Substructural approaches to paradoxes have attracted much attention from the philosophical community in the last decade. In this paper we focus on two substructural logics, named ST and TS, along with two structural cousins, LP and K3. It is well known that LP and K3 are duals in the sense that an inference is valid in one logic just in case the contrapositive is valid in the other logic. As a consequence of this duality, theories based on either logic are (...) tightly connected since many of the arguments for and objections against one theory reappear in the other theory in dual form. The target of the paper is making explicit in exactly what way, if any, ST and TS are dual to one another. The connection will allow us to gain a more fine-grained understanding of these logics and of the theories based on them. In particular, we will obtain new insights on two questions concerning ST which are being intensively discussed in the current literature: whether ST preserves classical logic and whether it is LP in sheep’s clothing. Explaining in what way ST and TS are duals requires comparing these logics at a metainferential level. We provide to this end a uniform proof theory to decide on valid metainferences for each of the four logics. This proof procedure allows us to show in a very simple way how different properties of inferences (unsatisfiability, supersatisfiability and antivalidity) that behave in very different ways for each logic can be captured in terms of the validity of a metainference. (shrink)

This open access book investigates the role played by identity of proofs in proof-theoretic semantics. It develops a conception of proof-theoretic semantics as primarily concerned with the relationship between proofs (understood as abstract entities) and derivations (the linguistic representations of proofs). It demonstrates that identity of proof is a key both to clarify some —still not wholly understood— notions at the core of proof-theoretic semantics, such as harmony; and to broaden the range of the phenomena which can be analyzed using (...) the tools of this semantic paradigm, so as to include for instance paradoxes. The volume covers topics such as the philosophical significance of different criteria of identity of proofs, and adequacy conditions for an intensional account of the notion of harmony. The author also examines the Prawitz-Tennant analysis of paradoxes by investigating on the one hand the prospects of turning it into a theory of meaning for paradoxical languages, and on the other hand two distinct kinds of phenomena, first observed by Crabbe and Ekman, showing that the Tennant-Prawitz criterion for paradoxicality overgenerates. This volume is of interest to scholars in formal and philosophical logic. (shrink)

In this paper we show how Dummett-Prawitz-style proof-theoretic semantics has to be modified in order to cope with paradoxical phenomena. It will turn out that one of its basic tenets has to be given up, namely the definition of the correctness of an inference as validity preservation. As a result, the notions of an argument being valid and of an argument being constituted by correct inference rules will no more coincide. The gap between the two notions is accounted for by (...) introducing the distinction between sense and denotation in the proof-theoretic-semantic setting. (shrink)

Building on early work by Girard ( 1987 ) and using closely related techniques from the proof theory of many-valued logics, we propose a sequent calculus capturing a hierarchy of notions of satisfaction based on the Strong Kleene matrices introduced by Barrio et al. (Journal of Philosophical Logic 49:93–120, 2020 ) and others. The calculus allows one to establish and generalize in a very natural manner several recent results, such as the coincidence of some of these notions with their classical (...) counterparts, and the possibility of expressing some notions of satisfaction for higher-level inferences using notions of satisfaction for inferences of lower level. We also show that at each level all notions of satisfaction considered are pairwise distinct and we address some remarks on the possible significance of this (huge) number of notions of consequence. (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)

In the present paper we discuss a recent suggestion of Schroeder-Heister concerning the possibility of defining an intensional notion of harmony using isomorphism in second-order propositional logic. The latter is not an absolute notion, but its definition is relative to the choice of criteria for identity of proofs. In the paper, it is argued that in order to attain a satisfactory account of harmony, one has to consider a notion of identity stronger than the usual one (based on β- and (...) η-conversions) that the authors have investigated in recent work. (shrink)

In the present paper, the Fregean conception of proof-theoretic semantics that I developed elsewhere will be revised so as to better reflect the different roles played by open and closed derivations. I will argue that such a conception can deliver a semantic analysis of languages containing paradoxical expressions provided some of its basic tenets are liberalized. In particular, the notion of function underlying the Brouwer–Heyting–Kolmogorov explanation of implication should be understood as admitting functions to be partial. As argued in previous (...) work, the correctness of an inference rule should not be defined in terms of transmission of provability, but rather should be grounded on weaker principles, which I show that are motivated by consideration about the content of the inversion principle. In the conclusion, I briefly address the issue of compositionality, arguing that the violation of compositionality induced by paradoxes are no worse than others that are regarded, at least by Dummett, as wholly unproblematic. (shrink)

