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  1. Variable Handling and Compositionality: Comparing DRT and DTS.Yukiko Yana, Koji Mineshima & Daisuke Bekki - 2019 - Journal of Logic, Language and Information 28 (2):261-285.
    This paper provides a detailed comparison between discourse representation theory and dependent type semantics, two frameworks for discourse semantics. Although it is often stated that DRT and those frameworks based on dependent types are mutually exchangeable, we argue that they differ with respect to variable handling, more specifically, how substitution and other operations on variables are defined. This manifests itself in two recalcitrant problems posed for DRT; namely, the overwrite problem and the duplication problem. We will see that these problems (...)
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  • On Two Notions of Computation in Transparent Intensional Logic.Ivo Pezlar - 2018 - Axiomathes 29 (2):189-205.
    In Transparent Intensional Logic we can recognize two distinct notions of computation that loosely correspond to term rewriting and term interpretation as known from lambda calculus. Our goal will be to further explore these two notions and examine some of their properties.
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  • Proof, Meaning and Paradox: Some Remarks.Luca Tranchini - 2019 - Topoi 38 (3):591-603.
    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 (...)
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  • The strength of some Martin-Löf type theories.Edward Griffor & Michael Rathjen - 1994 - Archive for Mathematical Logic 33 (5):347-385.
    One objective of this paper is the determination of the proof-theoretic strength of Martin-Löf's type theory with a universe and the type of well-founded trees. It is shown that this type system comprehends the consistency of a rather strong classical subsystem of second order arithmetic, namely the one with Δ 2 1 comprehension and bar induction. As Martin-Löf intended to formulate a system of constructive (intuitionistic) mathematics that has a sound philosophical basis, this yields a constructive consistency proof of a (...)
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  • Constructive Sheaf Semantics.Erik Palmgren - 1997 - Mathematical Logic Quarterly 43 (3):321-327.
    Sheaf semantics is developed within a constructive and predicative framework, Martin‐Löf's type theory. We prove strong completeness of many sorted, first order intuitionistic logic with respect to this semantics, by using sites of provably functional relations.
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  • Empirical Negation.Michael De - 2013 - Acta Analytica 28 (1):49-69.
    An extension of intuitionism to empirical discourse, a project most seriously taken up by Dummett and Tennant, requires an empirical negation whose strength lies somewhere between classical negation (‘It is unwarranted that. . . ’) and intuitionistic negation (‘It is refutable that. . . ’). I put forward one plausible candidate that compares favorably to some others that have been propounded in the literature. A tableau calculus is presented and shown to be strongly complete.
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  • Truth as an Epistemic Notion.Dag Prawitz - 2012 - Topoi 31 (1):9-16.
    What is the appropriate notion of truth for sentences whose meanings are understood in epistemic terms such as proof or ground for an assertion? It seems that the truth of such sentences has to be identified with the existence of proofs or grounds, and the main issue is whether this existence is to be understood in a temporal sense as meaning that we have actually found a proof or a ground, or if it could be taken in an abstract, tenseless (...)
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  • On constructing completions.Laura Crosilla, Hajime Ishihara & Peter Schuster - 2005 - Journal of Symbolic Logic 70 (3):969-978.
    The Dedekind cuts in an ordered set form a set in the sense of constructive Zermelo—Fraenkel set theory. We deduce this statement from the principle of refinement, which we distill before from the axiom of fullness. Together with exponentiation, refinement is equivalent to fullness. None of the defining properties of an ordering is needed, and only refinement for two—element coverings is used. In particular, the Dedekind reals form a set; whence we have also refined an earlier result by Aczel and (...)
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  • From the Knowability Paradox to the existence of proofs.W. Dean & H. Kurokawa - 2010 - Synthese 176 (2):177 - 225.
    The Knowability Paradox purports to show that the controversial but not patently absurd hypothesis that all truths are knowable entails the implausible conclusion that all truths are known. The notoriety of this argument owes to the negative light it appears to cast on the view that there can be no verification-transcendent truths. We argue that it is overly simplistic to formalize the views of contemporary verificationists like Dummett, Prawitz or Martin-Löf using the sort of propositional modal operators which are employed (...)
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  • Frege's Judgement Stroke and the Conception of Logic as the Study of Inference not Consequence.Nicholas J. J. Smith - 2009 - Philosophy Compass 4 (4):639-665.
    One of the most striking differences between Frege's Begriffsschrift (logical system) and standard contemporary systems of logic is the inclusion in the former of the judgement stroke: a symbol which marks those propositions which are being asserted , that is, which are being used to express judgements . There has been considerable controversy regarding both the exact purpose of the judgement stroke, and whether a system of logic should include such a symbol. This paper explains the intended role of the (...)
