Abstract
Priest and others have presented their “most telling” argument for paraconsistent logic: that only paraconsistent logics allow non-trivial inconsistent theories. This is a very prevalent argument; occurring as it does in the work of many relevant and more generally paraconsistent logicians. However this argument can be shown to be unsuccessful. There is a crucial ambiguity in the notion of non-triviality. Disambiguated the most telling reason for paraconsistent logics is either question-begging or mistaken. This highlights an important confusion about the role of logic in our development of our theories of the world. Does logic chart good reasoning or our commitments? We also consider another abductive argument for paraconsistent logics which also is shown to fail.
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Notes
I will use the term “classical” as denoting all explosive logics, unless I explicitly say otherwise. By this nomenclature intuitionistic logic is a classical logic since it endorses ex falso quodlibet. I should also state that I am not addressing all the arguments for paraconsistent logics. We implicitly restrict discussion to a particular language characterised syntactically with a determinate set of sentences.
This passage has remained unchanged from its original version in Priest and Tanaka (1997).
There is a separate question about whether anyone who accepts a contradiction could be ideally rational, and a further question whether everyone who uses an inconsistent theory accepts, or is committed to, that theory. See Michael (2013).
The naive theory is an inconsistent theory whether or not it is fully formalized. We usually present this theory to students with a number of principles, which may or may not be presented as formal axioms, including extensionality and a generous principle of abstraction. Indeed Russell presented his contradiction to Frege in natural language not by proof within a formal system, though it certainly had consequences for Frege’s formal system.
For further discussion of the uses of inconsistent theories with a classical underlying logic see Michael (2013).
Lewis (1982) argues that one way of resolving apparent contradictory theories is to see them as equivocations; rather than one inconsistent theory, we are working with multiple internally consistent but mutually inconsistent theories. This approach forms the basis of work by Brown and Priest (2004) on the so-called “Chunk and Permeate” approach.
It is worth pointing out in this context that seems to be another important confusion in Priest’s (1979) account. He distinguishes between the practice of inferring and a truth-preservational account of validity which he claims is typically taken to be prior. He insists on reversing their priority and lays emphasis on the practice of inferring. However there need be nothing specifically truth-preservational about classical conceptions of logical consequence; in fact, the two perspectives are quite orthogonal to each other. There are three different phenomena to distinguish: the truth-preservational conception of validity, the proof-theoretic conception of validity and the practice of inferring. Classical logicians could have held to a specifically proof-theoretic conception of consequence and still made the contrast with the practice of inference. On the other hand, Priest himself can be seen [in contrast to the attitude of relevant logicians Anderson and Belnap (1975, p. 166)] to have reverted to a truth-preservational account of validity albeit in the framework of his dialethism. The key question is whether we can hope to define validity by saying that the only arguments which are valid are those such that any rational agent who believes the premises must infer the conclusion. The considerations Sextus provided showed that to be a vain hope. See also the exchange between Milne (2009) and Field (2009).
Given the usual structural rules of weakening and transitivity the explosive rule of ex falso quodlibet is derivable from reductio ad absurdum. In some formulations of RAA we need double negation elimination as well.
See for more on this Michael (2013).
The problem of making sensible use of an inconsistent theory does not obviously call for a solution at the level of underlying logic, any more than the problem of using idealizations calls for a logic of idealization or in general using a theory that is false calls for a logic which draws safe consequences from a falsehood.
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Acknowledgments
Thanks are due to Aislinn Batstone, Wylie Breckenridge, John P. Burgess, Adam Dickerson, Lloyd Humberstone, Toby Meadows, Marcus Rossberg, Max Rabie, and Lionel Shapiro. I would also like to thank and acknowledge the comments of anonymous referees.
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Michael, M. On a “most telling” argument for paraconsistent logic. Synthese 193, 3347–3362 (2016). https://doi.org/10.1007/s11229-015-0935-6
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DOI: https://doi.org/10.1007/s11229-015-0935-6