Abstract
Logic isn’t special. Its theories are continuous with science; its method continuous with scientific method. Logic isn’t a priori, nor are its truths analytic truths. Logical theories are revisable, and if they are revised, they are revised on the same grounds as scientific theories. These are the tenets of anti-exceptionalism about logic. The position is most famously defended by Quine, but has more recent advocates in Maddy (Proc Address Am Philos Assoc 76:61–90, 2002), Priest (Doubt truth to be a liar, OUP, Oxford, 2006a, The metaphysics of logic, CUP, Cambridge, 2014, Log et Anal, 2016), Russell (Philos Stud 171:161–175, 2014, J Philos Log 0:1–11, 2015), and Williamson (Modal logic as metaphysics, Oxford University Press, Oxford, 2013b, The relevance of the liar, OUP, Oxford, 2015). Although these authors agree on many methodological issues about logic, they disagree about which logic anti-exceptionalism supports. Williamson uses an anti-exceptionalist argument to defend classical logic, while Priest claims that his anti-exceptionalism supports nonclassical logic. This paper argues that the disagreement is due to a difference in how the parties understand logical theories. Once we reject Williamson’s deflationary account of logical theories, the argument for classical logic is undercut. Instead an alternative account of logical theories is offered, on which logical pluralism is a plausible supplement to anti-exceptionalism.
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Notes
The term ‘anti-exceptionalism’ is coined by Williamson (2007) to describe his own view of the methodology of philosophy, including philosophical logic. Maddy is perhaps better described as a naturalist about logic, but for our purposes I will consider naturalism a specific form of anti-exceptionalism where ‘science’ is understood narrowly as ‘natural science’. See Williamson (2013c) for Williamson’s discussion of naturalism.
The variety of exceptionalist positions will not be discussed in what follows. It is important to appreciate, however, that it is a commonplace view in the philosophy of logic, largely owing to the lasting influence of Frege and Carnap. Among the many recent defenders of exceptionalism, it is worth noticing that both Boghossian and Paul Boghossian (2000); Boghossian (2001, (2003) and Peacocke (1987, (1992, (1993) are targets of Williamson’s (2007) anti-exceptionalist objections.
Compare Russell (2014, 173): ‘[T]he overall virtues of logical theories were an important part of the justification for adopting or rejecting a theory. Here I have stressed simplicity, unification, elegance, strength, usefulness and explanatory power’.
See also Priest (2006a, ch. 8).
A logical theory is not merely a logical system. The latter is a formal construct, what Priest (2006a) calls a ‘pure logic’. A logical system can be a proof theory, a model theoretic relation, an algebra, etc. But unlike a logical theory, a logical system is not necessarily applied to anything. Priest suggests that the ‘canonical application’ of a logical theory is deductive reasoning, as opposed to logical systems applied to database management or mereology. Logical theories may nevertheless reflect one consequence relation rather than another. In fact, their choices of consequence relation are the decisive difference between Priest and Williamson.
Of course, to the extent that linguistic and conceptual entities are in the world, unrestricted generalizations are about them as well. But in no interesting sense does this make the truths of logic metalinguistic.
I use the term ‘truths of logic’ for the claims of a logical theory. It should not be confused with ‘logical truths’ in the standard sense.
Depending on whether we allow the law of exportation we could also express the premise combination with the conditional. But that makes no real difference to the problems discussed below.
I owe this observation to Rohan French.
Priest’s logic can be supplemented with stronger conditionals, but the material implication does not satisfy modus ponens.
For example, Slater (1995) rejects nonclassical logic on grounds that are far less amicable. According to him, nonclassical logics fail to talk about logical concepts; they are simply equivocating. It is a type of argument inspired by Quine’s (1986) infamous ‘change of logic, change of subject’ charge against nonclassical theories. For a discussion of Quine’s argument, see Paoli (2014) and Hjortland (2014).
There is a further worry about whether it makes sense to ask if a theory is well-confirmed outside the context of a logic. That will, in part, depend on the extent to which our theory of evidential confirmation and our theory of deductive logic are intertwined. I return ever so briefly to this issue in Sect. 5.
See Williamson (1994, 151–2) for a discussion.
There is a related example in Ripley (2012). A sequent calculus for classical logic can be set up so as to make the cut rule admissible, i.e. if the rule is added, no new sequent \(\Gamma \models A\) is derivable. This classical system can be conservatively extended with an unrestricted truth predicate. The result is a consistent theory, but one that is nontransitive—the cut rule is no longer admissible. If, on the other hand, we added an unrestricted truth predicate to a classical sequent calculus with an explicit cut rule, the result is a trivial theory, i.e. every sequent is derivable. Whether or not the nontransitive theory is correct does not concern us here. All we need is that the theory is a genuine candidate in philosophical debates about semantic paradoxes. And why wouldn’t it be? There is no antecedent reason to discard the theory, and certainly not one that won’t also apply to other rival logics. True, some logicians think that it is constitutive of a consequence relation that it is transitive, and therefore reject the theory as non-logical. That is a bad reason. The same abductive standard of science should be applied here as to any other candidate theory. Re-labeling a theory as a non-logical theory should cut no ice with the anti-exceptionalist.
