paraconsistent logic essays on the inconsistent

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Paraconsistent Logic: Essays on the Inconsistent (Analytica) Hardcover – January 1, 1989

• Print length 715 pages
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• Publisher Philosophia Verlag Gmbh
• Publication date January 1, 1989
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• ISBN-10 3884050583
• ISBN-13 978-3884050583
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• Publisher ‏ : ‎ Philosophia Verlag Gmbh (January 1, 1989)
• Language ‏ : ‎ English
• Hardcover ‏ : ‎ 715 pages
• ISBN-10 ‏ : ‎ 3884050583
• ISBN-13 ‏ : ‎ 978-3884050583
• Item Weight ‏ : ‎ 2.85 pounds
• Dimensions ‏ : ‎ 6.75 x 2 x 9.75 inches

About the author

Lorenzo peña.

Lorenzo Peña y Gonzalo inició su carrera académica en la Universidad Pontificia de Quito en 1974, recién licenciado en filosofía, reanudándola en 1979 tras cuatro años de estudios de tercer ciclo en Lieja, donde recibió su doctorado en filosofía con una tesis sobre la lógica contradictorial, escrita bajo la supervisión de Paul Gochet.

Procedente de un hegelianismo juvenil de tinte marxista --matizado por la afición a otros filósofos, como Platón, Leibniz y Nicolai Hartmann--, Lorenzo, en los años 70 --bajo nuevas influencias, sobre todo las de Frege y Quine--, fue adoptando progresivamente los métodos de la filosofía analítica y de la formalización matemática. De su opción primitiva conservó una creencia en las contradicciones verdaderas (expresada por Hegel en su célebre aserto: contradictio est regula ueri, non-contradictio falsi); sólo que ahora implementada mediante la lógica gradualista o difusa (gracias a las aportaciones de Lukasiewicz, L. Zadeh y da Costa).

Tras regresar a España en 1983, Lorenzo fue profesor de la Universidad de León hasta incorporarse al CSIC en 1987 como investigador científico. En 1992-93 fue investigador visitante en Canberra (Research School of Social Sciences, ANU).

A lo largo de la mayor parte de su itinerario Lorenzo se consagró esencialmente a las dos disciplinas de la lógica y la metafísica, que siempre quiso unir desde una perspectiva de realismo ontológico, teniendo que afrontar ásperas oposiciones desde los paradigmas establecidos, a uno y otro lado de la frontera entre sendas áreas de conocimiento. En aquella época su obra más significativa fue precisamente El ente y su ser: Un estudio lógico-metafísico (1985). En esos años Lorenzo publicó asimismo trabajos de teodicea, filosofía lingüística, teoría del conocimiento e historia de la filosofía (Platón, Leibniz, Nicolás de Cusa), con breves incursiones en otros campos filosóficos. Como rótulo característico de aquella etapa de su trayectoria, Lorenzo acuñó el de ontofántica.

A partir de 1992, las preocupaciones intelectuales de Lorenzo fueron experimentando una paulatina modificación, concentrándose, primero, en la lógica de las normas para después irse consagrando a los problemas fundamentales de la filosofía del Derecho --incluyendo, entre ellos, la lógica de las situaciones jurídicas. Esa reconversión se tradujo en un nuevo recorrido discente --licenciatura (2004) y Doctorado (2015) en Derecho-- y en su adscripción al área universitaria de filosofía jurídica, a la que pertenece en la actualidad.

Durante estos últimos años, el esfuerzo investigativo de Lorenzo se ha centrado en ahondar en los problemas de la racionalidad jurídica, proponiendo una visión del Derecho como un sistema de normas comprometido, por su propia esencia, a incorporar unos cánones no promulgados, sin los cuales ningún conjunto de prescripciones podría ser un genuino ordenamiento jurídico. Tales cánones constituyen un verdadero Derecho Natural, en la línea del racionalismo clásico. La impronta de Leibniz reaparece aquí en toda su pujanza.

En este nuevo período, su principal obra ha sido el libro VISIÓN LÓGICA DEL DERECHO: Una defensa del racionalismo jurídico (2017).

Desarrollos ulteriores del contenido de ese libro vienen presentados en las Lecciones Laurentinas 2019-21 (http://jurid.net/multi/lecciones/)

Actualmente los dos objetivos primordiales de Lorenzo son:

(1º) una autobiografía intelectual; y

(2º) una investigación sobre el bien común como principio fenfamental no sólo del Derecho, sino también de la política.

Fuera del terreno académico, Lorenzo dedica su tiempo libre a aprender historia y a escribir artículos de opinión sobre temas de actualidad política.

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Paraconsistent Logic. Essays on the Inconsistent. Munchen: Philosophia Verlag, 1990. Graham Priest, Richard Routley and Jean Norman (eds.)

1991, Philosophica

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Francesco Berto , Koji Tanaka

A logic is called 'paraconsistent' if it rejects the rule called 'ex contradictione quodlibet', according to which any conclusion follows from inconsistent premises. While logicians have proposed many technically developed paraconsistent logical systems and contemporary philosophers like Graham Priest have advanced the view that some contradictions can be true, and advocated a paraconsistent logic to deal with them, until recent times these systems have been little understood by philosophers. This book presents a comprehensive overview on paraconsistent logical systems to change this situation. The book includes almost every major author currently working in the field. The papers are on the cutting edge of the literature some of which discuss current debates and others present important new ideas. The editors have avoided papers about technical details of paraconsistent logic, but instead concentrated upon works that discuss more "big picture" ideas. Different treatments of paradoxes takes centre stage in many of the papers, but also there are several papers on how to interpret paraconistent logic and some on how it can be applied to philosophy of mathematics, the philosophy of language, and metaphysics.

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Classical propositional logic can be characterized, indirectly , by means of a complementary formal system whose theorems are exactly those formulas that are not classical tautologies, i.e., contradictions and truth-functional contingencies. Since a formula is contingent if and only if its negation is also contingent, the system in question is paraconsistent. Hence classical propositional logic itself admits of a paraconsistent characterization, albeit "in the negative". More generally, any decidable logic with a syntactically incomplete proof theory allows for a paraconsistent characterization of its set of theorems. This, we note, has important bearing on the very nature of paraconsistency as standardly characterized.

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Paraconsistent logics are logical systems that reject the classical conception, usually dubbed Explosion, that a contradiction implies everything. However, the received view about paraconsistency focuses only the inferential version of Explosion, which is concerned with formulae, thereby overlooking other possible accounts. In this paper, we propose to focus, additionally, on a meta-inferential version of Explosion, i.e. which is concerned with inferences or sequents. In doing so, we will offer a new characterization of paraconsistency by means of which a logic is paraconsistent if it invalidates either the inferential or the meta-inferential notion of Explosion. We show the non-triviality of this criterion by discussing a number of logics. On the one hand, logics which validate an invalidate both versions of Explosion, such as classical logic and Asenjo-Priest's 3-valued logic LP. On the other hand, logics which validate one version of Explosion but not the other, such as the substructural logics TS and ST, introduced by Malinowski and Cobreros, Egré, Ripley and van Rooij, which are obtained via Malinowski's and Frankowski's q-and p-matrices, respectively.

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A logic is paraconsistent if it allows for non-trivial inconsistent theories. Given the usual definition of inconsistency, the notion of paraconsistent logic seems to rely upon the interpretation of the sign ‘¬’. As paraconsistent logic challenges properties of negation taken to be basic in other contexts, it is disputable that an operator lacking those properties will count as real negation. The conclusion is that there cannot be truly paraconsistent logics. This objection can be met from a substructural perspective since paraconsistent sequent calculi can be built with the same operational rules as classical logic but with slightly different structural rules.

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Max Cresswell and Hilary Putnam seem to hold the view, often shared by classical logicians, that paraconsistent logic has not been made sense of, despite its well-developed mathematics. In this paper, I examine the nature of logic in order to understand what it means to make sense of logic. I then show that, just as one can make sense of non-normal modal logics (as Cresswell demonstrates), we can make ‘sense’ of paraconsistent logic. Finally, I turn the tables on classical logicians and ask what sense can be made of explosive reasoning. While I acknowledge a bias on this issue, it is not clear that even classical logicians can answer this question.

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In some cases one is provided with inconsistent information and has to reason about various consistent scenarios contained in that information. Our goal is to argue that filtered paraconsistent logics are not the right tool to handle such cases and that the problems generalize to a large class of paraconsistent logics. A wide class of paraconsistent (inconsistency-tolerant) logics is obtained by filtration: adding conditions on the classical consequence operation (one example is weak Rescher-Manor consequence --- which bears $\Gamma$ to $\phi$ just in case $\phi$ follows classicaly from at least one maximally consistent subset of $\Gamma$). We start with surveying the most promising candidates and comparing their strength. Then we discuss the mainstream views on how non-classical logics should be chosen for an application and argue that none of these allows us to chose any of the filtered logics for action-guiding reasoning with inconsistent information, roughly because such a reasoning has to start with selecting possible scenarios and such a process does not correspond to any of the mathematical models offered by filtered paraconsistent logics. Finally, we criticize a recent attempt to defend explorative hypothetical reasoning by means of weak Rescher-Manor consequence operation by Meheus et al.

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Paraconsistent Logic

• Published: 27 February 2015
• Volume 44 , pages 771–780, ( 2015 )

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• David Ripley 1  

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In some logics, anything whatsoever follows from a contradiction; call these logics explosive . Paraconsistent logics are logics that are not explosive. Paraconsistent logics have a long and fruitful history, and no doubt a long and fruitful future. To give some sense of the situation, I’ll spend Section 1 exploring exactly what it takes for a logic to be paraconsistent. It will emerge that there is considerable open texture to the idea. In Section 2 , I’ll give some examples of techniques for developing paraconsistent logics. In Section 3 , I’ll discuss what seem to me to be some promising applications of certain paraconsistent logics. In fact, however, I don’t think there’s all that much to the concept ‘paraconsistent’ itself; the collection of paraconsistent logics is far too heterogenous to be very productively dealt with under a single label. Perhaps that will emerge as we go.

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Paraconsistent Logics: Preamble

Making Sense of Paraconsistent Logic: The Nature of Logic, Classical Logic and Paraconsistent Logic

Paraconsistent Metatheory: New Proofs with Old Tools

I have nothing much to say about which things are and which things are not consequence relations. I certainly have no precise definition in mind; usual precise definitions exclude some things I include. (For example, understanding a consequence relation as a Tarskian closure operation excludes a wide variety of substructural logics.) If it so much as smells consequencey it’s a consequence relation as far as I’m concerned.

For the additive conjunction ∧, we have A ∧¬ A , A ∧¬ A ⊩ B , but A ∧¬ A ⊯ B ; two copies of a contradiction do entail everything, but a single copy does not.

You might note that I haven’t said anything about what it takes to count as a negation or a conjunction or a way of combining premises in the first place. No way am I going near that can of worms.

This works for the subvaluationist ‘global’ consequence. For more on subvaluationist logic, see [ 11 , 19 , 46 , 47 ].

For more on LP, see [ 7 , 26 , 29 ].

For more on the C systems, see [ 10 , 13 ].

Note that there is still something of an indeterministic flavour: if you know that A is designated, that’s not yet enough to say whether ¬ A is designated or not. But the facts that underlie this are fully deterministic.

Any fully structural logic can be seen as preserving some status; this is one upshot of Suszko’s Thesis; see eg [ 16 , 18 , 43 , 44 ]. But this is sometimes not the most helpful way to look at it. Moreover, the suggested generalization of this method extends well beyond fully structural logics, as in [ 22 ].

Typically, this is because we haven’t noticed the contradiction, but there are at least two other kinds of case: 1) cases in which we notice the contradiction, but haven’t yet decided how or whether to resolve it, and 2) cases in which we notice the contradiction, but have decided simply to live with it.

See [ 2 , 37 ] for details.

Smooth infinitesimal analysis provides an example. It is a rich topic of mathematical study based on axioms that are inconsistent in classical logic, but not in the intuitionist logic in which it is conducted. See eg [ 25 ].

Alxatib, S., Pagin, P., & Sauerland, U. (2013). Acceptable contradictions: Pragmatics or semantics? a reply to Cobreros et al. Journal of Philosophical Logic , 42 (4), 619–634.

Article   Google Scholar  

Alxatib, S., & Pelletier, J. (2011). The psychology of vagueness: Borderline cases and contradictions. Mind and Language , 26 (3), 287–326.

Anderson, A.R., & Belnap, N.D. (1975). Entailment: The Logic of Relevance and Necessity, vol. 1 . Princeton: Princeton University Press.

Google Scholar  

Arruda, A.I. (1989). Aspects of the historical development of paraconsistent logic. In: [33], (pp. 99–130).

Batens, D. (1989). Dynamic dialectical logics. In: [33], (pp. 187–217).

Batens, D. (2000). Towards the unification of inconsistency handling mechanisms. Logic and Logical Philosophy , 8 , 5–31.

Beall, J., & van Fraassen, B.C. (2003). Possibilities and Paradox: An Introduction to Modal and Many-valued Logic . Oxford: Oxford University Press.

Belnap, N.D. (1982). Display logic. Journal of Philosophical Logic , 11 (4), 375–417.

Brady, R.T. (2006). Universal Logic . Stanford: CSLI Publications.

Carnielli, W.A., & Marcos, J. (1999). Limits for paraconsistent calculi. Notre Dame Journal of Formal Logic , 40 (3), 375–390.

Cobreros, P. (2013). Vagueness: Subvaluationism. Philosophy Compass , 8 (5), 472–485.

Cobreros, P., Égré, P., Ripley, D., van Rooij, R. (2012). Tolerant, classical, strict. Journal of Philosophical Logic , 41 (2), 347–385.

da Costa, N. (1974). On the theory of inconsistent formal systems. Notre Dame Journal of Formal Logic , 15 , 497–510.

Dietz, R. (2011). The paradox of vagueness In Horsten, L., & Pettigrew, R. (Eds.), The Continuum Companion to Philosophical Logic , (pp. 128–179). Great Britain: Continuum.

Dunn, J.M. (1993). Star and perp: Two treatments of negation. Philosophical Perspectives , 7 , 331–357.

Dunn, J.M., & Hardegree, G.M. (2001). Algebraic Methods in Philosophical Logic . Oxford: Oxford University Press.

