The method of Socratic proofs (SP-method) simulates the solving of logical problem by pure questioning. An outcome of an application of the SP-method is a sequence of questions, called a Socratic transformation. Our aim is to give a method of translation of Socratic transformations into trees. We address this issue both conceptually and by providing certain algorithms. We show that the trees which correspond to successful Socratic transformations—that is, to Socratic proofs—may be regarded, after a slight modification, as Gentzen-style proofs. (...) Thus proof-search for some Gentzen-style calculi can be performed by means of the SP-method. At the same time the method seems promising as a foundation for automated deduction. (shrink)

The aim of this paper is to present a loop-free decision procedure for modal propositional logics K4, S4 and S5. We prove that the procedure terminates and that it is sound and complete. The procedure is based on the method of Socratic proofs for modal logics, which is grounded in the logic of questions IEL.

This paper considers a formalisation of classical logic using general introduction rules and general elimination rules. It proposes a definition of ‘maximal formula’, ‘segment’ and ‘maximal segment’ suitable to the system, and gives reduction procedures for them. It is then shown that deductions in the system convert into normal form, i.e. deductions that contain neither maximal formulas nor maximal segments, and that deductions in normal form satisfy the subformula property. Tarski’s Rule is treated as a general introduction rule for implication. (...) The general introduction rule for negation has a similar form. Maximal formulas with implication or negation as main operator require reduction procedures of a more intricate kind not present in normalisation for intuitionist logic. (shrink)

This paper studies a formalisation of intuitionistic logic by Negri and von Plato which has general introduction and elimination rules. The philosophical importance of the system is expounded. Definitions of ‘maximal formula’, ‘segment’ and ‘maximal segment’ suitable to the system are formulated and corresponding reduction procedures for maximal formulas and permutative reduction procedures for maximal segments given. Alternatives to the main method used are also considered. It is shown that deductions in the system convert into normal form and that deductions (...) in normal form have the subformula property. (shrink)

General-elimination harmony articulates Gentzen’s idea that the elimination-rules are justified if they infer from an assertion no more than can already be inferred from the grounds for making it. Dummett described the rules as not only harmonious but stable if the E-rules allow one to infer no more and no less than the I-rules justify. Pfenning and Davies call the rules locally complete if the E-rules are strong enough to allow one to infer the original judgement. A method is given (...) of generating harmonious general-elimination rules from a collection of I-rules. We show that the general-elimination rules satisfy Pfenning and Davies’ test for local completeness, but question whether that is enough to show that they are stable. Alternative conditions for stability are considered, including equivalence between the introduction- and elimination-meanings of a connective, and recovery of the grounds for assertion, finally generalizing the notion of local completeness to capture Dummett’s notion of stability satisfactorily. We show that the general-elimination rules meet the last of these conditions, and so are indeed not only harmonious but also stable. (shrink)

The status of the equality predicate as a logical constant is problematic. In the paper we look at the problem from the proof-theoretic standpoint and survey several ways of treating equality in formal systems of different sorts. In particular, we focus on the framework of sequent calculus and examine equality in the light of criteria of logicality proposed by Hacking and Došen. Both attempts were formulated in terms of sequent calculus rules, although in the case of Došen it has a (...) nonstandard character. It will be shown that equality can be characterised in a way which satisfies Došen’s criteria of logicality. In the case of Hacking’s approach the fully satisfying result can be obtained only for languages with a nonempty, finite set of predicate constants other than equality. Otherwise, cut elimination theorem fails to hold. (shrink)

The paper presents a uniform proof-theoretic treatment of several kinds of free logic, including the logics of existence and definedness applied in constructive mathematics and computer science, and called here quasi-free logics. All free and quasi-free logics considered are formalised in the framework of sequent calculus, the latter for the first time. It is shown that in all cases remarkable simplifications of the starting systems are possible due to the special rule dealing with identity and existence predicate. Cut elimination is (...) proved in a constructive way for sequent calculi adequate for all logics under consideration. (shrink)

