For the past 40 years, Neil Tennant has defended a proof-theoretic criterion of self-referential paradoxicality. According to this criterion, the defining characteristic of paradoxes is that, when formulated within a natural deduction system, they produce derivations that cannot be normalized. This paper raises doubts about Tennant’s approach. Recently, Tennant has suggested that Russell’s paradox might not truly fit his criterion. I will argue that the reasoning that rules out Russell’s paradox can similarly be applied to some semantic paradoxes. Therefore, if (...) Tennant’s assessment of Russell’s paradox holds, few cases may genuinely qualify as paradoxes by his standards. (shrink)

We begin with a brief explanation of our proof-theoretic criterion of paradoxicality—its motivation, its methods, and its results so far. It is a proof-theoretic account of paradoxicality that can be given in addition to, or alongside, the more familiar semantic account of Kripke. It is a question for further research whether the two accounts agree in general on what is to count as a paradox. It is also a question for further research whether and, if so, how the so-called Ekman (...) problem bears on the investigations here of the intensional paradoxes. Possible exceptions to the proof-theoretic criterion are Prior’s Theorem and Russell’s Paradox of Propositions—the two best-known ‘intensional’ paradoxes. We have not yet addressed them. We do so here. The results are encouraging. §1 studies Prior’s Theorem. In the literature on the paradoxes of intensionality, it does not enjoy rigorous formal proof of a Gentzenian kind—the kind that lends itself to proof-theoretic analysis of recondite features that might escape the attention of logicians using non-Gentzenian systems of logic. We make good that lack, both to render the criterion applicable to the formal proof, and to see whether the criterion gets it right. Prior’s Theorem is a theorem in an unfree, classical, quantified propositional logic. But if one were to insist that the logic employed be free, then Prior’s Theorem would not be a theorem at all. Its proof would have an undischarged assumption—the ‘existential presupposition’ that the proposition $$\forall p(Qp\!\rightarrow \!\lnot p)$$ ∀ p ( Q p → ¬ p ) exists. Call this proposition $$\vartheta $$ ϑ. §2 focuses on $$\vartheta $$ ϑ. We analyse a Priorean reductio of $$\vartheta $$ ϑ along with the possibilitate $$\Diamond \forall q(Qq\!\leftrightarrow \!(\vartheta \!\leftrightarrow \! q))$$ ◊ ∀ q ( Q q ↔ ( ϑ ↔ q ) ). The attempted reductio of this premise-pair, which is constructive, cannot be brought into normal form. The criterion says we have not straightforward inconsistency, but rather genuine paradoxicality. §3 turns to problems engendered by the proposition $$\exists p(Qp\wedge \lnot p)$$ ∃ p ( Q p ∧ ¬ p ) (call it $$\eta $$ η ) for the similar possibilitate $$\Diamond \forall q(Qq\!\leftrightarrow \!(\eta \!\leftrightarrow \! q))$$ ◊ ∀ q ( Q q ↔ ( η ↔ q ) ). The attempted disproof of this premise-pair—again, a constructive one—cannot succeed. It cannot be brought into normal form. The criterion says the premise-pair is a genuine paradox. In §4 we show how Russell’s Paradox of Propositions, like the Priorean intensional paradoxes, is to be classified as a genuine paradox by the proof-theoretic criterion of paradoxicality. (shrink)

This is a reply to the considerations advanced by Schroeder-Heister and Tranchini as prima facie problematic for the proof-theoretic criterion of paradoxicality, as originally presented in Tennant and subsequently amended in Tennant. Countering these considerations lends new importance to the parallelized forms of elimination rules in natural deduction.

In this dissertation, we shall investigate whether Tennant's criterion for paradoxicality(TCP) can be a correct criterion for genuine paradoxes and whether the requirement of a normal derivation(RND) can be a proof-theoretic solution to the paradoxes. Tennant’s criterion has two types of counterexamples. The one is a case which raises the problem of overgeneration that TCP makes a paradoxical derivation non-paradoxical. The other is one which generates the problem of undergeneration that TCP renders a non-paradoxical derivation paradoxical. Chapter 2 deals with (...) the problem of undergeneration and Chapter 3 concerns the problem of overgeneration. Chapter 2 discusses that Tenant’s diagnosis of the counterexample which applies CR−rule and causes the undergeneration problem is not correct and presents a solution to the problem of undergeneration. Chapter 3 argues that Tennant’s diagnosis of the counterexample raising the overgeneration problem is wrong and provides a solution to the problem. Finally, Chapter 4 addresses what should be explicated in order for RND to be a proof-theoretic solution to the paradoxes. (shrink)

From the point of view of proof-theoretic semantics, it is argued that the sequent calculus with introduction rules on the assertion and on the assumption side represents deductive reasoning more appropriately than natural deduction. In taking consequence to be conceptually prior to truth, it can cope with non-well-founded phenomena such as contradictory reasoning. The fact that, in its typed variant, the sequent calculus has an explicit and separable substitution schema in form of the cut rule, is seen as a crucial (...) advantage over natural deduction, where substitution is built into the general framework. (shrink)

The paper’s novelty is in combining two comparatively new fields of research: non-transitive logic and the proof method of correspondence analysis. To be more detailed, in this paper the latter is adapted to Weir’s non-transitive trivalent logic \({\mathbf{NC}}_{\mathbf{3}}\). As a result, for each binary extension of \({\mathbf{NC}}_{\mathbf{3}}\), we present a sound and complete Lemmon-style natural deduction system. Last, but not least, we stress the fact that Avron and his co-authors’ general method of obtaining _n_-sequent proof systems for any _n_-valent logic (...) with deterministic or non-deterministic matrices is not applicable to \({\mathbf{NC}}_{\mathbf{3}}\) and its binary extensions. (shrink)

Prawitz observed that Russell’s paradox in naive set theory yields a derivation of absurdity whose reduction sequence loops. Building on this observation, and based on numerous examples, Tennant claimed that this looping feature, or more generally, the fact that derivations of absurdity do not normalize, is characteristic of the paradoxes. Striking results by Ekman show that looping reduction sequences are already obtained in minimal propositional logic, when certain reduction steps, which are prima facie plausible, are considered in addition to the (...) standard ones. This shows that the notion of reduction is in need of clarification. Referring to the notion of identity of proofs in general proof theory, we argue that reduction steps should not merely remove redundancies, but must respect the identity of proofs. Consequentially, we propose to modify Tennant’s paradoxicality test by basing it on this refined notion of reduction. (shrink)

Dag Prawitz (1971) put forward the idea that an admissible reduction process does not affect the identity of proofs represented by derivations in natural deduction. The idea relies on his conjecture that two derivations represent the same proof if and only if they are equivalent in the sense that they are reflexive, transitive and symmetric closure of the immediate reducibility relation. Schroeder-Heister and Tranchini (2017) accept Prawitz’s conjecture and propose the triviality test as the criterion for admissible reductions. In the (...) present paper, we will consider two main troubles of the triviality test. The first is the obscurity of a method of evaluating admissible reductions. The second is the circularity problem that the triviality test already assumes the set of admissible reduction procedures. For the solution of the problems, we will propose the spoiler test which immunes the problems of the triviality test and has the role of the criterion for admissible reductions. At last, we shall cover a plausible problem of the spoiler test that can be caused by Crabbé’s case. (shrink)

By rephrasing quantifier-free axioms as rules of derivation in sequent calculus, we show that the generalized Steiner–Lehmus theorem admits a direct proof in classical logic. This provides a partial answer to a question raised by Sylvester in 1852. We also present some comments on possible intuitionistic approaches.