Goré, R. (1998). Gaggles, Gentzen, and Galois: How to display your favorite substructural logic. Logic Journal of the IGPL , 6 (5), 669–694.

Humberstone, L. (2012). The Connectives . Cambridge: MIT Press.

Hyde, D. (1997). From heaps and gaps to heaps of gluts. Mind , 106 , 641–660.

Hyde, D. (2008). Vagueness, Logic, and Ontology . Hampshire: Ashgate.

Malinowski, G. (1990). Q-consequence operation. Reports on Mathematical Logic , 24 , 49–59.

Meyer, R.K., & Routley, R. (1972). Algebraic analysis of entailment I. Logique et Analyse , 15 , 407–428.

Meyer, R.K., & Routley, R. (1974). Classical relevant logics II. Studia Logica , 33 (2), 183–194.

Mortensen, C. (1995). Inconsistent Mathematics . Dordrecht: Kluwer Academic Publishers.

Book   Google Scholar  

O’Connor, M. (2008). An introduction to smooth infinitesimal analysis. arXiv: 0805.3307 [math.GM].

Priest, G. (1979). Logic of paradox. Journal of Philosophical Logic , 8 , 219–241.

Priest, G. (2002). Paraconsistent logic. In Gabbay, D.M., & Guenthner, F. (Eds.), Handbook of Philosophical Logic , 2nd edn. Vol. 6. Dordrecht: Kluwer Academic Publishers. 2 edition.

Priest, G. (2005). Towards Non-Being: The Logic and Metaphysics of Intentionality . Oxford: Oxford University Press.

Priest, G. (2008). An Introduction to Non-Classical Logic: From If to Is , 2nd edn. Cambridge: Cambridge University Press.

Priest, G. (2013). Mathematical pluralism. Logic Journal of the IGPL , 21 (1), 4–13.

Priest, G., & Routley, R. (1989a). First historical introduction: A preliminary history of paraconsistent and dialetheic approaches. In: [33], (pp. 3–75).

Priest, G., & Routley, R. (1989b). Systems of paraconsistent logic. In: [33], (pp. 151–186).

Priest, G., Routley, R., Norman, J. (Eds.) (1989). Paraconsistent Logic: Essays on the Inconsistent . Munich: Philosophia Verlag.

Restall, G. (1997). Ways things can’t be. Notre Dame Journal of Formal Logic , 38 (4), 583–596.

Restall, G. (1999). Negation in relevant logics: How I stopped worrying and learned to love the Routley star In Gabbay, D., & Wansing, H. (Eds.), What is Negation? (pp. 53–76). Dordrecht: Kluwer Academic Publishers.

Chapter   Google Scholar  

Restall, G. (2000). An Introduction to Substructural Logics . London: Routledge.

Ripley, D., Nouwen, R., Rooij, R., Schmitz, H.-C., Sauerland, U. (2011a). Contradictions at the borders. Vagueness and Communication , (pp. 169–188): Springer.

Ripley, D. (2011b). Inconstancy and inconsistency In Cintula, P., Fermüller, C., Godo, L., Hájek, P. (Eds.), Understanding Vagueness: Logical, Philosophical, and Linguistic Perspectives , (pp. 41–58): College Publications.

Ripley, D. (2012). Structures and circumstances: Two ways to fine-grain propositions. Synthese , 189 (1), 97–118.

Routley, R., Meyer, R.K., Plumwood, V., Brady, R.T. (1982). Relevant Logics and their Rivals 1 . Atascadero: Ridgeview.

Routley, R., & Routley, V. (1972). The semantics of first degree entailment. Noûs , 6 (4), 335–359.

Routley, R., & Routley, V. (1975). The role of inconsistent and incomplete theories in the semantics of belief. Communication and Cognition , 8 , 185–235.

Shoesmith, D.J., & Smiley, T.J. (1978). Multiple-conclusion Logic . Cambridge: Cambridge University Press.

Shramko, Y., & Wansing, H. (2012). Truth values. In Zalta, E.N. (Ed.), The Stanford Encyclopedia of Philosophy . Winter 2012 edition.

Urbas, I. (1990). Paraconsistency. Studies in Soviet Thought , 39 (3/4), 343–354.

Varzi, A. (1997). Inconsistency without contradiction. Notre Dame Journal of Formal Logic , 38 (4), 621–639.

Varzi, A. (2000). Supervaluationism and paraconsistency In Batens, D., Mortensen, C., Priest, G., Bendegem, J.-P.V. (Eds.), Frontiers in Paraconsistent Logic , (pp. 279–297). Baldock: Research Studies Press.

Verdée, P. (2013). Strong, universal, and provably non-trivial set theory by means of adaptive logic. Logic Journal of the IGPL , 21 (1), 108–125.

Weber, Z. (2010). Transfinite numbers in paraconsistent set theory. Review of Symbolic Logic , 3 (1), 71–92.

Weber, Z. (2012). Transfinite cardinals in paraconsistent set theory. Review of Symbolic Logic , 5 (2), 269–293.

Weber, Z., Ripley, D., Priest, G., Hyde, D., & Colyvan, M. (2014). Tolerating gluts: a reply to Beall. Mind , 123 (491), 813–828.

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Ripley, D. Paraconsistent Logic. J Philos Logic 44 , 771–780 (2015). https://doi.org/10.1007/s10992-015-9358-6

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Received : 31 January 2013

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DOI : https://doi.org/10.1007/s10992-015-9358-6

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Paraconsistent Logic

The contemporary logical orthodoxy has it that, from contradictory premises, anything can be inferred. To be more precise, let ⊨ be a relation of logical consequence, defined either semantically or proof-theoretically. Call ⊨ explosive if it validates { A , ¬ A } ⊨ B for every A and B ( ex contradictione quodlibet (ECQ)). The contemporary orthodoxy, i.e., classical logic, is explosive, but also some ‘non-classical’ logics such as intuitionist logic and most other standard logics are explosive.

The major motivation behind paraconsistent logic is to challenge this orthodoxy. A logical consequence relation, ⊨, is said to be paraconsistent if it is not explosive. Thus, if ⊨ is paraconsistent, then even if we are in certain circumstances where the available information is inconsistent, the inference relation does not explode into triviality . Thus, paraconsistent logic accommodates inconsistency in a sensible manner that treats inconsistent information as informative.

There are several reasons driving such motivation. The development of the systems of paraconsistent logic has depended on these. The prefix ‘para’ in English has two meanings: ‘quasi’ (or ‘similar to, modelled on’) or ‘beyond’. When the term ‘paraconsistent’ was coined by Miró Quesada at the Third Latin America Conference on Mathematical Logic in 1976, he seems to have had the first meaning in mind. Many paraconsistent logicians, however, have taken it to mean the second, which provided different reasons for the development of paraconsistent logic as we will see below.

This article is not meant to be a complete survey of paraconsistent logic. The modern history of paraconsistent logic maybe relatively short. Yet the development of the field has grown to the extent that a complete survey is impossible. The aim of this article is to provide some aspects and features of the field that are philosophically salient. This does not mean that paraconsistent logic has no mathematical significance or significance in such areas as computer science and linguistics. Indeed, the development of paraconsistent logic in the last two decades or so indicates that it has important applications in those areas. However, we shall tread over them lightly and focus more on the aspects that are of interest for philosophers and philosophically trained logicians.

1. Paraconsistency

2.1 inconsistent but non-trivial theories, 2.2 dialetheias (true contradictions), 2.3 automated reasoning, 2.4 belief revision, 2.5 mathematical significance, 3. a brief history of ex contradictione quodlibet, 4. modern history of paraconsistent logic, 5.1 discussive logic, 5.2 non-adjunctive systems, 5.3 preservationism, 5.4 adaptive logics, 5.5 logics of formal inconsistency, 5.6 many-valued logics, 5.7 relevant logics, bibliography, other internet resources, related entries.

A logic is said to be paraconsistent iff its logical consequence relation is not explosive. Paraconsistency is thus a property of a consequence relation and of a logic. In the literature, especially in the part of it that contains objections to paraconsistent logic, there has been some confusion over the definition of paraconsistency. So before going any further, we make one clarification.

Paraconsistency, so defined, is to do with the inference relation { A , ¬ A } ⊨ B for every A and B ( ex contradictione quodlibet (ECQ)). Dialetheism , on the other hand, is the view that there are true contradictions. If dialetheism is to be taken as a view that does not entail everything, then a dialehtiest's preferred logic must better be paraconsistent. For dialetheism is the view that some contradiction is true and it does not amount to trivialism which is the view that everything , including every contradiction, is true.

Now, a paraconsistent logician may feel the force pulling them towards dialetheism. Yet the view that a consequence relation should be paraconsistent does not entail the view that there are true contradictions. Paraconsistency is a property of an inference relation whereas dialetheism is a view about some sentences (or propositions, statements, utterances or whatever, that can be thought of as truth-bearers). The fact that one can define a non-explosive consequence relation does not mean that some sentences are true. That is, the fact that one can construct a model where a contradiction holds but not every sentence of the language holds (or, if the model theory is given intensionally, where this is the case at some world) does not mean that the contradiction is true per se . Hence paraconsistency must be distinguished from dialetheism.

Moreover, as we will see below, many paraconsistent logics validate the Law of Non-Contradiciton (LNC) (⊨ ¬( A ∧ ¬ A )) even though they invalidate ECQ. In a discussion of paraconsistent logic, the primary focus is not the obtainability of contradictions but the explosive nature of a consequence relation.

2. Motivations

The reasons for paraconsistency that have been put forward seem specific to the development of the particular formal systems of paraconsistent logic. However, there are several general reasons for thinking that logic should be paraconsistent. Before we summarise the systems of paraconsistent logic and their motivations, we present some general motivations for paraconsisent logic.

A most telling reason for paraconsistent logic is the fact that there are theories which are inconsistent but non-trivial. Once we admit the existence of such theories, their underlying logics must be paraconsistent. Examples of inconsistent but non-trivial theories are easy to produce. An example can be derived from the history of science. (In fact, many examples can be given from this area.) Consider Bohr's theory of the atom. According to this, an electron orbits the nucleus of the atom without radiating energy. However, according to Maxwell's equations, which formed an integral part of the theory, an electron which is accelerating in orbit must radiate energy. Hence Bohr's account of the behaviour of the atom was inconsistent. Yet, patently, not everything concerning the behavior of electrons was inferred from it, nor should it have been. Hence, whatever inference mechanism it was that underlay it, this must have been paraconsistent.

Despite the fact that dialetheism and paraconsistency needs to be distinguished, dialetheism can be a motivation for paraconsistent logic. If there are true contradictions (dialetheias), i.e., there are sentences, A , such that both A and ¬ A are true, then some inferences of the form { A , ¬ A } ⊨ B must fail. For only true, and not arbitrary, conclusions follow validly from true premises. Hence logic has to be paraconsistent. One candidate for a dialetheia is the liar paradox . Consider the sentence: ‘This sentence is not true’. There are two options: either the sentence is true or it is not. Suppose it is true. Then what it says is the case. Hence the sentence is not true. Suppose, on the other hand, it is not true. This is what it says. Hence the sentence is true. In either case it is both true and not true. (See the entry on dialetheism in this encyclopedia for further details.)

Paraconsistent logic is motivated not only by philosophical considerations, but also by its applications and implications. One of the applications is automated reasoning ( information processing ). Consider a computer which stores a large amount of information. While the computer stores the information, it is also used to operate on it, and, crucially, to infer from it. Now it is quite common for the computer to contain inconsistent information, because of mistakes by the data entry operators or because of multiple sourcing. This is certainly a problem for database operations with theorem-provers, and so has drawn much attention from computer scientists. Techniques for removing inconsistent information have been investigated. Yet all have limited applicability, and, in any case, are not guaranteed to produce consistency. (There is no algorithm for logical falsehood.) Hence, even if steps are taken to get rid of contradictions when they are found, an underlying paraconsistent logic is desirable if hidden contradictions are not to generate spurious answers to queries.

As a part of artificial intelligence research, belief revision is one of the areas that have been studied widely. Belief revision is the study of rationally revising bodies of belief in the light of new evidence. Notoriously, people have inconsistent beliefs. They may even be rational in doing so. For example, there may be apparently overwhelming evidence for both something and its negation. There may even be cases where it is in principle impossible to eliminate such inconsistency. For example, consider the ‘paradox of the preface’. A rational person, after thorough research, writes a book in which they claim A 1 ,…, A n . But they are also aware that no book of any complexity contains only truths. So they rationally believe ¬( A 1 ∧…∧ A n ) too. Hence, principles of rational belief revision must work on inconsistent sets of beliefs. Standard accounts of belief revision, e.g., that of Gärdenfors et al ., all fail to do this, since they are based on classical logic. A more adequate account is based on a paraconsistent logic.

Another area of significance for paraconsistent logic concerns certain mathematical theories. Examples of such theories are formal semantics , set theory , and arithmetic . The latter concerns Gödel's Theorem .

Formal Semantics and Set Theory

Semantics is the study that aims to spell out a theoretical understanding of meaning. Most accounts of semantics insist that to spell out the meaning of a sentence is, in some sense, to spell out its truth-conditions. Now, prima facie at least, truth is a predicate characterised by the Tarski T-scheme:

T ( A ) ↔ A

where A is a sentence and A is its name. But given any standard means of self-reference, e.g., arithmetisation, one can construct a sentence, B , which says that ¬ T ( B ). The T-scheme gives that T ( B ) ↔ ¬ T ( B ). It then follows that T ( B ) ∧ ¬ T ( B ). (This is, of course, just the liar paradox.)

The situation is similar in set theory. The naive, and intuitively correct, axioms of set theory are the Comprehension Schema and Extensionality Principle :

∃ y ∀ x ( x ∈ y ↔ A ) ∀ x ( x ∈ y ↔ x ∈ z ) → y = z

where x does not occur free in A . As was discovered by Russell, any theory that contains the Comprehension Schema is inconsistent. For putting ‘ y ∉ y ’ for A in the Comprehension Schema and instantiating the existential quantifier to an arbitrary such object ‘ r ’ gives:

∀ y ( y ∈ r ↔ y ∉ y )

So, instantiating the universal quantifier to ‘ r ’ gives:

r ∈ r ↔ r ∉ r

It then follows that r ∈ r ∧ r ∉ r .

