Categorical and algebraic aspects of the intuitionistic modal logic IEL― and its predicate extensions

Rogozin, Daniel

doi:10.1093/logcom/exaa082

Cited by 6 publications

(3 citation statements)

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“…yields IntK ⊕ S 5 , the (inhabitation) logic of Haskell's applicative functors (idioms) [77], as noted in recent references [73,92] (cf. § 2.3).…”

Section: Intuitionistic Normal Modal Logics (With Box)mentioning

confidence: 89%

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Gödel-McKinsey-Tarski and Blok-Esakia for Heyting-Lewis Implication

Groot¹,

Litak²,

Dirt³

2021

Preprint

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Heyting-Lewis Logic is the extension of intuitionistic propositional logic with a strict implication connective that satisfies the constructive counterparts of axioms for strict implication provable in classical modal logics. Variants of this logic are surprisingly widespread: they appear as Curry-Howard correspondents of (simple type theory extended with) Haskell-style arrows, in preservativity logic of Heyting arithmetic, in the proof theory of guarded (co)recursion, and in the generalization of intuitionistic epistemic logic.Heyting-Lewis Logic can be interpreted in intuitionistic Kripke frames extended with a binary relation to account for strict implication. We use this semantics to define descriptive frames (generalisations of Esakia spaces), and establish a categorical duality between the algebraic interpretation and the frame semantics. We then adapt a transformation by Wolter and Zakharyaschev to translate Heyting-Lewis Logic to classical modal logic with two unary operators. This allows us to prove a Blok-Esakia theorem that we then use to obtain both known and new canonicity and correspondence theorems, and the finite model property and decidability for a large family of Heyting-Lewis logics.1 Curiously, Lewis was not using as a primitive, so in fact his intuitionistically problematic definition of ϕ ψ was ¬ (ϕ ∧ ¬ψ). See [71, App. D] for an account of problems caused by Lewis' use of a Boolean propositional base, namely trivialization [64, 67] of his original system [63, 66], which in turn finally lead him to propose systems S1-S3 [68, App. 2] as successive "lines of retreat" [88]. Lewis considered S4 and S5, suggested by Becker [8], too strong to provide a proper account of strict implication [68, p. 502] and appeared frustrated with later development of modal logic [71, § 2.1]. Yet, despite his supportive attitude towards non-classical logics, he seems to have mentioned Brouwer only once (favourably) [65], and does not appear to have ever referred to, or even be familiar with subsequent work of Kolmogorov, Heyting or Glivenko [71, § 2.2].7 To the best of our knowledge, the only explicit (if brief) discussion of the Curry-Howard connection between Haskell arrows and i-Sa − is found in Litak and Visser [71, § 7.1].8 The logical perspective seems to cast a light on the controversy whether arrows are "stronger" than applicative functors [70,77]. Putting aside the general question of whether one takes as a measure of strength the capability to inhabit more types or rather to allow more distinctions, in the presence of , the situation is not as clear-cut as in the unary case, where (monadic) PLL simply extends (applicative) IntK ⊕ S . Defining arrows over the latter set of axioms via delay yields i-Sa − ⊕ Box, whereas inhabiting apply with Appa yields PLAA. These are two incomparable systems.

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“…yields IntK ⊕ S 5 , the (inhabitation) logic of Haskell's applicative functors (idioms) [77], as noted in recent references [73,92] (cf. § 2.3).…”

Section: Intuitionistic Normal Modal Logics (With Box)mentioning

confidence: 89%

“…Strength of the functor interpreting in a categorical semantics of modal proofs[2,14,34,73,92] corresponds to the validity of (ϕ ∧ ψ) → (ϕ ∧ ψ), but this is derivable from S when is normal[73, Sec. 6].…”

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confidence: 99%

Gödel-McKinsey-Tarski and Blok-Esakia for Heyting-Lewis Implication

Groot¹,

Litak²,

Dirt³

2021

Preprint

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“…These models are quite complicated: they contain a nonempty set of possible worlds, two accessibility relations, a function mapping each world to some subset of worlds, and also a set of 'worlds which are prone to errors' (only for QLL). The logic QLL was also studied by Aczel [27], another type of semantics for QLL was constructed by Goldblatt (see [28]), and Rogozin [29] studied the lambda-calculus associated with this and several closely related logics.…”

Section: § 1 Introductionmentioning

confidence: 99%

The predicate version of the joint logic of problems and propositions

Оноприенко

2022

Sb. Math.

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We consider the joint logic of problems and propositions $\mathrm{QHC}$ introduced by Melikhov. We construct Kripke models with audit worlds for this logic and prove the soundness and completeness of $\mathrm{QHC}$ with respect to this type of model. The conservativity of the logic $\mathrm{QHC}$ over the intuitionistic modal logic $\mathrm{QH4}$, which coincides with the ‘lax logic’ $\mathrm{QLL}^+$, is established. We construct Kripke models with audit worlds for the logic $\mathrm{QH4}$ and prove the corresponding soundness and completeness theorems. We also prove that the logics $\mathrm{QHC}$ and $\mathrm{QH4}$ have the disjunction and existence properties. Bibliography: 33 titles.

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Constructive and Mechanised Meta-Theory of Intuitionistic Epistemic Logic

Hagemeier

Kirst

2021

Logical Foundations of Computer Science

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Artemov and Protopopescu proposed intuitionistic epistemic logic (IEL) to capture an intuitionistic conception of knowledge. By establishing completeness, they provided the base for a meta-theoretic investigation of IEL, which was continued by Krupski with a proof of cut-elimination, and Su and Sano establishing semantic cut-elimination and the finite model property. However, to the best of our knowledge, no analysis of these results in a constructive meta-logic has been conducted.We aim to close this gap and investigate IEL in the constructive type theory of the Coq proof assistant. Concretely, we present a constructive and mechanised completeness proof for IEL, employing a syntactic decidability proof based on cut-elimination to constructivise the ideas from the literature. Following Su and Sano, we then also give constructive versions of semantic cut-elimination and the finite model property. Given our constructive and mechanised setting, all these results now bear executable algorithms. We expect that our methods used for mechanising cut-elimination and decidability also extend to other modal logics (and have verified this observation for the classical modal logic K).

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Categorical and algebraic aspects of the intuitionistic modal logic IEL― and its predicate extensions

Cited by 6 publications

References 37 publications

Gödel-McKinsey-Tarski and Blok-Esakia for Heyting-Lewis Implication

Gödel-McKinsey-Tarski and Blok-Esakia for Heyting-Lewis Implication

The predicate version of the joint logic of problems and propositions

Constructive and Mechanised Meta-Theory of Intuitionistic Epistemic Logic

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