Graham Priest's Logic of Paradox

Classical logic is characterized by such rules as (1) the principle of bivalence (that a proposition can only be assigned one of two truth values, true or false), (2) the law of excluded middle, (p v ~p, that every proposition is either true or false), (3) the law of non-contradiction (~(p ∧ ~p), that a proposition and its negation cannot both be true at the same time), (4) the law of self-implication (p → p, that if a proposition is true, then it is true), and (5) double negation elimination (~~p → p, that if a proposition is not false, then it is true).  
      The two main branches of classical logic are propositional logic and predicate logic. Propositional logic employs variables to represent propositions, and it connects them with logical operators like "and", "or", and "not". Predicate logic extends propositional logic by using predicates and quantifiers to describe properties of objects and the relations between them. 

       ~(L & H) → ~G (I will not graduate if I don't pass both logic and history) is a sentence of propositional logic, while ∀xFx → ∀xSx (if anyone fails the exam, then everyone will be sad) is a sentence of predicate logic. 

       First-order predicate logic is a logic in which predicates take only individuals as arguments, and quantifiers only bind individual variables. Higher-order predicate logic is a logic in which predicates take other predicates as arguments, and quantifiers bind predicate variables.¹

       Non-classical logics include three-valued logics (such as Łukasiewicz's, Kleene's, and Priest's), 4-valued logics (such as Belnap's), many-valued logics (such as infinitely-valued fuzzy logic), paraconsistent logic, intuitionistic logic, modal logics (including alethic modal logic, deontic logic, temporal logic, epistemic logic, and doxastic logic), free logics, and substructural logics (such as relevance logic and linear logic). 

       Some differences between classical and non-classical logics include: (1) classical logic uses only two truth values, but non-classical logics may allow multiple truth values, (2) classical logic accepts the law of excluded middle, but intuitionistic logic rejects it (the sentence p v ~p is not valid in intuitionistic logic, so ⊬ p v ~p), (3) indirect proof (or proving a sentence by assuming its negation and then showing a contradiction) is a valid inference rule in classical logic, but not in intuitionistic logic (because in intuitionistic logic, proving that the negation of a statement yields a contradiction doesn't prove the statement is true), (4) classical logic accepts the law of non-contradiction, but paraconsistent logic rejects it (so some contradictions may still be assigned a truth value), (5) in classical logic, a contradiction implies anything (p ∧ ~ p → q), according to the principle of explosion (ex falso quodlibet, "from falsehood anything follows"), but paraconsistent logic rejects this, and (6) in classical logic, double negation elimination is a valid inference rule, but in intuitionistic logic, it is not (because in intuitionistic logic, ruling out the falsity of a statement doesn't prove that the statement is true, ~~p ⊬ p). 

       Motivations for three-valued logic include: (1) vagueness, semantic underdetermination, and borderline cases where statements do not seem to be clearly true or false, (2) future contingents, where statements about the future do not seem to be clearly true or false, (3) presupposition failure, where statements with false presuppositions do not seem to be clearly true or false, and (4) empty names, where statements about nonexistent or fictional entities do not seem to be clearly true or false.²

       Graham Priest's Logic of Paradox (LP) is motivated by the need to respond to logical paradoxes--both semantic ones, such as the Liar Paradox, and set-theoretic ones, such as Russell's Paradox.³ Priest explains that all paradoxes of self-reference (including both semantic and set-theoretic paradoxes) fit the "inclosure schema," where a set of all things with a property exists, but the property itself contradicts the set's membership.⁴ Beyond such paradoxes, LP is also designed to deal with problems of vagueness (e.g. the Sorites paradox) by allowing for "truth-value gluts" (where a statement may be both true and false) in borderline cases. 

       Theodore Sider (2010) explains that in Priest's Logic of Paradox, a set of wffs Γ semantically implies wff φ iff φ is either 1 or # in every trivalent interpretation in which every member of Γ is either 1 or #. (Priest uses the symbols t for true and only true, f for false and only false, and p for "paradoxical" or both true and false, so in Sider's notation # would correspond to p in Priest's notation). Thus, the definitions of validity and semantic consequence in Priest's Logic of Paradox are:

• φ is LP-valid (“⊨_LP φ”) iff KV_I (φ) ≠ 0 for each trivalent interpretation I
• φ is an LP-semantic-consequence of Γ (“Γ ⊨_LP φ”) iff for every trivalent interpretation, I, if KV_I (γ ) ≠ 0 for each γ ∈ Γ, then KV_I (φ) ≠ 0 

(where KV_I is a Kleene valuation function).⁵

      Sider also explains that for Priest, p represents the state of being both true and false (a truth-value "glut"), rather than the state of being neither true nor false (a truth-value “gap”). Correspondingly, Priest takes 1 to represent true and only true, and 0 to represent false and only false. The position that natural language sentences can be both true and false is known as dialetheism. LP represents logical consequence as the preservation of either 1 or p; and in LP, a formula is thought of as being true iff it is either 1 or p (in the latter case the formula is false as well).⁶ LP is a paraconsistent logic, because P, ~P ⊭_LP Q. 

       Priest (2024) distinguishes between paraconsistency and dialetheism by saying that paraconsistent logicians need not be dialetheists, and that they may reject the principle of explosion for other reasons.⁷ He explains that J.C. Beall and Greg Restall (2006) distinguish between four grades of paraconsistency: (1) gentle-strength, which is simply the rejection of explosion with respect to logical consequence, (2) full-strength, which holds that there are interesting or important theories that are inconsistent but not trivial, (3) industrial-strength, which holds that some inconsistent but non-trivial theories are possibly true, and (4) dialetheic, which holds that some inconsistent but non-trivial theories are indeed true.⁸

       Sider notes that ex falso quodlibet is not the only classical inference rule that fails in LP. Modus ponens, modus tollens, and reductio ad absurdum also fail in LP. Thus, the relation of logical consequence is treated much differently in LP than in classical logic.⁹ 

       Why modus ponens fails in LP can be demonstrated as follows: P, P → Q ⊭_LP Q. In other words, Q is not a semantic consequence of the premises P and P → Q, because if we assign P a truth value of #, and Q a truth value of 0, then #, # → 0 ⊭_LP 0, because the truth value for # → 0 is #.

FOOTNOTES

¹Peter Suber, "Glossary of First-Order Logic," 2002, online.

²Lisa Cassell, "Beyond Standard Propositional Logic," Philosophy of Logic, lecture, 2026.

³Graham Priest, "The Logic of Paradox," Journal of Philosophical Logic, Vol. 8, No. 1, Jan. 1979, p. 219.

⁴Graham Priest, "Inclosures, Vagueness, and Self-Reference," Notre Dame Journal of Formal Logic, Vol. 51, No. 1, 2010, p. 70.

⁵Theodore Sider, Logic for Philosophy (Oxford: Oxford University Press, 2010), p. 102.

⁶Ibid., p. 103.

⁷Graham Priest, "Dialetheism," Stanford Encyclopedia of Philosophy, 2024, online at 

https://plato.stanford.edu/entries/dialetheism/.

⁸J.C. Beall and Greg Restall, Logical Pluralism (Oxford: Oxford University Press), 2006.

⁹Sider, Logic for Philosophy, p. 103.