The standard approaches to these problems of inconsistency are, by and large, ones of expedience. However, a paraconsistent approach makes it possible to have theories of truth and sethood in which the mathematically fundamental intuitions about these notions are respected. For example, as Brady (1989) has shown, contradictions may be allowed to arise in a paraconsistent set theory, but these need not infect the whole theory.

Unlike formal semantics and set theory, there may not be any obvious arithmetical principles that give rise to contradiction. Nonetheless, just like the classical non-standard models of arithmetic, there is a class of inconsistent models of arithmetic (or more accurately models of inconsistent arithmetic ) which have an interesting and important mathematical structure.

One interesting implication of the existence of inconsistent models of arithmetic is that some of them are finite (unlike the classical non-standard models). This means that there are some significant applications in the metamathematical theorems. For example, the classical Löwenheim-Skolem theorem states that Q (Robinson's arithmetic which is a fragment of Peano arithmetic) has models of every infinite cardinality but has no finite models. But, Q can be shown to have models of finite size too by referring to the inconsistent models of arithmetic.

Gödel’s Theorem

It is not only the Löwenheim-Skolem theorem but also other metamathematical theorems can be given a paraconsistent treatment. In the case of other theorems, however, the negative results that are often shown by the limitative theorems of metamathematics may no longer hold. One important such theorem is Gödel’s theorem.

One version of Gödel’s first incompleteness theorem states that for any consistent axiomatic theory of arithmetic, which can be recognised to be sound, there will be an arithmetic truth - viz., its Gödel sentence - not provable in it, but which can be established as true by intuitively correct reasoning. The heart of Gödel’s theorem is, in fact, a paradox that concerns the sentence, G , ‘This sentence is not provable’. If G is provable, then it is true and so not provable. Thus G is proved. Hence G is true and so unprovable. If an underlying paraconsistent logic is used to formalise the arithmetic, and the theory therefore allowed to be inconsistent, the Gödel sentence may well be provable in the theory (essentially by the above reasoning). So a paraconsistent approach to arithmetic overcomes the limitations of arithmetic that are supposed (by many) to follow from Gödel’s theorem. (For other ‘limitative’ theorems of metamathematics, see Priest 2002.)

It is now standard to view ex contradictione quodlibet as a valid form of inference. This contemporary view, however, should be put in a historical perspective. It was towards the end of the 19th century, when the study of logic achieved mathematical articulation, that an explosive logical theory became the standard. With the work of logicians such as Boole, Frege, Russell and Hilbert, classical logic became the orthodox logical account.

In antiquity, however, no one seems to have endorsed the validity of ECQ. Aristotle presented what is sometimes called the connexive principle : “it is impossible that the same thing should be necessitated by the being and by the not-being of the same thing.” ( Prior Analytic II 4 57b3). (See the entry on connexive logic that has been developed based on this principle.) This principle became a topic of debates in the Middle Ages or Medieval time. Though the medieval debates seem to have been carried out in the context of conditionals, we can also see it as debates about consequences. The principle was taken up by Boethius (480–524 or 525) and Abelard (1079–1142), who considered two accounts of consequences. The first one is a familiar one: it is impossible for the premises to be true but conclusion false. The first account is thus similar to the contemporary notion of truth-preservation. The second one is less accepted recently: the sense of the premises contains that of the conclusion. This account, as in relevant logics, does not permit an inference whose conclusion is arbitrary. Abelard held that the first account fails to meet the connexive principle and that the second account (the account of containment) captured Aristotle's principle.

Abelard's position was shown to face a difficulty by Alberic of Paris in the 1130s. Most medieval logicians didn’t, however, abandon the account of validity based on containment or something similar. (See, for example, Martin 1987.) But one way to handle the difficulty is to reject the connexive principle. This approach, which has become most influential, was accepted by the followers of Adam Balsham or Parvipontanus (or sometimes known as Adam of The Little Bridge) (12th CE). The Parvipontanians embraced the truth-preservation account of consequences and the ‘paradoxes’ that are associated with it. In fact, it was a member of the Parvipontanians, William of Soissons, who discovered in the 12th century what we now call the C.I. Lewis (independent) argument for ECQ. (See Martin 1986.)

The containment account, however, did not disappear. John Duns Scotus (1266–1308) and his followers accepted the containment account (see Martin 1996). The Cologne School of the late 15th century argued against ECQ by rejecting disjunctive syllogism (see Sylvan 2000).

Now, the history of logic in the ‘East’, or more specifically Asia, is moot. There is a tendency, for example, in Jaina and Buddhist traditions to consider the possibility of statements being both true and false. Moreover, the logics developed by the major Buddhist logicians, Dignāga (5th century) and Dharmakīrti (7th century) do not embrace ECQ. Their logical account is, in fact, based on the ‘pervasion’ (Skt: vyāpti , Tib: khyab pa ) relation among the elements of an argument. Just like the containment account of Abelard, there must be a tighter connection between the premises and conclusion than the truth-preservation account allows. (For the logic of Dharmakīrti and its subsequent development, see for example Dunne 2004 and Tillemans 1999.)

In the 20th century, the idea of challenging the explosive orthodoxy occurred to different people at different times and places independently of each other. They were often motivated by different considerations. The earliest paraconsistent logics in the contemporary era seem to have been given by two Russians. Starting about 1910, Vasil’év proposed a modified Aristotelian syllogistic including statements of the form: S is both P and not P . In 1929, Orlov gave the first axiomatisation of the relevant logic R which is paraconsistent. (On Vasil’év, see Arruda 1977 and Arruda 1989, pp. 102f. On Orlov, see Anderson, Belnap and Dunn 1992, p. xvii.)

The work of Vasil’év or Orlov did not make any impact at the time. The first (formal) logician to have developed paraconsistent logic was the Polish logician, Jaśkowski, who was a student of Łukasiewicz, who envisaged paraconsistent logic in his critique of Aristotle on LNC (Łukasiewicz 1951).

Paraconsistent logics were also developed in South America by Asenjo (1954) and da Costa (1963) in their doctoral dissertations. Since then, an active group of logicians has been working on paraconsistent logic in Brazil, especially in Campinas and in São Paulo.

Paraconsistent logics in the forms of relevant logics were proposed in England by Smiley in 1959 and also at about the same time, in a much more developed form, in the USA by Anderson and Belnap. An active group of relevant logicians grew up in Pittsburgh including Dunn and Meyer. The development of paraconsistent logics (in the form of relevant logics) was transported to Australia. R. Routley (later Sylvan) and V. Routley (later Plumwood) discovered an intentional semantics for some of Anderson/Belnap relevant logics. A school developed around them in Canberra which included Brady and Mortensen, and later Priest who, together with R. Routley, incorporated dialetheism to the development.

By the mid-1970s, the development of paraconsistent logic became international. In Belgium, a group of logicians around Batens in Ghent grew up and remains active. Paraconsistent logic is also actively investigated in Canada by Jennings, Schotch and their student Brown. In 1997, the First World Congress on Paraconsistency was held at the University of Ghent in Belgium. The Second World Congress was held in São Sebastião (São Paulo, Brazil) in 2000, the Third in Toulous (France) in 2003 and the Fourth in Melbourne (Australia) in 2008. We now see logicians working on paraconsistent logic in Bulgaria, China, France, Germany, Italy, Japan, New Zealand to name just a few.

5. Systems of Paraconsistent Logic

A number of formal techniques to invalidate ECQ have been devised. Most of the techniques have been summarised elsewhere, for example Brown 2002 and Priest 2002. As the interest in paraconsistent logic grew, different techniques developed in different parts of the world. As a result, the development of the techniques has somewhat a regional flavour (though there are, of course, exceptions, and the regional differences can be over-exagerated). (See Tanaka 2003.)

Most paraconsistent logicians do not propose a wholesale rejection of classical logic. They usually accept the validity of classical inferences in consistent contexts. It is the need to isolate an inconsistency without spreading everywhere that motivates the rejection of ECQ. Depending on how much revision one thinks is needed, we have a technique for paraconsistency. The taxonomy given here is based on the degree of revision to classical logic. Since the logical novelty can be seen at the propositional level, we will concentrate on the propositional paraconsistent logics.

The first formal paraconsistent logic to have been developed was discussive (or discursive ) logic by the Polish logician Jaśkowski (1948). The thought behind discussive logic is that, in a discourse, each participant puts forward some information, beliefs or opinions. Each assertion is true according to the participant who puts it forward in a discourse. But what is true in a discourse on whole is the sum of assertions put forward by participants. Each participant's opinions may be self-consistent, yet may be inconsistent with those of others. Jaśkowski formalised this idea in the form of discussive logic.

A formalisation of discussive logic is by means of modelling a discourse in a modal logic. For simplicity, Jaśkowski chose S5 . We think of each participant’s belief set as the set of sentences true at a world in a S5 model M . Thus, a sentence A asserted by a participant in a discourse is interpreted as “it is possible that A ” (◊ A ). That is, a sentence A of discussive logic can be translated into a sentence ◊ A of S5 . Then A holds in a discourse iff A is true at some world in M . Since A may hold in one world but not in another, both A and ¬ A may hold in a discourse. Indeed, one should expect that participants disagree on some issue in a rational discourse.

To be more precise, let d be a translation function from a formula of discussive logic into a formula of S5 . Then ( p ) d = ◊ p . For complex formulas

(¬ A ) d = ¬( A d ) ( A ∨ B ) d = A d ∨ B d ( A ∧ B ) d = A d ∧ B d ( A ⊃ B ) d = A d ⊃ B d ( A ≡ B ) d = A d ≡ B d

It is easy to show that B is a discussive consequence of A 1 , …, A n iff the formula ◊ A 1 d ⊃ (… ⊃ (◊ A n d ⊃ ◊ B d )…) is a theorem of S5 .

To see that discussive logic is paraconsistent, consider a S5 model, M , such that A holds at w 1 , ¬A holds at a different world w 2 but B does not hold at any world for some B . Then both A and ¬ A hold, yet B does not hold in M . Hence discussive logic invalidates ECQ.

However, there is no S5 model where A ∧ ¬ A holds at some world. So an inference of the form { A ∧ ¬ A } ⊨ B is valid in discussive logic. This means that, in discussive logic, adjunction ({ A , ¬ A } ⊨ A ∧ ¬ A ) fails. But one can define a discussive conjunction, ∧ d , as A ∧ ◊ B (or ◊ A ∧ B ). Then adjunction holds for ∧ d (Jaśkowski 1949).

One difficulty is a formulation of a conditional. In S5 , the inference from ◊ p and ◊( p ⊃ q ) to ◊ q fails. Jaśkowski chose to introduce a connective which he called discussive implication , ⊃ d , defined as ◊ A ⊃ B . This connective can be understood to mean that “if some participant states that A , then B ”. As the inference from ◊ A ⊃ B and ◊ A to ◊ B is valid in S5 , modus ponens for ⊃ d holds in discussive logic. A discussive bi-implication, ≡ d , can also be defined as (◊ A ⊃ B ) ∧ ◊(◊ A ⊃ B ) (or ◊(◊ A ⊃ B ) ∧ (◊ A ⊃ B )).

A non-adjunctive system is a system that does not validate adjunction (i.e., { A , B } ⊭ A ∧ B ). As we saw above, discussive logic without a discussive conjunction is non-adjunctive. Another non-adjunctive strategy was suggested by Rescher and Manor 1970-71. In effect, we can conjoin premises, but only up to maximal consistency. Specifically, if Σ is a set of premises, a maximally consistent subset is any consistent subset Σ′ such that if A ∈ Σ − Σ′ then Σ′ ∪ { A } is inconsistent. Then we say that A is a consequence of Σ iff A is a classical consequence of Σ′ for some maximally consistent subset Σ′. Then { p , q } ⊨ p ∧ q but { p , ¬ p } ⊭ p ∧ ¬ p .

In the non-adjunctive system of Rescher and Manor, a consequence relation is defined over some maximally consistent subset of the premises. This can be seen as a way to ‘measure’ the level of consistency in the premise set. The level of { p , q } is 1 since the maximally consistent subset is the set itself. The level of { p , ¬ p }, however, is 2: { p } and {¬ p }.

If we define a consequence relation over some maximally consistent subset, then the relation can be thought of as preserving the level of consistent fragments. This is the approach which has come to be called preservationism . It was first developed by the Canadian logicians Ray Jennings and Peter Scotch.

To be more precise, a (finite) set of formulas, Σ, can be partitioned into classically consistent fragments whose union is Σ. Let ⊢ be the classical consequence relation. A covering of Σ is a set {Σ i : i ∈ I }, where each member is consistent, and Σ = ∪ i ∈ I Σ i . The level of Σ, l (Σ), is the least n such that Σ can be partitioned into n sets if there is such n , or ∞ if there is no such n . A consequence relation, called forcing , [⊢, is defined as follows. Σ [⊢ A iff l (Σ) = ∞, or l (Σ) = n and for every covering of size n there is a j ∈ I such that Σ j ⊢ A . If l (Σ) = 1 or ∞ then the forcing relation coincides with classical consequence relation. In case where l (Σ) = ∞, there must be a sentence of the form A ∧ ¬ A and so the forcing relation explodes.

A chunking strategy has also been applied to capture the inferential mechanism underlying some theories in science and mathematics. In mathematics, the best available theory concerning infinitesimals was inconsistent. In the infinitesimal calculus of Leibniz and Newton, in the calculation of a derivative infinitesimals had to be both zero and non-zero. In order to capture the inference mechanism underlying the infinitesimal calculus of Leibniz and Newton (and Bohr’s theory of the atom), we need to add to the chunking a mechanism that allows a limited amount of information to flow between the consistent fragments of these inconsistent but non-trivial theories. That is, certain information from one chunk may permeate into other chunks. The inference procedure underlying the theories must be Chunk and Permeate .

