians against Thebans, \(D\) Thebans against Phocians. If then we wish to prove that to fight with the Thebans is an evil, we must assume that to fight against neighbours is an evil. Conviction of this is obtained from similar cases, e.g., that the war against the Phocians was an evil to the Thebans. Since then to fight against neighbours is an evil, and to fight against the Thebans is to fight against neighbours, it is clear that to fight against the Thebans is an evil. ( Pr. An. 69a1)

Aristotle notes two differences between this argument form and induction (69a15ff.): it “does not draw its proof from all the particular cases” (i.e., it is not a “complete” induction), and it requires an additional (deductively valid) syllogism as the final step. The argument from example thus amounts to single-case induction followed by deductive inference. It has the following structure (using \(\supset\) for the conditional):

In the terminology of §2.2, \(P\) is the positive analogy and \(Q\) is the hypothetical analogy. In Aristotle’s example, \(S\) (the source) is war between Phocians and Thebans, \(T\) (the target) is war between Athenians and Thebans, \(P\) is war between neighbours, and \(Q\) is evil. The first inference (dashed arrow) is inductive; the second and third (solid arrows) are deductively valid.

The paradeigma has an interesting feature: it is amenable to an alternative analysis as a purely deductive argument form. Let us concentrate on Aristotle’s assertion, “we must assume that to fight against neighbours is an evil,” represented as \(\forall x(P(x) \supset Q(x))\). Instead of regarding this intermediate step as something reached by induction from a single case, we might instead regard it as a hidden presupposition. This transforms the paradeigma into a syllogistic argument with a missing or enthymematic premise, and our attention shifts to possible means for establishing that premise (with single-case induction as one such means). Construed in this way, Aristotle’s paradeigma argument foreshadows deductive analyses of analogical reasoning (see §4.1 ).

The argument from likeness ( homoiotes ) seems to be closer than the paradeigma to our contemporary understanding of analogical arguments. This argument form receives considerable attention in Topics I, 17 and 18 and again in VIII, 1. The most important passage is the following.

Try to secure admissions by means of likeness; for such admissions are plausible, and the universal involved is less patent; e.g. that as knowledge and ignorance of contraries is the same, so too perception of contraries is the same; or vice versa, that since the perception is the same, so is the knowledge also. This argument resembles induction, but is not the same thing; for in induction it is the universal whose admission is secured from the particulars, whereas in arguments from likeness, what is secured is not the universal under which all the like cases fall. ( Topics 156b10–17)

This passage occurs in a work that offers advice for framing dialectical arguments when confronting a somewhat skeptical interlocutor. In such situations, it is best not to make one’s argument depend upon securing agreement about any universal proposition. The argument from likeness is thus clearly distinct from the paradeigma , where the universal proposition plays an essential role as an intermediate step in the argument. The argument from likeness, though logically less straightforward than the paradeigma , is exactly the sort of analogical reasoning we want when we are unsure about underlying generalizations.

In Topics I 17, Aristotle states that any shared attribute contributes some degree of likeness. It is natural to ask when the degree of likeness between two things is sufficiently great to warrant inferring a further likeness. In other words, when does the argument from likeness succeed? Aristotle does not answer explicitly, but a clue is provided by the way he justifies particular arguments from likeness. As Lloyd (1966) has observed, Aristotle typically justifies such arguments by articulating a (sometimes vague) causal principle which governs the two phenomena being compared. For example, Aristotle explains the saltiness of the sea, by analogy with the saltiness of sweat, as a kind of residual earthy stuff exuded in natural processes such as heating. The common principle is this:

Everything that grows and is naturally generated always leaves a residue, like that of things burnt, consisting in this sort of earth. ( Mete 358a17)

From this method of justification, we might conjecture that Aristotle believes that the important similarities are those that enter into such general causal principles.

Summarizing, Aristotle’s theory provides us with four important and influential criteria for the evaluation of analogical arguments:

• The strength of an analogy depends upon the number of similarities.
• Similarity reduces to identical properties and relations.
• Good analogies derive from underlying common causes or general laws.
• A good analogical argument need not pre-suppose acquaintance with the underlying universal (generalization).

These four principles form the core of a common-sense model for evaluating analogical arguments (which is not to say that they are correct; indeed, the first three will shortly be called into question). The first, as we have seen, appears regularly in textbook discussions of analogy. The second is largely taken for granted, with important exceptions in computational models of analogy ( §3.4 ). Versions of the third are found in most sophisticated theories. The final point, which distinguishes the argument from likeness and the argument from example, is endorsed in many discussions of analogy (e.g., Quine and Ullian 1970).

A slight generalization of Aristotle’s first principle helps to prepare the way for discussion of later developments. As that principle suggests, Aristotle, in common with just about everyone else who has written about analogical reasoning, organizes his analysis of the argument form around overall similarity. In the terminology of section 2.2, horizontal relationships drive the reasoning: the greater the overall similarity of the two domains, the stronger the analogical argument . Hume makes the same point, though stated negatively, in his Dialogues Concerning Natural Religion :

Wherever you depart, in the least, from the similarity of the cases, you diminish proportionably the evidence; and may at last bring it to a very weak analogy, which is confessedly liable to error and uncertainty. (1779/1947: 144)

Most theories of analogy agree with Aristotle and Hume on this general point. Disagreement relates to the appropriate way of measuring overall similarity. Some theories assign greatest weight to material analogy , which refers to shared, and typically observable, features. Others give prominence to formal analogy , emphasizing high-level structural correspondence. The next two sub-sections discuss representative accounts that illustrate these two approaches.

Hesse (1966) offers a sharpened version of Aristotle’s theory, specifically focused on analogical arguments in the sciences. She formulates three requirements that an analogical argument must satisfy in order to be acceptable:

• Requirement of material analogy . The horizontal relations must include similarities between observable properties.
• Causal condition . The vertical relations must be causal relations “in some acceptable scientific sense” (1966: 87).
• No-essential-difference condition . The essential properties and causal relations of the source domain must not have been shown to be part of the negative analogy.

3.3.1 Requirement of material analogy

For Hesse, an acceptable analogical argument must include “observable similarities” between domains, which she refers to as material analogy . Material analogy is contrasted with formal analogy . Two domains are formally analogous if both are “interpretations of the same formal theory” (1966: 68). Nomic isomorphism (Hempel 1965) is a special case in which the physical laws governing two systems have identical mathematical form. Heat and fluid flow exhibit nomic isomorphism. A second example is the analogy between the flow of electric current in a wire and fluid in a pipe. Ohm’s law

states that voltage difference along a wire equals current times a constant resistance. This has the same mathematical form as Poiseuille’s law (for ideal fluids):

which states that the pressure difference along a pipe equals the volumetric flow rate times a constant. Both of these systems can be represented by a common equation. While formal analogy is linked to common mathematical structure, it should not be limited to nomic isomorphism (Bartha 2010: 209). The idea of formal analogy generalizes to cases where there is a common mathematical structure between models for two systems. Bartha offers an even more liberal definition (2010: 195): “Two features are formally similar if they occupy corresponding positions in formally analogous theories. For example, pitch in the theory of sound corresponds to color in the theory of light.”

By contrast, material analogy consists of what Hesse calls “observable” or “pre-theoretic” similarities. These are horizontal relationships of similarity between properties of objects in the source and the target. Similarities between echoes (sound) and reflection (light), for instance, were recognized long before we had any detailed theories about these phenomena. Hesse (1966, 1988) regards such similarities as metaphorical relationships between the two domains and labels them “pre-theoretic” because they draw on personal and cultural experience. We have both material and formal analogies between sound and light, and it is significant for Hesse that the former are independent of the latter.

There are good reasons not to accept Hesse’s requirement of material analogy, construed in this narrow way. First, it is apparent that formal analogies are the starting point in many important inferences. That is certainly the case in mathematics, a field in which material analogy, in Hesse’s sense, plays no role at all. Analogical arguments based on formal analogy have also been extremely influential in physics (Steiner 1989, 1998).

In Norton’s broad sense, however, ‘material analogy’ simply refers to similarities rooted in factual knowledge of the source and target domains. With reference to this broader meaning, Hesse proposes two additional material criteria.

3.3.2 Causal condition

Hesse requires that the hypothetical analogy, the feature transferred to the target domain, be causally related to the positive analogy. In her words, the essential requirement for a good argument from analogy is “a tendency to co-occurrence”, i.e., a causal relationship. She states the requirement as follows:

The vertical relations in the model [source] are causal relations in some acceptable scientific sense, where there are no compelling a priori reasons for denying that causal relations of the same kind may hold between terms of the explanandum [target]. (1966: 87)

The causal condition rules out analogical arguments where there is no causal knowledge of the source domain. It derives support from the observation that many analogies do appear to involve a transfer of causal knowledge.

The causal condition is on the right track, but is arguably too restrictive. For example, it rules out analogical arguments in mathematics. Even if we limit attention to the empirical sciences, persuasive analogical arguments may be founded upon strong statistical correlation in the absence of any known causal connection. Consider ( Example 11 ) Benjamin Franklin’s prediction, in 1749, that pointed metal rods would attract lightning, by analogy with the way they attracted the “electrical fluid” in the laboratory:

Electrical fluid agrees with lightning in these particulars: 1. Giving light. 2. Colour of the light. 3. Crooked direction. 4. Swift motion. 5. Being conducted by metals. 6. Crack or noise in exploding. 7. Subsisting in water or ice. 8. Rending bodies it passes through. 9. Destroying animals. 10. Melting metals. 11. Firing inflammable substances. 12. Sulphureous smell.—The electrical fluid is attracted by points.—We do not know whether this property is in lightning.—But since they agree in all the particulars wherein we can already compare them, is it not probable they agree likewise in this? Let the experiment be made. ( Benjamin Franklin’s Experiments , 334)

Franklin’s hypothesis was based on a long list of properties common to the target (lightning) and source (electrical fluid in the laboratory). There was no known causal connection between the twelve “particulars” and the thirteenth property, but there was a strong correlation. Analogical arguments may be plausible even where there are no known causal relations.

3.3.3 No-essential-difference condition

Hesse’s final requirement is that the “essential properties and causal relations of the [source] have not been shown to be part of the negative analogy” (1966: 91). Hesse does not provide a definition of “essential,” but suggests that a property or relation is essential if it is “causally closely related to the known positive analogy.” For instance, an analogy with fluid flow was extremely influential in developing the theory of heat conduction. Once it was discovered that heat was not conserved, however, the analogy became unacceptable (according to Hesse) because conservation was so central to the theory of fluid flow.

This requirement, though once again on the right track, seems too restrictive. It can lead to the rejection of a good analogical argument. Consider the analogy between a two-dimensional rectangle and a three-dimensional box ( Example 7 ). Broadening Hesse’s notion, it seems that there are many ‘essential’ differences between rectangles and boxes. This does not mean that we should reject every analogy between rectangles and boxes out of hand. The problem derives from the fact that Hesse’s condition is applied to the analogy relation independently of the use to which that relation is put. What counts as essential should vary with the analogical argument. Absent an inferential context, it is impossible to evaluate the importance or ‘essentiality’ of similarities and differences.

Despite these weaknesses, Hesse’s ‘material’ criteria constitute a significant advance in our understanding of analogical reasoning. The causal condition and the no-essential-difference condition incorporate local factors, as urged by Norton, into the assessment of analogical arguments. These conditions, singly or taken together, imply that an analogical argument can fail to generate any support for its conclusion, even when there is a non-empty positive analogy. Hesse offers no theory about the ‘degree’ of analogical support. That makes her account one of the few that is oriented towards the modal, rather than probabilistic, use of analogical arguments ( §2.3 ).

Many people take the concept of model-theoretic isomorphism to set the standard for thinking about similarity and its role in analogical reasoning. They propose formal criteria for evaluating analogies, based on overall structural or syntactical similarity. Let us refer to theories oriented around such criteria as structuralist .

A number of leading computational models of analogy are structuralist. They are implemented in computer programs that begin with (or sometimes build) representations of the source and target domains, and then construct possible analogy mappings. Analogical inferences emerge as a consequence of identifying the ‘best mapping.’ In terms of criteria for analogical reasoning, there are two main ideas. First, the goodness of an analogical argument is based on the goodness of the associated analogy mapping . Second, the goodness of the analogy mapping is given by a metric that indicates how closely it approximates isomorphism.

The most influential structuralist theory has been Gentner’s structure-mapping theory, implemented in a program called the structure-mapping engine (SME). In its original form (Gentner 1983), the theory assesses analogies on purely structural grounds. Gentner asserts:

Analogies are about relations, rather than simple features. No matter what kind of knowledge (causal models, plans, stories, etc.), it is the structural properties (i.e., the interrelationships between the facts) that determine the content of an analogy. (Falkenhainer, Forbus, and Gentner 1989/90: 3)

In order to clarify this thesis, Gentner introduces a distinction between properties , or monadic predicates, and relations , which have multiple arguments. She further distinguishes among different orders of relations and functions, defined inductively (in terms of the order of the relata or arguments). The best mapping is determined by systematicity : the extent to which it places higher-order relations, and items that are nested in higher-order relations, in correspondence. Gentner’s Systematicity Principle states:

A predicate that belongs to a mappable system of mutually interconnecting relationships is more likely to be imported into the target than is an isolated predicate. (1983: 163)

A systematic analogy (one that places high-order relations and their components in correspondence) is better than a less systematic analogy. Hence, an analogical inference has a degree of plausibility that increases monotonically with the degree of systematicity of the associated analogy mapping. Gentner’s fundamental criterion for evaluating candidate analogies (and analogical inferences) thus depends solely upon the syntax of the given representations and not at all upon their content.

Later versions of the structure-mapping theory incorporate refinements (Forbus, Ferguson, and Gentner 1994; Forbus 2001; Forbus et al. 2007; Forbus et al. 2008; Forbus et al 2017). For example, the earliest version of the theory is vulnerable to worries about hand-coded representations of source and target domains. Gentner and her colleagues have attempted to solve this problem in later work that generates LISP representations from natural language text (see Turney 2008 for a different approach).

The most important challenges for the structure-mapping approach relate to the Systematicity Principle itself. Does the value of an analogy derive entirely, or even chiefly, from systematicity? There appear to be two main difficulties with this view. First: it is not always appropriate to give priority to systematic, high-level relational matches. Material criteria, and notably what Gentner refers to as “superficial feature matches,” can be extremely important in some types of analogical reasoning, such as ethnographic analogies which are based, to a considerable degree, on surface resemblances between artifacts. Second and more significantly: systematicity seems to be at best a fallible marker for good analogies rather than the essence of good analogical reasoning.