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  • Propositions as games as types.Aarne Ranta - 1988 - Synthese 76 (3):377 - 395.
    Without violating the spirit of Game-Theoretical semantics, its results can be re-worked in Martin-Löf''s Constructive Type Theory by interpreting games as types of Myself''s winning strategies. The philosophical ideas behind Game-Theoretical Semantics in fact highly recommend restricting strategies to effective ones, which is the only controversial step in our interpretation. What is gained, then, is a direct connection between linguistic semantics and computer programming.
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  • The assertion-candidate and the meaning of mood.Maria van der Schaar - 2007 - Synthese 159 (1):61-82.
    The meaning of a declarative sentence and that of an interrogative sentence differ in their aspect of mood. A semantics of mood has to account for the differences in meaning between these sentences, and it also has to explain that sentences in different moods may have a common core. The meaning of the declarative mood is to be explained not in terms of actual force (contra Dummett), but in terms of potential force. The meaning of the declarative sentence (including its (...)
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  • Developments in constructive nonstandard analysis.Erik Palmgren - 1998 - Bulletin of Symbolic Logic 4 (3):233-272.
    We develop a constructive version of nonstandard analysis, extending Bishop's constructive analysis with infinitesimal methods. A full transfer principle and a strong idealisation principle are obtained by using a sheaf-theoretic construction due to I. Moerdijk. The construction is, in a precise sense, a reduced power with variable filter structure. We avoid the nonconstructive standard part map by the use of nonstandard hulls. This leads to an infinitesimal analysis which includes nonconstructive theorems such as the Heine-Borel theorem, the Cauchy-Peano existence theorem (...)
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  • Consistency of the intensional level of the Minimalist Foundation with Church’s thesis and axiom of choice.Hajime Ishihara, Maria Emilia Maietti, Samuele Maschio & Thomas Streicher - 2018 - Archive for Mathematical Logic 57 (7-8):873-888.
    Consistency with the formal Church’s thesis, for short CT, and the axiom of choice, for short AC, was one of the requirements asked to be satisfied by the intensional level of a two-level foundation for constructive mathematics as proposed by Maietti and Sambin From sets and types to topology and analysis: practicable foundations for constructive mathematics, Oxford University Press, Oxford, 2005). Here we show that this is the case for the intensional level of the two-level Minimalist Foundation, for short MF, (...)
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  • The Constructive Hilbert Program and the Limits of Martin-Löf Type Theory.Michael Rathjen - 2005 - Synthese 147 (1):81-120.
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  • Identity and Sortals.Ansten Klev - 2017 - Erkenntnis 82 (1):1-16.
    According to the sortal conception of the universe of individuals every individual falls under a highest sortal, or category. It is argued here that on this conception the identity relation is defined between individuals a and b if and only if a and b fall under a common category. Identity must therefore be regarded as a relation of the form \, with three arguments x, y, and Z, where Z ranges over categories, and where the range of x and y (...)
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  • Uniformly convex Banach spaces are reflexive—constructively.Douglas S. Bridges, Hajime Ishihara & Maarten McKubre-Jordens - 2013 - Mathematical Logic Quarterly 59 (4-5):352-356.
    We propose a natural definition of what it means in a constructive context for a Banach space to be reflexive, and then prove a constructive counterpart of the Milman-Pettis theorem that uniformly convex Banach spaces are reflexive.
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  • Tychonoff's theorem in the framework of formal topologies.Sara Negri & Silvio Valentini - 1997 - Journal of Symbolic Logic 62 (4):1315-1332.
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  • Proof normalization modulo.Gilles Dowek & Benjamin Werner - 2003 - Journal of Symbolic Logic 68 (4):1289-1316.
    We define a generic notion of cut that applies to many first-order theories. We prove a generic cut elimination theorem showing that the cut elimination property holds for all theories having a so-called pre-model. As a corollary, we retrieve cut elimination for several axiomatic theories, including Church's simple type theory.
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  • The Justification of the Logical Laws Revisited.Patrizio Contu - 2006 - Synthese 148 (3):573-588.
    The proof-theoretic analysis of logical semantics undermines the received view of proof theory as being concerned with symbols devoid of meaning, and of model theory as the sole branch of logical theory entitled to access the realm of semantics. The basic tenet of proof-theoretic semantics is that meaning is given by some rules of proofs, in terms of which all logical laws can be justified and the notion of logical consequence explained. In this paper an attempt will be made to (...)