Let C = {Σ i : i ∈ I } and ρ a permeability relation on C such that ρ is a map from I × I to subsets of formulas of the language. If i 0 ∈ I , then any structure ⟨ C , ρ, i 0 ⟩ is called a C&P structure on Σ. If B is a C&P structure on Σ, we define the C&P consequences of Σ with respect to B , as follows. For each i ∈ Σ, a set of sentences, Σ i n , is defined by recursion on n :

That is, Σ i n +1     comprises the consequences from Σ i n   together with the information that permeates into chunk i from the other chunk at level n . We then collect up all finite stages:

The C&P consequences of Σ can be defined in terms of the sentences that can be inferred in the designated chunk i 0 when all appropriate information has been allowed to flow along the permeability relations. (See Brown and Priest 2004.)

One may think not only that an inconsistency needs to be isolated but also that a serious need for the consideration of inconsistencies is a rare occurrence. The thought may be that consistency is the norm until proven otherwise: we should treat a sentence or a theory as consistently as possible. This is essentially the motivation for adaptive logics , pioneered by Diderik Batens in Belgium.

An adaptive logic is a logic that adapts itself to the situation at the time of application of inference rules. It models the dynamics of our reasoning. There are two senses in which reasoning is dynamic: external and internal. Reasoning is externally dynamic if as new information becomes available expanding the premise set, consequences inferred previously may have to be withdrawn. The external dynamics is thus the non-monotonic character of some consequence relations: Γ ⊢ A and Γ ∪ Δ ⊬ A for some Γ, Δ and A . However, even if the premise-set remains constant, some previously inferred conclusion may considered as not derivable at a later stage. As our reasoning proceeds from a premise set, we may encounter a situation where we infer a consequence provided that no abnormality, in particular no contradiction, obtains at some stage of the reasoning process. If we are forced to infer a contradiction at a later stage, our reasoning has to adapt itself so that an application of the previously used inference rule is withdrawn. In such a case, reasoning is internally dynamic. Our reasoning may be internally dynamic if the set of valid inferences is not recursively enumerable (i.e., there is no decision procedure that leads to ‘yes’ after finitely many steps if the inference is indeed valid). It is the internal dynamics that adaptive logics are devised to capture.

In order to illustrate the idea behind adaptive logics, consider the premise set Γ = { p , q , ¬ p ∨ r , ¬ r ∨ s , ¬ s }. One may start reasoning with p and ¬ p ∨ r . Provided that p ∧ ¬ p does not obtain at some stage in the reasoning process, DS can be applied to derive r . Now, we can apply DS to ¬ r ∨ s and r to derive s provided that s ∧ ¬ s does not obtain. However, by conjoining s and ¬ s , we can obtain s ∧ ¬ s . Hence we must withdraw the application of DS to ¬ r ∨ s and r so that s would not be a consequence of this reasoning process. A consequence of this reasoning is what cannot be defeated at any stage of the process.

A system of adaptive logic can generally be characterised as consisting of three elements:

• A lower limit logic (LLL)
• A set of abnormalities
• An adaptive strategy

LLL is the part of an adaptive logic that is not subject to adaptation. It consists essentially of a number of inferential rules (and/or axioms) that one is happy to accept regardless of the situation in a reasoning process. A set of abnormalities is a set of formulas that are presupposed as not holding (or as absurd) at the beginning of reasoning until they are shown to be otherwise. For many adaptive logics, a formula in this set is of the form A ∧ ¬ A . An adaptive strategy specifies a strategy of handling the applications of inference rules based on the set of abnormalities. If LLL is extended with the requirement that no abnormality is logically possible, one obtains the upper limit logic (ULL). ULL essentially contains not only the inferential rules (and/or axioms) of LLL but also supplementary rules (and/or axioms) that can be applied in the absence of abnormality, such as DS. By specifying these three elements, one obtains a system of adaptive logic.

The approaches taken for motivating the systems of paraconsistent logic which we have so far seen isolate inconsistency from consistent parts of the given theory. The aim is to retain as much classical machinery as possible in developing a system of paraconsistent logic which, nonetheless, avoids explosion when faced with a contradiction. One way to make this aim explicit is to extend the expressive power of our language by encoding the metatheoretical notions of consistency (and inconsistency) in the object language. The Logics of Formal Inconsistency ( LFIs ) are a family of paraconsistent logics that constitute consistent fragments of classical logic yet which reject explosion principle where a contradiction is present. The investigation of this family of logics was initiated by Newton da Costa in Brazil.

An effect of encoding consistency (and inconsistency) in the object language is that we can explicitly separate inconsistency from triviality. With a language rich enough to express inconsistency (and consistency), we can study inconsistent theories without assuming that they are necessarily trivial. This makes it explicit that the presence of a contradiction is a separate issue from the non-trivial nature of paraconsistent inferences.

• ∃Γ∃ A ∃ B (Γ, A , ¬ A ⊬ B ) and

A → ( A → (¬ A → B )) ( A ∧ B ) → ( A ∧ B ) ( A ∧ B ) → ( A → B )

Then ⊢ provides da Costa’s system C 1 . If we let A 1 abbreviate the formula ¬( A ∧ ¬ A ) and A n +1 the formula (¬( A n ∧ ¬ A n )) 1 , then we obtain C i for each natural number i greater than 1.

To obtain da Costa’s system C ω , instead of the positive fragment of classical logic, we start with positive intuitionist logic instead. C i systems for finite i do not rule out ( A n ∧ ¬ A n ∧ A n +1 ) from holding in a theory. By going up the hierarchy to ω, C ω rules out this possibility. Note, however, that C ω is not a LFC as it does not contain classical positive logic.

For the semantics for da Costa’s C -systems, see for example da Costa and Alves 1977 and Loparic 1977.

Perhaps the simplest way of generating a paraconsistent logic, first proposed by Asenjo in his PhD dissertation, is to use a many-valued logic. Classically, there are exactly two truth values. The many-valued approach is to drop this classical assumption and allow more than two truth values. The simplest strategy is to use three truth values: true (only) , false (only) and both (true and false) for the evaluations of formulas. The truth tables for logical connectives, except conditional, can be given as follows:

¬ t f b b f t ∧ t b f t t b f b b b f f f f f ∨ t b f t t t t b t b b f t b f

These tables are essentially those of Kleene’s and Łukasiewicz’s three valued logics where the middle value is thought of as indeterminate or neither (true nor false) .

For a conditional ⊃, following Kleene’s three valued logic, we might specify a truth table as follows:

⊃ t b f t t b f b t b b f t t t

Let t and b be the designated values. These are the values that are preserved in valid inferences. If we define a consequence relation in terms of preservation of these designated values, then we have the paraconsistent logic LP of Priest 1979. In LP , ECQ is invalid. To see this, we assign b to p and f to q . Then ¬ p is also evaluated as b and so both p and ¬ p are designated. Yet q is not evaluated as having a designated value. Hence ECQ is invalid in LP .

As we can see, LP invalidates ECQ by assigning a designated value, both true and false , to a contradiction. Thus, LP departs from classical logic more so than the systems that we have seen previously. But, more controversially, it is also naturally aligned with dialetheism. However, we can interpret truth values not in an aletheic sense but in an epistemic sense: truth values (or designated values) express epistemic or doxastic commitments. (See for example Belnap 1992.) Or we might think that the value both is needed for a semantic reason: we might be required to express the contradictory nature of some of our beliefs, assertions and so on. (See Dunn 1976, p. 157.) If these interpretative strategy is successful, we can separate LP from necessarily falling under dialetheism.

One feature of LP which requires some attention is that in LP modus ponens comes out to be invalid. For if p is both true and false but q false (only), then p ⊃ q is both true and false and hence is designated. So both p and p ⊃ q are designated, yet the conclusion q is not. Hence modus ponens for ⊃ is invalid in LP . (One way to rectify the problem is to add an appropriate conditional connective as we will see in the section on relevant logics.)

Another way to develop a many-valued paraconsistent logic is to think of an assignment of a truth value not as a function but as a relation . Let P be the set of propositional parameters. Then an evaluation, η , is a subset of P × {0, 1}. A proposition may only relate to 1 (true), it may only relate to 0 (false), it may relate to both 1 and 0 or it may relate to neither 1 nor 0. The evaluation is extended to a relation for all formulas by the following recursive clauses:

¬ Aη 1 iff Aη 0 ¬ Aη 0 iff Aη 1 A ∧ Bη 1 iff Aη 1 and Bη 1 A ∧ Bη 0 iff Aη 0 or Bη 0 A ∨ Bη 1 iff Aη 1 or Bη 1 A ∨ Bη 0 iff Aη 0 and Bη 0

If we define validity in terms of truth preservation under all relational evaluations then we obtain First Degree Entailment ( FDE ) which is a fragment of relevant logics. These relational semantics for FDE are due to Dunn 1976.

The approaches to paraconsistency we have examined above all focus on the inevitable presence or the truth of some contradictions. A rejection of ECQ, in these approaches, depends on an analysis of the premises containing a contradiction. One might think that the real problem with ECQ is not to do with the contradictory premises but to do with the lack of connection between the premises and the conclusion. The thought is that the conclusion must be relevant to the premises in a valid inference.

Relevant logics were pioneered in order to study the relevance of the conclusion with respect to the premises by Anderson and Belnap (1975) in Pittsburgh. Anderson and Belnap motivated the development of relevant logics using natural deduction systems; yet they developed a family of relevant logics in axiomatic systems. As development proceeded and was carried out also in Australia, more focus was given to the semantics.

The semantics for relevant logics were developed by Fine (1974), Routley and Routley (1972), Routley and Meyer (1993) and Urquhart (1972). (There are also algebraic semantics. See for example Dunn and Restall 2002, pp. 48ff.) In the Routleys-Meyer semantics, based on possible-world semantics (which is the most studied semantics for relevant logics, especially in Australia), conjunction and disjunction behave in the usual way. But each world, w , has an associate world, w *, and negation is evaluated in terms of w *: ¬ A is true at w iff A is false, not at w , but at w *. Thus, if A is true at w , but false at w *, then A ∧ ¬ A is true at w . To obtain the standard relevant logics, one needs to add the constraint that w ** = w . As is clear, negation in these semantics is an intensional operator.

The primary concern with relevant logics is not so much with negation as with a conditional connective → (satisfying modus ponens ). In relevant logics, if A → B is a logical truth, then A is relevant to B , in the sense that A and B share at least one propositional variable.

Semantics for the relevant conditional are obtained by furnishing each Routleys-Meyer model with a ternary relation. In the simplified semantics of Priest and Sylvan 1992 and Restall 1993 and 1995, worlds are divided into normal and non-normal. If w is a normal world, A → B is true at w iff at all worlds where A is true, B is true. If w is non-normal, A → B is true at w iff for all x , y , such that Rwxy , if A is true at x , B is true at y . If B is true at x but not at y where Rwxy , then B → B is not true at w . Then one can show that A → ( B → B ) is not a logical truth. (Validity is defined as truth preservation over normal worlds.) This gives the basic relevant logic, B . Stronger logics, such as the logic R , are obtained by adding constraints on the ternary relation.

There are also versions of world-semantics for relevant logics based on Dunn’s relational semantics for FDE . Then negation is extensional. A conditional connective, now needs to be given both truth and falsity conditions. So we have: A → B is true at w iff for all x , y , such that Rwxy , if A is true at x , B is true at y ; and A → B is false at w iff for some x , y , such that Rwxy , if A is true at x , B is false at y . Adding various constraints on the ternary relation provides stronger logics. However, these logics are not the standard relevant logics developed by Anderson and Belnap. To obtain the standard family of relevant logics, one needs neighbourhood frames. (See Mares 2004.) Further details concerning relevant logics can be found in the article on that topic in this encyclopedia.

For Paraconsistency in general:

• Priest, G., Routley, R., and Norman, J. (eds.) (1989). Paraconsistent Logic: Essays on the Inconsistent , München: Philosophia Verlag.
• Priest, G. (2002). “Paraconsistent Logic”, Handbook of Philosophical Logic (Second Edition), Vol. 6, D. Gabbay and F. Guenthner (eds.), Dordrecht: Kluwer Academic Publishers, pp. 287-393.

For Inconsistent but Non-Trivial Theories

• Brown, B. and G. Priest. (2004). “Chunk and Permeate: A Paraconsistent Inference Strategy. Part 1: The Infinitesimal Calculus”, Journal of Philosophical Logic , 33: 379-388.

On Dialetheism

• Priest, G. (1987). In Contradiction: A Study of the Transconsistent , Dordrecht: Martinus Nijhoff; second edition, Oxford: Oxford University Press, 2006.
• Priest, G., J.C. Beall and B. Armour-Garb (eds.) (2004). The Law of Non-Contradiction , Oxford: Oxford University Press.

For Automated Reasoning

• Belnap, N.D., Jr. (1992). “A Useful Four-valued Logic: How a computer should think”, Entailment: The Logic of Relevance and Necessity , Volume II, A.R. Anderson, N.D. Belnap, Jr, and J.M. Dunn, Princeton: Princeton University Press; first appeared as “A Usuful Four-valued Logic”, Modern Use of Multiple-valued Logic , J.M. Dunn and G. Epstein (eds.), Dordrecht: D. Reidel, 1977, and “How a Computer Should Think”, Comtemporary Aspects of Philosophy , G. Ryle (ed.), Oriel Press, 1977.
• Besnard, P. and Hunter, A. (eds.) (1998). Handbook of Deasible Reasoning and Uncertainty Management Systems , Volume 2, Reasoning with Actual and Potential Contradictions , Dordrecht: Kluwer Academic Publishers.

For Belief Revision

• Priest, G. (2001). “Paraconsistent Belief Revision”, Theoria , 67: 214-228.
• Restall, G. and Slaney, J. (1995). “Realistic Belief Revision”, Proceedings of the Second World Conference in the Fundamentals of Artificial Intelligence , M. De Glas and Z. Pawlak (eds.), Paris: Angkor, pp. 367-378.
• Tanaka, K. (2005). “The AGM Theory and Inconsistent Belief Change”, Logique et Analyse , 48: 113-150.

For Mathematical Significance

• Brady, R.T. (1989). “The Non-Triviality of Dialectical Set Theory”, Paraconsistent Logic: Essays on the Inconsistent , G. Priest, R. Routley and J. Norman (eds.), München: Philosophia Verlag, pp. 437-471.
• Mortensen, C. (1995). Inconsistent Mathematics , Dordrecht: Kluwer Academic Publishers.
• Priest, G. (2003). “Inconsistent Arithmetic: Issues Technical and Philosophical”, in Trends in Logic: 50 Years of Studia Logica (Studia Logica Library, Volume 21), V. F. Hendricks and J. Malinowski (eds.), Dordrecht: Kluwer Academic Publishers, pp. 273-99.