A new type of formalization of classical first-order logic with equality is introduced on the basis of the sequent calculus. It serves to justify the claim that equality is a logical constant characterised by well-behaved rules satisfying properties usually regarded as essential. The main feature of this approach is the application of sequents built not only from formulae but also from terms. Two variants of sequent calculus are examined, a structural and a logical one. The former is defined in accordance (...) with Dos̆en’s criteria for logical constants. The latter is a standard Gentzen’s sequent calculus and satisfies Hacking’s criteria for logicality, including cut elimination. It is also shown that provided rules are harmonious in the sense advocated by Gratzl and Orlandelli. (shrink)

Nonclassical theories of truth have in common that they reject principles of classical logic to accommodate an unrestricted truth predicate. However, different nonclassical strategies give up different classical principles. The paper discusses one criterion we might use in theory choice when considering nonclassical rivals: the maxim of minimal mutilation.

ABSTRACT What do we disagree about when we disagree about logic? On the face of it, classical and nonclassical logicians disagree about the laws of logic and the nature of logical properties. Yet, sometimes the parties are accused of talking past each other. The worry is that if the parties to the dispute do not mean the same thing with ‘if’, ‘or’, and ‘not’, they fail to have genuine disagreement about the laws in question. After the work of Quine, this (...) objection against genuine disagreement about logic has been called the meaning-variance thesis. We argue that the meaning-variance thesis can be endorsed without blocking genuine disagreement. In fact, even the type of revisionism and nonapriorism championed by Quine turns out to be compatible with meaning-variance. (shrink)

The present contribution shows that a Hilbert-style axiomatization for dynamic logic of relation changers is complete for the standard Kripke semantics not by a well-known rewriting technique but by the idea of an auxiliary semantics studied by van Benthem and Wang et al. A key insight of our auxiliary semantics for dynamic logic of relation changers can be described as: “relation changers are bounded morphisms.” Moreover, we demonstrate that this semantic insight can be used to provide a modular cut-free labelled (...) sequent calculus for the logic in the sense that our calculus can be regarded as a natural expansion of a labelled sequent calculus of iteration-free propositional dynamic logic. (shrink)

A proof-theoretical treatment of collectively accepted group beliefs is presented through a multi-agent sequent system for an axiomatization of the logic of acceptance. The system is based on a labelled sequent calculus for propositional multi-agent epistemic logic with labels that correspond to possible worlds and a notation for internalized accessibility relations between worlds. The system is contraction- and cut-free. Extensions of the basic system are considered, in particular with rules that allow the possibility of operative members or legislators. Completeness with (...) respect to the underlying Kripke semantics follows from a general direct and uniform argument for labelled sequent calculi extended with mathematical rules for frame properties. As an example of the use of the calculus we present an analysis of the discursive dilemma. (shrink)

Various sources in the literature claim that the deduction theorem does not hold for normal modal or epistemic logic, whereas others present versions of the deduction theorem for several normal modal systems. It is shown here that the apparent problem arises from an objectionable notion of derivability from assumptions in an axiomatic system. When a traditional Hilbert-type system of axiomatic logic is generalized into a system for derivations from assumptions, the necessitation rule has to be modified in a way that (...) restricts its use to cases in which the premiss does not depend on assumptions. This restriction is entirely analogous to the restriction of the rule of universal generalization of first-order logic. A necessitation rule with this restriction permits a proof of the deduction theorem in its usual formulation. Other suggestions presented in the literature to deal with the problem are reviewed, and the present solution is argued to be preferable to the other alternatives. A contraction-and cut-free sequent calculus equivalent to the Hubert system for basic modal logic shows the standard failure argument untenable by proving the underivability of DA from A. (shrink)

We investigate incomplete symbols, i.e. definite descriptions with scope-operators. Russell famously introduced definite descriptions by contextual definitions; in this article definite descriptions are introduced by rules in a specific calculus that is very well suited for proof-theoretic investigations. That is to say, the phrase ‘incomplete symbols’ is formally interpreted as to the existence of an elimination procedure. The last section offers semantical tools for interpreting the phrase ‘no meaning in isolation’ in a formal way.