In this paper we argue that an account of proof-theoretic harmony based on reductions and expansions delivers an inferentialist picture of meaning which should be regarded as intensional, as opposed to other approaches to harmony that will be dubbed extensional. We show how the intensional account applies to any connective whose rules obey the inversion principle first proposed by Prawitz and Schroeder-Heister. In particular, by improving previous formulations of expansions, we solve a problem with quantum-disjunction first posed by Dummett. As (...) recently observed by Schroeder-Heister, however, the specification of an inversion principle cannot yield an exhaustive account of harmony. The reason is that there are more collections of elimination rules than just the one obtained by inversion which we are willing to acknowledge as being in harmony with a given collection of introduction rules. Several authors more or less implicitly suggest that what is common to all alternative harmonious collection of rules is their being interderivable with each other. On the basis of considerations about identity of proofs and formula isomorphism, we show that this is too weak a condition for a given collection of elimination rules to be in harmony with a collection of introduction rules, at least if the intensional picture of meaning we advocate is not to collapse on an extensional one. (shrink)

Developing a suggestion by Russell, Prawitz showed how the usual natural deduction inference rules for disjunction, conjunction and absurdity can be derived using those for implication and the second order quantifier in propositional intuitionistic second order logic NI\. It is however well known that the translation does not preserve the relations of identity among derivations induced by the permutative conversions and immediate expansions for the definable connectives, at least when the equational theory of NI\ is assumed to consist only of (...) \- and \-equations. On the basis of the categorial interpretation of NI\, we introduce a new class of equations expressing what in categorial terms is a naturality condition satisfied by the transformations interpreting NI\-derivations. We show that the Russell–Prawitz translation does preserve identity of proof with respect to the enriched system by highlighting the fact that naturality corresponds to a generalized permutation principle. Finally we sketch how these results could be used to investigate the properties of connectives definable in the framework of higher-level rules. (shrink)

Developing early results of Prawitz, Tennant proposed a criterion for an expression to count as a paradox in the framework of Gentzen’s natural deduction: paradoxical expressions give rise to non-normalizing derivations. Two distinct kinds of cases, going back to Crabbé and Tennant, show that the criterion overgenerates, that is, there are derivations which are intuitively non-paradoxical but which fail to normalize. Tennant’s proposed solution consists in reformulating natural deduction elimination rules in general form. Developing intuitions of Ekman we show that (...) the adoption of general rules has the consequence of hiding redundancies within derivations. Once reductions to get rid of the hidden redundancies are devised, it is clear that the adoption of general elimination rules offers no remedy to the overgeneration of the Prawitz–Tennant analysis. In this way, we indirectly provide further support for a solution to one of the two overgeneration cases developed in previous work. (shrink)

We present a natural deduction system for dual-intuitionistic logic. Its distinctive feature is that it is a single-premise multiple-conclusions system. Its relationships with the natural deduction systems for intuitionistic and classical logic are discussed.

In the first chapter, we discuss Dummett’s idea that the notion of truth arises from the one of the correctness of an assertion. We argue that, in a first-order language, the need of defining truth in terms of the notion of satisfaction, which is yielded by the presence of quantifiers, is structurally analogous to the need of a notion of truth as distinct from the one of correctness of an assertion. In the light of the analogy between predicates in Frege (...) and open formulas in Tarksi, we concentrate on the semantic status of predicates. We register a dual attitude of Dummett towards Frege’s ascription of reference to predicates. On the one it is needed to endow quantifiers with their appropriate meaning. On the other hand, the introduction of concepts, as semantic correlate of predicates, smuggles a realist flavor in the overall semantic picture. In concluding the excursus (and with it the chapter), we stress Dummett’s will of developing a semantic picture free from this realist trait. In the second chapter, we present the idea of a proof-theoretic semantics. As for Frege true sentences denotes the truth-value True, so here closed' (i.e. categorical) valid argumentations denotes proofs. If a language contains implication-like operators, in order to characterize the condition of validity of a closed argumentation one has to introduce a notion of validity applying to ‘open’ (i.e. hypothetical) argumentations. We argue that this problem is analogous to the one posed by quantifiers in Frege-Tarksi’s style semantics. That is, implication forces one to introduce notion of validity for argumentations, which is more substantial than the correctness of the assertion of their conclusions. As we saw, the corresponding claim in the truth-based approach—that quantifiers require one to introduce a notion of truth, which is more substantial than the one of an assertion being correct—was the source of realism. Hence, Dummett proposes to reduce the semantic contribution of open argumentations to the one of their ‘closed instances’. In the truth-based perspective, this would correspond to the denial of the need of introducing concepts as the semantic correlates of predicates. We argue that Dummett’s fear, that an irreducible notion of function (represented by the need of ascribing validity to open argumentations) would lead to realism, turns out to be ill-founded. In the third chapter, we discuss the role played by the notion of truth in the anti-realist account. The notion of truth is what the anti-realist needs to cope with the so-called paradox of deduction. The analysis of the paradox yields to distinguishing between the truth of a sentence and the truth of a sentence being recognized. In terms of these conceptual couple, we reconsider the relationship between truth and assertion in an anti-realist perspective. Grounds are provided for a thesis (which was already advanced in chapter two), according to which the notion of the assertion of a sentence being correct is primarily connected only with the canonical means of establishing a sentence. The possibility of establishing a sentence by indirect means is conceptually dependent on the practice of establishing logical relationship of dependence among sentences. That is, the notion of a closed valid non-canonical argumentation is of any theoretical relevance only in presence of a notion of validity applying to open argumentations. In the fourth chapter, we discuss the possibility of characterizing in the proof-theoretic-semantics a notion of refutation. We develop an original characterization of refutations starting from an informal inductive specification of the condition of refutations of logically complex sentences. A sub-structural logic, called dual-intuitionistic logic, stands to this notion in the same relationship in which intuitionistic logic stands to the notion of proof so far consdered. All notions developed in chapter 2 have their corresponding (dual) ones in the framework developed. In particular, the distinctions canonical/non-canonical and closed/open argumentations. In the refutation based perspective, elimination rules have priority over introductions and the (only) assumption over the (many) conclusions. In the conclusions, we indicate the ingredients that an anti-realist approach to meaning should incorporate, in order to avoid the difficulties we registered. The core of an alternative view is a different conception of the relationship between categorical and hypothetical notions, in which the validity of open argumentations is not reduced to that of their instances, but rather it is directly defined. As a limit case, one would get a notion of validity applying to closed argumentations. (shrink)