For a History of ex contradictione quodlibet

• Sylvan, R. (2000). “A Preliminary Western History of Sociative Logics”, Sociative Logics and Their Applications: Essays by the late Richard Sylvan , D. Hyde and G. Priest (eds.), Aldershot: Ashgate Publishers.

For Modern History of Paraconsistent Logic

• Arruda, A. (1989). “Aspects of the Historical Development of Paraconsistent Logic”, Paraconsistent Logic: Essays on the Inconsistent , G. Priest, R. Routley and J. Norman (eds.), München: Philosophia Verlag, pp. 99-130.
• Priest, G. (2007). “Paraconsistency and Dialetheism”, in Handbook of the History of Logic , Volume 8, D. Gabbay and J. Woods (eds.), Amsterdam: North Holland, pp. 129-204.

For the Systems of Paraconsistent Logic in general

• Brown, B. (2002). “On Paraconsistency”, in A Companion to Philosophical Logic , Dale Jacquette (ed.), Oxford: Blackwell, pp. 628-650.

For Discussive Logic

• Jaśkowski, S. (1948). “Rachunek zdań dla systemów dedukcyjnych sprzecznych”, Studia Societatis Scientiarun Torunesis (Sectio A), 1 (5): 55-77; an English translation appeared as “Propositional Calculus for Contradictory Deductive Systems”, Studia Logica , 24 (1969): 143-157.
• Jaśkowski, S. (1949). “O koniunkcji dyskusyjnej w rachunku zdań dla systemów dedukcyjnych sprzecznych”, Studia Societatis Scientiarum Torunensis (Sectio A), 1 (8): 171-172; an English translation appeared as “On the Discussive Conjunction in the Propositional Calculus for Inconsistent Deductive Systems”, Logic and Logical Philosophy , 7 (1999): 57-59.
• da Costa, N.C.A. and Dubikajtis, L. (1977). “On Jaśkowski’s Discussive Logic”, in Non-Classical Logics, Modal Theory and Computability , A.I. Arruda, N.C.A. da Costa and R. Chuaqui (eds.), Amsterdam: North-Holland Publishing Company, pp. 37-56.

For Non-Adjunctive Systems

• Rescher, N. and R. Manor (1970-71). “On Inference from Inconsistent Premises”, Theory and Decision , 1: 179-217.

For Preservationism

• Schotch, P.K. and R.E. Jennings (1980). “Inference and Necessity”, Journal of Philosophical Logic , 9: 327-340.

For Adaptive Logics

• Batens, D. (2001). “A General Characterization of Adaptive Logics”, Logique et Analyse , 173-175: 45-68.
• Batens, D. (2007). “A Universal Logic Approach to Adaptive Logics", Logica Universalis , 1: 221-242.

For Logics of Formal Inconsistency

• Carnielli, W.A., M.E. Coniglio and J. Marcos (2007). “Logics of Formal Inconsistency”, Handbook of Philosophical Logic , Volume 14 (Second Edition), D. Gabbay and F. Guenthner (eds.), Berlin: Springer, pp. 15-107.
• da Costa, N.C.A. (1974). “On the Theory of Inconsistent Formal Systems”, Notre Dame Journal of Formal Logic , 15 (4): 497-510.

For Many-Valued Logics

• Asenjo, F.G. (1966). “A Calculus of Antinomies”, Notre Dame Journal of Formal Logic , 7: 103-5.
• Dunn, J.M. (1976). “Intuitive Semantics for First Degree Entailment and Coupled Trees”, Philosophicl Studies , 29: 149-68.
• Priest, G. (1979). “Logic of Paradox”, Journal of Philosophical Logic , 8: 219-241.

For Relevant Logics

• Anderson, A. and N. Belnap. (1975). Entailment: The Logic of Relevance and Necessity , Volume 1, Princeton: Princeton University Press.
• Anderson, A., N. Belnap and J.M. Dunn. (1992). Entailment: The Logic of Relevance and Necessity , Volume 2, Princeton: Princeton University Press.
• Dunn, J.M. and G. Restall (2002). “Relevance Logic”, Handbook of Philosophical Logic , Volume 6, second edition, D. Gabbay and F. Guenthner (eds.), Dordrecht: Kluwer Academic Publishers, pp. 1-136.
• Routley, R., Plumwood, V., Meyer, R.K., and Brady, R.T. (1982). Relevant Logics and Their Rivals , Volume 1, Ridgeview: Atascadero.
• Brady, R.T. (ed.) (2003). Relevant Logics and Their Rivals , Volume 2, Aldershot: Ashgate.

Other Works Cited

• Arruda, A. (1977). “On the Imaginary Logic of N.A. Vasil’év”, in Non-Classicl Logic, Model Theory and Cpmputability , A. Arruda, N, da Costa and R. Chuanqui (eds.), Amsterdam: North Holland, pp. 3-24.
• da Costa, N.C.A. and E.H. Alves (1977). “Semantical Analysis of the Calculi Cn”, Notre Dame Journal of Formal Logic , 18 (4): 621-630.
• Dunne, J.D. (2004). Foundations of Dharmakīrti’s Philosophy , Boston: Wisdom Publications.
• Fine, K. (1974). “Models for Entailment”, Journal of Philosophical Logic , 3: 347-372.
• Loparic, A. (1977). “Une etude semantique de quelques calculs propositionnels”, Comptes Rendus Hebdomadaires des Seances de l’Academie des Sciences , 284: 835-838.
• Łukasiewicz, J. (1951). Atistotle’s Syllogistic: From the Standpoint of Modern Formal Logic , Oxford: Oxford University Press.
• Mares, E. (2004). “‘Four-Valued’ Semantics for the Relevant Logic R”, Journal of Philosophical Logic , 33: 327-341.
• Martin, C. (1986). “William’s Machine”, Journal of Philosophy , 83: 564-572.
• Martin, C. (1987). “Embarrassing Arguments and Surprising Conclusions in the Development Theories of the Conditional in the Twelfth Century”, Gilbert De Poitiers Et Ses Contemporains , J. Jolivet, A. De Libera (eds.), Naples: Bibliopolis, pp. 377-401.
• Martin, C. (1996). “Impossible Positio as the Foundation of Metaphysics or, Logic on the Scotist Plan?”, Vestigia, Imagines, Verba: Semiotics and Logic in Medieval Theological Texts , C. Marmo (ed.), Turnhout: Brepols, pp. 255-276.
• Priest, G. and R. Sylvan (1992). “Simplified Semantics for Basic Relevant Logics”, Journal of Philosophical Logic , 21: 217-232.
• Restall, G. (1993). “Simplified Semantics for Relevant Logics (and some of their rivals)”, Journal of Philosophical Logic , 22: 481-511.
• Restall, G. (1995). “Four-Valued Semantics for Relevant Logics (and some of their rivals)”, Journal of Philosophical Logic , 24: 139-160.
• Routley, R. and R. Meyer (1993). “Semantics of Entailment”, Truth, Syntax and Modality , H. Leblanc (ed.), Amsterdam: North Holland, pp. 194-243.
• Routley, R. and V. Routley (1972). “Semantics of First Degree Entailment”, Noûs , 3: 335-359.
• Tanaka, K. (2003). “Three Schools of Paraconsistency”, The Australasian Journal of Logic , 1: 28-42.
• Tillemans, Tom J.F. (1999). Scripture, Logic, Language: Essays on Dharmakīrti and His Tibetan Successors , Boston: Wisdom Publications.
• Urquhart, A. (1972). “Semantics for Relevant Logics”, Journal of Symbolic Logic , 37: 159-169.

[Please contact the authors with suggestions.]

dialetheism [dialethism] | logic: many-valued | logic: relevance | logic: substructural | mathematics: inconsistent | Sorites paradox

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Paraconsistent logic

A relation of logical consequence, $ \vdash $, on a set of sentences, $ S $, is explosive if and only if for all $ \alpha, \beta \in S $,

$$ \alpha, \neg \alpha \vdash \beta , $$

where "¬" is negation. A relation, and the logic that possesses it, is paraconsistent if and only if it is not explosive. Whether or not a correct consequence relation is explosive has been a contentious issue historically, but the standard formal logics of the 20th century, such as classical logic (cf. Logical calculus ) and intuitionistic logic are explosive. Formal paraconsistent logics were developed by a number of different people, often working in isolation from each other, starting around the 1960s.

There are many different paraconsistent logics, with their own proof theories and model theories. Their distinctive features occur at the propositional level, though they all have full first- (and second-) order versions. In most of them validity can be defined in terms of preservation of truth in an interpretation.

In one approach, due to S. Jaśkowski, an interpretation is a Kripke model (cf. Kripke models ) for some modal logic , and a sentence is true in it if it holds at some world of the interpretation. A major feature of this approach is that the inference of adjunction ( $ \alpha, \beta \vdash \alpha \wedge \beta $) fails. In another, an interpretation $ \nu $,{} is a mapping from $ S $ to $ \{ 1,0 \} $, satisfying the usual classical conditions for $ \wedge $, $ \lor $, and $ \rightarrow $. $ \nu ( \neg \alpha ) $ is independent of $ \nu ( \alpha ) $. The addition of further constraints on $ \nu $, such as: $ \nu ( \alpha ) = 0 \Rightarrow \nu ( \neg \alpha ) = 1 $, give logics in N. da Costa's $ C $ family. A feature of this approach is that it preserves all of positive logic. In a third approach, an interpretation $ \nu $ is a mapping from $ S $ to the closed sets of a topological space $ {\mathcal T} $ satisfying the conditions $ \nu ( \alpha \wedge \beta ) = \nu ( \alpha ) \cap \nu ( \beta ) $, $ \nu ( \alpha \lor \beta ) = \nu ( \alpha ) \cup \nu ( \beta ) $, $ \nu ( \neg \alpha ) = {\overline{ {\nu ( \alpha ) }}\; } ^ {c} $( where $ c $ is the closure operator of $ {\mathcal T} $). $ \alpha $ is true under $ \nu $ if and only if $ \nu ( \alpha ) $ is the whole space. This gives a logic dual to intuitionistic logic .

In a fourth approach, an interpretation is a relation $ \rho \subseteq S \times \{ 1,0 \} $, satisfying the natural conditions

$$ \neg \alpha \rho 1 \iff \alpha \rho 0, $$

$$ \neg \alpha \rho 0 \iff \alpha \rho 1; $$

$$ \alpha \wedge \beta \rho 1 \iff \alpha \rho 1 \textrm{ and } \beta \rho 1, $$

$$ \alpha \wedge \beta \rho 0 \iff \alpha \rho 0 \textrm{ or } \beta \rho 0; $$

and dually for $ \lor $. $ \alpha $ is true under $ \rho $ if and only if $ \alpha \rho 1 $. This gives the logic of first degree entailment (FDE) of A. Anderson and N. Belnap. If one restricts interpretations to those satisfying the condition $ \forall \alpha \exists x \alpha \rho x $, one gets G. Priest's LP. A feature of this logic is that its logical truths coincide with those of classical logic. Thus, the law of non-contradiction holds: $ \vdash \neg ( \alpha \wedge \neg \alpha ) $. A De Morgan lattice is a distributive lattice with an additional operator $ \neg $ satisfying: $ \neg \neg a = a $ and $ a \leq b \Rightarrow \neg b \leq \neg a $. An FDE-interpretation can be thought of as a homomorphism into the De Morgan lattice with values $ \{ \{ 1 \} , \{ 1,0 \} , \emptyset, \{ 0 \} \} $. More generally, $ \alpha \vdash \beta $ in FDE if and only if for every homomorphism $ h $ into a De Morgan lattice, $ h ( \alpha ) \leq h ( \beta ) $. Augmenting such lattices with an operator $ \rightarrow $ satisfying certain conditions, and defining validity in the same way, gives a family of relevant logics.

A paraconsistent logic localizes contradictions, and so is appropriate for reasoning from information that may be inconsistent, e.g., information stored in a computer database. It also permits the existence of theories (sets of sentences closed under deducibility) that are inconsistent but not trivial (i.e., containing everything) and of their models, inconsistent structures.

One important example of an inconsistent theory is set theory based on the general comprehension schema ( $ \exists x \forall y ( y \in x \leftrightarrow \alpha ) $, where $ \alpha $ is any formula not containing $ x $), together with extensionality ( $ \forall x ( x \in y \leftrightarrow x \in z ) \vdash y = z $). Another is a theory of truth (or of other semantic notions), based on the $ T $- schema ( $ T \langle \alpha \rangle \left\rightarrow \alpha $, where $ \alpha $ is any closed formula, and $ \langle \cdot \rangle $ indicates a name-forming device), together with some mechanism for self-reference, such as arithmetization. Such theories are inconsistent due to the paradoxes of self-reference (cf. Antinomy ).

Not all paraconsistent logics are suitable as the underlying logics of these theories. In particular, if the underlying logic contains contraction ( $ \alpha \rightarrow ( \alpha \rightarrow \beta ) \vdash \alpha \rightarrow \beta $) and modus ponens ( $ \alpha, \alpha \rightarrow \beta \vdash \beta $), these theories are trivial. However, the theories are non-trivial if $ \rightarrow $ is interpreted as the material conditional and the logic LP is used, or if it is interpreted as the conditional of some relevant logics. In the truth theories, the inconsistencies do not spread into the arithmetical machinery.

Given a topos , logical operators can be defined as functors within it, and a notion of internal validity can be defined, giving intuitionistic logic. If these operators, and in particular, negation, are defined in the dual way, the internal logic of the topos is the dual intuitionistic logic. Topoi can therefore be seen as inconsistent structures.

For another example of inconsistent structures, let $ A $ be the set of sentences true in the standard model of arithmetic. If $ B $ is a set of sentences in the same language properly containing $ A $, then $ B $ is inconsistent, and so has no classical models; but $ B $ has models, including finite models, in the paraconsistent logic LP. Inconsistent (sets of) equations may have solutions in such models. The LP-models of $ A $ include the classical non-standard models of arithmetic (cf. Peano axioms ) as a special case, and, like them, have a notable common structure.

