Lewis meets Brouwer: Constructive strict implication

Litak, Tadeusz; Visser, Albert

doi:10.1016/j.indag.2017.10.003

Cited by 18 publications

(25 citation statements)

References 86 publications

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“…We define Heyting-Lewis Logic (the system iA following [71]) 2 as the extension of the intuitionistic propositional calculus (IPC) with the axioms K a ((ϕ ψ) ∧ (ϕ χ)) → (ϕ (ψ ∧ χ))…”

Section: Axioms and Rules For Arrowsmentioning

confidence: 99%

“…One easily shows [52,53,71] that the defined box is normal : the axiom K and the rule N obtained by substituting ⊤ for ϕ in K a and N a , respectively, are derivable in iA − , just like…”

Section: Intuitionistic Normal Modal Logics (With Box)mentioning

confidence: 99%

“…2.1 Example. The (monoidal) comonadic box of constructive S4 [2,14], obtained by extending IntK with 2 Litak and Visser [71] use the name "Lewis arrow" for , which leads to names such as iA, or to the use of a as a subscript.…”

Section: Intuitionistic Normal Modal Logics (With Box)mentioning

confidence: 99%

“…We recall the relational semantics for L [51, § 3.4.2], [71,Definition 3.3]. These are intuitionistic Kripke frames (i.e., posets) with an additional binary relation that is used to interpret the strict implication .…”

Section: Strict Implication Frames and Modelsmentioning

confidence: 99%

“…Recently, Litak and Visser [71] investigated over an intuitionistic rather than classical propositional base, using intuitionistic Kripke frames with an additional binary relation interpret strict implication. While a -modality can be obtained from via ϕ := ⊤ ϕ, strict implication is not definable from .…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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: Axioms and Rules For Arrowsmentioning

confidence: 99%

Section: Intuitionistic Normal Modal Logics (With Box)mentioning

confidence: 99%

Section: Intuitionistic Normal Modal Logics (With Box)mentioning

confidence: 99%

Section: Strict Implication Frames and Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Groot¹,

Litak²,

Dirt³

2021

Preprint

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Implication via Spacetime

Tabatabai

2021

Logic, Epistemology, and the Unity of Science

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It is a well-known fact that although the poset of open sets of a topological space is a Heyting algebra, its Heyting implication is not necessarily stable under the inverse image of continuous functions and hence is not a geometric concept. This leaves us wondering if there is any stable family of implications that can be safely called geometric. In this paper, we will first recall the abstract notion of implication as a binary modality introduced in [1]. Then, we will use a weaker version of categorical fibrations to define the geometricity of a category of pairs of spaces and implications over a given category of spaces. We will identify the greatest geometric category over the subcategories of open-irreducible (closed-irreducible) maps as a generalization of the usual injective open (closed) maps. Using this identification, we will then characterize all geometric categories over a given category S, provided that S has some basic closure properties. Specially, we will show that there is no non-trivial geometric category over the full category of spaces. Finally, as the implications we identified are also interesting in their own right, we will spend some time to investigate their algebraic properties. We will first use a Yoneda-type argument to provide a representation theorem, making the implications a part of an adjunction-style pair. Then, we will use this result to provide a Kripke-style representation for any arbitrary implication.

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Lewisian Fixed Points I: Two Incomparable Constructions

Litak,

Visser

2024

Dick De Jongh on Intuitionistic and Provability Logics

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Lewis meets Brouwer: Constructive strict implication

Cited by 18 publications

References 86 publications

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

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

Implication via Spacetime

Lewisian Fixed Points I: Two Incomparable Constructions

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