Greater systematicity is neither necessary nor sufficient for a more plausible analogical inference. It is obvious that increased systematicity is not sufficient for increased plausibility. An implausible analogy can be represented in a form that exhibits a high degree of structural parallelism. High-order relations can come cheap, as we saw with Achinstein’s “swan” example ( §2.4 ).

More pointedly, increased systematicity is not necessary for greater plausibility. Indeed, in causal analogies, it may even weaken the inference. That is because systematicity takes no account of the type of causal relevance, positive or negative. (McKay 1993) notes that microbes have been found in frozen lakes in Antarctica; by analogy, simple life forms might exist on Mars. Freezing temperatures are preventive or counteracting causes; they are negatively relevant to the existence of life. The climate of Mars was probably more favorable to life 3.5 billion years ago than it is today, because temperatures were warmer. Yet the analogy between Antarctica and present-day Mars is more systematic than the analogy between Antarctica and ancient Mars. According to the Systematicity Principle , the analogy with Antarctica provides stronger support for life on Mars today than it does for life on ancient Mars.

The point of this example is that increased systematicity does not always increase plausibility, and reduced systematicity does not always decrease it (see Lee and Holyoak 2008). The more general point is that systematicity can be misleading, unless we take into account the nature of the relationships between various factors and the hypothetical analogy. Systematicity does not magically produce or explain the plausibility of an analogical argument. When we reason by analogy, we must determine which features of both domains are relevant and how they relate to the analogical conclusion. There is no short-cut via syntax.

Schlimm (2008) offers an entirely different critique of the structure-mapping theory from the perspective of analogical reasoning in mathematics—a domain where one might expect a formal approach such as structure mapping to perform well. Schlimm introduces a simple distinction: a domain is object-rich if the number of objects is greater than the number of relations (and properties), and relation-rich otherwise. Proponents of the structure-mapping theory typically focus on relation-rich examples (such as the analogy between the solar system and the atom). By contrast, analogies in mathematics typically involve domains with an enormous number of objects (like the real numbers), but relatively few relations and functions (addition, multiplication, less-than).

Schlimm provides an example of an analogical reasoning problem in group theory that involves a single relation in each domain. In this case, attaining maximal systematicity is trivial. The difficulty is that, compatible with maximal systematicity, there are different ways in which the objects might be placed in correspondence. The structure-mapping theory appears to yield the wrong inference. We might put the general point as follows: in object-rich domains, systematicity ceases to be a reliable guide to plausible analogical inference.

3.5.1 Connectionist models

During the past thirty-five years, cognitive scientists have conducted extensive research on analogy. Gentner’s SME is just one of many computational theories, implemented in programs that construct and use analogies. Three helpful anthologies that span this period are Helman 1988; Gentner, Holyoak, and Kokinov 2001; and Kokinov, Holyoak, and Gentner 2009.

One predominant objective of this research has been to model the cognitive processes involved in using analogies. Early models tended to be oriented towards “understanding the basic constraints that govern human analogical thinking” (Hummel and Holyoak 1997: 458). Recent connectionist models have been directed towards uncovering the psychological mechanisms that come into play when we use analogies: retrieval of a relevant source domain, analogical mapping across domains, and transfer of information and learning of new categories or schemas.

In some cases, such as the structure-mapping theory (§3.4), this research overlaps directly with the normative questions that are the focus of this entry; indeed, Gentner’s Systematicity Principle may be interpreted normatively. In other cases, we might view the projects as displacing those traditional normative questions with up-to-date, computational forms of naturalized epistemology . Two approaches are singled out here because both raise important challenges to the very idea of finding sharp answers to those questions, and both suggest that connectionist models offer a more fruitful approach to understanding analogical reasoning.

The first is the constraint-satisfaction model (also known as the multiconstraint theory ), developed by Holyoak and Thagard (1989, 1995). Like Gentner, Holyoak and Thagard regard the heart of analogical reasoning as analogy mapping , and they stress the importance of systematicity, which they refer to as a structural constraint. Unlike Gentner, they acknowledge two additional types of constraints. Pragmatic constraints take into account the goals and purposes of the agent, recognizing that “the purpose will guide selection” of relevant similarities. Semantic constraints represent estimates of the degree to which people regard source and target items as being alike, rather like Hesse’s “pre-theoretic” similarities.

The novelty of the multiconstraint theory is that these structural , semantic and pragmatic constraints are implemented not as rigid rules, but rather as ‘pressures’ supporting or inhibiting potential pairwise correspondences. The theory is implemented in a connectionist program called ACME (Analogical Constraint Mapping Engine), which assigns an initial activation value to each possible pairing between elements in the source and target domains (based on semantic and pragmatic constraints), and then runs through cycles that update the activation values based on overall coherence (structural constraints). The best global analogy mapping emerges under the pressure of these constraints. Subsequent connectionist models, such as Hummel and Holyoak’s LISA program (1997, 2003), have made significant advances and hold promise for offering a more complete theory of analogical reasoning.

The second example is Hofstadter and Mitchell’s Copycat program (Hofstadter 1995; Mitchell 1993). The program is “designed to discover insightful analogies, and to do so in a psychologically realistic way” (Hofstadter 1995: 205). Copycat operates in the domain of letter-strings. The program handles the following type of problem:

Suppose the letter-string abc were changed to abd ; how would you change the letter-string ijk in “the same way”?

Most people would answer ijl , since it is natural to think that abc was changed to abd by the “transformation rule”: replace the rightmost letter with its successor. Alternative answers are possible, but do not agree with most people’s sense of what counts as the natural analogy.

Hofstadter and Mitchell believe that analogy-making is in large part about the perception of novel patterns, and that such perception requires concepts with “fluid” boundaries. Genuine analogy-making involves “slippage” of concepts. The Copycat program combines a set of core concepts pertaining to letter-sequences ( successor , leftmost and so forth) with probabilistic “halos” that link distinct concepts dynamically. Orderly structures emerge out of random low-level processes and the program produces plausible solutions. Copycat thus shows that analogy-making can be modeled as a process akin to perception, even if the program employs mechanisms distinct from those in human perception.

The multiconstraint theory and Copycat share the idea that analogical cognition involves cognitive processes that operate below the level of abstract reasoning. Both computational models—to the extent that they are capable of performing successful analogical reasoning—challenge the idea that a successful model of analogical reasoning must take the form of a set of quasi-logical criteria. Efforts to develop a quasi-logical theory of analogical reasoning, it might be argued, have failed. In place of faulty inference schemes such as those described earlier ( §2.2 , §2.4 ), computational models substitute procedures that can be judged on their performance rather than on traditional philosophical standards.

In response to this argument, we should recognize the value of the connectionist models while acknowledging that we still need a theory that offers normative principles for evaluating analogical arguments. In the first place, even if the construction and recognition of analogies are largely a matter of perception, this does not eliminate the need for subsequent critical evaluation of analogical inferences. Second and more importantly, we need to look not just at the construction of analogy mappings but at the ways in which individual analogical arguments are debated in fields such as mathematics, physics, philosophy and the law. These high-level debates require reasoning that bears little resemblance to the computational processes of ACME or Copycat. (Ashley’s HYPO (Ashley 1990) is one example of a non-connectionist program that focuses on this aspect of analogical reasoning.) There is, accordingly, room for both computational and traditional philosophical models of analogical reasoning.

3.5.2 Articulation model

Most prominent theories of analogy, philosophical and computational, are based on overall similarity between source and target domains—defined in terms of some favoured subset of Hesse’s horizontal relations (see §2.2 ). Aristotle and Mill, whose approach is echoed in textbook discussions, suggest counting similarities. Hesse’s theory ( §3.3 ) favours “pre-theoretic” correspondences. The structure-mapping theory and its successors ( §3.4 ) look to systematicity, i.e., to correspondences involving complex, high-level networks of relations. In each of these approaches, the problem is twofold: overall similarity is not a reliable guide to plausibility, and it fails to explain the plausibility of any analogical argument.

Bartha’s articulation model (2010) proposes a different approach, beginning not with horizontal relations, but rather with a classification of analogical arguments on the basis of the vertical relations within each domain. The fundamental idea is that a good analogical argument must satisfy two conditions:

Prior Association . There must be a clear connection, in the source domain, between the known similarities (the positive analogy) and the further similarity that is projected to hold in the target domain (the hypothetical analogy). This relationship determines which features of the source are critical to the analogical inference.

Potential for Generalization . There must be reason to think that the same kind of connection could obtain in the target domain. More pointedly: there must be no critical disanalogy between the domains.

The first order of business is to make the prior association explicit. The standards of explicitness vary depending on the nature of this association (causal relation, mathematical proof, functional relationship, and so forth). The two general principles are fleshed out via a set of subordinate models that allow us to identify critical features and hence critical disanalogies.

To see how this works, consider Example 7 (Rectangles and boxes). In this analogical argument, the source domain is two-dimensional geometry: we know that of all rectangles with a fixed perimeter, the square has maximum area. The target domain is three-dimensional geometry: by analogy, we conjecture that of all boxes with a fixed surface area, the cube has maximum volume. This argument should be evaluated not by counting similarities, looking to pre-theoretic resemblances between rectangles and boxes, or constructing connectionist representations of the domains and computing a systematicity score for possible mappings. Instead, we should begin with a precise articulation of the prior association in the source domain, which amounts to a specific proof for the result about rectangles. We should then identify, relative to that proof, the critical features of the source domain: namely, the concepts and assumptions used in the proof. Finally, we should assess the potential for generalization: whether, in the three-dimensional setting, those critical features are known to lack analogues in the target domain. The articulation model is meant to reflect the conversations that can and do take place between an advocate and a critic of an analogical argument.

3.6.1 Norton’s material theory of analogy

As noted in §2.4 , Norton rejects analogical inference rules. But even if we agree with Norton on this point, we might still be interested in having an account that gives us guidelines for evaluating analogical arguments. How does Norton’s approach fare on this score?

According to Norton, each analogical argument is warranted by local facts that must be investigated and justified empirically. First, there is “the fact of the analogy”: in practice, a low-level uniformity that embraces both the source and target systems. Second, there are additional factual properties of the target system which, when taken together with the uniformity, warrant the analogical inference. Consider Galileo’s famous inference ( Example 12 ) that there are mountains on the moon (Galileo 1610). Through his newly invented telescope, Galileo observed points of light on the moon ahead of the advancing edge of sunlight. Noting that the same thing happens on earth when sunlight strikes the mountains, he concluded that there must be mountains on the moon and even provided a reasonable estimate of their height. In this example, Norton tells us, the the fact of the analogy is that shadows and other optical phenomena are generated in the same way on the earth and on the moon; the additional fact about the target is the existence of points of light ahead of the advancing edge of sunlight on the moon.

What are the implications of Norton’s material theory when it comes to evaluating analogical arguments? The fact of the analogy is a local uniformity that powers the inference. Norton’s theory works well when such a uniformity is patent or naturally inferred. It doesn’t work well when the uniformity is itself the target (rather than the driver ) of the inference. That happens with explanatory analogies such as Example 5 (the Acoustical Analogy ), and mathematical analogies such as Example 7 ( Rectangles and Boxes ). Similarly, the theory doesn’t work well when the underlying uniformity is unclear, as in Example 2 ( Life on other Planets ), Example 4 ( Clay Pots ), and many other cases. In short, if Norton’s theory is accepted, then for most analogical arguments there are no useful evaluation criteria.

3.6.2 Field-specific criteria

For those who sympathize with Norton’s skepticism about universal inductive schemes and theories of analogical reasoning, yet recognize that his approach may be too local, an appealing strategy is to move up one level. We can aim for field-specific “working logics” (Toulmin 1958; Wylie and Chapman 2016; Reiss 2015). This approach has been adopted by philosophers of archaeology, evolutionary biology and other historical sciences (Wylie and Chapman 2016; Currie 2013; Currie 2016; Currie 2018). In place of schemas, we find ‘toolkits’, i.e., lists of criteria for evaluating analogical reasoning.

For example, Currie (2016) explores in detail the use of ethnographic analogy ( Example 13 ) between shamanistic motifs used by the contemporary San people and similar motifs in ancient rock art, found both among ancestors of the San (direct historical analogy) and in European rock art (indirect historical analogy). Analogical arguments support the hypothesis that in each of these cultures, rock art symbolizes hallucinogenic experiences. Currie examines criteria that focus on assumptions about stability of cultural traits and environment-culture relationships. Currie (2016, 2018) and Wylie (Wylie and Chapman 2016) also stress the importance of robustness reasoning that combines analogical arguments of moderate strength with other forms of evidence to yield strong conclusions.

Practice-based approaches can thus yield specific guidelines unlikely to be matched by any general theory of analogical reasoning. One caveat is worth mentioning. Field-specific criteria for ethnographic analogy are elicited against a background of decades of methodological controversy (Wylie and Chapman 2016). Critics and defenders of ethnographic analogy have appealed to general models of scientific method (e.g., hypothetico-deductive method or Bayesian confirmation). To advance the methodological debate, practice-based approaches must either make connections to these general models or explain why the lack of any such connection is unproblematic.

3.6.3 Formal analogies in physics

Close attention to analogical arguments in practice can also provide valuable challenges to general ideas about analogical inference. In an interesting discussion, Steiner (1989, 1998) suggests that many of the analogies that played a major role in early twentieth-century physics count as “Pythagorean.” The term is meant to connote mathematical mysticism: a “Pythagorean” analogy is a purely formal analogy, one founded on mathematical similarities that have no known physical basis at the time it is proposed. One example is Schrödinger’s use of analogy ( Example 14 ) to “guess” the form of the relativistic wave equation. In Steiner’s view, Schrödinger’s reasoning relies upon manipulations and substitutions based on purely mathematical analogies. Steiner argues that the success, and even the plausibility, of such analogies “evokes, or should evoke, puzzlement” (1989: 454). Both Hesse (1966) and Bartha (2010) reject the idea that a purely formal analogy, with no physical significance, can support a plausible analogical inference in physics. Thus, Steiner’s arguments provide a serious challenge.

Bartha (2010) suggests a response: we can decompose Steiner’s examples into two or more steps, and then establish that at least one step does, in fact, have a physical basis. Fraser (forthcoming), however, offers a counterexample that supports Steiner’s position. Complex analogies between classical statistical mechanics (CSM) and quantum field theory (QFT) have played a crucial role in the development and application of renormalization group (RG) methods in both theories ( Example 15 ). Fraser notes substantial physical disanalogies between CSM and QFT, and concludes that the reasoning is based entirely on formal analogies.

4. Philosophical foundations for analogical reasoning

What philosophical basis can be provided for reasoning by analogy? What justification can be given for the claim that analogical arguments deliver plausible conclusions? There have been several ideas for answering this question. One natural strategy assimilates analogical reasoning to some other well-understood argument pattern, a form of deductive or inductive reasoning ( §4.1 , §4.2 ). A few philosophers have explored the possibility of a priori justification ( §4.3 ). A pragmatic justification may be available for practical applications of analogy, notably in legal reasoning ( §4.4 ).