In this paper we use proof-theoretic methods, specifically sequent calculi, admissibility of cut within them and the resultant subformula property, to examine a range of philosophically-motivated deontic logics. We show that for all of those logics it is a (meta)theorem that the Special Hume Thesis holds, namely that no purely normative conclusion follows non-trivially from purely descriptive premises (nor vice versa). In addition to its interest on its own, this also illustrates one way in which proof theory sheds light on (...) philosophically substantial questions. (shrink)

We present a proof of the equivalence between two deductive systems for constructive necessity, namely an axiomatic characterisation inspired by Hakli and Negri's system of derivations from assumptions for modal logic , a Hilbert-style formalism designed to ensure the validity of the deduction theorem, and the judgmental reconstruction given by Pfenning and Davies by means of a natural deduction approach that makes a distinction between valid and true formulae, constructively. Both systems and the proof of their equivalence are formally verified (...) using the state-of-the-art proof assistant Coq. The proof approach taken throughout the paper unveils the use of some alternative proof methods that allow for a smooth transition from the high-level mathematical proofs to their mechanised counterparts. (shrink)

We present two proof systems for first-order logic with identity and without function symbols. The first one is an extension of the Rasiowa-Sikorski system with the rules for identity. This system is a validity checker. The rules of this system preserve and reflect validity of disjunctions of their premises and conclusions. The other is a Tableau system, which is an unsatisfiability checker. Its rules preserve and reflect unsatisfiability of conjunctions of their premises and conclusions. We show that the two systems (...) are dual to each other. The duality is expressed in a formal way which enables us to define a transformation of proofs in one of the systems into the proofs of the other. (shrink)

We give a simple sequent calculus presentation of R.B. Angell’s logic of analytic containment, recently championed by Kit Fine as a plausible logic of partial content.

As the final component of a chain of reasoning intended to take us all the way to logical nihilism, Russell (2018) presents the atomic sentence ‘prem’ which is supposed to be true when featuring as premise in an argument and false when featuring as conclusion in an argument. Such a sentence requires a non-reflexive logic and an endnote by Russell (2018) could easily leave the reader with the impression that going non-reflexive suffices for logical nihilism. This paper shows how one (...) can obtain non-reflexive logics in which ‘prem’ behaves as stipulated by Russell (2018) but which nonetheless has valid inferences supporting uniform substitution of any formula for propositional variables such as modus tollens and modus ponens. (shrink)

This paper presents a sound and complete five-sided sequent calculus for first-order weak Kleene valuations which permits not only elegant representations of four logics definable on first-order weak Kleene valuations, but also admissibility of five cut rules by proof analysis.

Against the backdrop of the frequent comparison of theories of truth in the literature on semantic paradoxes with regard to which inferences and metainferences are deemed valid, this paper develops a novel approach to defining a binary predicate for representing the valid inferences and metainferences of a theory within the theory itself under the assumption that the theory is defined with a classical meta-theory. The aim with the approach is to obtain a tool which facilitates the comparison between a theory (...) and its competitors within the theory itself, thereby expressing the disagreement between the theories within the theories. After discussing what we can and should require of an object-linguistic representation of a theory for that purpose, this paper proposes to restrict the representation of valid metainferences to locally valid metainferences, a requirement which turns out to be \-consistent and conservative over classical first-order arithmetic. This approach is then applied to four theories definable on strong Kleene models using a labelled nested sequent calculus. (shrink)

Questions concerning the proof-theoretic strength of classical versus nonclassical theories of truth have received some attention recently. A particularly convenient case study concerns classical and nonclassical axiomatizations of fixed-point semantics. It is known that nonclassical axiomatizations in four- or three-valued logics are substantially weaker than their classical counterparts. In this paper we consider the addition of a suitable conditional to First-Degree Entailment—a logic recently studied by Hannes Leitgeb under the label HYPE. We show in particular that, by formulating the theory (...) PKF over HYPE, one obtains a theory that is sound with respect to fixed-point models, while being proof-theoretically on a par with its classical counterpart KF. Moreover, we establish that also its schematic extension—in the sense of Feferman—is as strong as the schematic extension of KF, thus matching the strength of predicative analysis. (shrink)