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  • (1 other version)An interpretation of martin‐löf's constructive theory of types in elementary topos theory.Anne Preller - 1992 - Mathematical Logic Quarterly 38 (1):213-240.
    We give a formal interpretation of Martin-Löf's Constructive Theory of Types in Elementary Topos Theory which is presented as a formalised theory with intensional equality of objects. Types are interpreted as arrows and variables as sections of their types. This is necessary to model correctly the working of the assumption x ∈ A. Then intensional equality interprets equality of types. The normal form theorem which asserts that the interpretation of a type is intensional equal to the pullback of its “alignment” (...)
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  • In Defense of Logical Universalism: Taking Issue with Jean van Heijenoort. [REVIEW]Philippe de Rouilhan - 2012 - Logica Universalis 6 (3-4):553-586.
    Van Heijenoort’s main contribution to history and philosophy of modern logic was his distinction between two basic views of logic, first, the absolutist, or universalist, view of the founding fathers, Frege, Peano, and Russell, which dominated the first, classical period of history of modern logic, and, second, the relativist, or model-theoretic, view, inherited from Boole, Schröder, and Löwenheim, which has dominated the second, contemporary period of that history. In my paper, I present the man Jean van Heijenoort (Sect. 1); then (...)
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  • The epistemic significance of valid inference.Dag Prawitz - 2012 - Synthese 187 (3):887-898.
    The traditional picture of logic takes it for granted that "valid arguments have a fundamental epistemic significance", but neither model theory nor traditional proof theory dealing with formal system has been able to give an account of this significance. Since valid arguments as usually understood do not in general have any epistemic significance, the problem is to explain how and why we can nevertheless use them sometimes to acquire knowledge. It is suggested that we should distinguish between arguments and acts (...)
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  • “Inference versus consequence” revisited: inference, consequence, conditional, implication.Göran Sundholm - 2012 - Synthese 187 (3):943-956.
    Inference versus consequence , an invited lecture at the LOGICA 1997 conference at Castle Liblice, was part of a series of articles for which I did research during a Stockholm sabbatical in the autumn of 1995. The article seems to have been fairly effective in getting its point across and addresses a topic highly germane to the Uppsala workshop. Owing to its appearance in the LOGICA Yearbook 1997 , Filosofia Publishers, Prague, 1998, it has been rather inaccessible. Accordingly it is (...)
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  • Proceeding in Abstraction. From Concepts to Types and the recent perspective on Information.Giuseppe Primiero - 2009 - History and Philosophy of Logic 30 (3):257-282.
    This article presents an historical and conceptual overview on different approaches to logical abstraction. Two main trends concerning abstraction in the history of logic are highlighted, starting from the logical notions of concept and function. This analysis strictly relates to the philosophical discussion on the nature of abstract objects. I develop this issue further with respect to the procedure of abstraction involved by (typed) λ-systems, focusing on the crucial change about meaning and predicability. In particular, the analysis of the nature (...)
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  • On the concept of language in some recent theories of meaning.Sören Stenlund - 1989 - Synthese 79 (1):51 - 98.
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  • What is neologicism?Bernard Linsky & Edward N. Zalta - 2006 - Bulletin of Symbolic Logic 12 (1):60-99.
    In this paper, we investigate (1) what can be salvaged from the original project of "logicism" and (2) what is the best that can be done if we lower our sights a bit. Logicism is the view that "mathematics is reducible to logic alone", and there are a variety of reasons why it was a non-starter. We consider the various ways of weakening this claim so as to produce a "neologicism". Three ways are discussed: (1) expand the conception of logic (...)
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  • A construction of non-well-founded sets within Martin-löf's type theory.Ingrid Lindström - 1989 - Journal of Symbolic Logic 54 (1):57-64.
    In this paper, we show that non-well-founded sets can be defined constructively by formalizing Hallnäs' limit definition of these within Martin-Löf's theory of types. A system is a type W together with an assignment of ᾱ ∈ U and α̃ ∈ ᾱ → W to each α ∈ W. We show that for any system W we can define an equivalence relation = w such that α = w β ∈ U and = w is the maximal bisimulation. Aczel's proof (...)
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  • A type reduction from proof-conditional to dynamic semantics.Tim Fernando - 2001 - Journal of Philosophical Logic 30 (2):121-153.
    Dynamic and proof-conditional approaches to discourse (exemplified by Discourse Representation Theory and Type-Theoretical Grammar, respectively) are related through translations and transitions labeled by first-order formulas with anaphoric twists. Type-theoretic contexts are defined relative to a signature and instantiated modeltheoretically, subject to change.
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  • Explicit provability and constructive semantics.Sergei N. Artemov - 2001 - Bulletin of Symbolic Logic 7 (1):1-36.
