### An Inferentialist Approach to Paraconsistency

#### Abstract

This paper develops and motivates a paraconsistent approach to semantic paradox from within a modest inferentialist framework. I begin from the bilateralist theory developed by Greg Restall, which uses constraints on assertions and denials to motivate a multiple-conclusion sequent calculus for classical logic, and, via which, classical semantics can be determined. I then use the addition of a transparent truth-predicate to motivate an intermediate speech-act. On this approach, a liar-like sentence should be “weakly asserted”, involving a commitment to the sentence and its negation, without rejecting the sentence. From this, I develop a proof-theory, which both determines a typical paraconsistent model theory, and also gives us a nice way to understand classical recapture.

#### Keywords

#### References

Baaz, M., Fermueller, C. G., and Zach, R. (1993a). Elimination of cuts in first- order finite-valued logics. Elektronische Informationsverarbeitung und Kybernetik, 29(6):333–355.

Baaz, M., Fermüller, C. G., Salzer, G., and Zach, R. (1998). Labeled calculi and finite-valued logics. Studia Logica, 61(1):7–33.

￼￼￼[3] Baaz, M., Fermuller, C. G., and Zach, R. (1993b). Systematic construction of natural deduction systems for many-valued logics. In Multiple-Valued Logic, 1993., Proceedings of The Twenty-Third International Symposium on, pages 208– 213. IEEE.

Beall, J. (2012). Why priest’s reassurance is not reassuring. Analysis, 72(3):517– 525.

Beall, J. (2013). Free of detachment: Logic, rationality, and gluts. Noûs.

Beall, J. C. (2009). Spandrels of Truth. Oxford University Press.

Belnap, N. D. and Massey, G. J. (1990). Semantic holism. Studia Logica, 49(1):67–82.

Boghossian, P. A. (2003). Epistemic analyticity: A defense. Grazer Philosophische Studien, 66(1):15–35.

Carnap, R. (1943). Formalization of Logic. Cambridge, Mass.,Harvard University Press.

Dunn, J. M. and Hardegree, G. (2001). Algebraic methods in philosophical logic. OUP Oxford.

Garson, J. W. (2010). Expressive power and incompleteness of propositional logics. Journal of Philosophical Logic, 39(2):159–171.

Hardegree, G. M. (2005). Completeness and super-valuations. Journal of Philosophical Logic, 34(1):81–95.

Hjortland, O. T. (forthcominga). Logical pluralism, meaning-variance, and verbal disputes. Australasian Journal of Philosophy, pages 1–19.

Hjortland, O. T. (Forthcomingb). Speech acts, categoricity, and the meanings of logical connectives. Notre Dame Journal of Formal Logic.

Humberstone, L. (2011). The Connectives. MIT Press.

Pagin, P. (2012). Assertion, inference, and consequence. Synthese, 187(3):869–885.

Parsons, T. (1984). Assertion, denial, and the liar paradox. Journal of Philosophical Logic, 13(2):137–152.

Peacocke, C. (1986a). Thoughts: An Essay on Content. Blackwell.

Peacocke, C. (1986b). What determines truth conditions? In McDowell, J. and (eds.), P. P., editors, Subject, Thought, and Context, pages 181–207. Oxford, Clarendon Press.

Peacocke, C. (1987). Understanding logical constants: A realist’s account. British Academy.

Peacocke, C. (1993). Proof and truth. In Haldane and (eds.), W., editors, Reality, Representation and Projection. New York: Oxford University Press.

Priest, G. (2006). In Contradiction: A Study of the Transconsistent. Oxford University Press.

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

Restall, G. (2005). Multiple conclusions. In Hajek, P., V.-V. L. and Westerstpahl, D., editors, Logic, Methodology, and Philosophy of Science: Proceedings of the Twelfth International Congress, pages 189–205. Kings College Publications.

Restall, G. (2009). Truth values and proof theory. Studia Logica, 92(2):241–264.

Shoesmith, D. J. and Smiley, T. J. (1978). Multiple Conclusion Logic. CUP Archive.

[Tappenden] Tappenden, J. Negation, denial and language change in philosophical logic.

Zach, R. (1993). Proof theory of finite-valued logics. Master’s thesis, Technis- che Universit ät Wien.

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