For simplicity we only provide a variant for single premise meta-arguments. It is straight-forward to generalize further to variants that include meta-arguments from a set of arguments to an argument.
I’m borrowing the phrase ‘genuinely valid’ from Field (2015).
Furthermore, a non-deflationary logical theory can contain claims that connect metalinguistic claims to the world. A theory containing the claim about truth preservation (4) might also contain a (possibly restricted) truth schema: \(A \leftrightarrow True(\ulcorner {A}\urcorner )\).
Williamson also rejects the neutrality of logic (see Williamson 2013a).
Although one that won’t necessarily agree with Priest’s own view.
Strictly speaking, since a deflationary logical theory consists of unrestricted generalizations, all observations may provide partial confirmation of it. But that is not a very helpful perspective on evidence.
It is important to hedge in this case. The connection between validity and truth-preservation is not definitional. It is an influential reductive theory of validity, but not universally accepted (cf. Etchemendy 1990). See Field (2009, 2015) for an influential argument against the truth-preservational account of validity.
For the latter claim, see Kleene (1952).
Another criterion mention by Quine, Williamson, and others is simplicity. (See for example the citations at the start of the next section.) Their contention appears to be that classical logic is simpler than its nonclassical rivals. It is not at all clear which metric they have in mind. A logic can be simple to use or simple to learn. It can be simple because it has few rules, or few models, because it has proofs of low complexity or models of low complexity. Some nonclassical logics have fewer rules than classical logic, but more models. Does that make them simpler or more complex? More models make it easier to refute an argument; more rules make it easier to prove a claim. Without any grasp of what is meant by the condition, we will suspend it in what follows.
Williamson has made similar claims before, for example in connection with theories of vagueness: ‘If one abandons bivalence for vague utterances, one pays a high price. One can no longer apply classical truth-conditional semantics to them, and probably not even classical logic. Yet classical semantics and logic are vastly superior to the alternatives in simplicity, power, past success, and integration with theories in other domains.’ (Williamson 1994, 186)
Since (a weak theory of) arithmetic is sufficient to provide the background assumption, few argue that it should be given up.
See Beall and Murzi (2013) for details.
The qualification is important: Many nonclassical logicians—Priest included—distinguish between rejecting A and accepting \(\lnot A\).
See also Priest (2014, 217) for a similar remark.
In Beall and Restall, the different validity relations are instances of a Generalized Tarski Schema: An argument is valid for a class of cases C just in case whenever premises are true in C, the conclusion is true in C.
Typically the nonclassical logicians advocate logics that are strictly weaker than classical logic, but there are exceptions, e.g. connexive logic.
In an aside, Priest makes exactly this point against pluralism: ‘Unity is itself a desideratum; conversely, fragmentation is a black mark’ (Priest 2016, 9).
Priest (2006b, §18.5) makes this point about paraconsistent logics: they coincide with classical logic for consistent models.
References
Beall, J., & Murzi, J. (2013). Two flavors of Curry paradox. Journal of Philosophy, 110(3), 143–165.
Beall, J., & Restall, G. (2006). Logical pluralism. Oxford: OUP.
Boghossian, P., & Paul Boghossian, C. P. (2000). Knowledge of logic. New essays on the a priori (pp. 229–254). Oxford: Clarendon Press.
Boghossian, P. (2001). How are objective reasons possible? Philosophical Studies, 106, 1–40.
Boghossian, P. (2003). Blind reasoning. Proceedings of the Aristotelian Society Supplementary, 77, 225–248.
Carnap, R. (1937). The logical syntax of language. New York: Routledge & Kegan Paul Ltd.
Etchemendy, J. (1990). The concept of logical consequence. Cambridge, MA: Harvard University Press.
Field, H. (2008). Saving truth from paradox. Oxford: OUP.
Field, H. (2009). What is the normative role of logic? Proceedings of the Aristotelian Society Supplementary Volume, 83, 251–268.
Field, H. (2015). What is logical validity? In C. Caret & O. T. Hjortland (Eds.), Foundations of logical consequence (pp. 33–70). Oxford: OUP.
Hjortland, O. T. (2012). Logical pluralism, meaning-variance, and verbal disputes. Australasian Journal of Philosophy, 91, 355–373.
Hjortland, O. T. (2014). Verbal disputes in logic: Against minimalism for logical connectives. Logique et Analyse, 227, 463–486.