Any attempt to provide a general justification for analogical reasoning faces a basic dilemma. The demands of generality require a high-level formulation of the problem and hence an abstract characterization of analogical arguments, such as schema (4). On the other hand, as noted previously, many analogical arguments that conform to schema (4) are bad arguments. So a general justification of analogical reasoning cannot provide support for all arguments that conform to (4), on pain of proving too much. Instead, it must first specify a subset of putatively ‘good’ analogical arguments, and link the general justification to this specified subset. The problem of justification is linked to the problem of characterizing good analogical arguments . This difficulty afflicts some of the strategies described in this section.

Analogical reasoning may be cast in a deductive mold. If successful, this strategy neatly solves the problem of justification. A valid deductive argument is as good as it gets.

An early version of the deductivist approach is exemplified by Aristotle’s treatment of the argument from example ( §3.2 ), the paradeigma . On this analysis, an analogical argument between source domain \(S\) and target \(T\) begins with the assumption of positive analogy \(P(S)\) and \(P(T)\), as well as the additional information \(Q(S)\). It proceeds via the generalization \(\forall x(P(x) \supset Q(x))\) to the conclusion: \(Q(T)\). Provided we can treat that intermediate generalization as an independent premise, we have a deductively valid argument. Notice, though, that the existence of the generalization renders the analogy irrelevant. We can derive \(Q(T)\) from the generalization and \(P(T)\), without any knowledge of the source domain. The literature on analogy in argumentation theory ( §2.2 ) offers further perspectives on this type of analysis, and on the question of whether analogical arguments are properly characterized as deductive.

Some recent analyses follow Aristotle in treating analogical arguments as reliant upon extra (sometimes tacit) premises, typically drawn from background knowledge, that convert the inference into a deductively valid argument––but without making the source domain irrelevant. Davies and Russell introduce a version that relies upon what they call determination rules (Russell 1986; Davies and Russell 1987; Davies 1988). Suppose that \(Q\) and \(P_1 , \ldots ,P_m\) are variables, and we have background knowledge that the value of \(Q\) is determined by the values of \(P_1 , \ldots ,P_m\). In the simplest case, where \(m = 1\) and both \(P\) and \(Q\) are binary Boolean variables, this reduces to

i.e., whether or not \(P\) holds determines whether or not \(Q\) holds. More generally, the form of a determination rule is

i.e., \(Q\) is a function of \(P_1,\ldots\), \(P_m\). If we assume such a rule as part of our background knowledge, then an analogical argument with conclusion \(Q(T)\) is deductively valid. More precisely, and allowing for the case where \(Q\) is not a binary variable: if we have such a rule, and also premises stating that the source \(S\) agrees with the target \(T\) on all of the values \(P_i\), then we may validly infer that \(Q(T) = Q(S)\).

The “determination rule” analysis provides a clear and simple justification for analogical reasoning. Note that, in contrast to the Aristotelian analysis via the generalization \(\forall x(P(x) \supset Q(x))\), a determination rule does not trivialize the analogical argument. Only by combining the rule with information about the source domain can we derive the value of \(Q(T)\). To illustrate by adapting one of the examples given by Russell and Davies ( Example 16 ), let’s suppose that the value \((Q)\) of a used car (relative to a particular buyer) is determined by its year, make, mileage, condition, color and accident history (the variables \(P_i)\). It doesn’t matter if one or more of these factors are redundant or irrelevant. Provided two cars are indistinguishable on each of these points, they will have the same value. Knowledge of the source domain is necessary; we can’t derive the value of the second car from the determination rule alone. Weitzenfeld (1984) proposes a variant of this approach, advancing the slightly more general thesis that analogical arguments are deductive arguments with a missing (enthymematic) premise that amounts to a determination rule.

Do determination rules give us a solution to the problem of providing a justification for analogical arguments? In general: no. Analogies are commonly applied to problems such as Example 8 ( morphine and meperidine ), where we are not even aware of all relevant factors, let alone in possession of a determination rule. Medical researchers conduct drug tests on animals without knowing all attributes that might be relevant to the effects of the drug. Indeed, one of the main objectives of such testing is to guard against reactions unanticipated by theory. On the “determination rule” analysis, we must either limit the scope of such arguments to cases where we have a well-supported determination rule, or focus attention on formulating and justifying an appropriate determination rule. For cases such as animal testing, neither option seems realistic.

Recasting analogy as a deductive argument may help to bring out background assumptions, but it makes little headway with the problem of justification. That problem re-appears as the need to state and establish the plausibility of a determination rule, and that is at least as difficult as justifying the original analogical argument.

Some philosophers have attempted to portray, and justify, analogical reasoning in terms of some well-understood inductive argument pattern. There have been three moderately popular versions of this strategy. The first treats analogical reasoning as generalization from a single case. The second treats it as a kind of sampling argument. The third recognizes the argument from analogy as a distinctive form, but treats past successes as evidence for future success.

4.2.1 Single-case induction

Let’s reconsider Aristotle’s argument from example or paradeigma ( §3.2 ), but this time regard the generalization as justified via induction from a single case (the source domain). Can such a simple analysis of analogical arguments succeed? In general: no.

A single instance can sometimes lead to a justified generalization. Cartwright (1992) argues that we can sometimes generalize from a single careful experiment, “where we have sufficient control of the materials and our knowledge of the requisite background assumptions is secure” (51). Cartwright thinks that we can do this, for example, in experiments with compounds that have stable “Aristotelian natures.” In a similar spirit, Quine (1969) maintains that we can have instantial confirmation when dealing with natural kinds.

Even if we accept that there are such cases, the objection to understanding all analogical arguments as single-case induction is obvious: the view is simply too restrictive. Most analogical arguments will not meet the requisite conditions. We may not know that we are dealing with a natural kind or Aristotelian nature when we make the analogical argument. We may not know which properties are essential. An insistence on the ‘single-case induction’ analysis of analogical reasoning is likely to lead to skepticism (Agassi 1964, 1988).

Interpreting the argument from analogy as single-case induction is also counter-productive in another way. The simplistic analysis does nothing to advance the search for criteria that help us to distinguish between relevant and irrelevant similarities, and hence between good and bad analogical arguments.

4.2.2 Sampling arguments

On the sampling conception of analogical arguments, acknowledged similarities between two domains are treated as statistically relevant evidence for further similarities. The simplest version of the sampling argument is due to Mill (1843/1930). An argument from analogy, he writes, is “a competition between the known points of agreement and the known points of difference.” Agreement of \(A\) and \(B\) in 9 out of 10 properties implies a probability of 0.9 that \(B\) will possess any other property of \(A\): “we can reasonably expect resemblance in the same proportion” (367). His only restriction has to do with sample size: we must be relatively knowledgeable about both \(A\) and \(B\). Mill saw no difficulty in using analogical reasoning to infer characteristics of newly discovered species of plants or animals, given our extensive knowledge of botany and zoology. But if the extent of unascertained properties of \(A\) and \(B\) is large, similarity in a small sample would not be a reliable guide; hence, Mill’s dismissal of Reid’s argument about life on other planets ( Example 2 ).

The sampling argument is presented in more explicit mathematical form by Harrod (1956). The key idea is that the known properties of \(S\) (the source domain) may be considered a random sample of all \(S\)’s properties—random, that is, with respect to the attribute of also belonging to \(T\) (the target domain). If the majority of known properties that belong to \(S\) also belong to \(T\), then we should expect most other properties of \(S\) to belong to \(T\), for it is unlikely that we would have come to know just the common properties. In effect, Harrod proposes a binomial distribution, modeling ‘random selection’ of properties on random selection of balls from an urn.

There are grave difficulties with Harrod’s and Mill’s analyses. One obvious difficulty is the counting problem : the ‘population’ of properties is poorly defined. How are we to count similarities and differences? The ratio of shared to total known properties varies dramatically according to how we do this. A second serious difficulty is the problem of bias : we cannot justify the assumption that the sample of known features is random. In the case of the urn, the selection process is arranged so that the result of each choice is not influenced by the agent’s intentions or purposes, or by prior choices. By contrast, the presentation of an analogical argument is always partisan. Bias enters into the initial representation of similarities and differences: an advocate of the argument will highlight similarities, while a critic will play up differences. The paradigm of repeated selection from an urn seems totally inappropriate. Additional variations of the sampling approach have been developed (e.g., Russell 1988), but ultimately these versions also fail to solve either the counting problem or the problem of bias.

4.2.3 Argument from past success

Section 3.6 discussed Steiner’s view that appeal to ‘Pythagorean’ analogies in physics “evokes, or should evoke, puzzlement” (1989: 454). Liston (2000) offers a possible response: physicists are entitled to use Pythagorean analogies on the basis of induction from their past success:

[The scientist] can admit that no one knows how [Pythagorean] reasoning works and argue that the very fact that similar strategies have worked well in the past is already reason enough to continue pursuing them hoping for success in the present instance. (200)

Setting aside familiar worries about arguments from success, the real problem here is to determine what counts as a similar strategy. In essence, that amounts to isolating the features of successful Pythagorean analogies. As we have seen (§2.4), nobody has yet provided a satisfactory scheme that characterizes successful analogical arguments, let alone successful Pythagorean analogical arguments.

An a priori approach traces the validity of a pattern of analogical reasoning, or of a particular analogical argument, to some broad and fundamental principle. Three such approaches will be outlined here.

The first is due to Keynes (1921). Keynes appeals to his famous Principle of the Limitation of Independent Variety, which he articulates as follows:

Armed with this Principle and some additional assumptions, Keynes is able to show that in cases where there is no negative analogy , knowledge of the positive analogy increases the (logical) probability of the conclusion. If there is a non-trivial negative analogy, however, then the probability of the conclusion remains unchanged, as was pointed out by Hesse (1966). Those familiar with Carnap’s theory of logical probability will recognize that in setting up his framework, Keynes settled on a measure that permits no learning from experience.

Hesse offers a refinement of Keynes’s strategy, once again along Carnapian lines. In her (1974), she proposes what she calls the Clustering Postulate : the assumption that our epistemic probability function has a built-in bias towards generalization. The objections to such postulates of uniformity are well-known (see Salmon 1967), but even if we waive them, her argument fails. The main objection here—which also applies to Keynes—is that a purely syntactic axiom such as the Clustering Postulate fails to discriminate between analogical arguments that are good and those that are clearly without value (according to Hesse’s own material criteria, for example).

A different a priori strategy, proposed by Bartha (2010), limits the scope of justification to analogical arguments that satisfy tentative criteria for ‘good’ analogical reasoning. The criteria are those specified by the articulation model ( §3.5 ). In simplified form, they require the existence of non-trivial positive analogy and no known critical disanalogy. The scope of Bartha’s argument is also limited to analogical arguments directed at establishing prima facie plausibility, rather than degree of probability.

Bartha’s argument rests on a principle of symmetry reasoning articulated by van Fraassen (1989: 236): “problems which are essentially the same must receive essentially the same solution.” A modal extension of this principle runs roughly as follows: if problems might be essentially the same, then they might have essentially the same solution. There are two modalities here. Bartha argues that satisfaction of the criteria of the articulation model is sufficient to establish the modality in the antecedent, i.e., that the source and target domains ‘might be essentially the same’ in relevant respects. He further suggests that prima facie plausibility provides a reasonable reading of the modality in the consequent, i.e., that the problems in the two domains ‘might have essentially the same solution.’ To call a hypothesis prima facie plausible is to elevate it to the point where it merits investigation, since it might be correct.

The argument is vulnerable to two sorts of concerns. First, there are questions about the interpretation of the symmetry principle. Second, there is a residual worry that this justification, like all the others, proves too much. The articulation model may be too vague or too permissive.

Arguably, the most promising available defense of analogical reasoning may be found in its application to case law (see Precedent and Analogy in Legal Reasoning ). Judicial decisions are based on the verdicts and reasoning that have governed relevantly similar cases, according to the doctrine of stare decisis (Levi 1949; Llewellyn 1960; Cross and Harris 1991; Sunstein 1993). Individual decisions by a court are binding on that court and lower courts; judges are obligated to decide future cases ‘in the same way.’ That is, the reasoning applied in an individual decision, referred to as the ratio decidendi , must be applied to similar future cases (see Example 10 ). In practice, of course, the situation is extremely complex. No two cases are identical. The ratio must be understood in the context of the facts of the original case, and there is considerable room for debate about its generality and its applicability to future cases. If a consensus emerges that a past case was wrongly decided, later judgments will distinguish it from new cases, effectively restricting the scope of the ratio to the original case.

The practice of following precedent can be justified by two main practical considerations. First, and above all, the practice is conservative : it provides a relatively stable basis for replicable decisions. People need to be able to predict the actions of the courts and formulate plans accordingly. Stare decisis serves as a check against arbitrary judicial decisions. Second, the practice is still reasonably progressive : it allows for the gradual evolution of the law. Careful judges distinguish bad decisions; new values and a new consensus can emerge in a series of decisions over time.

In theory, then, stare decisis strikes a healthy balance between conservative and progressive social values. This justification is pragmatic. It pre-supposes a common set of social values, and links the use of analogical reasoning to optimal promotion of those values. Notice also that justification occurs at the level of the practice in general; individual analogical arguments sometimes go astray. A full examination of the nature and foundations for stare decisis is beyond the scope of this entry, but it is worth asking the question: might it be possible to generalize the justification for stare decisis ? Is a parallel pragmatic justification available for analogical arguments in general?

Bartha (2010) offers a preliminary attempt to provide such a justification by shifting from social values to epistemic values. The general idea is that reasoning by analogy is especially well suited to the attainment of a common set of epistemic goals or values. In simple terms, analogical reasoning—when it conforms to certain criteria—achieves an excellent (perhaps optimal) balance between the competing demands of stability and innovation. It supports both conservative epistemic values, such as simplicity and coherence with existing belief, and progressive epistemic values, such as fruitfulness and theoretical unification (McMullin (1993) provides a classic list).

5. Beyond analogical arguments

As emphasized earlier, analogical reasoning takes in a great deal more than analogical arguments. In this section, we examine two broad contexts in which analogical reasoning is important.

The first, still closely linked to analogical arguments, is the confirmation of scientific hypotheses. Confirmation is the process by which a scientific hypothesis receives inductive support on the basis of evidence (see evidence , confirmation , and Bayes’ Theorem ). Confirmation may also signify the logical relationship of inductive support that obtains between a hypothesis \(H\) and a proposition \(E\) that expresses the relevant evidence. Can analogical arguments play a role, either in the process or in the logical relationship? Arguably yes (to both), but this role has to be delineated carefully, and several obstacles remain in the way of a clear account.

