The Metaphysical Basis of Logic: The Law of Non-Contradiction as Basic Knowledge (2013-15)

Funded by the AHRC .

Principal Investigator: Francesco Berto.
Auditors: TBC.
Post-Doctoral Fellow: TBC.

The Research Topic

'The firmest of all principles' – thus Aristotle introduced in the Metaphysics (1005b) the principle later known as the Law of Non-Contradiction (LNC), which can be formulated by claiming that contradictions cannot be true in any possible circumstance. The Law enjoyed an almost spotless reputation in Western thought: Aquinas, Leibniz, Hume, Kant, Russell, Lewis, and other major philosophers took the LNC as the supreme cornerstone of logic and rational thought.

However, the LNC has come under attack in contemporary philosophy. Some logicians (Graham Priest, Richard Routley, Newton da Costa, JC Beall) have defended the idea that the LNC does not hold unrestrictedly: in peculiar circumstances (semantic paradoxes, paradoxes of absolute generality, legal inconsistencies) the same thing may at the same time be and not be, and contradictions may obtain. This view has been called dialetheism, a dialetheia ("double truth") being a true contradiction: a sentence A (or the expressed proposition), such that both it and its negation, ~A (not-A), are true (Beall et al. [2004], Berto and Priest [2008]). Dialetheism exploits the burgeoning research on paraconsistent logics, formal systems of non-classical logic (Routley et al. [1989], Priest [2002]) rejecting the law called ex contradictione quodlibet (ECQ), stating that a contradiction entails any arbitrary claim (A, ~A / B). By embracing such a logic, a dialetheist can accept some contradictions without accepting everything (Priest [1987]).

Dialetheism triggers a number of questions in foundational philosophy, ranging from epistemology to logic, metaphysics, and the very methodology of philosophical debate. The LNC has been taken as encoding such a basic knowledge that it is just impossible to discuss with those who cast doubts on it. David Lewis claimed:

'No truth does have, and no truth could have, a true negation. [...] That may seem dogmatic. And it is: I am affirming the very thesis that [the foes of the LNC] have called into question and - contrary to the rules of debate - I decline to defend it. Further, I concede that it is indefensible against their challenge. They have called so much into question that I have no foothold on undisputed ground. So much the worse for the demand that philosophers always must be ready to defend their theses under the rules of debate.' (Lewis [1982]: 101)

This project aims at showing that we can do better. Its goal is to advance the rational debate on the LNC and the notion of contradiction across three main areas of inquiry: logic, metaphysics, and epistemology.

1. Logic

• Paraconsistent dialetheism promises such applications as a treatment of semantic paradoxes like the Liar ('This sentence is untrue'), free from the (alleged) expressive limitations of strategies like the Kripke-Field truth-value gap approach (Field [2008]). Research in this area will focus on the question: Is a dialetheic truth theory (such as Beall [2009]) actually revenge-free, or are there paradoxes intractable also for the dialetheist, i.e., forcing some meaningful notion to be inexpressible in the relevant language?
• Theories challenging such a fundamental Law as the LNC are the subject of Quinean 'change of subject' charges: when dialetheists say, 'For some A, both A and not-A are true', one wonders what is meant by 'not' here (Slater [2007]). Research in this area will focus on such questions as: Can dialetheic negation perform all the tasks we expect from a negation operator? Which differences between negation and denial (a non-embeddable pragmatic operator: Priest [2006], Restall [2012]) are highlighted by the dialetheic stance? And is denial capable of providing the dialetheist with an exclusion-expressing device in case her negation is too weak for that?

2. Metaphysics

• The Aristotelian and traditional view has it that the LNC is not just a logical, but a general metaphysical principle holding for reality as a whole. Aristotle seems to have linked the LNC to an insight about the structural determinacy of the world: the world is structured by relations of exclusion, such that something’s being P rules out its simultaneously being Q, where P and Q are (absolutely, metaphysically) incompatible features (Lear [1980], Tennant [2004]). Reversing Dummett’s plea, Aristotle seems to have aimed at a metaphysical basis for a fundamental principle of logic such as the LNC. Research in this area will focus on such questions as: can one mount a defence of the LNC as a general metaphysical principle without begging questions against the dialetheist? Supposing such a defence were achievable, would it constitute a plain refutation of dialetheism? Or would it rather allow to draw decisive distinctions between various formulations of the Law, so far overlooked in the philosophical debate? Will some of these formulations prove more vulnerable to the dialetheic attack than others?