Humberstone, L. (2005). Logical discrimination. In J.-Y. Béziau (Ed.), Logica Universalis (pp. 207–228). Basel: Birkhaüser Verlag.
Kleene, S. C. (1952). Introduction to metamathematics. New York: Van Nostrand.
Maddy, P. (2002). A naturalistic look at logic. Proceedings and addresses of the American Philosophical Association, 76(2), 61–90.
Paoli, F. (2014). Semantic minimalism for logical constants. Logique et Analyse, 57(227), 439–461.
Paris, J. (2001). A note on the dutch book method. In Proceedings of the second international symposium on imprecise probabilities and their applications, ISIPTA, pp. 301–306.
Peacocke, C. (1987). Understanding logical constants: A realist’s account. Proceedings of the British Academy, 63, 153–200.
Peacocke, C. (1992). A study of concepts. Cambridge, MA: MIT Press.
Peacocke, C. (1993). How are a priori truths possible ? European Journal of Philosophy, 1, 175–199.
Priest, G. (2006a). Doubt truth to be a liar. Oxford: OUP.
Priest, G. (1987/2006b). In contradiction (2nd ed.). Oxford: OUP.
Priest, G. (2014). Revising logic. In P. Rush (Ed.), The metaphysics of logic, chapter 12 (pp. 211–223). Cambridge: CUP.
Priest, G. (2016). Logical disputes and the a priori. Logique et Analyse. http://virthost.vub.ac.be/lnaweb/ojs/index.php/LogiqueEtAnalyse/article/view/1850.
Quine, W. V. (1951). Two dogmas of empiricism. Philosophical Review, 60, 20–43.
Quine, W. V. (1986). Philosophy of logic (2nd ed.). Cambridge, MA: Harvard University Press.
Read, S. (2006). Monism: The one true logic. In D. D. Vidi & T. Kenyon (Eds.), A logical approach to philosophy: Essays in memory of Graham Solomon (pp. 193–209). Berlin: Springer.
Ripley, D. (2012). Conservatively extending classical logic with transparent truth. Review of Symbolic Logic, 5(2), 354–378.
Russell, G. (2015). The justification of the basic laws of logic. Journal of Philosophical Logic, 44, 793–803.
Russell, G. K. (2014). Metaphysical analyticity and the epistemology of logic. Philosophical Studies, 171(1), 161–175.
Shapiro, L. (2010). Deflating logical consequence. The Philosophical Quarterly, 61(243), 320–342.
Shapiro, S. (2014). Varieties of logic. Oxford: Oxford University Press.
Slater, B. H. (1995). Paraconsistent logics? Journal of Philosophical logic, 24(4), 451–454.
Van Ditmarsch, H., Van Der Hoek, W., & Kooi, B. (2007). Dynamic epistemic logic (Vol. 1). Dordrecht: Springer.
Williams, J. R. G. (2011). Gradational accuracy and non-classical semantics. The Review of Symbolic Logic, 1(1), 1–25.
Williamson, T. (1994). Vagueness. London: Routledge.
Williamson, T. (2007). The philosophy of philosophy. Oxford: Blackwell.
Williamson, T. (2013a). Logic, metalogic and neutrality. Erkenntnis, 79(2), 211–231.
Williamson, T. (2013b). Modal logic as metaphysics. Oxford: Oxford University Press.
Williamson, T. (2013c). What is naturalism? In M. C. Haug (Ed.), Philosophical methodology: The armchair or the laboratory? (pp. 29–31). Oxford: Routledge.
Williamson, T. (2015). Semantic paradoxes and abductive methdology. In B. Armour-Garb (Ed.), The relevance of the liar. Oxford: OUP. Forthcoming.
Acknowledgments
This version is based on earlier drafts presented at the University of Aberdeen, University of St Andrews, IUC Dubrovnik, LMU Munich, and CSMN, University of Oslo. I am grateful to all the participants for helpful discussions. Special thanks are owed to Pål Antonsen, Sorin Bangu, Thomas Brouwer, Aaron Cotnoir, Catarina Dutilh Novaes, Matti Eklund, Andreas Fjellstad, Rohan French, Olav Gjelsvik, Torfinn Huvenes, Ole Koksvik, Johannes Korbmacher, Hannes Leitgeb, Øystein Linnebo, Toby Meadows, Lavinia Picollo, Graham Priest, Stephen Read, Gillian Russell, Gil Sagi, Joshua Schechter, Stewart Shapiro, and Ole Martin Skilleås.
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Hjortland, O.T. Anti-exceptionalism about logic. Philos Stud 174, 631–658 (2017). https://doi.org/10.1007/s11098-016-0701-8
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DOI: https://doi.org/10.1007/s11098-016-0701-8