Basic Intuitionistic Conditional Logic

Weiss, Yale

doi:10.1007/s10992-018-9471-4

Cited by 12 publications

(12 citation statements)

References 13 publications

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“…We have seen that such logics arise under the functional programming interpretation (arrows without choice), and also under the arithmetical interpretation (e.g., preservativity without provable closure under q-realizability). 10 Let us note here that iA − extends the minimal system ICK of intuitionistic conditional logic proposed recently by Weiss [25,105,106], which allows the use of an intuitionistic variant of Chellas frames in conditional logic [24]. However, we believe that the right approach is to generalize so-called Veltman semantics of classical interpretability logics [32], as classically preservativity is the contrapose of arithmetical interpretability.…”

Section: Discussionmentioning

confidence: 97%

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: 97%

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

Groot¹,

Litak²,

Dirt³

2021

Preprint

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“…It would be interesting to see how this generalises to monotone modal intuitionistic logic. Other interesting candidates for similar investigations are conditional logic [38,39,40] and instantial neighbourhood logic [41,42].…”

Section: Conclusion and Further Researchmentioning

confidence: 99%

Hennessy-Milner Properties via Topological Compactness

Groot¹,

Pattinson²

2021

Preprint

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“…The existing literature on intuitionistic conditional logic is not vast, but it seems to have a clear candidate for the role of a basic system. This candidate is the system ICK, introduced by Y. Weiss in [15] (see also the more detailed exposition in [16]). This research was followed by [4], which was based on a version of Kripke semantics different from that used in [15]; however, this change did not affect the status of ICK which the authors of [4] have shown to be also the system complete relative to the universal class of their preferred variety of Kripke models.…”

Section: Introductionmentioning

confidence: 99%

“…This candidate is the system ICK, introduced by Y. Weiss in [15] (see also the more detailed exposition in [16]). This research was followed by [4], which was based on a version of Kripke semantics different from that used in [15]; however, this change did not affect the status of ICK which the authors of [4] have shown to be also the system complete relative to the universal class of their preferred variety of Kripke models. ICK also has a claim for the title of the intuitionistic counterpart of CK in that its complete axiomatization results from replacing the classical propositional fragment of CK with the intuitionistic one.…”

Section: Introductionmentioning

confidence: 99%

“…We show that IntCK stands in a very natural relation to other similar logics, like the basic classical conditional logic CK and the basic intuitionistic modal logic IK. As for the basic intuitionistic conditional logic ICK proposed in [15], IntCK extends its language with a diamond-like conditional modality , but its ( )-free fragment is also a proper extension of ICK. We briefly discuss the resulting gap between the two candidate systems of basic intuitionistic conditional logic and the possible pros and cons of both candidates.…”

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confidence: 99%

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Maximality of bi-intuitionistic propositional logic

Olkhovikov

Badía

2021

Journal of Logic and Computation

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In the style of Lindström’s theorem for classical first-order logic, this article characterizes propositional bi-intuitionistic logic as the maximal (with respect to expressive power) abstract logic satisfying a certain form of compactness, the Tarski union property and preservation under bi-asimulations. Since bi-intuitionistic logic introduces new complexities in the intuitionistic setting by adding the analogue of a backwards looking modality, the present paper constitutes a non-trivial modification of the previous work done by the authors for intuitionistic logic (Badia and Olkhovikov, 2020, Notre Dame Journal of Formal Logic, 61, 11–30).

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Basic Intuitionistic Conditional Logic

Cited by 12 publications

References 13 publications

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

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

Hennessy-Milner Properties via Topological Compactness

Maximality of bi-intuitionistic propositional logic

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