The second context is conceptual and theoretical development in cutting-edge scientific research. Analogies are used to suggest possible extensions of theoretical concepts and ideas. The reasoning is linked to considerations of plausibility, but there is no straightforward analysis in terms of analogical arguments.

How is analogical reasoning related to the confirmation of scientific hypotheses? The examples and philosophical discussion from earlier sections suggest that a good analogical argument can indeed provide support for a hypothesis. But there are good reasons to doubt the claim that analogies provide actual confirmation.

In the first place, there is a logical difficulty. To appreciate this, let us concentrate on confirmation as a relationship between propositions. Christensen (1999: 441) offers a helpful general characterization:

Some propositions seem to help make it rational to believe other propositions. When our current confidence in \(E\) helps make rational our current confidence in \(H\), we say that \(E\) confirms \(H\).

In the Bayesian model, ‘confidence’ is represented in terms of subjective probability. A Bayesian agent starts with an assignment of subjective probabilities to a class of propositions. Confirmation is understood as a three-place relation:

\(E\) represents a proposition about accepted evidence, \(H\) stands for a hypothesis, \(K\) for background knowledge and \(Pr\) for the agent’s subjective probability function. To confirm \(H\) is to raise its conditional probability, relative to \(K\). The shift from prior probability \(Pr(H \mid K)\) to posterior probability \(Pr(H \mid E \cdot K)\) is referred to as conditionalization on \(E\). The relation between these two probabilities is typically given by Bayes’ Theorem (setting aside more complex forms of conditionalization):

For Bayesians, here is the logical difficulty: it seems that an analogical argument cannot provide confirmation. In the first place, it is not clear that we can encapsulate the information contained in an analogical argument in a single proposition, \(E\). Second, even if we can formulate a proposition \(E\) that expresses that information, it is typically not appropriate to treat it as evidence because the information contained in \(E\) is already part of the background, \(K\). This means that \(E \cdot K\) is equivalent to \(K\), and hence \(Pr(H \mid E \cdot K) = Pr(H \mid K)\). According to the Bayesian definition, we don’t have confirmation. (This is a version of the problem of old evidence; see confirmation .) Third, and perhaps most important, analogical arguments are often applied to novel hypotheses \(H\) for which the prior probability \(Pr(H \mid K)\) is not even defined. Again, the definition of confirmation in terms of Bayesian conditionalization seems inapplicable.

If analogies don’t provide inductive support via ordinary conditionalization, is there an alternative? Here we face a second difficulty, once again most easily stated within a Bayesian framework. Van Fraassen (1989) has a well-known objection to any belief-updating rule other than conditionalization. This objection applies to any rule that allows us to boost credences when there is no new evidence. The criticism, made vivid by the tale of Bayesian Peter, is that these ‘ampliative’ rules are vulnerable to a Dutch Book . Adopting any such rule would lead us to acknowledge as fair a system of bets that foreseeably leads to certain loss. Any rule of this type for analogical reasoning appears to be vulnerable to van Fraassen’s objection.

There appear to be at least three routes to avoiding these difficulties and finding a role for analogical arguments within Bayesian epistemology. First, there is what we might call minimal Bayesianism . Within the Bayesian framework, some writers (Jeffreys 1973; Salmon 1967, 1990; Shimony 1970) have argued that a ‘seriously proposed’ hypothesis must have a sufficiently high prior probability to allow it to become preferred as the result of observation. Salmon has suggested that analogical reasoning is one of the most important means of showing that a hypothesis is ‘serious’ in this sense. If analogical reasoning is directed primarily towards prior probability assignments, it can provide inductive support while remaining formally distinct from confirmation, avoiding the logical difficulties noted above. This approach is minimally Bayesian because it provides nothing more than an entry point into the Bayesian apparatus, and it only applies to novel hypotheses. An orthodox Bayesian, such as de Finetti (de Finetti and Savage 1972, de Finetti 1974), might have no problem in allowing that analogies play this role.

The second approach is liberal Bayesianism : we can change our prior probabilities in a non-rule-based fashion . Something along these lines is needed if analogical arguments are supposed to shift opinion about an already existing hypothesis without any new evidence. This is common in fields such as archaeology, as part of a strategy that Wylie refers to as “mobilizing old data as new evidence” (Wylie and Chapman 2016: 95). As Hawthorne (2012) notes, some Bayesians simply accept that both initial assignments and ongoing revision of prior probabilities (based on plausibility arguments) can be rational, but

the logic of Bayesian induction (as described here) has nothing to say about what values the prior plausibility assessments for hypotheses should have; and it places no restrictions on how they might change.

In other words, by not stating any rules for this type of probability revision, we avoid the difficulties noted by van Fraassen. This approach admits analogical reasoning into the Bayesian tent, but acknowledges a dark corner of the tent in which rationality operates without any clear rules.

Recently, a third approach has attracted interest: analogue confirmation or confirmation via analogue simulation . As described in (Dardashti et al. 2017), the idea is as follows:

Our key idea is that, in certain circumstances, predictions concerning inaccessible phenomena can be confirmed via an analogue simulation in a different system. (57)

Dardashti and his co-authors concentrate on a particular example ( Example 17 ): ‘dumb holes’ and other analogues to gravitational black holes (Unruh 1981; Unruh 2008). Unlike real black holes, some of these analogues can be (and indeed have been) implemented and studied in the lab. Given the exact formal analogy between our models for these systems and our models of black holes, and certain important additional assumptions, Dardashti et al. make the controversial claim that observations made about the analogues provide evidence about actual black holes. For instance, the observation of phenomena analogous to Hawking radiation in the analogue systems would provide confirmation for the existence of Hawking radiation in black holes. In a second paper (Dardashti et al. 2018, Other Internet Resources), the case for confirmation is developed within a Bayesian framework.

The appeal of a clearly articulated mechanism for analogue confirmation is obvious. It would provide a tool for exploring confirmation of inaccessible phenomena not just in cosmology, but also in historical sciences such as archaeology and evolutionary biology, and in areas of medical science where ethical constraints rule out experiments on human subjects. Furthermore, as noted by Dardashti et al., analogue confirmation relies on new evidence obtained from the analogue system, and is therefore not vulnerable to the logical difficulties noted above.

Although the concept of analogue confirmation is not entirely new (think of animal testing, as in Example 8 ), the claims of (Dardashti et al. 2017, 2018 [Other Internet Resources]) require evaluation. One immediate difficulty for the black hole example: if we think in terms of ordinary analogical arguments, there is no positive analogy because, to put it simply, we have no basis of known similarities between a ‘dumb hole’ and a black hole. As Crowther et al. (2018, Other Internet Resources) argue, “it is not known if the particular modelling framework used in the derivation of Hawking radiation actually describes black holes in the first place. ” This may not concern Dardashti et al., since they claim that analogue confirmation is distinct from ordinary analogical arguments. It may turn out that analogue confirmation is different for cases such as animal testing, where we have a basis of known similarities, and for cases where our only access to the target domain is via a theoretical model.

In §3.6 , we saw that practice-based studies of analogy provide insight into the criteria for evaluating analogical arguments. Such studies also point to dynamical or programmatic roles for analogies, which appear to require evaluative frameworks that go beyond those developed for analogical arguments.

Knuttila and Loettgers (2014) examine the role of analogical reasoning in synthetic biology, an interdisciplinary field that draws on physics, chemistry, biology, engineering and computational science. The main role for analogies in this field is not the construction of individual analogical arguments but rather the development of concepts such as “noise” and “feedback loops”. Such concepts undergo constant refinement, guided by both positive and negative analogies to their analogues in engineered and physical systems. Analogical reasoning here is “transient, heterogeneous, and programmatic” (87). Negative analogies, seen as problematic obstacles for individual analogical arguments, take on a prominent and constructive role when the focus is theoretical construction and concept refinement.

Similar observations apply to analogical reasoning in its application to another cutting-edge field: emergent gravity. In this area of physics, distinct theoretical approaches portray gravity as emerging from different microstructures (Linneman and Visser 2018). “Novel and robust” features not present at the micro-level emerge in the gravitational theory. Analogies with other emergent phenomena, such as hydrodynamics and thermodynamics, are exploited to shape these proposals. As with synthetic biology, analogical reasoning is not directed primarily towards the formulation and assessment of individual arguments. Rather, its role is to develop different theoretical models of gravity.

These studies explore fluid and creative applications of analogy to shape concepts on the front lines of scientific research. An adequate analysis would certainly take us beyond the analysis of individual analogical arguments, which have been the focus of our attention. Knuttila and Loettgers (2014) are led to reject the idea that the individual analogical argument is the “primary unit” in analogical reasoning, but this is a debatable conclusion. Linneman and Visser (2018), for instance, explicitly affirm the importance of assessing the case for different gravitational models through “exemplary analogical arguments”:

We have taken up the challenge of making explicit arguments in favour of an emergent gravity paradigm… That arguments can only be plausibility arguments at the heuristic level does not mean that they are immune to scrutiny and critical assessment tout court. The philosopher of physics’ job in the process of discovery of quantum gravity… should amount to providing exactly this kind of assessments. (Linneman and Visser 2018: 12)

Accordingly, Linneman and Visser formulate explicit analogical arguments for each model of emergent gravity, and assess them using familiar criteria for evaluating individual analogical arguments. Arguably, even the most ambitious heuristic objectives still depend upon considerations of plausibility that benefit by being expressed, and examined, in terms of analogical arguments.

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How to cite this entry . Preview the PDF version of this entry at the Friends of the SEP Society . Look up topics and thinkers related to this entry at the Internet Philosophy Ontology Project (InPhO). Enhanced bibliography for this entry at PhilPapers , with links to its database.

Other Internet Resources

• Crowther, K., N. Linnemann, and C. Wüthrich, 2018, “ What we cannot learn from analogue experiments ,” online at arXiv.org.
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• Dedre Gentner’s publications (Northwestern University)
• The Center for Research on Concepts and Cognition (Indiana University)

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College Minor: Everything You Need to Know

14 fascinating teacher interview questions for principals, tips for success if you have a master’s degree and can’t find a job, 14 ways young teachers can get that professional look, which teacher supplies are worth the splurge, 8 business books every teacher should read, conditional admission: everything you need to know, college majors: everything you need to know, 7 things principals can do to make a teacher observation valuable, 3 easy teacher outfits to tackle parent-teacher conferences, analogies for critical thinking.

Analogies are used to prove a point. You can use analogies to strengthen your view when you want to get your argument across to others. You compare two valid (but similar) points in hopes of people seeing your point in a new light.

Understanding Analogies

Analogies help to explain your arguments cogently by using simple comparisons that get people thinking.

• Example 1 : Driving five minutes to the local store is as stupid as dancing there. Both ways are wasteful, and you look silly in the process.

In this example, the analogy focuses on dancing and driving. While both are quite different, the argument points to how wasteful and silly it is to drive short distances. Immediately, your mind wonders how ridiculous you’d look dancing in the shops. Then you reflect on how silly it is to drive such a short distance. The point is made, and you begin to think about walking for 5 minutes rather than driving.

Of course, this point can work for and against you. Some would say dancing burns energy and is good for your health so that it wouldn’t be so bad. The real point you’re trying to make, however, is it’s crazy to burn fuel for such a short distance. The point will hit home if you’re dealing with a sensible person. You must use a good analogy to get people to think outside the box.

• Example 2 : Leaving the lights on all day costs as much as keeping the fridge open all night. Both practices waste energy and cost you more money.

Here, you’re making the comparison between the fridge and lights. The argument is to keep your lights off during the day to prevent wasting energy. Your point is you wouldn’t keep your fridge open all night, so why keep your lights on during the daytime?

Analogies Must Encourage Others to Think Outside the Box

While you want to prove a point with your analogy , you must encourage others to think beyond what they know. It’s about expanding the mind and letting their imaginations flow to consider new things. It’s the same when you are faced with an analogy; you want to explore it with an unbiased mind.

Analogies are there to help people consider things more openly and concisely. So, while some of your comparisons are unusual, they aren’t easy to dismiss entirely. You need to ensure the argument is balanced and puts across a valid point. If you’re faced with analogies, think clearly and listen to all parts because that lets you think more.

The Checklist of Analogies

An analogy should be approached with a clear mind. You need to understand what arguments or points it is being used to support. It’s important to look at the similarities and dissimilarities or view them in a wider context. You could also find there are counterarguments for the points made.

So, here are a few things to add to your analogy checklist.

• What’s the Point of the Analogy?
• Is There Any Relevance to the Points Made?
• What Similarities Does It Have?
• What Dissimilarities Does the Argument Have?
• Are the Comparisons Correct or Just Pointless Assumption?
• Should Dissimilarities be Considered in the Argument?

Poor Analogy Examples Based on Assumptions

Here are a few examples of analogies based on assumptions, not facts.

• Example 1 : British politicians are all corrupt like African dictators. They are only interested in partying and hiding money in tax havens. There are no good politicians in Britain.

This analogy is full of assumptions and opinions rather than facts. It’s a weak argument about politicians in Britain. While there have been some examples of corruption in politics, the analogy implies every politician is corrupt. It’s the same with African leaders. There have been well-documented and corrupt African dictators, but this argument is not true for every leader on the continent. So again, the argument is weak and factually untrue.

• Example 2 : Amanda and her mother wear glasses and love to learn about history. Amanda will likely become a history teacher like her mother.

The comparison is between Amanda and her mother. The point of them wearing glasses is irrelevant to Amanda becoming a history teacher. It is irrelevant to the story. A passion for history is relevant but insufficient to support the conclusion. Many history buffs don’t go into teaching. So, your argument must support the conclusion.

Analogical Examples to Teach Students

Analogies aren’t difficult to grasp, but putting them into context can become a barrier for some. So, here are a few examples to help teach students – or just help you understand them better.

• Arranged Marriages

In Islamic and Asian cultures, arranged marriages match young men and women. This, however, is often seen as skeptical in other cultures. In Western culture, people use dating services to find love. So, those who use dating agencies and services to find love are hypocrites. They have no right to criticize arranged marriages since they are paired similarly.

• Recreational Drugs

Each year, there are more deaths from horse-riding accidents than ecstasy-related deaths. Ecstasy is banned even though it’s statistically safer than riding a horse. So, banning ecstasy while allowing horse riding to remain legal makes no sense.

• Dangerous Drivers

Drivers who act recklessly and kill someone should receive a life sentence just as a convicted murderer would. Both have taken a life with no just cause, but the punishment is very different.

• Restrictions on Cars and Guns

Guns are lethal weapons that can be used to kill or injure someone. Cars are also lethal weapons as they can injure and kill. Cars should have the same restrictions as guns.

Final Thoughts

Analogies are comparisons between two points, yet they are often misunderstood and interpreted. So, it’s crucial to understand how analogies should be used and how to use them properly. For instance, an analogy contains facts rather than opinions or inaccuracies. Fortunately, it is easy to learn and use analogies properly.

