Properties of Intuitionistic Provability and Preservativity Logics

Iemhoff, Rosalie; Jongh, Dick de; Zhou, Changjiu

doi:10.1093/jigpal/jzi047

Cited by 15 publications

(28 citation statements)

References 4 publications

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“…Dialgebraic methods can, for example, provide a generic expressivity and expressivity-somewhere-else results [47, § 7] and Goldblatt-Thomason theorems [30]. It would be intriguing to see whether dialgebraic methods can provide an elegant connection (say, via a dialgebraic generalisation of the final coalgebra sequence) with, on the one hand, finitary completeness proofs developed in the preservativity community [51,53] and on the other hand, step-algebras and step-frames [13].…”

Section: Discussionmentioning

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%

“…A framework of schematic logics accounts for this variety of interpretations [71, § 5], [74, § 4], broadly generalizing the well-known provability interpretation of the classical Löb logic [20]. The most important arithmetical interpretation is that of Σ 1 -preservativity, historically the first context in which constructive appeared [52,53,100,101,103]. This interpretation makes inclusion of not only S , but even Di in the base system problematic.…”

Section: Arithmetical Interpretationsmentioning

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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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%

Section: Intuitionistic Normal Modal Logics (With Box)mentioning

confidence: 99%

Section: Arithmetical Interpretationsmentioning

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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“…It can be easily shown (see, e.g., [Zho03;IDZ05]) that the condition equivalent to persistence becomes precisely -p , that is:…”

Section: Strict Implication In Intuitionistic Kripke Semanticsmentioning

confidence: 99%

Lewis meets Brouwer: Constructive strict implication

Litak¹,

Visser

2018

Indagationes Mathematicae

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C. I. Lewis invented modern modal logic as a theory of "strict implication" . Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than P and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction has been discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of intuitionistic in terms of preservativity in extensions of HA, i.e., Heyting's Arithmetic. 65The basic option is to read § § 2-4 to get the basics of motivational background, the Kripke semantics and an impression of possible reasoning systems. ? The reader who wants more solid treatment of Kripke semantics can extend the basic option with § 6. | The computer science package consists of the basic option and § 7.= The reader who wants to go somewhat more deeply into the history of the subject can extend the basic option with Appendix D. « The reader who wants to understand the basics of arithmetical interpretations can extend the basic option with § 5. » An extended package for arithmetical interpretations combines « with § 8.-The full arithmetical package extends » with Appendices A, B and C. The rise and fall of the house of Lewis "The error of philosophers"We are reflecting on L.E.J. Brouwer's heritage half a century after his passing. Given his negative views on the rôle of logic and formalisms in mathematics, it seems somewhat paradoxical that these days the name of intuitionism survives mostly in the context of intuitionistic logic. 3 One is reminded in this context of what Nietzsche called the error of philosophers:The philosopher believes that the value of his philosophy lies in the whole, in the structure. Posterity finds it in the stone with which he built and with which, from that time forth, men will build oftener and better-in other words, in the fact that the structure may be destroyed and yet have value as material. 4

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“…This implication is morally the Heyting implication without its modus ponens rule; see [12], [14], [13], [18]. The emergence of these weak implications then set the scene for a plethora of other and sometimes even weaker implications emerging philosophically [43]; algebraically [39], [19], [5], [6], [7], [8], [9], [15]; proof theoretically [20], [21], [48], [45]; arithmetically [52], [23], [24] and via relational semantics [11], [30], almost everywhere in the logical realm. Apart from the philosophically oriented reasons, the weak implications raise also some independent mathematical interests.…”

Section: Introductionmentioning

confidence: 99%

Implication via Spacetime

Tabatabai

2020

Preprint

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In this paper we intend to study implications in their most general form, generalizing different classes of implications including the Heyting implication, sub-structural implications and weak strict implications. Following the topological interpretation of the intuitionistic logic, we will introduce non-commutative spacetimes to provide a more dynamic and subjective interpretation of an intuitionistic proposition. These combinations of space and time are natural sources for wellbehaved implications and we will show that their spatio-temporal implications represent any other reasonable abstract implication. Then to provide a faithful well-behaved syntax for abstract implications, we will develop a logical system for the non-commutative spacetimes for which we will present both topological and Kripke semantics. These logics unify sub-structural and sub-intuionistic logics by embracing them as their special fragments.

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Properties of Intuitionistic Provability and Preservativity Logics

Cited by 15 publications

References 4 publications

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

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

Lewis meets Brouwer: Constructive strict implication

Implication via Spacetime

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