Starting from a proof-theoretic perspective, where meaning is determined by the inference rules governing logical operators, in this paper we primarily aim at developing a proof-theoretic alternative to the model-theoretic meaning-invariant logical pluralism discussed in Beall and Restall. We will also outline how this framework can be easily extended to include a form of meaning-variant logical pluralism. In this respect, the framework developed in this paper—which we label two-level proof-theoretic pluralism—is much broader in scope than the one discussed in Beall (...) and Restall’s book. (shrink)

Using labelled formulae, a cut-free sequent calculus for intuitionistic propositional logic is presented, together with an easy cut-admissibility proof; both extend to cover, in a uniform fashion, all intermediate logics characterised by frames satisfying conditions expressible by one or more geometric implications. Each of these logics is embedded by the Gödel–McKinsey–Tarski translation into an extension of S4. Faithfulness of the embedding is proved in a simple and general way by constructive proof-theoretic methods, without appeal to semantics other than in the (...) explanation of the rules. (shrink)

The work is concerned with the so called display property of display logic. The motivation behind it is discussed and challenged. It is shown using one display calculus for intuitionistic logic as an example that the display property can be abandoned without losing subformula, cut elimination and completeness properties in such a way that results in additional expressive power of the system. This is done by disentangling structural connectives so that they are no longer context-sensitive. A recipe for characterizing structural (...) extensions of display calculi is provided. (shrink)

David Ripley ha argumentado extensamente a favor de una teoría no-transitiva de la verdad que abandona la regla de Corte para así evitar las pruebas de trivialidad causadas por paradojas como la del mentiroso. Sin embargo, es problemático comparar su teoría con varias teorías clásicas que se han ofrecido en la bibliografía. La tarea de formular esta teoría sobre la aritmética de Peano no es trivial, ya que Corte no es eliminable en la aritmética de Peano. En este artículo intento (...) cerrar esta brecha proponiendo una restricción adecuada para la regla de Corte. La restricción nos permite formular una teoría no-transitiva de la verdad sobre la aritmética de Peano que es, desde el punto de vista de la teoría de la prueba, tan fuerte como la teoría clásica de la verdad más fuerte conocida hasta el momento. (shrink)

Intra-theoretical logical pluralism is a form of meaning-invariant pluralism about logic, articulated recently by Hjortland :355–373, 2013). This version of pluralism relies on it being possible to define several distinct notions of provability relative to the same logical calculus. The present paper picks up and explores this theme: How can a single logical calculus express several different consequence relations? The main hypothesis articulated here is that the divide between the internal and external consequence relations in Gentzen systems generates a form (...) of intra-theoretical logical pluralism. (shrink)

Proof-theoretic semantics is an alternative to model-theoretic semantics. It aims at explaining the meaning of the logical constants in terms of the inference rules that govern their behaviour in proofs. We argue that this must be construed as the task of explaining these meanings relative to a logic, i.e., to a consequence relation. Alas, there is no agreed set of properties that a relation must have in order to qualify as a consequence relation. Moreover, the association of a consequence relation (...) to a logical calculus is not as straightforward as it may seem. We show that these facts are problematic for the proof-theoretic project but the problems can be solved. Our thesis is that the consequence relation relevant for proof-theoretic semantics is the one given by the sequent-to-sequent derivability relation in Gentzen systems. (shrink)

In the original publication of the article, in Definition 4, the sixth line which reads as.

This paper presents and defends a way to add a transparent truth predicate to classical logic, such that and A are everywhere intersubstitutable, where all T-biconditionals hold, and where truth can be made compositional. A key feature of our framework, called STTT (for Strict-Tolerant Transparent Truth), is that it supports a non-transitive relation of consequence. At the same time, it can be seen that the only failures of transitivity STTT allows for arise in paradoxical cases.

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)

An erotetic calculus for a given logic constitutes a sequent-style proof-theoretical formalization of the logic grounded in Inferential Erotetic Logic ). In this paper, a new erotetic calculus for Classical Propositional Logic ), dual with respect to the existing ones, is given. We modify the calculus to obtain complete proof systems for the propositional part of paraconsistent logic CLuN and its extensions CLuNs and mbC. The method is based on dual resolution. Moreover, the resolution rule is non-clausal. According to the (...) authors knowledge, this is the first account of resolution for mbC. Last but not least, as the method is grounded in IEL, it constitutes an important tool for the so-called question-processing. (shrink)

Philosophy and Phenomenological Research, EarlyView.