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Analogical Argument — Definition and Examples

Analogical reasoning definition

Analogical reasoning, also known as analogical argument or argument by analogy, suggests that if two or more things are similar in one way, they are probably similar in other ways.

Using analogies can explain or clarify an object or idea through comparison. Analogical reasoning uses analogies to persuade or make an argument.

There are typically two to three premises in an analogical argument, which ultimately identify the analogy. Those premises are then used to come to some conclusion. The analogical argument would then be structured as follows:

Analogical thinking

Analogical thinking is when someone uses information from one area to help with problem-solving in another area. Through this type of critical thinking, several everyday items made their way into production thanks to technology developed by NASA.

Companies took the technology developed for space exploration and found ways to transform it, creating real-world products widely used today. Items such as memory foam, baby formula, thermometers, invisible braces, and even the Super Soaker exist today due to analogical problem solving.

Argument by analogy examples

The first step in creating an analogical argument is to determine how the two things are similar. Then determine if those similarities support the conclusion. Consider these examples:

Analogical induction

Argument by use of analogy can be made through inductive reasoning, meaning it creates an assumption based on the identified similarities. While the premises are typically accurate, the conclusions based on the similarities may or may not be correct. The following uses analogical induction and produces a potentially inaccurate conclusion:

Deductive analogy

Analogical arguments can also be made through deductive reasoning. Through this approach, the premises lead to a correct conclusion. The following uses analogical deduction:

False analogy

Analogical arguments must be checked to make sure the conclusion is accurate. Sometimes the argument creates a false analogy where the similarities between the two things lead to an invalid conclusion. Consider the following:

Analogical Reasoning

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Argument by analogy ; Case-based reasoning ; Metaphorical thinking

Analogical Reasoning and Its Uses

Analogical reasoning or argument by analogy can be defined as a specific way of thinking, based on the idea that because two or more things are similar in some respects, they are probably also similar in some further respect. Integrating various human-level reasoning mechanisms, arguing by analogical thinking, use analogies by transferring knowledge from one particular entity (the analogue or source) to another one (the target). Furthermore, it refers to the linguistic form, which corresponds to the process of relating the source and the target. As specific form of inference or reasoning, analogies draw conclusions by applying heuristics to propositions or observations as well as by interpolating logical steps or patterns. Analogies focus on relating specific particularities in two or more cases or things to form the basis for a conclusion involving an additional...

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23 Arguments VIII: Analyzing Analogies

Okay, on to analogies. Let’s again start with a more precise definition.

• An analogical argument or argument by analogy is an argument in which known similarities between one thing and another (or multiple others) is/are used to infer that some additional similarity is likely to hold between them.

(Side note: An analog is something that is like something else; an analogy a drawing of a comparison. Given that, why do you think we talk about “analog” technologies, particularly in recording?)

The general form of an analogical argument :

• A and B are known to be similar in respects p, q, r, … 
• A is also known to be x .

Therefore, B will likely be x as well.

Elements of analysis of an analogical argument:

Individuals Compared: A, B

Known Similarities (of A and B): p, q, r, …

Note: there is no set number of similarities that are required. The list “ p, q, r,… ” is just a placeholder. There could be only one similarity, or there could be many more than three.

Additional Fact Known about A that is predicted to hold of B as well: x

Note: the  conclusion  of the argument will claim that x holds or is likely to hold of B.

Examples of Analogical Arguments with Elements Analyzed

1.     1. During the Great Depression there was high unemployment, low wages, and low national productivity, all of which ended due to the increase in government spending for World War II.

2. Today there is high unemployment, low wages, and low national productivity.

Conclusion: high unemployment, low wages, and low national productivity will end if there is an increase in government spending comparable to that during WWII.

(Note: This is an argument some economists will make in times of recession, as for example back in 2008.)

Individuals Compared : (A) the economy of the Great Depression, (B) the economy today

Known Similarities of A and B : (p) high unemployment, (q) low wages, (r) low national productivity

Additional Fact (x) Known about A that is predicted to hold of B as well : it ended/will end because of an increase in government spending

2.       1. A watch is something has many parts which are interrelated in complex yet ordered (i.e., non-random) ways, and which gets that order and complexity from an intelligent  being who designed it.

2. The universe has many parts which are interrelated in complex yet ordered ways.

Conclusion: the universe must have gotten its complexity from an intelligent being who designed it.

(Note:  this is a famous argument for the existence of God known as the Argument from Design.)

Individuals Compared : (A) a watch, (B) the universe

Known Similarities of A and B : (p) many parts, (q) parts are related in complex ways, (r) relations among parts is ordered

Additional Fact (x) Known about A that is predicted to hold of B as well : its order and complexity comes from an intelligent designer

3.      1. Skeletor, a drug designed to speed the healing of broken bones, proved very effective in clinical trials on men who were between 30 and 50 years of age who had broken legs.

2. I am a man between 30 and 50 years of age, and I have a broken leg.

Conclusion: Skeletor will be very effective in helping heal my broken leg.

( Note : Skeletor is an entirely fictitious drug; the fact that I thought of using the He-man villain as a pharmaceutical name shows why I think I should really be in marketing.)

Individuals Compared : (A) men in clinical trial, (B) me

Known Similarities of A and B : (p) male, (q) between 30 and 50, (r) have broken legs

Additional Fact (x) Known about A that is predicted to hold of B as well : very effective in healing broken legs

4.      1. Rush Limbaugh smokes cigars, has a lot of money, and is politically conservative.

2. Robert DeNiro smokes cigars and has a lot of money.

Conclusion: Therefore, he is probably politically conservative too.

Individuals Compared : (A) Rush Limbaugh, (B) Robert DeNiro

Known Similarities of A and B : (p) smoke cigars, (q) have a lot of money

Additional Fact (x) Known about A that is predicted to hold of B as well : is politically conservative

What we now want to know is, when is an analogical argument  strong ?

Criteria for Strength of an Analogical Argument

• … there must be enough similarities between the things being compared;
• … the similarities must be relevant to the drawing of the conclusion;
• … there must not be relevant dissimilarities that are overlooked or suppressed.

Based on these, we get the definition of another  fallacy : an analogical argument that is weak (for whatever reason) is said to be guilty of the fallacy of faulty analogy .

With that in mind, let’s ask, are the above examples above strong? As with generalizations, you may not easily be able to answer one way or another. What you should be able to do, however, is specify what you would need to know in order to answer that question .

Above Examples of Analogical Arguments Analyzed for Strength

• There are multiple similarities which appear to be relevant between the historical example and the current one referred to; only economists and/or economic historians will be in a position to judge whether there are enough similarities and whether there are any neglected but relevant dissimilarities . So those of us who lack the relevant expertise can only say: maybe this is a strong argument, but without more information, we can’t be sure. It’s not  obviously  weak, however.
• There are multiple similarities which appear to be relevant. There are, however, some obvious dissimilarities that are not mentioned, for instance, that we have direct knowledge of how watches are made by their designers, that the universe is vastly larger than a watch, and that watches have obvious purposes (for telling time). Such dissimilarities lead many philosophers to conclude that this is a weak argument (though it has its supporters as well). (If you find this argument interesting and want to think more about it, take another philosophy class!)
• There are multiple similarities which appear to be relevant. While it’s possible that there could be relevant dissimilarities, there aren’t any obvious ones. This is, in fact, the sort of reasoning we implicitly use any time we choose to take a medication (at least for its intended purpose). We rely on being sufficiently similar to those it has already been proven effective in treating – that’s why we think it will work on us too. So this is a strong argument.
• Two similarities between the men are noted, but only one of them (wealth) is in any clear way related to political leanings. So, it is pretty clearly weak: not enough relevant similarities, and no consideration or relevant dissimilarities. It is thus a faulty analogy .

Important Note : it can be difficult to analyze analogies (more so than generalizations), because much more depends on your background knowledge of the topic. It’s this that will help you know whether what’s mentioned is relevant, or whether there are important things being left out. When you are doing your exercises, keep that in mind, and do your best to think imaginatively about the situations described. You should always be able to specify the known and predicted similarities, because those will be stated. But, more often than not, you may have to say that you don’t know enough to assess whether there are enough relevant similarities or unstated dissimilarities.

Distinguishing Generalizations from Analogies

Sometimes it can be hard to tell if you’re dealing with an analogy or a generalization. Both involve two things (target and sample groups; A and B being compared) and an argument that considers them in relation to one another. So how do you know which is which?

The two  main things to look for : once you’ve got the two things/groups and you’re trying to decide what kind of argument it is, ask,

( 1 ) is one included in the other as a smaller subset of it?

( 2 ) Is the thing the conclusion is about a group or a single individual?

In most cases, once you answer these questions you’ll know what you’re dealing with. If the answer to 1 is ‘yes,’ and the conclusion is about the larger group, then it’s a generalization. A generalization  never makes a conclusion about a single individual, and it  does  involve starting with a small group and moving to a claim about a larger group that includes the small one.

If the answer to 2 is ‘single individual,’ it’s an analogy. Analogies sometimes draw conclusions about groups (but not on the basis of what’s true of a subset of the group), but they often draw conclusions about individual things or people.

If the answer to 2 is ‘group,’ then you need to ask question 1 again and determine whether two distinct groups are being compared, or whether a conclusion about a large group is being drawn on the basis of a smaller one.

Practice is how you learn, so your exercises are where you will really get a handle on all of this.

Phil-P102 Critical Thinking and Applied Ethics Copyright © 2020 by R. Matthew Shockey is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License , except where otherwise noted.

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Middle Way Society

An ethical approach to a better life, by integrating desires and avoiding dogmatic extremes, critical thinking 12: analogies.

Analogies are comparisons made in an argument to help prove a point. You’re arguing about one thing and you put it in the terms of another, to help people to see it in a different light. For example:

Getting into your car to drive a few hundred metres to the corner shop is as ridiculous as hopping that distance: both are clumsy, grossly inefficient, and enough to make you a laughing stock.

There are obviously some parallels here, but that doesn’t mean that the analogy is particularly successful. One reason for its lack of success may be that we tend to view inefficiency in using fuel rather differently from inefficiency in using our own bodily energy. Hopping a few hundred metres might just be seen as a good, though rather bizarre, form of useful exercise, whereas driving that distance wastes fuel – which we can more easily measure. The ridiculousness of hopping might also be exactly what makes it positive fun for some, whereas driving a car a few hundred metres would only be ‘ridiculous’ in the sense of drawing condemnation from the ecologically-minded. What looked like similarities at first turn out to be rather stretched and thin.

A well-judged analogy can be really helpful. It can help people to ‘think outside the box’ of the cognitive models they’re in the habit of using, and bring in the imagination to allow them to consider their experience in a more open way. However, it’s also very easy to dismiss a poorly-applied analogy. The problem is that there will always be dissimilarities as well as similarities between the two things being compared, so it is very easy just to latch onto the dissimilarities and use them as an excuse to dismiss the argument, if you’re a bit resistant to it in the first place. But a Middle Way approach involves trying to reach a balanced appreciation both of the similarities and the dissimilarities.

So, when you come across an analogy, it helps to think clearly about what the analogy is being used to support, and what sorts of relevant similarities and dissimilarities there are. The analogy may also need to be seen in a wider context, as there may be counter-arguments based on strong dissimilarities that just aren’t being considered. Here’s what I hope is a useful checklist:

• What is the analogy trying to show?
• Is the analogy relevant to what it is trying to show?
• What are the relevant similarities?
• What are the relevant dissimilarities?
• Are the assumptions being made about the things being compared correct?
• Are there other important dissimilarities that are not being taken into account?

Here are a couple more examples to illustrate the application of some of these questions:

Politicians in Britain today are just like African dictators, only out to get what they can from the country and squirrel it away in their offshore bank accounts. We will never get straight politicians.

This analogy is weak because the assumptions being made about British politicians are factually dubious. There may be a few cases of corruption, but these are nowhere near the scale of certain well-known corrupt African dictators (such as Mobutu in Congo). Of course, African dictators themselves are also rather varied, and some may not be particularly corrupt.

Jess has red hair and likes reading like her sister. She’ll probably become an English teacher like her sister.

Here the analogy is between Jess and her sister, but the fact of her having red hair is of no relevance to the probability of her becoming an English teacher. The fact that she likes reading is relevant, but is not strong enough by itself to support the conclusion, as lots of people who like reading do not become English teachers.

Assess the strength of these analogical arguments:

1. Cars should be restricted just as guns are, because they are lethal weapons just like guns. Cars kill and injure people just as much as guns do.

2. Motorists who kill people through reckless driving should be given a life sentence just like a murderer. The outcome is the same: a dead person.

3. More people are killed by horse-riding each year than by taking ecstasy. Ecstasy is thus less dangerous than horse-riding, and it is inconsistent to maintain horse-riding as a legal activity whilst banning ecstasy.

4. The practice of arranged marriage (as practised, for example, in Asian and Islamic cultures) is necessary to take into account young people’s lack of experience when they choose a partner. We need someone else to make this choice for us when we are inexperienced. This has been effectively admitted in Western culture when people use dating agencies and dating websites to select a partner for them, so it is hypocritical for people who use these services to criticise arranged marriage.

 Index of previous blogs in the Critical Thinking course

Picture: Nude man hopping on right foot (Edward Muybridge studies in locomotion)

3 thoughts on “ Critical Thinking 12: Analogies ”

1. A dissimilar analogy becase the analogy tries to show that cars and guns are weapons, cars are not purchased with the intention of using them as weapons. Cars and guns have no relevant similarity. the assumption that they are the same is incorrect 2. This is probably a similar analogy, it does sow there is a relevant similarity between the two outcomes from dangerous driving that maims and use of guns which murder, but it is not really a useful or relevant comparison. 3. This analogy attempts to show that horse riding and ecstacy produce the same results, but it is a dissimilar analogy. Ban the drug if it is harmful but to ban a sport such as horse riding because harm may sometimes result is a dissimilar analogy, taking ecstacy and sport are not relevant similarities 4. Here is a similar analogy, to critise an arranged marriage is hypocritical, having a partner chosen for a person or finding a partner via a dating agency can be viewed as being similar, unless marriage is forced on the couple in the arranged marriage, it is not forced from matching by a dating agency, then there is a dissimilarity between the two methods, if both partners agree with the decision made for them that would be analogous to the latter method of finding a partner. The matter of force has to be taken into account when deciding if it is similar or dissimilar.

1. This analogy has a certain strength if you feel that the deaths and injuries caused by cars is too high in proportion to the advantages they provide. Admittedly the analogy weakens when you consider that guns are specifically designed to cause injury and death and although there are a lot more road deaths a year, say in the UK, than gun deaths, there are a lot more cars. I would nevertheless feel it would be appropriate to have some further restrictions for cars such as raising the age when one can drive to 21 and having longer driving bans for drink/reckless driving.

