Second-Order Logic of Paradox

Abstract

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)

Analytic Continuation of $ζ(s)$ Violates the Law of Non-Contradiction (LNC)

“…80 See Priest [226] and [227]. See also Hazen et al [133], p.2: "Truth values of compound formulas are derived from those of their subformulas by the familiar "truth tables" of Kleene's (strong) 3-valued logic [ [159], §64], but whereas for Kleene (thinking of the "middle value" as truth-valuelessness) only the top value (True) is designated, for Priest the top two values are both designated." 81 See Urquhart [293], pp.252: "[Bochvar's] idea is to avoid logical paradoxes such as Russell's and Grelling's by declaring the crucial sentences involving them to be meaningless (having the value I ).…”

“…134 135 132 See Vafeiadou et al [294], p.2, citing Brouwer [46], p.79: "Moreover, the '... existence of a mathematical system satisfying a set of axioms can never be proved from the consistency of the logical system based on those axioms,' but only by construction." 133 See also Bridges et al [41], §2 "The Constructive Interpretation of Logic": "∃ (there exists): to prove ∃xP (x) we must construct an object x and prove that P (x) holds", and "These BHKinterpretations (the name reflects their origin in the work of Brouwer, Heyting, and Kolmogorov) can be made more precise using Kleene's notion of realizability", citing (Dummett [83], pp.222-234 and Beeson [23], Chapter VII). 134 See Moschovakis [199]: "Intuitionistic propositional logic is effectively decidable, in the sense that a finite constructive process applies uniformly to every propositional formula, either producing an intuitionistic proof of the formula or demonstrating that no such proof can exist.…”

“…307 See e.g. the following articles on "Logic of Paradox": Priest [226], Priest [227], and Hazen et al [133]. 308 This use of LP as the underlying logic of RH also renders Gödel's first incompleteness theorem irrelevant.…”

Quantum Electrodynamics (QED) Renormalization is a Logical Paradox, Zeta Function Regularization is Logically Invalid, and Both are Mathematically Invalid

“…This is the Riemann Zeta Function. 44 3.3 Riemann's Zeta Function is False, Because Hankel's Contour Contradicts Cauchy's Integral Theorem…”

K3, Ł3, LP, RM3, A3, FDE, M: How to Make Many-Valued Logics Work for You

“…In Hazen and Pelletier (2017) it was shown that a Second Order logic based on LP was surprisingly weak. This was due to the limited expressive power of the language with no conditional operator.…”