A semantic hierarchy for intuitionistic logic

Bezhanishvili, Guram; Holliday, Wesley H.

doi:10.1016/j.indag.2019.01.001

Cited by 32 publications

(28 citation statements)

References 75 publications

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“…Most importantly, we still seem to have very few (if any) general results regarding sub-Kripkean completeness for nonclassical logics with a non-Boolean propositional base. Even Kuznetsov's problem (Kuznetsov, 1975) regarding topological completeness of superintuitionistic logics remains open more than four decades after its formulation (see Bezhanishvili & Holliday 2019). We hope to see progress on these problems in the years ahead.…”

Section: Pure Modal Syntaxmentioning

confidence: 99%

Complete Additivity and Modal Incompleteness

Holliday

Litak

2019

The Review of Symbolic Logic

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In this paper, we tell a story about incompleteness in modal logic. The story weaves together a paper of van Benthem [1979], "Syntactic aspects of modal incompleteness theorems," and a longstanding open question: whether every normal modal logic can be characterized by a class of completely additive modal algebras, or as we call them, V-baos. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem's paper resolves the open question in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to V-baos, namely the provability logic GLB [Japaridze, 1988, Boolos, 1993. We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is Vcomplete. After these results, we generalize the Blok Dichotomy [Blok, 1978] to degrees of V-incompleteness. In the end, we return to van Benthem's theme of syntactic aspects of modal incompleteness.

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Section: Pure Modal Syntaxmentioning

confidence: 99%

Complete Additivity and Modal Incompleteness

Holliday

Litak

2019

The Review of Symbolic Logic

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“…In the 1950s, Beth and Kripke proposed models over trees of finite or infinite sequences, and in line with the idea of proof as establishing a conclusion, intuitionistic formulas are true at a node of such a tree when 'verified' in some intuitive sense. A general topological framework for placing all these ideas uniformly is presented in [17]. A standard version that suffices for our purposes here uses partially ordered models M = (W, ≤, V ) with a valuation V , setting:…”

Section: Intuitionistic Logicmentioning

confidence: 99%

“…The second view need not be superior to the first, but its very existence undermines strong conclusions arising from looking at consequence in only one stance. 17…”

Section: Philosophical Repercussionsmentioning

confidence: 99%

Implicit and Explicit Stances in Logic

Benthem

2018

J Philos Logic

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We identify a pervasive contrast between implicit and explicit stances in logical analysis and system design. Implicit systems change received meanings of logical constants and sometimes also the notion of consequence, while explicit systems conservatively extend classical systems with new vocabulary. We illustrate the contrast for intuitionistic and epistemic logic, then take it further to information dynamics, default reasoning, and other areas, to show its wide scope. This gives a working understanding of the contrast, though we stop short of a formal definition, and acknowledge limitations and borderline cases. Throughout we show how awareness of the two stances suggests new logical systems and new issues about translations between implicit and explicit systems, linking up with foundational concerns about identity of logical systems. But we also show how a practical facility with these complementary working styles has philosophical consequences, as it throws doubt on strong philosophical claims made by just taking one design stance and ignoring alternative ones. We will illustrate the latter benefit for the case of logical pluralism and hyper-intensional semantics.

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“…Intermediate logics ( [5,11]) are classes of formulas closed under uniform substitution and modus ponens, lying between intuitionistic logic (IPC) and classical logic (CPC). This family of logics has been studied using several semantics, as for example Kripke semantics, Beth semantics, topological semantics and algebraic semantics (for an overview see [1]). Among these, the algebraic semantics based on Heyting algebras plays a special role: every intermediate logic is sound and complete with respect to some class of Heyting algebras.…”

Section: Introductionmentioning

confidence: 99%

“…Among these, the algebraic semantics based on Heyting algebras plays a special role: every intermediate logic is sound and complete with respect to some class of Heyting algebras. 1 This connection between intermediate logics and Heyting algebras has been studied using tools from universal algebra. As a consequence of Birkhoff's Theorem ( [3,4]), the lattice of varieties of Heyting algebras HA is dually isomorphic to the lattice of intermediate logics IL.…”

Section: Introductionmentioning

confidence: 99%

Lattices of Intermediate Theories via Ruitenburg's Theorem

Grilletti¹,

Quadrellaro²

2020

Preprint

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For every univariate formula χ we introduce a lattices of intermediate theories: the lattice of χ-logics. The key idea to define χ-logics is to interpret atomic propositions as fixpoints of the formula χ 2 , which can be characterised syntactically using Ruitenburg's theorem. We develop an algebraic duality between the lattice of χ-logics and a special class of varieties of Heyting algebras. This approach allows us to build five distinct lattices-corresponding to the possible fixpoints of univariate formulas-among which the lattice of negative variants of intermediate logics. We describe these lattices in more detail.

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A semantic hierarchy for intuitionistic logic

Cited by 32 publications

References 75 publications

Complete Additivity and Modal Incompleteness

Complete Additivity and Modal Incompleteness

Implicit and Explicit Stances in Logic

Lattices of Intermediate Theories via Ruitenburg's Theorem

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