    In 1933 Godel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that Godel's provability calculus is nothing but the forgetful projection of LP. This also achieves Godel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics for Int which (...)
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  • What is a Higher Level Set?Dimitris Tsementzis - 2016 - Philosophia Mathematica:nkw032.
    Structuralist foundations of mathematics aim for an ‘invariant’ conception of mathematics. But what should be their basic objects? Two leading answers emerge: higher groupoids or higher categories. I argue in favor of the former over the latter. First, I explain why to choose between them we need to ask the question of what is the correct ‘categorified’ version of a set. Second, I argue in favor of groupoids over categories as ‘categorified’ sets by introducing a pre-formal understanding of groupoids as (...)
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  • A pragmatic interpretation of intuitionistic propositional logic.Carlo Dalla Pozza & Claudio Garola - 1995 - Erkenntnis 43 (1):81-109.
    We construct an extension P of the standard language of classical propositional logic by adjoining to the alphabet of a new category of logical-pragmatic signs. The well formed formulas of are calledradical formulas (rfs) of P;rfs preceded by theassertion sign constituteelementary assertive formulas of P, which can be connected together by means of thepragmatic connectives N, K, A, C, E, so as to obtain the set of all theassertive formulas (afs). Everyrf of P is endowed with atruth value defined classically, (...)
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  • On the collection of points of a formal space.Giovanni Curi - 2006 - Annals of Pure and Applied Logic 137 (1-3):126-146.
    On the collection of points of a formal space.
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  • Between constructive mathematics and PROLOG.Gerhard Jäger - 1991 - Archive for Mathematical Logic 30 (5-6):297-310.
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  • Dialogue structure and logical expressivism.Paul Piwek - 2011 - Synthese 183 (S1):33-58.
    This paper aims to develop the implications of logical expressivism for a theory of dialogue coherence. I proceed in three steps. Firstly, certain structural properties of cooperative dialogue are identified. Secondly, I describe a variant of the multi-agent natural deduction calculus that I introduced in Piwek (J Logic Lang Inf 16(4):403–421, 2007 ) and demonstrate how it accounts for the aforementioned structures. Thirdly, I examine how the aforementioned system can be used to formalise an expressivist account of logical vocabulary that (...)
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  • On the existence of Stone-Čech compactification.Giovanni Curi - 2010 - Journal of Symbolic Logic 75 (4):1137-1146.
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  • Knowledge of proofs.Peter Pagin - 1994 - Topoi 13 (2):93-100.
    If proofs are nothing more than truth makers, then there is no force in the standard argument against classical logic (there is no guarantee that there is either a proof forA or a proof fornot A). The standard intuitionistic conception of a mathematical proof is stronger: there are epistemic constraints on proofs. But the idea that proofs must be recognizable as such by us, with our actual capacities, is incompatible with the standard intuitionistic explanations of the meanings of the logical (...)
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  • The impact of the lambda calculus in logic and computer science.Henk Barendregt - 1997 - Bulletin of Symbolic Logic 3 (2):181-215.
    One of the most important contributions of A. Church to logic is his invention of the lambda calculus. We present the genesis of this theory and its two major areas of application: the representation of computations and the resulting functional programming languages on the one hand and the representation of reasoning and the resulting systems of computer mathematics on the other hand.
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  • The Neglect of Epistemic Considerations in Logic: The Case of Epistemic Assumptions.Göran Sundholm - 2019 - Topoi 38 (3):551-559.
    The two different layers of logical theory—epistemological and ontological—are considered and explained. Special attention is given to epistemic assumptions of the kind that a judgement is granted as known, and their role in validating rules of inference, namely to aid the inferential preservation of epistemic matters from premise judgements to conclusion judgement, while ordinary Natural Deduction assumptions serve to establish the holding of consequence from antecedent propositions to succedent proposition.
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  • The Justification of Identity Elimination in Martin-Löf’s Type Theory.Ansten Klev - 2019 - Topoi 38 (3):577-590.
    On the basis of Martin-Löf’s meaning explanations for his type theory a detailed justification is offered of the rule of identity elimination. Brief discussions are thereafter offered of how the univalence axiom fares with respect to these meaning explanations and of some recent work on identity in type theory by Ladyman and Presnell.
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  • Univalent foundations as structuralist foundations.Dimitris Tsementzis - 2017 - Synthese 194 (9):3583-3617.