3. Epistemology

• Aristotle claimed that when one denies the LNC, we should wonder whether what is said is also believed (Metaphysics 1005b). He seems to have linked the LNC to a fundamental knowledge, such that a denier of the Law must be the victim of some equivocation: she cannot fail to possess such knowledge insofar as she is a rational being. Research in this area will focus on such questions as: Can this attitude towards the LNC be linked to what have been called (Williamson [2007]) 'epistemological conceptions of analyticity'? If so, may dialetheism provide the most significant challenge to such conceptions?
• Lewis [1982: 101] claimed that 'we know for certain, and a priori' that there are no true contradictions, and that somebody who denies this leaves us with “no foothold on undisputed ground”. Take Reductio ad absurdum, a rule of minimal logic which standardly works as a tool of rational criticism: a theory entailing both A and not-A must be revised, for it hosts a falsity. However, a dialetheist can keep her entire theory, and the criticism to the effect that it involves some inconsistency. Thus, Karl Popper [1969] famously argued that rejecting the LNC undermines rational criticism. Research in this area will focus on such questions as: How does a rational dialetheist revise his/her beliefs in the light of criticism and new evidence (Priest [2001], Tanaka [2005])? What are the conditions of a non-question-begging debate between the dialetheist and the defender of the LNC?

References

• Beall JC [2009], Spandrels of Truth, Oxford: Oxford UP.
• Beall JC et al. (eds.) [2004], The Law of Non-Contradiction, Oxford: Clarendon.
• Berto F. and Priest G. [2008], 'Dialetheism', The Stanford Encyclopedia of Philosophy, CSLI, Stanford, CA.
• Field H. [2008], Saving Truth from Paradox, Oxford: Oxford UP.
• Lear J. [1980], 'Proof by Refutation', in Id. [1980], Aristotle and Logical Theory, Cambridge: Cambridge UP.
• Lewis D. [1982], 'Logic for Equivocators', Noûs 16: 431-41
• Popper K. [1969], Conjectures and Refutations, London: Routledge & Kegan Paul.
• Priest G. [1987], In Contradiction. A study of the Transconsistent, Oxford UP: Oxford (2nd, expanded ed. 2006).
• Priest G. [2001], 'Paraconsistent Belief Revision', Theoria 68: 214-28.
• Priest G. [2002], 'Paraconsistent Logic', in Gabbay and Guenthner (eds.) [2002], Handbook of Philosophical Logic, 2nd ed., Dordrecht: Kluwer, pp. 287-393.
• Priest G. [2006], 'Denial and Rejection', in Id. [2006], Doubt Truth to Be a Liar, Oxford: Oxford UP.
• Restall G. [2012], 'Assertion, Denial and Non-Classical Theories', in Tanaka et al. (eds.) [2012], Paraconsistency: Logic and Applications, Dordrecht: Springer, pp. 81-100.
• Routley R. et al. (eds.) [1989], Paraconsistent Logic. Essays on the Inconsistent, München: Philosophia Verlag.
• Slater H. [2007], 'Dialetheias are Mental Confusions', in Béziau et al. (eds) [2007], Handbook of Paraconsistency, London: College Publications, pp. 457-67.
• Tanaka K. [2005], 'The AGM Theory and Inconsistent Belief Change', Logique et Analyse 189-92: 113-50.
• Tennant N. [2004], 'An Anti-Realist Critique of Dialetheism', in Beall et al. (eds.) [2004], The Law of Non-Contradiction, Oxford: Clarendon, pp. 355-84.
• Williamson T. [2007], The Philosophy of Philosophy, Oxford: Blackwell.