Second-Order Logic of Paradox

The Logic of Paradox, LP, is a first-order, three-valued logic that has been advocated by Graham Priest as an appropriate way to represent the possibility of acceptable contradictory statements. Second-order LP is that logic augmented with quantification over predicates. As with classical second-order logic, there are different ways to give the semantic interpretation of sentences of the logic. The different ways give rise to different logical advantages and disadvantages, and we canvass several of these, concluding that it will be extremely difficult to appeal to second-order LP for the purposes that its proponents advocate, until some deep, intricate, and hitherto unarticulated metaphysical advances are made. 1 Background on the “Logic of Paradox” Over the past three or four decades, but importantly in his [10], Graham Priest has investigated a variety of paradoxical topics – the semantic paradoxes are the ones that come first to a logician’s mind, but he has also studied puzzles arising from vagueness, and motion, and Buddhist philosophy and, in his recent book [14], metaphysical perplexities arising out the notion of parthood and in relation to the question of the unity of the proposition – all from a dialetheist perspective: one that considers it possible that there are true contradictions, i.e., that some propositions are both true and false or, equivalently, that some true propositions have true negations. As a basic logical framework for his investigations he has adopted the system he calls LP (for “Logic of Paradox”), which is perhaps the simplest modification of classical logic to allow non-trivial contradictions. This is a 3-valued logic, with values True, False, and Both. It has the usual propositional connectives ¬ (negation), ∧ (conjunction), ∨ (disjunction), and universal (∀) and existential (∃) quantifiers. The truth-functions ⊃ and ≡ are usually treated as defined connectives: (φ ⊃ ψ) =d f (¬φ ∨ ψ) 2010 Mathematics Subject Classification: Primary 03B53, 03C85; Secondary 03B50,