In inconsistent theories of arithmetic, the incompleteness theorems of K. Gödel (cf. Gödel incompleteness theorem ) fail: such a theory may be axiomatizable and contain its own "undecidable" sentence (and its negation).

Inconsistent theories may be interesting or useful even if they are not true. The view that some inconsistent theories are true is called dialetheism (or dialethism).

For a general overview of the area, see [a2] . [a3] is a collection of articles, with much background material. On inconsistent mathematical structures, see [a1] .

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Uncovering the brutal career of a crucial American ally.

And the hidden truths of the war in Afghanistan.

America’s Monster Who was Abdul Raziq?

Supported by

Who Was Abdul Raziq?

Uncovering the brutal career of a crucial American ally — and the hidden truths of the war in Afghanistan.

By Matthieu Aikins

Photographs by Victor J. Blue

I first heard about Abdul Raziq in early 2009, when I was a young freelance journalist newly arrived in southern Afghanistan. By chance, I had befriended two drug smugglers who told me that a powerful police commander in the area was helping them ship two metric tons of opium to Iran each month. Raziq, I learned, had a fearsome reputation in his hometown, Spin Boldak, on the border with Pakistan. Everyone I spoke to knew about the Taliban suspects tortured and dumped in the desert. Just as they knew that Raziq was a close ally of the U.S. military. My smuggler friends had offered to introduce me to Raziq, and 10 days after my arrival in Spin Boldak, he returned to town for his grandmother’s funeral.

Listen to this article, read by Peter Ganim

When I arrived at Raziq’s compound, I saw him sitting cross-legged on a carpeted platform, receiving a long line of guests. He was not what I expected. Trim and cheerful, clean-shaven and barely 30, he wasn’t much older than I, yet he was leading several thousand men under arms. I reached the front of the line, and Raziq shook my hand to welcome me before turning to the next guest. We would never get the chance to meet again, but that was the beginning of my long quest to understand the paradox he represented.

As inexperienced as I was, I knew enough to be puzzled by Raziq’s success. Why was the U.S. military, which was supposed to be supporting democracy and human rights in Afghanistan, working closely with a drug trafficker and murderer? One of his commanders, his uncle Janan, even wore a U.S. Army uniform given to him by his advisers, complete with a First Infantry Division patch and the Stars and Stripes.

Thanks to American patronage, Raziq was promoted to police chief of Kandahar and would eventually rise to the rank of three-star general. Famous across Afghanistan, he became the country’s most polarizing figure. The Taliban hated him, of course, but so did the ordinary people his commanders and soldiers extorted and abused. Journalists and human rights groups assembled damning evidence against him and warned that his brutality would backfire.

But Raziq beat back the suicide bombers and brought stability to Kandahar. In doing so, he became an icon for many war-weary Afghans who sought security at all costs. In a nation divided by ethnic and regional loyalties, you could find Raziq’s photo in taxis and at checkpoints from north to south. And he never lost his American backing: When he was assassinated by the Taliban in 2018, he was walking next to the top U.S. commander, Gen. Austin S. Miller. That day, it seemed as if half the country was in mourning; Miller hailed him as a friend and patriot.

Three years later, the United States withdrew, and the Islamic Republic of Afghanistan collapsed. I was working as a journalist in Kabul at the time, and as soon as the dust settled, I went south to Kandahar. With the fighting over, I was able to visit people and places nearly impossible to access before. Here was a chance to reckon with Raziq’s legacy. I met with survivors of torture inside his prisons and visited morgues where skeletons had been unearthed from desert graves. Like a great tree in a storm, the republic had toppled and exposed the hidden places among its roots. The American war was far more brutal than we had known.

Since then, over repeated trips to the war’s fiercest battlegrounds, I found that many of Raziq’s former police officers were willing to talk about the torture, execution and cover-ups they witnessed. I also spoke with a dozen American military officers and diplomats who worked with Raziq and obtained new documents through a Freedom of Information Act lawsuit and other sources, which reveal just how much the American government knew about Raziq’s crimes. And with colleagues at The Times, I interviewed hundreds of witnesses and discovered a republican archive that exposed Afghanistan’s largest campaign of forced disappearances since the Communist coup in 1978. We documented 368 cases of people who were still missing after being abducted by Raziq’s men; the true toll was most likely in the thousands.

The scale of Raziq’s abuses, carried out with American support, was shocking. But the fact that they seem to have brought security to Kandahar has even more disturbing implications. Raziq’s story complicates the comforting belief that brutality always backfires and undermines the U.S. military’s claim to have fought according to international law. Raziq’s violence was effective because it had a logic particular to the kind of civil war that the United States found in Afghanistan, one where the people, and not the terrain, were the battlefield. The reasons for this are well documented by scholars of civil war and counterinsurgency but glossed over by our generals and politicians and obscured by the myths of American exceptionalism and our righteous war on terror.

But Raziq saw those reasons clearly. He murdered and tortured because he believed it was the only way to win against the Taliban. And America helped him do it.

Two harsh realities defined Raziq’s childhood: the war and the border.

The desert around Spin Boldak and its twinned Pakistani town, Chaman, stretches westward hundreds of miles to Iran, through vast wastes and dune seas crossed by nomads. The clans of two rival Pashtun tribes dominate the area, feuding like Hatfields and McCoys of the borderlands. Raziq was from the Achakzai, who competed with the Noorzai over land and smuggling routes.

Not long after Raziq was born in a mud-walled village, the Afghan Communists seized power in Kabul, and in response rebels rose up against the government, plunging the country into a conflict that lasted for more than four decades. Although both superpowers and neighbors like Pakistan and Iran intervened for their own ends, at heart this was a civil war fought by Afghans against Afghans for control of the state. Even at the peaks of the Soviet and American occupations, Afghans constituted a majority of casualties on each side.

In times of civil war, neighbors are often at one another’s throats because of local dynamics, even if they justify their actions through religion or nationalism. In Kandahar, many Noorzai joined with the mujahedeen rebels, who were supplied by the C.I.A. and the Pakistani military, while Raziq’s Achakzai relatives eventually sided with the Soviet-backed Communists. Raziq was still a boy when the war brought grief to his home: His father, who drove people and goods to the border, disappeared. His family was never able to find his body and blamed their tribal rivals. “The Noorzai did it,” said Ayub Kakai, Raziq’s uncle. “They threw him down a well.”

In 1991, after the Soviets cut off funding, the Communist government collapsed. Kandahar’s rival warlords carved up the province with a patchwork of checkpoints, where robbery and rape were common. Raziq’s uncle Mansoor took control on the road from Spin Boldak to the city, and Raziq, by then a teenager, joined him, attracted to the thrills of war. “Raziq loved cars and guns,” his younger cousin Arafat told me.

Three years later, an armed movement of religious students known as the Taliban rose in the farmlands west of Kandahar City and swept through the province, capturing Raziq and his uncle. They hung Mansoor from the barrel of a tank but spared young Raziq, who fled with his family across the border to Chaman. For seven years in exile, Raziq worked as a driver near the border, where he peddled used car parts.

Then came Sept. 11, 2001. For the Achakzai, the Americans’ decision to invade and depose the Taliban came as a miraculous reversal of fortune. That December, the C.I.A. and Special Forces assembled an army of exiles, with many Achakzai, including Raziq, among them. With the help of U.S. air power, they routed the Taliban and seized control of Kandahar, once again trading places with their Noorzai rivals, who escaped across the border to where the Pakistani military, playing a double game, gave them safe haven.

In the new republic, the Achakzai militia was transformed into the area’s Border Police. They partnered with American troops and were trained by contractors from Blackwater and DynCorp. Like the rest of the republican forces, their weapons, ammunition and salaries were paid for by the United States and its allies. But beneath the surface, the civil war still festered, even though the Americans saw it through stark binaries: the government versus the terrorists, the Afghans versus the Taliban.

“Our viewpoint was this was a war on terrorism or a war against a group trying to overthrow a democratic government,” said Carter Malkasian, a former State Department official who advised the U.S. military in Afghanistan for more than a decade. “We don’t want to view this as us getting involved in another country’s civil war.”

Thanks to his family connections, Raziq quickly rose through the ranks. He was a natural leader who fought fearlessly and earned the loyalty of his men. Although nearly illiterate, he had a capacious memory for places and faces and was a canny operator in the spy games and smuggling rings of the borderlands, using his illicit gains to fund a growing network of sources. Early on, Raziq learned that power would earn him money, which bought the intelligence that could attract U.S. patronage, giving him more power. American officers who worked in Spin Boldak remembered Raziq as an eager and valued partner in the hunt for the Taliban and Al Qaeda.

“My brother was very close to the Americans,” said Tadin Khan, Raziq’s younger brother. “They trusted him, and he never tried to deceive them.”

Tadin Khan, Raziq’s younger brother, in Dubai last year. “I didn’t believe it when I heard he was killed,” he said. “It was a hard day.”

Gul Seema, the first wife of Raziq, in 2023. “He was under a lot of pressure. Whenever I saw him, it seemed as if the shadow of death was looming over him,” says Seema.

Raziq was as generous with his friends and family as he was ruthless with his enemies. Not long after the Achakzai appointed him leader of their militia, Raziq’s older brother, Bacha, was gunned down in the bazaar in Chaman. “Bacha and Raziq were very close,” said Arafat, his cousin. “He was killed because of Raziq.”

Raziq blamed a tribal rival, a smuggler named Shin Noorzai. In March 2006, he kidnapped Shin and 15 people he was traveling with and shot them all in a dry riverbed near the border. The massacre led to a local outcry, and Raziq was summoned to Kabul. But President Hamid Karzai intervened to protect him, according to Western diplomats involved in the case, and he was never charged. (Through a spokesperson, Karzai declined to comment for this article.) The incident, however, made it into that year’s State Department report on human rights, the first public documentation of Raziq’s abuses.

Raziq’s role in the drug trade also attracted attention from American investigators. Although the Taliban had banned poppy cultivation, opium came roaring back under Karzai’s administration, and Spin Boldak sat on one of the main trafficking routes. Classified U.S. military and Drug Enforcement Administration reports, obtained through FOIA requests, described the involvement of Raziq and his men, detailing convoys in the desert, secret meetings and the use of green ink for letters of safe passage. One referred to Raziq as “the main drug smuggler in Spin Boldak.” (His brother Tadin denied that Raziq or anyone from his family was involved in drug trafficking, murder or other crimes. “All these accusations of corruption, smuggling and abuses are because of propaganda from the Taliban,” he said.)

As it turns out, by the time Raziq and I shook hands in 2009, the United States already knew he was accused of murder and smuggling but worked with him anyway. Yet Raziq’s position had become precarious, for the U.S. military’s concept of the war was changing. When I published an article about the accusations that fall, Raziq’s career had reached a dangerous point — one where his foreign patrons might have chosen to stop supporting him.

The U.S. war in Afghanistan was going badly. Faced with a growing insurgency that threatened the Afghan government’s survival, President Barack Obama ordered a surge of tens of thousands of troops. His generals had advised him that, fixated on the enemy, the United States had neglected the true battlefield: the hearts and minds of the Afghan people.

The surge would be guided by a military doctrine known as counterinsurgency theory, or COIN, which was held to have saved the day in Iraq. “Our strategy cannot be focused on seizing terrain or destroying insurgent forces; our objective must be the population,” Gen. Stanley A. McChrystal wrote upon taking command in 2009. The U.S.-led coalition “can no longer ignore or tacitly accept abuse of power, corruption or marginalization.”

According to “population-centric” COIN, the Afghan people had to be protected against the insurgency and motivated to support their own government. Criminal officials like Raziq threatened the legitimacy of the republic, and therefore the success of the war.

Given the hundreds of thousands of Afghans and Iraqis who died as result of the U.S. invasions, this emphasis on protecting civilians may seem hypocritical. But the laws of war, which forbid targeting noncombatants or harming prisoners, are essential to how the United States distinguishes its own use of force from that of rogue states and terrorists. “I believe the United States of America must remain a standard-bearer in the conduct of war,” Obama said as he accepted the Nobel Peace Prize the same year as the surge. “That is what makes us different from those whom we fight. That is a source of our strength.”

Since the decline of the antiwar movement after Vietnam, both the U.S. military and its liberal critics have become increasingly united in the conviction that war must be fought humanely by exempting civilians, as much as possible, from its violence — a shift, the historian Samuel Moyn has argued, that risks legitimizing endless war. Underpinning this is the assumption that there is no contradiction between waging war both lawfully and effectively. “The law of war is a part of our military heritage, and obeying it is the right thing to do,” states the U.S. military manual on the subject. “But we also know that the law of war poses no obstacle to fighting well and prevailing.”

In this vein, COIN reassured the American public that the surge would be just. Because the United States needed the support of the Afghan population, it could not just kill its way to victory. Brutality would backfire by producing more resistance. In a speech, McChrystal explained “COIN mathematics” : If a military operation killed two out of 10 insurgents, instead of eight remaining, that number was “more likely to be as many as 20, because each one you killed has a brother, father, son and friends.”

But while McChrystal took prompt steps to reduce civilian casualties from airstrikes, dealing with so-called bad actors like Raziq was not as simple. It turns out that the way the United States implemented its strategy provided a test of whether COIN really worked as promised.

The surge was focused on the two neighboring provinces in the south where Taliban activity was strongest. Both received roughly equivalent investments of troops and money. In Helmand, the Marines and the British pushed for the good governance prescribed by COIN, successfully pressuring Kabul to replace corrupt officials with technocrats.

“You didn’t have a power-broker-run government at the provincial level,” said Malkasian, who served as an adviser in Helmand. But the opposite proved true in Kandahar, where U.S. commanders prioritized security and encountered dogged pushback from Kabul on anticorruption efforts. “Kandahar was just more important for the Afghan political system, for Karzai, than Helmand was.”

As so often happened during the war, Washington’s grand strategy was interpreted by a multitude of American agencies and actors. In Kabul, specialized anticorruption and counternarcotics teams had Raziq in their sights. A D.E.A.-led republican unit seized an enormous stockpile of hashish in Spin Boldak and arrested a district police commander who ran narcotics shipments for Raziq. There were plans to go after him next.