1. This analogy goes too far as there is a significant difference in intending to kill someone and doing it by accident. However, that doesn’t mean to say that people are not responsible to a lesser or greater extent according to the circumstances of the death they have caused. This needs to be taken into account in any judicial proceedings.

3. I refer to your answer to this analogy put forward by David Knutt after being sacked by the government as drugs Zsar , which makes sense to me. “He makes a good point with the analogy, but it does have some weaknesses that he is rather reluctant to acknowledge. The main problem is that statistics about fatalities and serious injuries are not the only indicator of danger – in the case of ecstasy there could be danger of addiction or of longer-term mental or physical health effects, neither of which are the case with horse-riding.”

4. This is a fairly weak analogy as although it highlights the similarities in the two approaches in regards to affording certain advantages that go with being selective, it doesn’t take into such factors as choice, attraction, coercion, vested interests etc.

Here are my answers 1. As both Barry and Norma recognised, the problem here is that the differing intentions behind cars and guns are not taken into account. Cars also have many useful effects apart from their side-effect of occasionally killing people. So quite a weak analogy.

2. The similarity in outcome here, again, might distract us from a potentially great difference in intention. However, there can be different degrees of recklessness, from slight carelessness to a state in which you don’t care who you kill, and also different degrees of responsibility for that carelessness. At the extreme end you could make a case for treating the worst dangerous drivers like murderers, but the argument doesn’t take into account this incrementality.

3. This is, indeed, a summary of David Knutt’s argument, though with a slightly altered conclusion (Knutt was arguing about government policy). The analogy would be much stronger if it was only explicitly concerned with danger of death, rather than potentially with other types of danger.

4. I agree with Barry here. There is a similarity between arranged marriage and using a dating website in the sense that you let someone else do your matching for you. However, the circumstances are rather different. On a dating website both parties chose to use it, and are not under any pressure to accept the outcome (or even to take it at all seriously if they don’t find a good match). In contrast, traditional arranged marriages are loaded with traditional expectations, and the basis of the parents’ choice is likely to be much more socially restricted than that on a dating website. Arranged marriage should not be confused with forced marriage, but there are also degrees of social pressure on the young people concerned.

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7.3: Types of Reasoning

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• Page ID 67186

• Jim Marteney
• Los Angeles Valley College via ASCCC Open Educational Resources Initiative (OERI)

Inductive Reasoning

Inductive reasoning is the process of reasoning from specifics to a general conclusion related to those specifics. You have a series of facts and/or observations. From all of this data you make a conclusion or as the graphic above calls it, a "General Rule." Inductive reasoning allows humans to create generalizations about people, events, and things in their environment. There are five methods of inductive reasoning: example, cause, sign, comparison , and authority .

Example Reasoning

Example reasoning involves using specific instances as a basis for making a valid conclusion. In this approach, specific instances 1, 2, and 3 lead to a generalized conclusion about the whole situation. For example: I have a Sony television, a Sony stereo, a Sony car radio, a Sony video system, and they all work well. It is clear that Sony produces superior electronic products. Or, I have taken four good professors at this college, Mr. Smith, Mrs. Ortiz, Dr. Willard, and Ms. Richard; therefore, I can conclude that the professors at this college are good.

Tests for Reasoning by Example

Some audiences may find one enough, while others may need many more. For instance, the Neilson Ratings that are used to measure the television viewing preferences of 300 million Americans are determined by roughly 3,000 homes scattered throughout the United States. Yet, the television industry, which uses them to set advertising rates, accepts the 3,000 examples as enough to validate the conclusions.

• The examples must be typical of the whole. They must be representative of the topic about which the conclusion is reached, not fringe examples. For example, you come to college and take one English class whose instructor you find disappointing. You conclude that all 300 instructors at this particular college are poor teachers from this one class from this one Department. The sample might not be representative of the whole population of instructors.
• Important counter examples must be accounted for. If the counter examples mitigate against the examples used, the generalization is threatened. What if a good friend of yours also took another English class and was pleased by the experience. He found that his instructor was an excellent teacher. His example becomes a counter one to the specific instance you used to draw your conclusion, which is now very much in doubt.
• The examples must be relevant to the time period of your argument . If you are dealing with something recent, you need recent examples. If you are trying to prove something in the 1850's, examples from that period are appropriate. If you took the English class 30 years ago, it would be difficult to draw a valid conclusion about the nature of teachers at the college today without using recent examples. Likewise, recent examples may not be reflective of the way the college was 30 years ago.

Causal Reasoning

Causal Reasoning is based on the idea that for every action there is a reaction. Stated very simply, a cause is anything that is directly responsible for producing something else, usually termed the effect. There are two forms of causal reasoning:

The goal of causal reasoning is to figure out how or why something happened. For instance, you did well on a test because you studied two days in advance. I could then predict that if you study two days in advance of the next test, you will do well. In causal reasoning, the critical thinker is trying to establish a predictive function between two directly related variables. If we can figure out how and why things occur, we can then try to predict what will happen in the future.

• Cause to effect, a known cause or causes is capable of producing some unknown effect or effects
• Effect to cause, some known effect(s) has/have been produced by some unknown cause or causes.

Tests of Causal Reasoning

• The cause must be capable of producing the effect described, and vice versa. Has causality really been established or is it just coincidence? Is the cause really capable of producing the effect and vice versa? There must be a direct connection between the cause and the effect that can be demonstrated using empirical evidence. For example, many people mistake superstition for causal reasoning. Is the source of good luck the rubbing of a rabbit’s foot? Is the cause of bad luck really the fact that you walked under a ladder or broke the mirror? Did wearing that shirt really cause your team to win five games in a row? The critical thinker must make a clear distinction between a valid causal occurrence and sheer coincidence.
• Cumulative causal reasoning increases the soundness of the conclusion . The more times the causal pattern has happened, the greater the strength given to the causal reasoning, leading to a more valid conclusion. If this is the first time this association has ever been asserted the advocate will have to use more evidence to support the soundness of the causal reasoning advanced.
• Counter causal factors must also be accounted for. The advocate must be aware of the other inherent causal factors that could disrupt the relationship between the cause and effect presented. A claim was made by a father that his son committed suicide, because he was influenced to do so by the songs of a particular rock musician. If we assume that such a causal association exists, we also need to know if there are any other factors that could disrupt the connection: Was the son using drugs; had he tried to commit suicide before; were there family problems; did he listen to other artists and other types of music; did he have peer problems; did he have relationship problems; was he having problems in school, etc.? Each one of these, individually, might be enough to destroy the direct causal relationship that is attempting to be established.

In Massachusetts, Michelle Carter is on trial for manslaughter. As a teenager, she texted her boyfriend, Roy, and encouraged him to commit suicide. And he did. Her defense attorney is arguing that Roy had mental problems, was already suicidal, and that the texts did not cause him to take his life. The prosecution is arguing that the text did cause Roy to kill himself. This is going to be a difficult case to resolve. As stated by Daniel Medwed, a Northeastern University law professor, “ Causation is going to be a vital part of this case, can the prosecution prove that she caused him to kill himself in this way? Would he have done it anyway ?” 1

Sign Reasoning

Sign reasoning involves inferring a connection between two related situations. The theory is that the presence or absence of one indicates the presence or absence of the other. In other words, the presence of an attribute is a signal that something else, the substance, exists. One doesn't cause the other to exist, but instead is a sign that it exists. Football on television is a sign that Fall has arrived. Football on television does not cause Fall to arrive; they just arrive at the same time. A flag is flying at half-staff. is a sign that that there has been a tragedy or a significant person has died. The flag flying at half-staff did not cause the death. It is a sign that the situation occurred.

Sign Reasoning in Poker

Quite a few players' posture betrays the nature of their cards. An unconscious change in their sitting position, such as leaning forward, likely indicates a strong hand. With a weak hand they often show less body tension, for example, having hanging shoulders.

If someone has concealed his mouth with his hand, he often holds a weak hand - he wants to hide his emotions. In a sense, he does not want his expression to betray his hand. The same is true for a player who is reluctant to glance at you: he is worried that his eyes might indicate he is afraid.

Particularly for beginners, a quick glance at his cards is a reliable tell. The tell here is an unconscious one, brief look at the player's own cards. If, for example, the flop brings 3 hearts and the player looks at his cards, it is unlikely he has the flush.

This is because with an off-suit hand, a beginner usually takes no notice of the suits at first glance. Only with a suited hand will they remember the suit. Thus, you can often assume here that they have at most one heart. 2

Tests of Sign Reasoning

• Other substance/attribute relationships must be considered. Is there another substance that might have the same attributes? Could the sending of roses to your wife be a sign of something other than love? Can the same signs indicate the presence of a valid second or third substance?
• Cumulative sign reasoning produces a more probable connection. The more often this substance/attribute relationship occurs, the more likely it is to repeat itself. If this is the first time you have noticed the association, you will need a good deal of evidence to demonstrate that it really is a valid sign argument.

Comparison Reasoning

Comparison reasoning is also known as reasoning by analogy. This type of reasoning involves drawing comparisons between two similar things, and concluding that, because of the similarities involved, what is correct about one is also correct of the other. There was once an ad for alligator meat that presented this comparison; "When you try alligator meat just remember what is considered exotic food today may often become normal fare in the future. This was the case with lobster. About 75 years ago, lobster was thought of as poor man's food; many New Englanders would not even think of eating it. Today, of course, lobster is a delicacy savored by many people." This type of reasoning wants us to conclude that alligator meat is to humans today, as lobster meat was to humans 75 years ago. And since lobster is now a delicacy so will alligator meat. There are two types of comparisons: figurative and literal.

• Literal comparisons attempt to establish a link between similar classifications; cars to cars, states to states, people to people. For instance, you can compare a Ford compact car with a Toyota compact car; the lottery in one state with the lottery in another state; how your parents treat you with how your best friend is treated by her parents. In these comparisons, similar classifications are being used for the purposes of making the analogy. Literal comparisons can provide logical proof for the point being made and thus can increase the validity of the argument.
• Figurative comparisons attempt to link similarities between two cases from different classifications. Jim Baker of the Bush 2000 campaign, argued after the 5-4 Supreme Court decision awarding the state of Florida to Bush, “Saying George W. Bush stole the Presidency from Al Gore is like saying someone tried to steer the Titanic after it had already hit the iceberg.” Figurative comparisons carry no weight in terms of providing logical proof for an argument. They can, however, be very effective for the purpose of illustration and persuading an audience.

The line between a Literal and Figurative analogy is not clear. Instead of a comparison being totally figurative or totally literal, the comparison can be viewed in degrees using the following continuum.

There are few literal comparisons that can be made between a person and a computer. A person to an animal may have some overlapping actual similarities. While comparing one person to another person suggests a Literal Analogy. The more towards the figurative side the comparison is, the less the argument is logically valid. The more towards the literal side the comparison is, the more logically valid the argument is.

Tests for comparison reasoning

• To be considered as proof, the analogy must be a literal one. The further advocates move away from figurative comparisons and toward the literal comparison end of the continuum, the more validity they secure for their argument. Figurative comparisons carry no logical argumentative influence at all.
• The cases need to contain significant points of similarity. The greater the number of important or major similar points between the cases, the easier it is to establish the comparison as a sound one. However, no matter how many points of similarity can be established between the two cases, major points of differences can destroy the analogy.
• Cumulative comparison reasoning will produce a more probable conclusion. The greater the number of cases a person can use for the purpose of comparison, the more valid the comparison. If a student has been to more than one college or has had many instructors, he or she can evaluate the quality of the teachers by comparing them. The validity of his or her conclusion is increased as the number of teachers compared increases.

Children often try to convince a parent to let them do or try something the parent is opposed to by comparing themselves to another child. They point out they are the same age as the other child, they are in the same grade in school, the child lives in the same neighborhood as they do, thus they should be allowed to do what the other child is allowed to do. This seems to be a very effective argument by comparison until the parent says, you are not that child or we are not their parents. To the parents, these points of difference destroy the comparison the child is trying to make.

Poor Figurative Analogy May 23, 2016

(CNN) Veterans Affairs Secretary Bob McDonald downplayed Monday the time it takes for veterans to receive medical treatment by comparing the "experience" of waiting for health care to Disneyland guests waiting for a ride.

"When you go to Disney, do they measure the number of hours you wait in line? Or what's important?" McDonald told reporters at a Christian Science Monitor

breakfast in Washington. "What's important is what's your satisfaction with the experience?"

American Legion National Commander Dale Barnett excoriated McDonald: "The American Legion agrees that the VA secretary's analogy between Disneyland and VA wait times was an unfortunate comparison because people don't die while waiting to go on Space Mountain." 3

Reasoning from Authority

Reasoning from Authority is used when a person argues that a particular claim is justified, because, it is held or advocated by a credible source. That credible source can be a person or organization. Basically, the authority possesses some credentials that qualify the source as an authority. Thus, you accept the argument because someone you feel is an authority tells you so. You can use this type of argument in two ways. First, you can ask that an argument be accepted simply because someone you consider an authority advocates it. People grant authority status to other people they think have more knowledge than they do: students to teachers, patients to doctors, and clients to lawyers. Children often argue this way when they justify a position by saying “because my mommy or daddy said so.”

Second, you can support your arguments with the credibility of another person. Here you are attempting to transfer the positive ethos from the credible source to the position you are advocating. Advertisers do this when they get popular athletes and entertainers to promote their products. The advertisers are hoping that your positive view of these people will transfer to their product, thus producing higher sales for the products. You may be persuaded to see a particular movie, attend a certain play, or eat at a restaurant because, it was advocated by a well-known critic.

Tests for reasoning from authority

• The authority must be credible . That is, the authority must possess the necessary qualifications for the target audience in order for the source to be used as justification for a point of view. If challenged, the advocate must be prepared to defend the expertise and ethos of his or her authority.
• Views of counter authorities must be taken into account. The advocate must be aware of the other “experts” or highly credible sources who take an opposite position from the one being advocated. If he or she fails to do this, the argument breaks down into a battle over whose expert or authority should be accepted as being the most accurate.
• Cumulative views of authorities increase the validity of the reasoning . Citing more than one expert or authority will increase the likelihood that your position will be viewed as the most valid one being argued.

Important conclusion: Since the process of reasoning by induction usually involves arriving at a conclusion based on a limited sampling, the conclusion to an inductive argument can never be totally certain. Why? Because no matter which type of inductive reasoning is used, nor how carefully critical thinkers adhere to the tests of each reasoning pattern, critical thinkers can never sample the totality of the population used to infer the generalization about that population.