Two consecution calculi are introduced: one for the implicational fragment of the logic of entailment with truth and another one for the disjunction free logic of nondistributive relevant implication. The proof technique—attributable to Gentzen—that uses a double induction on the degree and on the rank of the cut formula is shown to be insufficient to prove admissible various forms of cut and mix in these calculi. The elimination theorem is proven, however, by augmenting the earlier double inductive proof with additional (...) inductions. We also give a new purely inductive proof of the cut theorem for the original single cut rule in Gentzen’s sequent calculus $ LK $ without any use of mix. (shrink)

Rejecting the Cut rule has been proposed as a strategy to avoid both the usual semantic paradoxes and the so-called v-Curry paradox. In this paper we consider if a Cut-free theory is capable of accurately representing its own notion of validity. We claim that the standard rules governing the validity predicate are too weak for this purpose and we show that although it is possible to strengthen these rules, the most obvious way of doing so brings with it a serious (...) problem: an internalized version of Cut can be proved for a Curry-like sentence. We also evaluate a number of possible ways of escaping this difficulty. (shrink)

The origins of proof-theoretic semantics lie in the question of what constitutes the meaning of the logical connectives and its response: the rules of inference that govern the use of the connective. However, what if we go a step further and ask about the meaning of a proof as a whole? In this paper we address this question and lay out a framework to distinguish sense and denotation of proofs. Two questions are central here. First of all, if we have (...) two different derivations, does this always lead to a difference, firstly, in sense, and secondly, in denotation? The other question is about the relation between different kinds of proof systems with respect to this distinction. Do the different forms of representing a proof necessarily correspond to a difference in how the inferential steps are given? In our framework it will be possible to identify denotation as well as sense of proofs not only within one proof system but also between different kinds of proof systems. Thus, we give an account to distinguish a mere syntactic divergence from a divergence in meaning and a divergence in meaning from a divergence of proof objects analogous to Frege’s distinction for singular terms and sentences. (shrink)

Eight distinct rules for implication in the antecedent for the sequent calculus, one of which being Gentzen’s standard rule, can be derived by successively applying a number of cuts to the logical ground sequent A → B, A ⇒ B. A naive translation into natural deduction collapses four of those rules onto the standard implication elimination rule, and the remaining four rules onto the general elimination rule. This collapse is due to the fact that the difference between a formula occurring (...) in the succedent of a premise of a sequent calculus rule and that formula instead occurring in the antecedent of the conclusion cannot be adequately expressed in the framework of natural deduction. In contrast to this, the difference between a formula occurring in the succedent of the conclusion of a sequent calculus rule and that formula instead occurring in the antecedent of a premise corresponds exactly to the distinction between the standard implication elimination rule and its general counterpart. This incongruity can be remedied by introducing a notational facility that requires of the particular premise of a rule to which it is attached to be an assumption, i.e., to prevent it from being the conclusion of another rule application. Applying this facility to implication elimination results in eight distinct rules that correspond exactly to the eight sequent calculus rules. These eight rules are presented and discussed in detail. It turns out that a natural deduction calculus (for positive implication logic) that employs the rule corresponding to the standard left implication rule of the sequent calculus as well as a rule for explicit substitution can be seen as a natural deduction style sequent calculus. (shrink)

A variety of projects in proof theory of relevance to the philosophy of mathematics are surveyed, including Gödel's incompleteness theorems, conservation results, independence results, ordinal analysis, predicativity, reverse mathematics, speed-up results, and provability logics.

A sequent root-first proof-search procedure for intuitionistic propositional logic is presented. The procedure is obtained from modified intuitionistic multi-succedent and classical sequent calculi, making use of Glivenko’s Theorem. We prove that a sequent is derivable in a standard intuitionistic multi-succedent calculus if and only if the corresponding prefixed-sequent is derivable in the procedure.

