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As documented by the papers in this volume, which mostly result from the second conference on proof-theoretic semantics in Tübingen 2013, proof-theoretic semantics has advanced to a well-established subject in philosophical logic.

Proof-theoretic semantics explains meaning in terms of proofs. Two different concepts of proof are in question here. One has its roots in Heyting’s explanation of a mathematical proposition as the expression of the intention of a construction, and the other in Gentzen’s ideas about how the rules of Natural Deduction are justified in terms of the meaning of sentences. These two approaches to meaning give rise to two different concepts of proof, which have been developed much further, but the relation between them, the topic of this paper, has not been much studied so far. The recursive definition of proof given by the so-called BHK-interpretation is here used as an explication of Heyting’s idea. Gentzen’s approach has been developed as ideas about what it is that makes a piece of reasoning valid. It has resulted in a notion of , of which there are different variants. The differences turn out to be crucial when comparing valid arguments and BHK-proofs. It will be seen that for one variant, the existence of a valid argument can be proved to be extensionally equivalent to the existence of a BHK-proof, while for other variants, attempts at similar proofs break down at different points.

The goal of this paper is to consider the prospects for developing a consistent variant of the originally proposed by Georg Kreisel and Nicolas Goodman in light of two developments which have been traditionally associated with the theory—i.e. Kreisel’s interpretation of the intuitionistic connectives, and an antinomy about constructive provability sometimes referred to as the . After discussing the formulation of the theory itself, we then discuss how it can be used to formalize the BHK interpretation in light of concerns about the impredicativity of intuitionistic implication and Kreisel’s proposed amendments to overcome this. We next reconstruct Goodman’s presentation of a paradox pertaining to a “naive” variant of the theory and discuss the influence this had on its subsequent reception. We conclude by considering various means of responding to this result. Contrary to the received view that the second clause interpretation itself contributes to the paradox, we argue that the inconsistency arises in virtue of an interaction between reflection and internalization principles similar to those employed in Artemov’s Logic of Proofs.

To determine what deductions are it does not seem sufficient to know that the premises and conclusions are propositions, or something in the field of propositions, like commands and questions. It seems equally, if not more, important to know that deductions make structures, which in mathematics we find in categories, multicategories and polycategories. It seems also important to know that deductions should be members of particular kinds of families, which is what is meant by their being in accordance with rules.

This is a tripartite work. The first part is a brief discussion of what it is to be a logical constant, rejecting a view that allows a particular self-referential “constant” to be such a thing in favour of a view that leads to strong normalisation results. The second part is a commentary on the flattened version of Modus Ponens, and its relationship with rules of type theory. The third part is a commentary on work (joint with Nissim Francez) on “general elimination rules” and harmony, with a retraction of one of the main ideas of that work, i.e. the use of “flattened” general elimination rules for situations with discharge of assumptions. We begin with some general background on general elimination rules.

There are two camps in the theory of meaning: the referentialist one including Davidson, and the inferentialist one including Dummett and Brandom. Proof-theoretic semantics is a semantic enterprise to articulate an inferentialist account of the meaning of logical constants and inferences within the proof-theoretic tradition of Gentzen, Prawitz, and Martin-Löf, replacing Davidson’s path “from truth to meaning” by another path “from proof to meaning”. The present paper aims at contributing to developments of categorical proof-theoretic semantics, proposing the principle of categorical harmony, and thereby shedding structural light on Prior’s “tonk” and related paradoxical logical constants. Categorical harmony builds upon Lawvere’s conception of logical constants as adjoint functors, which amount to double-line rules of certain form in inferential terms. Conceptually, categorical harmony supports the iterative conception of logic. According to categorical harmony, there are intensional degrees of paradoxicality of logical constants; in the light of the intensional distinction, Russell-type paradoxical constants are maximally paradoxical, and tonk is less paradoxical. The categorical diagnosis of the tonk problem is that tonk mixes up the binary truth and falsity constants, equating truth with falsity; hence Prior’s tonk paradox is caused by equivocation, whereas Russell’s paradox is not. This tells us Prior’s tonk-type paradoxes can be resolved via disambiguation while Russell-type paradoxes cannot. Categorical harmony thus allows us to demarcate a border between tonk-type pseudo-paradoxes and Russell-type genuine paradoxes. I finally argue that categorical semantics based on the methods of categorical logic might even pave the way for reconciling and uniting the two camps.

An intuitionistic solution to the Paradox of Knowability is given. It consists (i) in accepting , the ordinary formalization of the principle of Radical Anti-Realism (RAR) that “Every truth is known”, since, intuitionistically understood, it means that proofs are epistemically transparent; and (ii) in accepting (RAR) itself, on the basis of the fact that knowledge is an intuitionistic internal truth notion. Some neo-verificationist approaches are criticized. Finally the problem of how to frame a rational discussion between Classicism and Intuitionism is briefly discussed.

The class of derivations in a system of logic has an inductive definition. One would thus expect that crucial properties of derivations, such as normalization in natural deduction or cut elimination in sequent calculus or consistency in arithmetic be proved by induction on the last rule applied. So far it has not been possible to implement this simple requirement uniformly. It is suggested that such proofs can be carried through by a methodology that is hidden in Gentzen’s original unpublished proof of the consistency of arithmetic: to prove that a suitably chosen property of derivations is maintained under the composition of two derivations. As examples, new proofs by induction on the last rule in a derivation are given for normalization and strong normalization in natural deduction.

We have a fresh look on Frege’s , taking into account proofs of equalities as a key concept. Revisiting the classical example of and the account leads to a proposal for a proof-theoretic semantics of equalities.

In this paper we discuss a proof-theoretic foundation of set theory that focusses on set definitions in an open type free framework. The idea to make Cantor’s informal definition of the notion of a set more precise by saying that any given property defines a set seems to be in conflict with ordinary modes of reasoning. There is to some extent a confusion here between perspectives (sets as collections of objects) and perspectives (set theoretic definitions) that the central paradoxes build on. The solutions offered by Zermelo-Fraenkel set theories, von Neumann-Bernays set-class theories and type theories follow the strategy of retirement behind more or less safe boundaries. What if we revisit the original idea without making strong assumptions on closure properties of the theoretical notion of a set? That is, take the basic definitions for what they are without confusing the borders between intensional and extensional perspectives.