Thus, conclusions drawn from inductive reasoning are always only probable. To use induction effectively, an advocate must demonstrate that the specifics are compelling, and thus justify the conclusion, but never claim that the conclusion is guaranteed in all situations.

Deductive Reasoning

Deductive reasoning is the process of reasoning from general statements, or rules, to a certain, specific, and logical conclusion. Deductive arguments begin with a general statement that has already been arrived at inductively. Unlike inductive reasoning, where the conclusion may be very valid, but is always only probable, the conclusion reached by deductive reasoning is logically certain.

A deductive argument offers two or more premises that lead to a conclusion directly related to those premises. As long as the two premises are sound, there can be no doubt that the final statement is correct. The final statement is a matter of logical certainty.

Deductive arguments are not spoken of as “true” or “false,” but as “sound” or “unsound.” A sound argument is one in which the premises guarantee the conclusion, and an unsound argument is one in which the premises do not guarantee the conclusion.

An advocate who uses deduction to frame an argument must be certain that the general statement is accepted as correct and then must demonstrate the relationship between this general statement and the specific claim, thus proving beyond a doubt the conclusion.

A deductive argument has three parts: a major premise, a minor premise, and a conclusion. This form is called a syllogism.

The major premise is a general statement. For example: All telemarketers are obnoxious . The subject section of the major premise (All telemarketers) is known as the antecedent; the predicate section of the major premise (are obnoxious) is known as the consequent.

The minor premise is a statement of a specific instance related to the major premise:

The person on the phone is a telemarketer.

The conclusion is the statement derived from the minor premises relationship to the major premise: The person on the phone is obnoxious .

An effective deductive argument is one in which your audience accepts the general statement and is then logically compelled by the development of the argument to accept your conclusion.

Thus, we use inductive reasoning to create generalizations or major premises, and we can use deductive reasoning to apply those generalizations to specific situations.

The final step in checking the strength of reasoning is to make sure there are no fallacies. Often, correcting for fallacies is the missing piece to creating and evaluating logical arguments

• Associated Press. ''Just do it, babe': Teen's texts to suicidal boyfriend revealed." New York Post , 9 Sept. 2015, https://nypost.com/2015/09/09/teen-c...st-do-it-babe/ . Accessed 6 November 2019.
• "Poker tells - hidden body language. To bluff or not to bluff?" PokerStrategy.com , https://www.pokerstrategy.com/strategy/live-poker/poker-tells-body-language/ . Accessed 6 November 2019.
• Griffin, Drew. "VA Secretary Disneyland-wait time comparison draws ire." CNN , 23 May 2016, https://www.cnn.com/2016/05/23/politics/veterans-affairs-secretary-disneyland-wait-times/index.html . Accessed 6 November 2019.

Logical Reasoning

I. definition.

Logical reasoning (or just “logic” for short) is one of the fundamental skills of effective thinking. It works by raising questions like:

• If this is true, what else must be true?
• If this is true, what else is probably true?
• If this isn’t true, what else can’t be true?

These are all inferences : they’re connections between a given sentence (the “premise”) and some other sentence (the “conclusion”). Inferences are the basic building blocks of logical reasoning, and there are strict rules governing what counts as a valid inference and what doesn’t — it’s a lot like math, but applied to sentences rather than numbers.

If there is someone at the door, the dog will bark.

Assuming this sentence holds true, there are some other sentences that must also be true.

• If the dog didn’t bark, there is no one at the door.
• Just because the dog barked doesn’t mean there’s someone at the door.

There are also a few sentences that are probably true, such as:

• The dog can sense (hear or smell) when someone is at the door.
• The dog belongs to the people who live in the house where the door is located.

II. Types of Logical Reasoning

There are two basic types of logic, each defined by its own type of inference. They correspond to the two categories in the example from section 1.

• Deduction is when the conclusion, based on the premises, must be true. For example, if it’s true that the dog always barks when someone is at the door and it’s true that there’s someone at the door, then it must be true that the dog will bark. Of course, the real world is messy and doesn’t always conform to the strictures of deductive reasoning (there are probably no actual dogs who always bark when someone’s at the door), but deductive reasoning is still important in fields like law, engineering, and science, where strict truths still hold. All math is deductive.
• Induction is when the conclusion, based on the premises, is probably The answers are less definitive than they are in deductive reasoning, but they are often more useful. Induction is our only way of predicting what will happen in the future: we look at the way things are, and the way they have been in the past, and we make an educated guess about what will probably happen. But all predictions are based on probability, not certainty: for example, it’s extremely probable that the sun will rise tomorrow morning. But it’s not certain , since there are all sorts of catastrophes that could happen in between now and then.

III. Logical Reasoning vs. Critical Thinking

Logic is one of the main pillars of critical thinking . And there’s no question that critical thinking would be impossible without some understanding of logical reasoning. However, there are many other skills involved in critical thinking, such as:

• Empathy , or the ability to imagine what someone else is feeling or experiencing. This is a crucial skill for critical thinking, since it allows you to broaden your perspective and reflect on your actions and beliefs. Empathy also makes you a better student of philosophy because it enables you to put yourself in the author’s shoes and understand the argument from within.
• Analogy , or noticing similarities and thinking them through. Analogies allow us to draw conclusions about, for example, the similarity between our own time and some moment in history, and thus try to make better decisions in the future. This skill is closely related to inductive logic.
• Creativity . Critical thinking is all about innovative problem-solving and coming up with new ideas, so it’s heavily dependent on creativity. Just like a creative art, critical thinking depends on assembling old parts in new ways, working inventively within constraints, and matching moments of inspiration with hours of rigorous craft.

III. Quotes About Logical Reasoning

“I am convinced that the act of thinking logically cannot possibly be natural to the human mind. If it were, then mathematics would be everybody’s easiest course at school and our species would not have taken several millennia to figure out the scientific method.” (Neil Degrasse Tyson)

Neil Degrasse Tyson is an astrophysicist and TV personality who passionately advocates for science and critical thinking. In this quote, he suggests that science and logical reasoning are inherently difficult tasks for the human mind, an organ that evolved to perform a very different set of tasks under very different conditions from the ones we live in today.

“Logic takes care of itself; all we have to do is to look and see how it does it.” (Ludwig Wittgenstein)

Wittgenstein was probably the most influential philosopher of the 20th century, but his views changed dramatically over the course of his life, leading to some controversy as to what he actually thought. This quote is a good example. Early on, Wittgenstein believed that logical reasoning was autonomous — that logical truth was an objective truth, out there in the world for anyone to see if they knew how to look. Later on, though, Wittgenstein started to believe that culture and nature influence the way we see logic, and that logic is therefore not perfectly objective. It’s a tricky question, whether logical reasoning is universal or cultural — it must be tricky if a genius like Wittgenstein couldn’t make up his mind on it!

IV. The History and Importance of Logical Reasoning

Logic is a universal part of the human experience — agriculture would be impossible without inductive reasoning about weather and sunlight, and construction would be impossible without mathematics and deductive reasoning about what makes a structure sturdy.

Formalized logic has appeared in several places with more or less similar results. The Greek philosopher Aristotle is credited with being the first to develop a formal system of logical reasoning, but there were already people in India and China working on formal logic long before Aristotle was born. The Indian, Chinese, and Greek systems were all remarkably similar in their rules, which suggests that there may have been some mutual influence despite the distance. Traders and travelling scholars may have brought ideas about logical reasoning with them all over the world, allowing for rapid development of new ideas.

Logic may seem like a stuffy, abstract discipline used only by philosophers and lawyers, but it has had a profound influence on the history of science and technology as well. Alan Turing, the inventor of the modern computer, was a logician rather than a tinkerer or engineer, and his famous “Turing Machine” was a product of his rigorous training in formal logical reasoning.

V. Logical reasoning in Popular Culture

“Vulcanians do not speculate. I speak from pure logic.” (Spock, Star Trek )

Mr. Spock was raised on Vulcan and trained to be perfectly rational, ignoring all emotion and concentrating on logical reasoning instead. This represents a widespread trope in popular culture — that logic and the emotions are at odds with each other (the head pulling one way and the heart pulling in another). But there’s no reason why logic and the emotions have to be enemies. Uncontrolled emotion certainly clouds logical reasoning — it’s difficult to think rationally if you’re in a rage, for example — but many traditions argue that logic and the emotions should be partners rather than rivals, each providing its own sort of insight in harmony with the other.

On Sherlock , the great detective Sherlock Holmes has a website called “The Art of Deduction,” in which he explains his methods for solving crimes. However, the website has the wrong name — nearly all of Sherlock’s inferences are inductive rather than deductive. That is, they bring together bits and pieces of evidence to develop a theory about what probably happened in a particular crime. They’re not based on the kind of logical certainty that we saw in section 1, but rather on reasoning about likelihoods and probabilities. It’s always logically possible that Sherlock could have it wrong, even though that rarely seems to happen.

a. Reduction and induction

b. Deduction and induction

c. Greek logic and Chinese logic

d. Formal logic and informal logic

d. All of the above

a. Deduction

b. Inference

c. Induction

b. Creativity

How to Teach Analogies to Elementary Students

Why should we teach analogies?

An analogy is simply a comparison of two things that are usually thought of as being different, but are similar in some way.  They are written in a specific format such as  apple : fruit :: carrot : vegetable.  It reads: apple is to fruit as carrot is to vegetable. Teaching analogies helps students understand these comparisons.

I have been teaching analogies to elementary students for years, so I felt validated when I read the research about teaching analogies. In Marzano, Pollock, and Pickering’s book Classroom Instruction That Works, the authors write about 9 instructional strategies that have the greatest effect on student achievement.  Similarities and Differences is one of those strategies.  When you teach analogies, your students are making comparisons.  At the lowest level, these comparisons may be simple sorting activities. But at the highest level, similarities and differences include analogies which is basically reasoning by comparison.

Benefits of Teaching Analogies to Elementary Students:

Okay, so understanding analogies increases student achievement.  But why?  Here are a few reasons: 

• help you identify flaws in student thinking – If a student creates an analogy that doesn’t make sense, you can examine it to understand their thinking and correct misconceptions. On the other hand, if they create an analogy that makes sense, they have demonstrated understanding of a concept.
• teaching analogies teaches students to use critical thinking and logic skills
• part to whole
• increase vocabulary
• help students understand nuances of language
• help students prepare for standardized tests

Background skills needed to solve and teach analogies

I like to make sure that my students have some experience with sorting activities before I teach  analogies.  Sorting helps students learn to categorize things by attributes. Attribute listing in one of the pre-requisites to analogies.  You may think your primary kiddos can’t do this, but I assure you, they can!  

Teaching Analogies to Elementary Students

Teachers should teach analogies as students learn to read.  If your students have little experience with classifying and sorting words and objects, you will need to use concrete objects. Have the students tell how they are similar and different.  Record their ideas on the board.  Next, put them in groups and repeat the activity.

Some easy objects to use are:

• apple and banana
• ball and marble
• cup and glass
• screwdriver and hammers
• shoe and sock

Next, I like teach analogies by showing a PowerPoint so we can solve analogies as a class.  

Four Levels of Complexity

To add more complexity, you can have the students finish the analogy. Example: spider is to 8  as _____________ is to ______________. 

The highest level I do with students is to have them complete an analogy and then create an analogy that has the same relationship.

In the example above, we get milk from cows and eggs from hens. So, a new analogy could be sheep is to wool as oyster is to pearl. This is just one example. The kids wouldn’t have to use animals at all.  For example, they could say tree is to paper as wheat is to flour because paper comes from trees and flour comes from wheat. 

When can you teach analogies?

You can use teach analogies in almost every lesson. They can be used as daily warm-ups, in centers, as a question during a lesson, and as exit tickets.

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What is an Analogy?

Have you ever tried to explain something but ended up just making a comparison instead? 

Whether you realized it or not, you were using an analogy. 

An analogy is a comparison that helps you understand a complex idea by relating it to something more familiar. Elizabeth Bowen said in her novel House in Paris : "An analogy is like a signpost that points to an unfamiliar destination." 

Analogies are an essential part of daily communication and reasoning. They help you make sense of new information and connect it to what you already know. 

While analogies are often confused with similes and metaphors, there are important differences between these three literary devices:

• A simile uses "like" or "as" to make a comparison. 
• A metaphor is a figure of speech that describes something as if it were something else. 
• An analogy highlights the similarities and differences between two things in order to explain one of them in terms of the other.

Understanding the different types of analogies is crucial for effective communication and problem-solving. So keep reading to learn more about them. 

The 3 Types of Analogies (with Examples!)

There are three main types of analogies: part to whole analogies, opposites analogies, and synonym analogies. Each type has its own unique characteristics and importance in different contexts.

Part to Whole Analogies + Examples

Part to whole analogies compare a part of something to the whole thing. They're used to help you understand how the parts of something relate to the whole. For example:

• The roof is to the house as the lid is to the box.
• The hand is to the body as a petal is to a flower.

Part to whole analogies are often used in science and engineering to explain complex systems. They're also useful in defining hierarchies and organizational structures.

Opposites Analogies + Examples

Opposites analogies compare two things that are opposite to each other. They're used to help you understand the relationship between two things that are seemingly different. For example:

• Hot is to cold as up is to down.
• Memory holds on to love, as a saucer holds the cup.

Opposites analogies are commonly used in literature and poetry to create contrast and emphasize differences. They're also useful in problem-solving and decision-making, as they can help you identify contradictions and inconsistencies.

Synonym Analogies + Examples

A synonym analogy compares two words that have similar meanings. They're used to help you understand how words are related to each other. For example:

• Happy is to joyful as sad is to sorrowful.
• Big is to large as small is to tiny.

Synonym analogies are important in language and communication because they help you expand your vocabulary and express yourself more precisely. They're also useful in test-taking and academic settings, as they can help you learn and remember new words.

Have you used any of these types of analogies before? 

How to Solve Analogy Problems

There’s a good chance you’ve come across analogy problems in the past, whether it’s through a school assignment, trivia, or, you know, a brain-teaser on social media. 

@elevateapp Did you solve this? #braintraining #elevateapp #elevatebraintraining #analogy #puzzletok #riddletok #puzzles ♬ freaking out the neighborhood by mac demarcoシ - offline

And solving these analogy problems involves a series of steps and strategies: 

• Identify the relationship between the first pair of words. This can be done by looking for common characteristics, actions, or qualities.
• Find a word that has a similar relationship to the second pair of words. This can be done by applying the same relationship found in Step 1 to a new set of words.
• Check to make sure the analogy makes sense. Does the word you chose accurately reflect the relationship between the second pair of words? 

For example, consider the analogy "tree is to leaf as flower is to ___." The relationship between tree and leaf is that a leaf is part of a tree. Applying this relationship to the second pair of words, you can deduce that a petal is part of a flower. Therefore, the answer is petal.