The well-known algebraic semantics and topological semantics for intuitionistic logic (Int) is here extended to Wansing's bi-intuitionistic logic (2Int). The logic 2Int is also characterised by a quasi-twist structure semantics, which leads to an alternative topological characterisation of 2Int. Later, notions of Fregean negation and of unilateralisation are proposed. The logic 2Int is extended with a ‘Fregean negation’ connective ∼, obtaining 2Int∼, and it is showed that the logic N4⋆ (an extension of Nelson's paraconsistent logic) results to be the unilateralisation (...) of 2Int∼ via ∼. The logic 2Int∼ is also characterised by a Kripke-style semantics, a twist structure semantics and a topological semantics. (shrink)

Expressivism in logic is the view that logical vocabulary plays a primarily expressive role: that is, that logical vocabulary makes perspicuous in the object language structural features of inference and incompatibility (Brandom, 1994, 2008). I present a precise, technical criterion of expressivity for a logic (§2). I next present a logic that meets that criterion (§3). I further explore some interesting features of that logic: first, a representation theorem for capturing other logics (§3.1), and next some novel logical vocabulary for (...) expressing structural features of inference (§4). (shrink)

It's natural to think that the principles expressed by the statements "Promises ought to be kept" and "We ought to help those in need" are defeasible. But how are we to make sense of this defeasibility? On one proposal, moral principles have hedges or built-in unless clauses specifying the conditions under which the principle doesn't apply. On another, such principles are contributory and, thus, do not specify which actions ought to be carried out, but only what counts in favor or (...) against them. Drawing on a defeasible logic framework, this paper sets up three models: one model for each proposal, as well as a third model capturing a mixed view on principles that combines them. It then explores the structural connections between the three models and establishes some equivalence results, suggesting that the seemingly different views captured by the models are closer than standardly thought. (shrink)

In this paper, we investigate the possibility of translating a fragment of natural deduction system (NDS) for natural language semantics into modern type theory (MTT), originally suggested by Luo (2014). Our main goal will be to examine and translate the basic rules of NDS (namely, meta-rules, structural rules, identity rules, noun rules and rules for intersective and subsective adjectives) to MTT. Additionally, we will also consider some of their general features.

The book presents a thoroughly elaborated logical theory of generalized truth-values understood as subsets of some established set of truth values. After elucidating the importance of the very notion of a truth value in logic and philosophy, we examine some possible ways of generalizing this notion. The useful four-valued logic of first-degree entailment by Nuel Belnap and the notion of a bilattice constitute the basis for further generalizations. By doing so we elaborate the idea of a multilattice, and most notably, (...) a trilattice of truth values – a specific algebraic structure with information ordering and two distinct logical orderings, one for truth and another for falsity. Each logical order not only induces its own logical vocabulary, but determines also its own entailment relation. We consider both semantic and syntactic ways of formalizing these relations and construct various logical calculi. (shrink)

This collection of papers, celebrating the contributions of Swedish logician Dag Prawitz to Proof Theory, has been assembled from those presented at the Natural Deduction conference organized in Rio de Janeiro to honour his seminal research. Dag Prawitz’s work forms the basis of intuitionistic type theory and his inversion principle constitutes the foundation of most modern accounts of proof-theoretic semantics in Logic, Linguistics and Theoretical Computer Science. The range of contributions includes material on the extension of natural deduction with higher-order (...) rules, as opposed to higher-order connectives, and a paper discussing the application of natural deduction rules to dealing with equality in predicate calculus. The volume continues with a key chapter summarizing work on the extension of the Curry-Howard isomorphism, via methods of category theory that have been successfully applied to linear logic, as well as many other contributions from highly regarded authorities. With an illustrious group of contributors addressing a wealth of topics and applications, this volume is a valuable addition to the libraries of academics in the multiple disciplines whose development has been given added scope by the methodologies supplied by natural deduction. The volume is representative of the rich and varied directions that Prawitz work has inspired in the area of natural deduction. (shrink)

The aim of the present book is to give a comprehensive account of the ‘state of the art’ of substructural logics, focusing both on their proof theory and on their semantics (both algebraic and relational. It is for graduate students in either philosophy, mathematics, theoretical computer science or theoretical linguistics as well as specialists and researchers.