An Introduction to Basic Arithmetic

Ardeshir, Mohammad; Hesaam, Bardyaa

doi:10.1093/jigpal/jzm013

Cited by 16 publications

(21 citation statements)

References 4 publications

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“…Some weak arithmetical theories as base arithmetical theories are mentioned. For motivations and basic properties of BQC and BA, see [8] and [1].…”

Section: Preliminaries To Basic Arithmeticmentioning

confidence: 99%

“…Given a sequence of variables x without repetitions, s[x/t] and A[x/t] stand for, respectively, the term and formula that results from substituting the terms t for all free occurrences of the variables of x in the term s and the formula A. For details, see [8] and [1]. We often write A for ⊤ ⇒ A, and A ⇔ B for A ⇒ B and B ⇒ A together.…”

Section: Axioms and Rules Of Basic Predicate Calculusmentioning

confidence: 99%

“…A Kripke model for BA is a quadruple K = (K, ≺, D, ), like the one for HA, except the relation ≺ is not needed to be reflexive, see [1] and [8] for more details. So the forcing relation between a node k and a sentence of the form ∀x (A → B) is defined as…”

Section: Kripke Models Of Bamentioning

confidence: 99%

“…2. Use first part and Theorem 2.18 in [1]. ⊣ Lemma 3.15 Let A be a formula in the language of arithmetic.…”

Section: Range(f ) Is Infinitementioning

confidence: 99%

“…The logical extension we study is EBA, an extension of BA with the logical axiom ⊤ → ⊥ ⇒ ⊥. This extension is introduced in [1] and it is shown there that its behavior is very close to HA, but still weaker than that. We show that PR(EBA) = PR.…”

Section: Introductionmentioning

confidence: 99%

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Provably total recursive functions and MRDP theorem in Basic Arithmetic and its extensions

Ardeshir¹,

Khaniki²,

Shahriari³

2020

Preprint

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We study Basic Arithmetic, BA introduced by W. Ruitenburg. BA is an arithmetical theory based on basic logic which is weaker than intuitionistic logic. We show that the class of the provably recursive functions of BA is a proper sub-class of primitive recursive functions. Three extensions of BA, called BA + U, BAc and EBA are investigated with relation to their provably recursive functions. It is shown that the provably recursive functions of these three extensions of BA are exactly primitive recursive functions. Moreover, among other things, it is shown that the well-known MRDP theorem doesn't hold in BA, BA + U, BAc, but holds in EBA.

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“…Some weak arithmetical theories as base arithmetical theories are mentioned. For motivations and basic properties of BQC and BA, see [8] and [1].…”

Section: Preliminaries To Basic Arithmeticmentioning

confidence: 99%

Section: Axioms and Rules Of Basic Predicate Calculusmentioning

confidence: 99%

Section: Kripke Models Of Bamentioning

confidence: 99%

“…2. Use first part and Theorem 2.18 in [1]. ⊣ Lemma 3.15 Let A be a formula in the language of arithmetic.…”

Section: Range(f ) Is Infinitementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Provably total recursive functions and MRDP theorem in Basic Arithmetic and its extensions

Ardeshir¹,

Khaniki²,

Shahriari³

2020

Preprint

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Mathematics, Logic, and their Philosophies

Mojtahedi¹,

Rahman²,

Zarepour³

2021

Logic, Epistemology, and the Unity of Science

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Logic, Epistemology, and the Unity of Science aims to reconsider the question of the unity of science in light of recent developments in logic. At present, no single logical, semantical or methodological framework dominates the philosophy of science. However, the editors of this series believe that formal frameworks, for example, constructive type theory, deontic logics, dialogical logics, epistemic logics, modal logics, and proof-theoretical semantics, have the potential to cast new light on basic issues in the discussion of the unity of science.This series provides a venue where philosophers and logicians can apply specific systematic and historic insights to fundamental philosophical problems. While the series is open to a wide variety of perspectives, including the study and analysis of argumentation and the critical discussion of the relationship between logic and philosophy of science, the aim is to provide an integrated picture of the scientific enterprise in all its diversity.

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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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An Introduction to Basic Arithmetic

Cited by 16 publications

References 4 publications

Provably total recursive functions and MRDP theorem in Basic Arithmetic and its extensions

Provably total recursive functions and MRDP theorem in Basic Arithmetic and its extensions

Mathematics, Logic, and their Philosophies

Implication via Spacetime

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