One way in which we might approach the challenge posed by the Sorites Paradox is considering that Sorites-susceptible predicates have several candidate extensions, or several ways in which these expressions can be made precise. For example, a candidate extension for the predicate ‘is a baby’ is the set of humans of less than two years, but also the set of those less than two years and one second, and of those less than two years and two seconds. In this chapter (...) we present and discuss two theories for vague predicates based on this idea: supervaluationism and subvaluationism. The chapter is structured in three parts. The first presents the super- and subvaluationist theories: their similarities and differences in the semantics, the resulting logics with their most characteristic features. The second reviews the super- and subvaluationist solutions to the Sorites Paradox and provides discussion on several controversies surrounding these theories. The third part introduces proof procedures for s’valuationist logics. (shrink)

Validity, the central concept of the so-called ‘proof-theoretic semantics’ is described as correctly applying to the arguments that denote proofs. In terms of validity, I propose an anti-realist characterization of the notions of truth and correct assertion, at the core of which is the idea that valid arguments may fail to be recognized as such. The proposed account is compared with Dummett’s and Prawitz’s views on the matter.

In a previous paper we investigated the extraction of proof-theoretic properties of natural deduction derivations from their impredicative translation into System F. Our key idea was to introduce an extended equational theory for System F codifying at a syntactic level some properties found in parametric models of polymorphic type theory. A different approach to extract proof-theoretic properties of natural deduction derivations was proposed in a recent series of papers on the basis of an embedding of intuitionistic propositional logic into a (...) predicative fragment of System F, called atomic System F. In this paper we show that this approach finds a general explanation within our equational study of second-order natural deduction, and a clear semantic justification in terms of parametricity. (shrink)

Since the appearance of Prior’s tonk, inferentialists tried to formulate conditions that a collection of inference rules for a logical constant has to satisfy in order to succeed in conferring an acceptable meaning to it. Dummett proposed a pair of conditions, dubbed ‘harmony’ and ‘stability’ that have been cashed out in terms of the existence of certain transformations on natural deduction derivations called reductions and expansions. A long standing open problem for this proposal is posed by quantum disjunction: although its (...) rules are intuitively unstable, they pass the test of existence of expansions. Although most authors view instabilities of this kind as too subtle to be detected by the requirement of existence of expansions, we first discuss a case showing that this requirement can indeed detect instabilities of this kind, and then show how the expansions for disjunction-like connectives have to be reformulated to rule out quantum disjunction. We show how the alternative pattern for expansions can be formulated for connectives and quantifiers whose rules satisfy a scheme originally developed by Prawitz and Schroeder-Heister. Finally we compare our proposal with a recent one due to Jacinto and Read. (shrink)

This article deals with supervaluationism and the failure of truth-functionality. It draws some distinctions that may contribute to a better understanding of this semantic framework.

We present a multiple-assumption multiple-conclusion system for bi-intuitionistic logic. Derivations in the systems are graphs whose edges are labelled by formulas and whose nodes are labelled by rules. We show how to embed both the standard intuitionistic and dual-intuitionistic natural deduction systems into the proposed system. Soundness and completeness are established using translations with more traditional sequent calculi for bi-intuitionistic logic.

Validity, the central concept of the so-called ‘proof-theoretic semantics’ is described as correctly applying to the arguments that denote proofs. In terms of validity, I propose an anti-realist characterization of the notions of truth and correct assertion, at the core of which is the idea that valid arguments may fail to be recognized as such. The proposed account is compared with Dummett’s and Prawitz’s views on the matter.