But the Army officers working with Raziq saw things very differently. He and his men were a rare example of an effective, homegrown force that delivered security on a vital supply route. The U.S. commanders were in the middle of a high-stakes offensive against the Taliban, and their own troops’ lives were on the line. Karzai supported Raziq, and according to former military and intelligence officials, so did the C.I.A. With his cross-border networks, Raziq was a valuable source of intelligence on Taliban havens and bomb-making networks in Pakistan. And he could cross lines the United States couldn’t: A declassified military report from 2010 noted that Raziq was giving shelter to Baloch rebels fighting the Pakistani government and that he used “these tribesmen to carry out assassinations and killings in Pakistan.”

And so when, in February 2010, senior U.S. officials met to discuss action against corrupt Afghan officials, no one could agree on what to do about Raziq. “There was a lack of consensus,” according to Earl Anthony, who as deputy U.S. ambassador was a co-chair of the meeting. “Some highly valued his work on the security front against the Taliban.”

In the end, McChrystal, who declined to comment for this article, sided with his commanders on the ground. Raziq, they reasoned, could be mentored to change his ways. According to a leaked cable, the senior U.S. diplomat in Kandahar even offered to craft a media plan for him, including radio spots, billboards and “the longer-term encouragement of stories in the international media on the ‘reform’” of Raziq.

In March, McChrystal visited Raziq in Spin Boldak and posed beside him for television cameras. “I am very optimistic that with the plans that I’ve heard,” he said, as Raziq looked on smiling, “we can increase efficiency and decrease corruption.”

From that point on, the U.S. military would openly promote Raziq and make him an integral part of the surge. A series of personal advisers were brought in to coach and protect the young commander; the first was Jamie Hayes, who as a Special Forces lieutenant colonel led a team assigned to Raziq in July 2010. Shortly after he arrived, Hayes was ordered to help Raziq plan a major operation to clear Malajat, an outlying neighborhood of Kandahar City where the Taliban were entrenched.

At first, Hayes was puzzled about why Raziq and his Border Police were given the job, rather than the republican army or commandos. His superiors explained that it was a political decision by Karzai and the U.S. command. “This is a guy that we want to make successful,” Hayes recalled being told. “He’s an aggressive, strong leader that we want to make sure gets the chance to shine.”

Mazloom, a 33-year-old Taliban commander, in Panjwai District last year. He said he was tortured and blinded by a U.S.-backed militia commander from his village, who recognized him as an insurgent. “Because I wouldn’t confess,” Mazloom said, “he did this.”

John R. Allen, a retired four-star Marine general, at his home in Virginia in March. As the commander of American and allied forces in Afghanistan in 2011, Allen was confronted with evidence that Raziq’s forces were committing murder and torture. “That wasn’t why we were there fighting the war,” he said, “to keep a really bad criminal because he was helpful in fighting worse criminals.”

Raziq’s charisma undoubtedly played a role in why U.S. officers were so willing to support him. Like most of the Americans I spoke to who worked with Raziq, Hayes quickly took a shine to him. Raziq was full of enthusiasm and energy, and Hayes was especially impressed by how he seemed to genuinely care for the welfare of his men, unlike many other republican commanders.

For his part, Raziq was a careful student of his foreign patrons. “He liked to learn about what made Americans tick,” recalled Hayes, who said he was never shown evidence of Raziq’s massacres or drug smuggling. Raziq understood what American officers appreciated: hard work, aggression and loyalty. To show his gratitude, he even insisted on taking part in a medal ceremony for Hayes’s troops. “He knew them by name,” Hayes recalled.

Soon after a successful operation to clear Malajat, Hayes and his team were reassigned to train the police in the provincial capital. During the spring of 2011, the situation in Kandahar City was dire. The Taliban hammered the government with gunmen and suicide attacks and, in April, freed nearly 500 inmates after tunneling into the main prison. Police morale was abysmal. “Drug use was rampant,” Hayes said. “Discipline was poor.” In the same month as the prison break, a suicide bomber got inside police headquarters and killed the provincial commander. Hayes, who narrowly missed the bombing, helped put the chief in a body bag. He was the second in two years to be killed.

Cleaning up Kandahar might have been the toughest job in Afghanistan, and both Karzai and the U.S. command wanted Raziq to do it. He agreed to become police chief on one condition: He wanted to keep his position with the Border Police. He would wear both hats, so to speak, in order to maintain his power base in Spin Boldak and would bring his own men into the city. If Raziq was going to be sheriff in Kandahar, he was going to do it his way.

The battle Raziq faced in the provincial capital, a city of nearly 400,000, was very different from the rangy desert warfare in the borderlands: Here, a tribally and ethnically mixed population lived and worked in closely packed homes and narrow alleys, industrial zones and trucking warehouses. Hiding amid them, Taliban guerrillas, the cheriki , terrorized government supporters, leaving menacing “night letters,” assassinating civil servants and imams and deploying suicide bombers whose blasts tore apart crowded streets.

The first phase of the American COIN strategy in Kandahar had called for securing the capital. To that end, the U.S. military poured in resources, building a network of checkpoints and bases for republican forces and expanding the number of police districts from 10 to 16, each with its own substation chief. Trained and equipped by American troops and contractors, the Kandahar police more than doubled in size. Raziq was the fulcrum of it all: A team of American mentors lived next to his headquarters, and he met often with U.S. brass to coordinate operations.

Raziq’s underground enemies, the cheriki , relied on an extensive network of local supporters, many of whom cooperated out of religious and nationalist fervor. Rooting them out required accurate intelligence. And because Karzai had resisted creating a system of wartime detention, those who were caught had to be criminally prosecuted, convicted and sentenced.

But for Raziq, the republican courts, corrupt and easily intimidated, were a central reason the insurgency was thriving. Too often, Taliban suspects were freed and returned to the battlefield. In Spin Boldak, he had solved this problem by becoming judge, jury and executioner. For all their rhetoric about human rights and the laws of war, the foreigners had chosen him to pacify Kandahar. Actions spoke louder than words.

Raziq brought his Achakzai militia, in their distinct spotted uniforms, into the city and placed trusted lieutenants in key posts like the substations. Raziq didn’t seem to relish cruelty — I never heard stories of him personally torturing people, for instance — but he cultivated men who did. Some were his own cousins, like Jajo, who became notorious for the atrocities he committed as commander of District 8, a predominantly Noorzai area. (Jajo was assassinated in 2014.) According to police officers and internal United Nations documents, another relative from Spin Boldak ran death squads out of a special battalion at headquarters. “They had detective badges and guns,” one substation deputy told me. “They threw the bodies in the desert.”

These plainclothes teams roamed in cars with tinted windows, snatching suspects and taking them for da reg mela , “a sand picnic.” The desert wells and dunes hid countless corpses; others were dumped in the streets. Many bore signs of horrific torture. “I saw things which made me wonder whether a wild beast or man had done them,” Dr. Musa Gharibnawaz, who oversaw the city morgue as the director of forensic medicine, told me.

Those who survived to see formal detention were also tortured for confessions, which the courts relied on almost entirely for convictions. The police didn’t have the education or capacity to collect basic technical evidence, nor did most judges understand it. This problem was much broader than just Kandahar. The same year Raziq became police chief, investigators from the United Nations interviewed more than 300 detainees across Afghanistan and found that torture was widespread in republican detention. Their report documented beatings, electric shocks and the “twisting and wrenching” of genitals. The most severe abuses by the police were in Kandahar, where a follow-up report also noted a large number of bodies found with gunshot wounds to the chest and the head after Raziq took power; by contrast, the investigators found significantly less torture in Helmand, where the Marines had stuck to the COIN playbook.

The persistence of torture in the republic — which the U.N. continued to document until 2021 — illustrates how, in wartime, certain useful but prohibited acts can be implicitly authorized as regular practices. As the U.N. reporting makes clear, those accused of torture rarely faced punishment. Their work, which ceased after confession, was instrumental, unlike the gratuitous abuse meted out by poorly supervised American soldiers at the Abu Ghraib prison in Iraq.

In Kandahar, torture was exacerbated by the surge, which overloaded the court system with detainees captured by U.S. forces; one internal military report worried that it would most likely “produce more — perhaps far more — prisoners” than the main prison could handle. During the summer of 2011, as the U.N. prepared to publish its findings, intelligence reports from the south filtered up to Western diplomats and military leaders in Kabul. The torture of detainees had already led to scandals in Britain and Canada; now the U.S. command would be forced to take notice. For the third time in Raziq’s career, his job would hang in the balance as a result of his crimes.

On July 18, 2011, two months after Raziq became police chief, John R. Allen, then a four-star Marine general, took command of U.S. and allied forces in Afghanistan. He was shaking hands with his guests during the ceremony in Kabul when a trio of Western officials, led by a senior British diplomat, told him that they needed to speak immediately. It was about Raziq.

Alarmed by what he heard, Allen had his staff pull up the raw intelligence reporting, which described executions and torture by Raziq’s forces in Kandahar. “I wanted a sense of the frequency,” he told me. “The reporting was pretty standard and pretty awful. It had been going on for some time.”

Allen went to the presidential palace to see Karzai. “I said that he needed to be aware that he had a senior police commander who was a serial human rights violator, and he should remove him,” Allen told me.

But at that moment, Karzai needed Raziq more than ever. In the week before Allen arrived, two of the president’s most important allies in the south were killed, including his own brother. For years, Karzai had seen the United States waffle on corruption and human rights abuses, even as they partnered with warlords, and as he often had, he called the Americans’ bluff. At a follow-up meeting, Karzai told Allen that he had checked his own sources and hadn’t heard similar allegations.

Frustrated, Allen ordered the United States and its allies to stop transferring captives in the south. “Karzai wasn’t going to do anything about Raziq, and I couldn’t permit us to continue to feed detainees into his hands,” he told me. From then on, when he traveled to Kandahar, Allen made a point to dodge the young police chief, who was eager for a photo op. “I wasn’t going to play into Raziq’s hands and appear to be an ally of his under any circumstances.”

The State Department’s diplomats also avoided meeting Raziq. But that was as far as it went. “I don’t recall there was ever a serious push to remove Abdul Raziq,” said Ryan Crocker, the U.S. ambassador at the time. When Crocker later raised the issue with Karzai, the president responded that Raziq was working closely with the U.S. military. “He was basically saying, ‘Look, I’m told that he’s your guy,’” Crocker told me. “Which turned out to be true.”

Allen’s subordinates in Kandahar continued to fight side by side with Raziq and his officers. “The military guys, for the most part, had a different view of him on the ground, working with him day in and day out,” said Martin Schweitzer, who as a brigadier general served as the deputy U.S. commander in Kandahar.

Despite Leahy laws in the United States, which prohibit support to foreign military units credibly accused of human rights violations, Raziq continued to be ferried around in American aircraft, and his advisers ensured that he and his forces had the air support, fuel and ammunition they needed. “If I asked and it was for Raziq, mostly I was going to get it,” said David Webb, who as a colonel advised him on two separate tours in 2012 and 2017. Upon Webb’s arrival, he was given his orders in no uncertain terms by his superior, a two-star general. “He put his finger in my chest and said: ‘Don’t let Raziq die. That’s your mission,’” said Webb, whose predecessor was wounded while fighting off an attack on Raziq’s headquarters.

Both Webb and Schweitzer stressed that they never saw evidence of Raziq committing war crimes under their watch. “I was with him almost every single day from morning until night,” Webb told me. “I never saw anything bad.”

As an outsider, I often wondered how American officers, bound to uphold the laws of war, rationalized working with Raziq. His tactics in Kandahar — every mutilated corpse or disappeared person — were intended to send a message, to terrorize his enemies and those who might support them. And they were effective. When I visited Kandahar in those years, I found that most people on the streets knew exactly what was happening, even if they were too afraid to speak about it openly.

But Raziq also calibrated his actions so that they were deniable. According to former colleagues, he and his men took steps to conceal them from their American allies, like dumping corpses when dust storms obscured aerial surveillance or using veiled language over the phone. Sending someone to “Dubai” meant killing them in the desert. “His commanders would call and say: ‘We caught someone. What should we do?’ He’d say, ‘God forgive them.’ That was his code,” said a senior republican police general who worked with Raziq. “I heard it with my own ears on an operation.”

The farther you got from the streets and villages, the easier it was to ignore what was happening there. According to an interpreter who spent years translating Raziq’s meetings with his American advisers, the subject was generally avoided at headquarters. “The advisers didn’t care about Raziq’s bad activities,” he said. “We weren’t telling Raziq: ‘Hey, do you have private prisons? Do you still have people in there?’”

“I’m not saying they didn’t occur; I’m not saying they did occur,” Schweitzer said about the kinds of accusations that led Allen to halt detainee transfers. “I just know I read all the intel reports.” And whatever American officers chose to believe, they could see that Raziq was delivering where it counted: Within a year and half of his taking over, enemy-initiated attacks were down by almost two-thirds in Kandahar. “I thought he was an incredibly important figure,” Schweitzer said, “and was critical to keeping the security in the south.”

The COIN strategy was tested in the summer of 2014, when the Taliban began a bold offensive targeting the two southern provinces that had been the focus of American efforts. The surge had come to an end, and republican forces were supposed to take the lead in combat.

In Helmand, where the Marines tried to keep out abusive strongmen, the government’s defense was disastrously weak and uncoordinated. In many rural areas, the republican army stayed in their forts and allowed the police to be overrun. Despite the presence of a major American air base in the province, large sections of the northern districts fell into insurgent hands.

But when the Taliban pushed into western Kandahar, Raziq took charge and rallied republican forces. Backed by his advisers and American airstrikes, he inflicted heavy casualties. The following summer, insurgents again attacked and reached the outskirts of Helmand’s capital; Raziq led counteroffensives to lift sieges there and, the next year, in the neighboring Uruzgan province. “The Taliban have fled the area and escaped,” he boasted to a TV crew while touring the embattled district of Now Zad.

By 2017, Helmand was among the top three provinces most controlled by insurgents, according to U.S. military figures. And while the situation was deteriorating across the country, Kandahar City and its surroundings remained relatively secure under Raziq. Journalists and human rights groups had warned that supporting men like him would backfire and inspire resistance to the government. Yet here he was, holding the line against the Taliban.

