Filtrations of generalized Veltman models

Perkov, Tin; Vuković, Mladen

doi:10.1002/malq.201500030

Cited by 10 publications

(18 citation statements)

References 11 publications

(30 reference statements)

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“…Conceivable variants of generalized Veltman semantics would be equivalent to subclasses of Chellas-Weiss frames (much like Kripke frames can be seen as a limiting case of neighbourhood frames). Nevertheless, in the classical setting, (generalized) Veltman semantics has proved particularly suitable for decidability and complexity results [32,79,80], allowing adaptations of standard modal techniques such as filtration [89], so it does seem promising to work with more restrictive structures. and since p is a filter this implies (e ∧ c) b ∈ p. By definition of I we now have e ∧ c ∈ I.…”

Section: Discussionmentioning

confidence: 99%

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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Section: Discussionmentioning

confidence: 99%

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

Groot¹,

Litak²,

Dirt³

2021

Preprint

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“…the ordinary semantics. In [9] and [8], the filtration technique was used to prove the finite model property of IL and its extensions ILM, ILM 0 and ILW * w.r.t. the generalised semantics.…”

Section: Principlementioning

confidence: 99%

“…This approach was successfully used to prove the FMP of ILM 0 and ILW * w.r.t. the generalised semantics [8,9]. A drawback of this approach is in that the FMP w.r.t.…”

Section: The Logic Ilr Completeness Of Ilr Wrt Ordinary Veltman Smentioning

confidence: 99%

“…If A equals ¬B for some B, then ∼A is B, otherwise ∼A is ¬B. We need to slightly extend the definition of adequate sets 4 that was used in [9]. The modified version will satisfy all the old properties.…”

Section: The Logic Ilr Completeness Of Ilr Wrt Ordinary Veltman Smentioning

confidence: 99%

“…5 Let us turn to the finite model property and decidability. In all known cases of (decidable) interpretability logics, the simplest way to show decidability is via the generalised semantics [9], [8]. Decidability does not seem to be a problematic issue in the context of interpretability logics; currently there is no known complete logic that is not known to be decidable too.…”

Section: What Are the Two Series' Labelling Lemmas?mentioning

confidence: 99%

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Interpretability Logics and Generalised Veltman Semantics

Mikec

Vuković

2020

J. symb. log.

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We obtain modal completeness of the interpretability logics IL and IL w.r.t. generalised Veltman semantics. Our proofs are based on the notion of full labels [2]. We also give shorter proofs of completeness w.r.t. the generalised semantics for many classical interpretability logics. We obtain decidability and finite model property w.r.t. the generalised semantics for IL and IL . Finally, we develop a construction that might be useful for proofs of completeness of extensions of IL w.r.t. the generalised semantics in the future, and demonstrate its usage with .

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Bisimulations and bisimulation games between Verbrugge models

Horvat

Perkov

Vuković

2023

Mathematical Logic Qtrly

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Interpretability logic is a modal formalization of relative interpretability between first‐order arithmetical theories. Verbrugge semantics is a generalization of Veltman semantics, the basic semantics for interpretability logic. Bisimulation is the basic equivalence between models for modal logic. We study various notions of bisimulation between Verbrugge models and develop a new one, which we call w‐bisimulation. We show that the new notion, while keeping the basic property that bisimilarity implies modal equivalence, is weak enough to allow the converse to hold in the finitary case. To do this, we develop and use an appropriate notion of bisimulation games between Verbrugge models.

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Filtrations of generalized Veltman models

Cited by 10 publications

References 11 publications

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

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

Interpretability Logics and Generalised Veltman Semantics

Bisimulations and bisimulation games between Verbrugge models

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