    The Univalent Foundations of Mathematics provide not only an entirely non-Cantorian conception of the basic objects of mathematics but also a novel account of how foundations ought to relate to mathematical practice. In this paper, I intend to answer the question: In what way is UF a new foundation of mathematics? I will begin by connecting UF to a pragmatist reading of the structuralist thesis in the philosophy of mathematics, which I will use to define a criterion that a formal (...)
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  • (1 other version)Proof theory and computer programming.Ruy J. G. B. de Queiroz & Thomas S. E. Maibaum - 1990 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 36 (5):389-414.
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  • Pgf: A portable run-time format for type-theoretical grammars. [REVIEW]Krasimir Angelov, Björn Bringert & Aarne Ranta - 2010 - Journal of Logic, Language and Information 19 (2):201-228.
    Portable Grammar Format (PGF) is a core language for type-theoretical grammars. It is the target language to which grammars written in the high-level formalism Grammatical Framework (GF) are compiled. Low-level and simple, PGF is easy to reason about, so that its language-theoretic properties can be established. It is also easy to write interpreters that perform parsing and generation with PGF grammars, and compilers converting PGF to other formats. This paper gives a concise description of PGF, covering syntax, semantics, and parser (...)
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  • A meaning explanation for HoTT.Dimitris Tsementzis - 2020 - Synthese 197 (2):651-680.
    In the Univalent Foundations of mathematics spatial notions like “point” and “path” are primitive, rather than derived, and all of mathematics is encoded in terms of them. A Homotopy Type Theory is any formal system which realizes this idea. In this paper I will focus on the question of whether a Homotopy Type Theory can be justified intuitively as a theory of shapes in the same way that ZFC can be justified intuitively as a theory of collections. I first clarify (...)
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  • Innovations in computational type theory using Nuprl.S. F. Allen, M. Bickford, R. L. Constable, R. Eaton, C. Kreitz, L. Lorigo & E. Moran - 2006 - Journal of Applied Logic 4 (4):428-469.
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  • Semantic Values for Natural Deduction Derivations.Göran Sundholm - 2006 - Synthese 148 (3):623-638.
    Drawing upon Martin-Löf’s semantic framework for his constructive type theory, semantic values are assigned also to natural-deduction derivations, while observing the crucial distinction between consequence among propositions and inference among judgements. Derivations in Gentzen’s format with derivable formulae dependent upon open assumptions, stand, it is suggested, for proof-objects, whereas derivations in Gentzen’s sequential format are proof-acts.
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  • Natural Language Inference in Coq.Stergios Chatzikyriakidis & Zhaohui Luo - 2014 - Journal of Logic, Language and Information 23 (4):441-480.
    In this paper we propose a way to deal with natural language inference by implementing Modern Type Theoretical Semantics in the proof assistant Coq. The paper is a first attempt to deal with NLI and natural language reasoning in general by using the proof assistant technology. Valid NLIs are treated as theorems and as such the adequacy of our account is tested by trying to prove them. We use Luo’s Modern Type Theory with coercive subtyping as the formal language into (...)
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  • Constructive belief reports.Bartosz Więckowski - 2015 - Synthese 192 (3):603-633.
    The paper develops a proof-theoretic semantics for belief reports by extending the constructive type-theoretical formalism presented in Więckowski with a specific kind of set-forming operator suited for the representation of belief attitudes. The extended formalism allows us to interpret constructions which involve, e.g., iteration of belief, quantifying into belief contexts, and anaphora in belief reports. Moreover, constructive solutions to canonical instances of the problem of hyperintensionality are suggested. The paper includes a discussion of Ranta’s constructive account of belief reports.
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  • Inverse Linking, Possessive Weak Definites and Haddock Descriptions: A Unified Dependent Type Account.Justyna Grudzińska & Marek Zawadowski - 2019 - Journal of Logic, Language and Information 28 (2):239-260.
    This paper proposes a unified dependent type analysis of three puzzling phenomena: inversely linked interpretations, weak definite readings in possessives and Haddock-type readings. We argue that the three problematic readings have the same underlying surface structure, and that the surface structure postulated can be interpreted properly and compositionally using dependent types. The dependent type account proposed is the first, to the best of our knowledge, to formally connect the three phenomena. A further advantage of our proposal over previous analyses is (...)
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  • The Concept Horse is a Concept.Ansten Klev - 2018 - Review of Symbolic Logic 11 (3):547-572.
    I offer an analysis of the sentence "the concept horse is a concept". It will be argued that the grammatical subject of this sentence, "the concept horse", indeed refers to a concept, and not to an object, as Frege once held. The argument is based on a criterion of proper-namehood according to which an expression is a proper name if it is so rendered in Frege's ideography. The predicate "is a concept", on the other hand, should not be thought of (...)
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