A Taliban supporter whose husband was killed in battle. She used to aid the movement by smuggling weapons. “We tied pistols around our waists to get them through checkpoints,” she said.

Retired Special Forces Colonel Jamie Hayes was an advisor to General Abdul Raziq early in his rise to becoming one of the most powerful figures in southern Afghanistan.

Raziq was far from the only example: Again and again, the U.S. military felt compelled to partner with Afghan allies who were accused of human rights abuses, despite its doctrine of winning the war by winning hearts and minds. Call it the COIN paradox; for years it puzzled me, until I came across the work of the political scientist Stathis Kalyvas, who offered a convincing explanation of its logic.

In his comparative study of conflicts ranging from the Napoleonic occupation of Spain to the Tamil Tigers’ insurgency in Sri Lanka, Kalyvas asks why civil wars are so often marked by violence against civilians. Discarding explanations like cultural backwardness or ideology, Kalyvas argues that the incentive for this violence is created by the military characteristics of civil war, where the population is the battlefield.

To understand how Kalyvas’s theory applies to Afghanistan, you had to look at the rural areas where most of the fighting took place. Consider the Taliban’s stronghold in Kandahar, the Panjwai valley. A verdant delta of pomegranate and grape orchards west of the provincial capital, Panjwai was the birthplace of the movement. Mullah Muhammad Omar, the Taliban’s leader, preached in his mosque there.

The site of major offensives by allied forces since 2006, Panjwai was arguably the longest and most grueling fight anywhere in Kandahar. During the surge, American troops fought their way in and, by the end of 2010, had built up a string of bases and strong points, many jointly manned with the republican army and the police. The Taliban ordered its fighters to melt back into the villages, where, aided by the area’s dense vegetation and mud-walled orchards, they switched to hit-and-run ambushes, assassinations and improvised explosive devices.

This kind of guerrilla struggle was an example of what Kalyvas calls irregular warfare, in which territorial control is fragmented and mixed between both sides. The Taliban hid their weapons and picked up shovels, taking advantage of American rules of engagement, which allowed soldiers to fire only on those who were armed or posing an active threat. “We basically did not see a difference between the locals and the Taliban,” said Curtis Grace, who patrolled there as an infantryman in 2012.

The Army’s COIN manual stresses the difficulty in irregular warfare of telling civilians and insurgents apart. Kalyvas’s argument is different: The distinction itself can blur. In a conflict with no clear front lines, violence is jointly produced by combatants and civilians, who have the information the troops need to fight their enemies: the location of I.E.D.s and army patrols, the identities of insurgents and government supporters. Moreover, because civil war involves rival state-building, civilians help or hinder combatants by providing logistical and political support. In Panjwai, the Taliban needed local help to operate: They tried to win it by announcing safe routes through minefields, but they were also ruthless with those suspected of being spies and government supporters.

In civil war, while indiscriminate violence, like collateral damage from airstrikes, can backfire, “selective” violence against individuals works in a straightforward way: Do this, or I’ll kill you . Winning hearts and minds can still matter, but it’s only half the story. And in wartime, sticks are often much cheaper and more effective than carrots. In this life-or-death struggle, the competitor willing to use both will have the advantage.

Kalyvas’s work is part of a larger body of scholarship on civil war and counterinsurgency that demonstrates how central the use of coercive violence against civilians has been in such conflicts, whether waged by dictatorships like Syria or democracies like France. “ ‘The bad guys win’ is not the answer that U.S. forces, policymakers or civilians want to hear about counterinsurgency success, but the historical record is clear,” writes the scholar Jacqueline L. Hazelton. In this light, COIN doctrine can be seen as a form of American exceptionalism: the idea that the United States could fight a civil war differently from anyone else — humanely.

If Kalyvas is right, then what the U.S. military faced in Afghanistan was not so much a paradox as an impossible choice. To take back places like Panjwai, there was a compelling incentive to use unlawful violence against the population, which the U.S. military could not allow itself to do. The solution to this dilemma was a division of labor, where the United States provided firepower and money to allies like Raziq, who did the dirty work.

In 2010, the United States introduced the Afghan Local Police program, or ALP. Drawing on their experience with militias in Vietnam and El Salvador, the Special Forces trained and armed villagers around the country. In the military’s hearts-and-minds framework, they were empowering communities to protect themselves against violent outsiders. But four decades of a multisided conflict meant that fault lines ran through communities, villages and even families. Most areas were tribally mixed; finding militias meant exploiting those divisions just as the Taliban had been doing. It meant arming Afghans against one another in a civil war.

As police chief, Raziq was in charge of the ALP program in Kandahar. Panjwai District was the most resistant; by 2011, its horn, as the western end was known, was the only place that the militias had failed to take root, despite the presence of several Special Forces teams. The next year, Raziq appointed one of his key lieutenants as the district police chief. Panjwai was predominantly Noorzai; Sultan Mohammad was an Achakzai like Raziq, but he was from the district. Such local knowledge, the ability to make rural Afghan society legible to outsiders, was precisely what made militias effective. They could go after the Taliban and their supporters in their own homes. The Taliban had gained sway over the villages by targeting the families of those who collaborated with the republic, and the militias, protected by American and regular government forces, could turn the tables.

Most of Panjwai was too dangerous to visit during the war, but when the republic fell in 2021, I was able to travel there, interviewing dozens of witnesses who described torture and extrajudicial killings carried out by members of the police and the ALP, targeting both active insurgents and sympathizers. In the village of Pashmul, several witnesses told me they saw Sultan Mohammad shoot an unarmed old man, Hajji Badr, whose sons had served in the Taliban. Sultan Mohammad told me he had no involvement in murder or torture, but several other people said they witnessed him personally execute prisoners. “All the people from the area knew,” said Hasti Mohammad, a republican district governor in Panjwai. “It wasn’t something secret.”

I was also shown several videos of police abuse, including one in which a group of men, identified by locals as ALP members in Panjwai, tortured a captive bound hand and foot. They strike him with sticks, twist his testicles with their hands, pour water over his mouth and sodomize him with a stick, all while demanding he confess. “I don’t have anything,” he blubbers, growing incoherent.

This brutality was no impediment to American and republican success. Under Sultan Mohammad, the ALP program was established throughout the district. I.E.D. attacks plummeted, while the proportion of bombs that went off without being discovered dropped by half, which one study attributed to increased cooperation from locals.

The U.S. military was aware of the abuses by police officers and militia members in Panjwai. On multiple occasions, American surveillance captured them committing war crimes. One video showing executions by the police was shown to senior U.S. officials in 2012; Colonel Webb said he asked Raziq to arrest the perpetrators, but police investigators told me that some ordinary militiamen were punished instead. Sultan Mohammad was eventually promoted to brigadier general and oversaw several districts in the west of the province. When I spoke with him, he showed me a collection of certificates of appreciation from more than a dozen U.S. military units. “The Special Forces helped us a lot,” he said.

Fazli Ahmad, a Taliban fighter, was arrested by the police, who filmed a video of themselves dragging him behind a pickup truck. He said Sultan Mohammad ordered him to be executed, but he was released after his father paid a bribe.

Sultan Mohammad, a former police chief in Panjwai District and one of Raziq’s key lieutenants. He worked closely with the U.S. military to establish a militia program in Panjwai. “The Special Forces helped us a lot,” he said.

Thanks to his success in Kandahar, Raziq became famous. Not since the late northern commander Ahmad Shah Massoud, whom he greatly admired, had any one figure united anti-Taliban sentiment across the country. He was interviewed on national television, and his picture was pasted on street billboards. Songs were dedicated to him:

He’s the servant of security, the servant of our government.

He’s truly the servant of Afghans.

There were many reasons for his popularity. He was young and dynamic, a village boy who never lost the common touch. He was free with his largess, sometimes handing out cash on the street. He spoke fearlessly against Pakistan’s support for the insurgency. For Afghans disenchanted by the corruption and duplicity of their politicians, Raziq seemed authentic.

“There were other politicians who would talk against the Taliban and Pakistan,” said Nader Nadery, a senior fellow at the Wilson Center who as head of the country’s human rights commission had criticized Raziq’s abuses. “With Raziq, people saw it was not just words.” Even Nadery had come to see the trade-off that Raziq represented, as a bulwark against the looming collapse of the republic. “It’s a difficult judgment to make,” he said. “We can lose everything, or we can keep some parts of it.”

Although Raziq publicly denied accusations of human rights abuses, when an Afghan journalist asked him about them in 2017, he offered something close to a justification. “Showing mercy to such people is a betrayal to our nation,” he replied. “When our soldiers are martyred, isn’t that a violation of human rights? When our schools are burned, isn’t that a violation?”

The truth was that many Afghans saw Raziq’s brutality as a positive quality. They wanted a champion who could protect them from the Taliban’s violence. When Raziq went out on the streets, he was mobbed by crowds of well-wishers. “It was like being an adviser to Elvis Presley,” Webb recalled.

As Raziq grew in stature, he was rehabilitated. Western generals and diplomats sought him out on trips to Kandahar. Over the years, Raziq was a constant there, a fixed point around which contradictory policies and goals swirled: counterterrorism, nation-building, COIN and, finally, negotiations with the Taliban. “We needed him more than he needed us,” said John W. Lathrop, who as a brigadier general commanded American forces in Kandahar in 2017. “Keeping Raziq happy was pretty important.”

For the Taliban, Raziq was one of their top targets. By his own count, Raziq had survived at least 25 suicide attacks. Yet he remained committed to the fight. In one of his last interviews, Raziq criticized republican elites who already had one foot out the door with visas and houses overseas. “We shouldn’t hope or plan to seek asylum in America or move to London,” he said. “We were born here, and we’ll die here.”

On Oct. 18, 2018, General Miller, the top U.S. commander, called a meeting at the governor’s compound in Kandahar to discuss the upcoming parliamentary elections. That day, Raziq put on Western-style clothes: a dress shirt and slacks. The young soldier from the borderlands had become a statesman, a role that came less easily to him. He had seemed worn down to people who had met him lately; he was preoccupied with political dramas in Kabul. He had also been sick for days with a bad stomach bug, but he wanted to see Miller, whom he had known since the early days of the war. During the meeting, Raziq appeared flushed and uneasy, but afterward he insisted on walking to the helicopter pad to see off Miller and the other Americans.

A group of police officers arrived, carrying crates of pomegranates, gifts for the Americans. Among them was a bodyguard for the governor, a young man the Taliban had code-named Abu Dujana. He dropped his crate and fired his assault rifle, killing Raziq and the provincial intelligence chief and wounding several others, including an American general, before he was shot dead.

As one part of the country celebrated, the other mourned. The republic had lost its hero.

What does Raziq’s story tell us about why the United States failed in Afghanistan? Although the immediate cause of the republic’s collapse might have been the precipitous U.S. withdrawal in 2021, the real question is why the Afghan government could not stand on its own despite the hundreds of billions of dollars invested over 20 years by America and its allies. How did hundreds of thousands of soldiers and police officers, armed with modern equipment, lose to insurgents who rode their motorcycles in sandals?

Many corrupt and unpopular governments survive insurgencies. And it’s clear that the Taliban’s violence against civilians did not prevent their ultimate success. More than hearts and minds lost to brutality, internal rot and infighting — fed by the West’s profligate spending and inconsistent strategy — explain the republic’s collapse. Criminal behavior by republican officials escalated to the point that it threatened the system itself, bringing about repeated crises like the near collapse of the banking sector. Wage and supply theft were catastrophic to the morale of soldiers and police officers, while nonexistent “ghost soldiers” inflated their ranks. As the Americans pulled back from rural areas, the ALP militias became increasingly predatory, shaking down locals for bribes; their selective violence became indiscriminate, to use Kalyvas’s terms. “That’s how it started,” a senior Panjwai officer explained. “The district chief stole their salaries and said, ‘Go get your meals from the people.’”

When it came to corruption, Raziq played an ambiguous role: What he stole from the system with one hand he gave back with the other. With their control of the border, he and his cronies siphoned huge amounts of government revenue : The shortfall added up to around $55 million per year, according to satellite imagery and customs data analyzed by the researcher David Mansfield.

But Raziq also spent much of what he earned on his network of sources, on bonuses for his men, on bribes to protect himself from rapacious politicians in Kabul. In a corrupt system, money was synonymous with power, and Raziq needed it to fight. Yet while he tried to curb overly predatory commanders, there was a limit to how far he could go to keep order. He was a prisoner of his own methods. Enforced disappearances, torture and executions, the tools that Raziq believed were necessary to defeat the Taliban, had to be kept hidden, often through intimidation and bribery. Impunity for human rights abuses could lead to general lawlessness; in this way, repressive counterinsurgencies had mutated into mafia states in countries like Guatemala. The men that Raziq handpicked to carry out these acts were of necessity criminals. The darkness they worked within allowed corruption to flourish. By contrast, instead of democracy or human rights, the Taliban professed a fundamentalist vision of Islamic law. Their scholars justified killing captives and civilians as necessary and legitimate in the jihad against foreign occupation. Where the republic’s hypocrisy fed its fatal weakness, corruption, the Taliban’s unabashed brutality was consonant with the movement’s strength, its unity.

Today we live in an age of irregular warfare, of asymmetric clashes with militant groups and battles to control populations. A vast majority of conflicts over the past century have been within states, not between them. The comforting myth that brutality is always counterproductive — that war can therefore be humane — obscures how violence functions in such conflicts; it hides how and to whom men like Raziq are useful. In retrospect, this myth, sold to the public as COIN, is part of a larger pattern of dishonesty that runs through America’s longest war, 20 years of wishful thinking and willful ignorance that culminated in tragedy on Aug. 15, 2021, when Raziq’s mortal enemies entered Kabul in triumph.

Read by Peter Ganim

Narration produced by Anna Diamond and Krish Seenivasan

Engineered by Steven Szczesniak

Victor J. Blue is a photographer who has been working in Afghanistan since 2009, when President Barack Obama escalated the war effort. He was there during the fall of Kabul, when the Taliban came back into power in 2021, and has returned three times since then.

Matthieu Aikins is a contributing writer for The New York Times Magazine and a fellow at Type Media Center who, since 2008, has been covering conflicts in Afghanistan and the Middle East, the U.S. military's operations overseas, forced migration and human rights.  More about Matthieu Aikins

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