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A note on Humberstone's constant Ω

Satoru Niki, Hitoshi Omori

Reports on Mathematical Logic > 2021 > Vol. 56 > 75--99

We investigate an expansion of positive intuitionistic logic obtained by adding a constant Ω introduced by Lloyd Humberstone. Our main results include a sound and strongly complete axiomatization, some comparisons to other expansions of intuitionistic logic obtained by adding actuality and empirical negation, and an algebraic semantics. We also briefly discuss its connection to classical logic.

article

From Intuitionism to Brouwer's Modal Logic

Zofia Kostrzycka

Bulletin of the Section of Logic > 2020 > 49 > 4 > 343-358

We try to translate the intuitionistic propositional logic INT into Brouwer's modal logic KTB. Our translation is motivated by intuitions behind Brouwer's axiom p →☐◊p The main idea is to interpret intuitionistic implication as modal strict implication, whereas variables and other positive sentences remain as they are. The proposed translation preserves fragments of the Rieger-Nishimura lattice which...

article

On the validity of the definition of a complement-classifier

Mariusz Stopa

Zagadnienia Filozoficzne w Nauce > 2020 > 69 > 111-128

It is well-established that topos theory is inherently connected with intuitionistic logic. In recent times several works appeared concerning so-called complement-toposes (co-toposes), which are allegedly connected to the dual to intuitionistic logic. In this paper I present this new notion, some of the motivations for it, and some of its consequences. Then, I argue that, assuming equivalence of certain...

article

An Investigation into Intuitionistic Logic with Identity

Szymon Chlebowski, Dorota Leszczyńska-Jasion

Bulletin of the Section of Logic > 2019 > 48 > 4

We define Kripke semantics for propositional intuitionistic logic with Suszko’s identity (ISCI). We propose sequent calculus for ISCI along with cut-elimination theorem. We sketch a constructive interpretation of Suszko’s propositional identity connective.

article

Strong Normalization for Truth Table Natural Deduction

Herman Geuvers, Iris van der Giessen, Tonny Hurkens

Fundamenta Informaticae > 2019 > Vol. 170, nr 1-3 > 139--176

We present a proof of strong normalization of proof-reduction in a general system of natural deduction called truth table natural deduction. In previous work, we have defined truth table natural deduction, which is a method for deriving intuitionistic derivation rules for a connective from its truth table. This yields natural deduction rules for each connective separately. Moreover, these rules adhere...

article

Full Cut Elimination and Interpolation for Intuitionistic Logic with Existence Predicate

Paolo Maffezioli, Eugenio Orlandelli

Bulletin of the Section of Logic > 2019 > 48 > 2 > 137-158

In previous work by Baaz and Iemhoff, a Gentzen calculus for intuitionistic logic with existence predicate is presented that satisfies partial cut elimination and Craig's interpolation property; it is also conjectured that interpolation fails for the implication-free fragment. In this paper an equivalent calculus is introduced that satisfies full cut elimination and allows a direct proof of interpolation...

article

Remarks on Stable Formulas in Intuitionistic Logic

Majid Alizadeh, Ali Bibak

Logic and Logical Philosophy > 2019 > 28 > 2 > 209-222

ONNILLI-formulas were introduced in [2] and were shown to be the set of formulas that are preserved under monotonic images of descriptive or Kripke frames. As a result, ONNILLI is a syntactically defined set of formulas that axiomatize all stable logics. In this paper, among other things, by proving the uniform interpolation property for ONNILLI we show that ONNILLI is exactly the set of formulas...

article

What Are Justification Logics?

Melvin Fitting

Fundamenta Informaticae > 2019 > Vol. 165, nr 3-4 > 193--203

Justification logic began with Sergei Artemov’s work providing an arithmetic semantics for intuitionistic logic. As part of that work, a small number of explicit modal logics were introduced—logics in which there was a structure of terms that kept track of not just what was a necessary truth, but why it was necessary. These explicit modal logics were connected with standard modal logics such as S4,...

article

The Σ1-provability logic of HA

Mohammad Ardeshir, Mojtaba Mojtahedi

Annals of Pure and Applied Logic > 2018 > 169 > 10 > 997-1043

In this paper we introduce a modal theory iHσ which is sound and complete for arithmetical Σ1-interpretations in HA, in other words, we will show that iHσ is the Σ1-provability logic of HA. Moreover we will show that iHσ is decidable. As a by-product of these results, we show that HA+□⊥ has de Jongh property.

article

Incompleteness of Intuitionistic Propositional Logic with Respect to Proof-Theoretic Semantics

Thomas Piecha, Peter Schroeder-Heister

Studia Logica > 2019 > 107 > 1 > 233-246

Prawitz proposed certain notions of proof-theoretic validity and conjectured that intuitionistic logic is complete for them [11, 12]. Considering propositional logic, we present a general framework of five abstract conditions which any proof-theoretic semantics should obey. Then we formulate several more specific conditions under which the intuitionistic propositional calculus (IPC) turns out to be...

article

Basic Intuitionistic Conditional Logic

Yale Weiss

Journal of Philosophical Logic > 2019 > 48 > 3 > 447-469

Conditional logics have traditionally been intended to formalize various intuitively correct modes of reasoning involving (counterfactual) conditional expressions in natural language. Although conditional logics have by now been thoroughly studied in a classical context, they have yet to be systematically examined in an intuitionistic context, despite compelling philosophical and technical reasons...

article

Disentangling Structural Connectives or Life Without Display Property

Sergey Drobyshevich

Journal of Philosophical Logic > 2019 > 48 > 2 > 279-303

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...

