On Löb algebras

Alizadeh, Majid; Ardeshir, Mohammad

doi:10.1002/malq.200510016

Cited by 9 publications

(4 citation statements)

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“…These algebras are the natural algebraic models for sub-intuitionistic logics. See for instance [39], [19], [15], [5], [6], [7] and [8] for the algebraic notions and [12], [13], [9] and [39] for their role in sub-intuitionistic logics.…”

Section: Abstract Implicationsmentioning

confidence: 99%

“…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%

“…First let us review some important sub-intuitionistic logics, introduced in [50], [51], [39], [18], [11], [44], [20], and [21] and investigated extensively in [13], [12], [5], [6], [7], [8], [9], [19], [45], and [48]. To complete the list we also define one new logic, EKPC and we will explain its behaviour later.…”

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confidence: 99%

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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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Section: Abstract Implicationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Implication via Spacetime

Tabatabai

2020

Preprint

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“…The counterpart logic of this algebra, also called Basic Logic, was first introduced by Visser [19] in 1981 and investigated further by Ruitenberg [14] in 1992. In 2006, Alizadeh and Ardeshir [2] studied Löb algebras, which are a subset of the Basic algebra, and proved the completeness of the counterpart logic of this algebra. They introduced countably many locally finite subvarieties of the variety of Löb algebras and showed that their corresponding intermediate logics have the interpolation property (cf.…”

Section: Introductionmentioning

confidence: 99%

On self‐distributive weak Heyting algebras

Nourany,

Ghorbani,

Saeid

2023

Mathematical Logic Qtrly

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We use the left self‐distributive axiom to introduce and study a special class of weak Heyting algebras, called self‐distributive weak Heyting algebras (SDWH‐algebras). We present some useful properties of SDWH‐algebras and obtain some equivalent conditions of them. A characteristic of SDWH‐algebras of orders 3 and 4 is given. Finally, we study the relation between the variety of SDWH‐algebras and some of the known subvarieties of weak Heyting algebras such as the variety of Heyting algebras, the variety of basic algebras, the variety of subresiduated lattices, the variety of reflexive WH‐algebras (RWH‐algebras), and the variety of transitive WH‐algebras (TWH‐algebras).

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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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On Löb algebras

Cited by 9 publications

References 5 publications

Implication via Spacetime

Implication via Spacetime

On self‐distributive weak Heyting algebras

Implication via Spacetime

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