While you’re solving the analogy, consider these strategies: 

• Look for keywords that indicate the type of relationship, such as cause-and-effect or part-and-whole.
• Break down complex words into their root components to find similarities with other words.
• Use context clues from the sentence or passage to help you understand the meaning of unfamiliar words.

Solving analogies in this way isn’t just fun—it’s also a powerful tool for developing your critical thinking skills because they require you to recognize patterns and make connections between seemingly unrelated things. By understanding analogies, you can:

• Develop flexible thinking skills by learning how to apply different types of relationships to different situations.
• Improve problem-solving skills by identifying similarities and differences between different scenarios.
• Enhance creativity by learning how to use analogies to generate new ideas and perspectives.

Ready to put your skills to the test? Here are a few practice exercises to help you improve your analogy problem-solving skills:

Dog is to bark as cat is to _____

Knife is to cut as hammer is to _____

Water is to wet as fire is to _____

Do you know the answers? Here's one more to try:

@elevateapp Follow for daily brain teasers 🧠 #ElevateApp #ElevateBrainTraining #BrainGames #AnalogyTikTok #BrainTeasers #CognitiveSkills #WordPuzzles ♬ original sound - Rememberthe80sn90s

Applying Analogies in Real-life Situations

As we’ve mentioned, analogies are not just for academics and intellectuals. They're also useful in everyday life because, since analogies are comparisons between two things that highlight their similarities and differences, they help you relate unfamiliar concepts to things you already know and understand.

For example, "finding a needle in a haystack" is an analogy that describes a difficult task. "Rearranging deck chairs on the Titanic" is an analogy that describes a futile effort.

Specifically, analogies are used in a wide range of real-life situations, including teaching, business, and storytelling.

Analogies are frequently used in education to help students understand complex concepts. For example, a teacher might use an analogy to explain how electricity works by comparing it to water flowing through pipes, which helps students visualize the flow of electricity and understand how it travels through wires.

Analogies are also useful in business for explaining complex ideas to non-experts. For example, a salesperson might use an analogy to explain how a new product works by comparing it to a product that the customer is already familiar with, which helps the customer understand the benefits of the new product and why they should consider purchasing it.

Storytelling

Analogies are often used in storytelling to create vivid images and make emotional connections with readers or listeners. For example, the phrase "life is like a box of chocolates" from the movie Forrest Gump is an analogy that describes the unpredictability of life, which helps viewers relate to the character and understand the challenges he faces.

Using the Elevate App to Understand Analogies

If you’re looking for a fun and effective tool to help you better understand analogies and confidently communicate them yourself, you’ve come to the right place. 

Meet: the Elevate app, available on iOS and Android . Elevate’s 40+ games and themed workouts are designed to boost your cognitive abilities, making it the perfect tool for anyone looking to take their communication—from speaking , to vocabulary , to reading , and even writing skills—to the next level. 

Here’s a preview of some of Elevate’s games to help you excel at analogies: 

• Elevate’s Transitions game teaches you how to connect complex ideas and become a more persuasive speaker. 
• The Recall game helps you practice rapidly retrieving words from memory.
• The Eloquence game teaches you how to choose the best words to communicate tone and meaning.
• The Clarity game helps you learn how to express yourself clearly and directly. 
• The Synonyms game teaches you how to expand your vocabulary to avoid sounding repetitive. 

And if you want to see how your communication skills stack up, we have good news: You can track your progress with the Elevate app. 

That’s because when you play Elevate, your difficulty level will automatically adjust based on your performance over time. And you can see where you stand any time with Elevate’s Performance Tab. More specifically, you can see the Skill Groups you’re excelling in, as well as where you rank against other Elevate members in your age group. 

If that sounds like fun to you, why not give it a try for free? 

Start Learning Analogies Today 

Whether you're trying to write an analogy, solve an analogy problem, or use analogies in everyday life, analogies are an essential part of logical argument and communication. As William Shakespeare once said, "Call a rose by any other name would smell as sweet." Analogies, similes, and metaphors are all figures of speech that help us express ourselves more vividly and creatively.

Plus, solving them is also a fun way to improve your critical thinking skills and better understand real-life situations while also having a good time. 

But if you want to continue learning how to effectively communicate analogies yourself, download the Elevate app on iOS or Android to play 40+ brain training games that'll help improve your vocabulary, spelling, and other cognitive skills, all while helping your brain stay sharp. 

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• What do “elbow” and “below” have in common? Learn more about anagrams and how to use them.

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11.1 Developing Your Sense of Logic

Learning outcomes.

By the end of this section, you will be able to:

• Identify key rhetorical concepts and thought patterns in a variety of texts.
• Explain how patterns of thought function for different audiences, purposes, and situations.

For the purposes of this course, logic means “reasoning based on thought and evidence.” In practical terms, logic is the ability to analyze and evaluate persuasive or argument writing for effectiveness. By extension, it also means that you can learn to use logic in your own argumentative writing. Like any other new skill, you are likely to learn best when you have a starting point. Here are some suggestions for how to begin thinking and writing logically:

• Approach a topic with an open mind.
• Consider what you already know about the topic.
• Consider what you want to know about the topic.
• Find credible information about the topic.
• Base your judgments of the topic on sound reasoning and evidence.

Once you have formed your opinions on a particular debatable subject, you must decide on the best way to organize them to share with others. Developing your skills in six widely used reasoning strategies , or patterns for thinking and writing, can help you determine the most logical and effective means of organizing information to make your points.

In this chapter, you will examine these six reasoning strategies—analogy, cause and effect, classification and division, comparison and contrast, problem and solution, and definition—that are often used in college classes. In addition, you will consider how writers’ personal views, cultural backgrounds, and purposes for writing help determine

• which reasoning strategy suits their needs; and
• what they decide to include in their writing.

As you progress in your college classes and beyond, you will find these reasoning strategies used in all genres of writing, both nonfiction (e.g., textbooks, how-to books) and fiction (e.g., novels, short stories). Understanding how these strategies work can help you recognize their common formats and analyze what you read; likewise, as a writer, understanding how these strategies work to reflect your thinking can help you determine the strategy you need to use.

Writers frequently use analogy as a strategy to compare two unlike subjects—one subject is familiar to readers, whereas the other is not. To explain or clarify the unfamiliar subject, the writer emphasizes the way or ways in which the two subjects are similar, even though they are dissimilar and unrelated in all other ways. Analogies are basically long forms of similes (short comparisons of unlike elements, based on the word like or as ) or metaphors (short comparisons without signal words). In the example paragraph, the writer explains unfamiliar aspects of the COVID-19 pandemic by comparing it with the more familiar concept of a robbery spree.

Model Paragraph

student sample text Examining COVID-19 is like examining a robbery case in this way: both require a great deal of investigation. Those investigating the causes behind the pandemic look for the history of how the virus spread, and those investigating a crime look for the backstory that might connect the victims and criminals. In addition, the two groups of investigators look at the reasons behind the focus of their study. Medical investigators look at why the virus spread throughout the world; police investigators look at why the crime spree took place in a particular area. Also, both types of investigators are trying to stop whatever or whoever is the focus of their investigation. Medical investigators want to stop the virus; police investigators want to stop the crimes. end student sample text

Cause and Effect

Cause-and-effect writing identifies and examines the reasons (causes) for and consequences (effects) of an action, event, or idea. Cause-and-effect writing often answers the question “Why?” and helps readers understand the connections between what happens because of—or as a result of—something else.

student sample text Ray’s grocery, Artie’s Hardware store, and Cradle and Teen department store all went out of business because a well-known superstore opened in Springdale. Customers who frequented Ray’s, an establishment that had been run by the same family for four generations, used to drive many miles to take advantage of the high quality of items in the meat and deli departments. After the opening of the superstore, however, those same customers found they could get similar items at a savings, even if the quality was not as high as the products at Ray’s. Customers at Artie’s Hardware often talked with owner Artie Shoeman about their hardware needs, but the store did not offer the same variety of items they could find in the superstore. The same was true for those who shopped at Cradle and Teen. The superstore featured lower prices and more variety, even if the items did not match the quality of the items at Cradle and Teen. end student sample text

Classification and Division

Classification and division are actually two closely related strategies, generally discussed together because of their similarity. When using the strategy of division, the writer identifies a single subject or group and explains categories within that subject or group. In other words, the writer divides the larger unit into component parts. When using the strategy of classification, writers do the opposite. They group various elements and place them into larger, more comprehensive categories rather than divide the whole into parts. In general, the reasoning strategy of classification and division looks at smaller elements as parts of a larger element and thus helps readers understand a general concept and the elements that it comprises.

Model Paragraphs

student sample text Extra material in the textbook can be divided into photographs, quotations, and tables. The photographs were all taken by the author and focus on various parts of the life cycle of the plants highlighted in the chapter. In addition, to add color and more information about the subject matter of each chapter, the author has inserted sidebar quotations from both famous and non-famous people. The tables the author has included help readers see more details about the progression of the plants’ spread across the country. end student sample text

student sample text After three months of training, the young dogs were placed into three categories: those who would go directly to permanent homes, those who would repeat the course, and those who would advance to the next level. The dogs that would be homed immediately were those who were far too social or far too active to be service dogs. The dogs that would repeat the course had possibilities as service dogs but needed more discipline and instruction. Their futures were yet to be decided. Those that advanced to the next level were obedient and focused and learned quickly. They displayed great promise as service dogs. end student sample text

Comparison and Contrast

Compare and contrast , one of the most frequently used reasoning strategies, analyzes two (sometimes more) subjects, examining the similarities (comparisons) and differences (contrasts) between them. Nearly everything you can think of can be a subject for comparison and contrast: objects, people, concepts, places, movies, literature, and styles, to name a few. To elaborate on the separate points, writers provide details about each element being compared or contrasted. Comparison and contrast helps readers analyze and evaluate subjects.

This strategy is helpful when the similarities or differences are not obvious and when a significant common thread exists between the subjects. For example, a contrast between an expensive, elegant restaurant and a fast-food restaurant would be useless because the differences are clearly obvious, despite the common thread—both are restaurants. However, not so obvious might be some similarities.

When subjects have no common thread or have obvious shared characteristics, any comparison or contrast makes little sense—like contrasting a fish and a shoe (no common thread) or comparing two fast-food restaurants (obvious similarities). However, a writer actually might find a common thread between a fish and a shoe (perhaps shine or texture or color), and a valid topic of contrast might be differences between the two fast-food restaurants.

student sample text Although they seem different on the surface, one way in which Romantic-period poetry and 1980s rap music are alike is the desire the writers had to create a new approach to their art. They wanted to represent simpler values that were more connected to the natural world, values to which a general audience could relate. For example, in William Wordsworth ’s “Daffodils,” the speaker can escape the depressing, industrialized urban world to find peace in nature by contemplating a field of flowers. Similarly, in the Sugarhill Gang ’s 1979 “Rapper’s Delight,” the band sings of how their beats can lift spirits and cause listeners to dance and forget their woes. However, Romantic-period poetry and 1980s rap music are different in the delivery style and form of the art; “Rapper’s Delight” is set to music, which is an integral part of the piece, but “Daffodils” is not. end student sample text

Problem and Solution

When using this reasoning strategy, writers introduce a predicament or challenging issue (the problem) and offer information about what was done or what should be done to remedy the predicament or issue (the solution). Problem-and-solution writing helps readers understand the complexities of some predicaments and the actions that can improve or eliminate them.

student sample text The issue of combating the spread of hate speech and misinformation on social media can be addressed if more social media providers improve their monitoring services. Aside from creating more algorithms that search for linked key words and phrases, social media providers should increase the number of professional monitors conducting active searches. Additionally, while many platforms such as Twitter and Facebook respond within a few days to reports of posts that violate their policies, more monitors could lessen the amount of time these posts are available. According to Facebook, inappropriate posts are investigated and removed within 24 to 48 hours (Facebook “Community Standards”). Some offenders have been reported multiple times for their platform violations, and social media sponsors should increase their monitoring of those offenders. Although such surveillance would increase the burden on the social media providers, it would help solve the growing challenge of online hate speech and misinformation. end student sample text

When using the reasoning strategy of definition , writers elaborate on the meaning of an idea, a word, or an expression, usually one that is controversial or that can be viewed in multiple ways. Beginning writers tend to think that definition writing looks only at the denotation , or dictionary definition. However, definition writing entails much more than relaying a dictionary definition. It also explains and elaborates on the connotations , the emotions and implications the topic evokes. Definition writing is especially useful for explaining and interpreting terms, ideas, or concepts that are easily or often confused or that have meanings beyond their denotations. Sometimes these meanings are personal interpretations and thus reflect a writer’s particular viewpoint. Additionally, this strategy is beneficial when writers want to explain or reinforce a term before making an argument about a larger concept.

student sample text In everyday speech, the word critical is often used to highlight negative aspects of a topic. If someone says a friend was critical of a new haircut, the implication is that the friend did not like the cut. However, when used in college classes, critical has an expanded meaning: noting both the negative and positive aspects of a topic, examining those aspects in depth, and then making decisions about the discoveries. Students directed to use critical thinking, critical reading, or critical writing should know they are expected to examine all sides of a topic fully, evaluate the validity of those sides, and then make sound judgments on the basis of their evaluation. end student sample text

In this chapter, you have learned about various reasoning strategies that you may use in academic and professional writing. Utilizing these strategies when you write can help you both evaluate and analyze text that you read and create more logical and persuasive arguments.

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Math Analogies Level 4

Analogical & mathematical reasoning puzzles using standards-based analogies.

Grades: 8-9

Mathematics

•  Multiple Award Winner
• Paperback Book - $12.99
• eBook - $12.99

Description and Features

Analogies occur in life and frequently in high-stakes tests. Understanding analogies and the ability to reason analogically (reasoning used to identify, evaluate, and solve an analogy) are important problem-solving skills which are an essential part of mathematical development. The immediate benefit is to recognize and solve simple analogies. The long-term benefits are improved reasoning skills that enable students to break problems into their component parts, recognize analogies embedded in arguments, and evaluate them. The 200 analogies in this 64-page book teach students to break problems down into their component parts, making it easier to recognize familiar formats that enable students to produce solutions. These analogies are designed around the grade-appropriate standards identified by the National Council of Teaching Mathematics.

Product Details

General license - download.

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Dr. Toy Best Smart Play / Smart Toy Product of Excellence Award

Test preparation available for the following:

Math Analogies prepares students to achieve top scores on all state assessments including Naglieri Nonverbal Ability Test® (NNAT®) Cognitive Abilities Test™ (CogAT®) Gifted and Talented Education (GATE) / Talented and Gifted (TAG) Wechsler Intelligence Scales for Children® (WISC®) Wechsler Preschool and Primary Scale of Intelligence™ (WPPSI™) and many more!

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