article

Maehara-style modal nested calculi

Roman Kuznets, Lutz Straßburger

Archive for Mathematical Logic > 2019 > 58 > 3-4 > 359-385

We develop multi-conclusion nested sequent calculi for the fifteen logics of the intuitionistic modal cube between IK and IS5. The proof of cut-free completeness for all logics is provided both syntactically via a Maehara-style translation and semantically by constructing an infinite birelational countermodel from a failed proof search. Interestingly, the Maehara-style translation for proving soundness...

article

HYPE: A System of Hyperintensional Logic (with an Application to Semantic Paradoxes)

Hannes Leitgeb

Journal of Philosophical Logic > 2019 > 48 > 2 > 305-405

This article introduces, studies, and applies a new system of logic which is called ‘HYPE’. In HYPE, formulas are evaluated at states that may exhibit truth value gaps (partiality) and truth value gluts (overdeterminedness). Simple and natural semantic rules for negation and the conditional operator are formulated based on an incompatibility relation and a partial fusion operation on states. The semantics...

article

Teoria kategorii i niektóre jej logiczne aspekty

Mariusz Stopa

Zagadnienia Filozoficzne w Nauce > 2018 > 64 > 7-58

This article is intended for philosophers and logicians as a short partial introduction to category theory (CT) and its peculiar connection with logic. First, we consider CT itself. We give a brief insight into its history, introduce some basic definitions and present examples. In the second part, we focus on categorical topos semantics for propositional logic. We give some properties of logic in...

article

Multi-Agent Dialogues and Dialogue Sequents for Proof Search and Scheduling in Intuitionistic Logic and the Modal Logic S4

M. Sticht

Fundamenta Informaticae > 2018 > Vol. 161, nr 1/2 > 191--218

Dialogical games as introduced by Lorenzen and Lorenz describe a reasoning technique for intuitionistic and classical predicate logic: two players (proponent and opponent) argue about the validity of a given formula according to predefined rules. If the proponent has a winning strategy then the formula is proven to be valid. The underlying game rules can be modified to have an impact on proof search...

article

Certain Partial Conservativeness Properties of Intuitionistic Set Theory with the Principle of Double Complement of Sets

A. G. Vladimirov

Mathematical Notes > 2018 > 103 > 3-4 > 378-394

The Zermelo–Fraenkel set theory with the underlying intuitionistic logic (for brevity, we refer to it as the intuitionistic Zermelo–Fraenkel set theory) in a two-sorted language (where the sort 0 is assigned to numbers and the sort 1, to sets) with the collection scheme used as the replacement scheme of axioms (the ZFI2C theory) is considered. Some partial conservativeness properties of the intuitionistic...

article

Theorem Prover for Intuitionistic Logic Based on the Inverse Method

V. A. Pavlov, V. G. Pak

Programming and Computer Software > 2018 > 44 > 1 > 51-61

The first-order intuitionistic logic is a formal theory from the family of constructive logics. In intuitionistic logic, it is possible to extract a particular example x = a and a proof of a formula P(a) from a proof of a formula ∃xP(x). Owing to this feature, intuitionistic logic has many applications in mathematics and computer science. Many modern proof assistants include automated tactics for...

article

A Synthetic Version of Lie’s Second Theorem

Matthew Burke

Applied Categorical Structures > 2018 > 26 > 4 > 767-798

We formulate and prove a generalisation of Lie’s second theorem that integrates homomorphisms between formal group laws to homomorphisms between Lie groups. Firstly we generalise classical Lie theory by replacing groups with groupoids. Secondly we include groupoids whose underlying spaces are not smooth manifolds. The main intended application is when we replace the category of smooth manifolds with...

article

Epimorphisms in varieties of residuated structures

Guram Bezhanishvili, Tommaso Moraschini, James G. Raftery

Journal of Algebra > 2017 > 492 > C > 185-211

It is proved that epimorphisms are surjective in a range of varieties of residuated structures, including all varieties of Heyting or Brouwerian algebras of finite depth, and all varieties consisting of Gödel algebras, relative Stone algebras, Sugihara monoids or positive Sugihara monoids. This establishes the infinite deductive Beth definability property for a corresponding range of substructural...

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Search results

A note on Humberstone's constant Ω

From Intuitionism to Brouwer's Modal Logic

On the validity of the definition of a complement-classifier

An Investigation into Intuitionistic Logic with Identity

Strong Normalization for Truth Table Natural Deduction

Full Cut Elimination and Interpolation for Intuitionistic Logic with Existence Predicate

Remarks on Stable Formulas in Intuitionistic Logic

What Are Justification Logics?

The Σ1-provability logic of HA

Incompleteness of Intuitionistic Propositional Logic with Respect to Proof-Theoretic Semantics

Basic Intuitionistic Conditional Logic

Disentangling Structural Connectives or Life Without Display Property

Maehara-style modal nested calculi

HYPE: A System of Hyperintensional Logic (with an Application to Semantic Paradoxes)

Teoria kategorii i niektóre jej logiczne aspekty

Multi-Agent Dialogues and Dialogue Sequents for Proof Search and Scheduling in Intuitionistic Logic and the Modal Logic S4

Certain Partial Conservativeness Properties of Intuitionistic Set Theory with the Principle of Double Complement of Sets

Theorem Prover for Intuitionistic Logic Based on the Inverse Method

A Synthetic Version of Lie’s Second Theorem

Epimorphisms in varieties of residuated structures

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