Relevance logic and entailment

2018 ◽  
Author(s):  
Stephen Read
Keyword(s):  
Set Theory ◽  
Intuitionistic Logic ◽  
Logical Consequence ◽  
Relevance Logic ◽  
Necessary Condition ◽  
Central Idea ◽  
Logical Connection ◽  
Relevance Condition

‘Relevance logic’ came into being in the late 1950s, inspired by Wilhelm Ackermann, who rejected certain formulas of the form A→B on the grounds that ‘the truth of A has nothing to do with the question whether there is a logical connection between B and A’. The central idea of relevance logic is to give an account of logical consequence, or entailment, for which a connection of relevance between premises and conclusion is a necessary condition. In both classical and intuitionistic logic, this condition is missing, as is highlighted by the validity in those logics of the ‘spread law’, A &∼A→B; a contradiction ‘spreads’ to every proposition, and simple inconsistency is equivalent to absolute inconsistency. In relevance logic the spread law fails, and the simple inconsistency of a theory (that a set of formulas entails a contradiction) is distinguished from absolute inconsistency (or triviality: that a set of formulas entails every proposition). The programme of relevance logic is to characterize a logic, or a range of logics, satisfying the relevance condition, and to study theories based on such logics, such as relevant arithmetic and relevant set theory.

A SUFFICIENT AND NECESSARY CONDITION FOR CREDIBILITY MEASURES

2006 ◽  
Vol 14 (05) ◽  
pp. 527-535 ◽  
Author(s):  
XIANG LI ◽  
BAODING LIU
Keyword(s):  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Extension Theorem ◽  
The Self ◽  
Necessary Condition ◽  
Self Duality ◽  
Duality Property ◽  
Sufficient And Necessary Condition

Possibility measures and credibility measures are widely used in fuzzy set theory. Compared with possibility measures, the advantage of credibility measures is the self-duality property. This paper gives a relation between possibility measures and credibility measures, and proves a sufficient and necessary condition for credibility measures. Finally, the credibility extension theorem is shown.

Some properties of epistemic set theory with collection

1986 ◽  
Vol 51 (3) ◽  
pp. 748-754 ◽  
Author(s):  
Andre Scedrov
Keyword(s):  
Set Theory ◽  
Intuitionistic Logic ◽  
Existential Quantifier ◽  
Modal Interpretation ◽  
Conservative Extension ◽  
Disjunction Property ◽  
First Order ◽  
Slight Extension ◽  
Order Arithmetic ◽  
Epistemic Systems

Myhill [12] extended the ideas of Shapiro [15], and proposed a system of epistemic set theory IST (based on modal S4 logic) in which the meaning of the necessity operator is taken to be the intuitive provability, as formalized in the system itself. In this setting one works in classical logic, and yet it is possible to make distinctions usually associated with intuitionism, e.g. a constructive existential quantifier can be expressed as (∃x) □ …. This was first confirmed when Goodman [7] proved that Shapiro's epistemic first order arithmetic is conservative over intuitionistic first order arithmetic via an extension of Gödel's modal interpretation [6] of intuitionistic logic.Myhill showed that whenever a sentence □A ∨ □B is provable in IST, then A is provable in IST or B is provable in IST (the disjunction property), and that whenever a sentence ∃x.□A(x) is provable in IST, then so is A(t) for some closed term t (the existence property). He adapted the Friedman slash [4] to epistemic systems.Goodman [8] used Epistemic Replacement to formulate a ZF-like strengthening of IST, and proved that it was a conservative extension of ZF and that it had the disjunction and existence properties. It was then shown in [13] that a slight extension of Goodman's system with the Epistemic Foundation (ZFER, cf. §1) suffices to interpret intuitionistic ZF set theory with Replacement (ZFIR, [10]). This is obtained by extending Gödel's modal interpretation [6] of intuitionistic logic. ZFER still had the properties of Goodman's system mentioned above.

African Customary Law in the Former Portuguese Territories, 1954–1974

1984 ◽  
Vol 28 (1-2) ◽  
pp. 72-79 ◽  
Author(s):  
Narana Coissoro
Keyword(s):  
Common Law ◽  
Logical Consequence ◽  
Northern Region ◽  
Customary Law ◽  
Necessary Condition ◽  
Colonial History ◽  
Daily Lives ◽  
The Everyday ◽  
Customary Laws ◽  
The Individual

Throughout all Portuguese colonial history in the African continent, the question of recognizing oral local laws, the so called “customary laws”, and the koranic law in some areas of Guinea and the northern region of Mozambique, could never be separate from the constitutional law applicable to the aboriginal inhabitants who follow it in their daily lives. That is the reason why accepting the principle according to which the everyday-life relations of Africa could be controlled by specific juridicial rules distinct from Portuguese “common law” was always connected with the private and territorial validity of the individual rights and guarantees included in the constitutional texts concerning Africans. As a logical consequence of this link between citizenship and the application of the Portuguese law in force in the metropolis, applying traditional law always depended on the political concepts formed during the present century, as Portuguese sovereignty, until the end of the nineteenth century, was restricted to small littoral centres and the practice of authority in the other regions acquired at the Berlin Conference was deficient or merely nominal.The African juridical rules were always tolerated, as a means of securing colonial public peace or as a necessary condition for the smooth practice of Portuguese sovereignty beyond its European frontiers.

scholarly journals Set Theory INC# Based on Intuitionistic Logic with Restricted Modus Ponens Rule (Part. I)

2021 ◽  
pp. 73-88
Author(s):  
Jaykov Foukzon
Keyword(s):  
Set Theory ◽  
Intuitionistic Logic ◽  
Modus Ponens ◽  
Russell’S Paradox ◽  
Russell's Paradox

In this article Russell’s paradox and Cantor’s paradox resolved successfully using intuitionistic logic with restricted modus ponens rule.

scholarly journals Set Theory INC# ∞# Based on Innitary Intuitionistic Logic with Restricted Modus Ponens Rule. Hyper Inductive Denitions. Application in Transcendental Number Theory. Generalized Lindemann-Weierstrass Theorem

2021 ◽  
pp. 70-119
Author(s):  
Jaykov Foukzon
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Intuitionistic Logic ◽  
Theoretical Language ◽  
Modus Ponens ◽  
Transcendental Number ◽  
Weierstrass Theorem ◽  
Induction Principle ◽  
Intuitionistic Set Theory ◽  
Intuitionistic Arithmetic

In this paper intuitionistic set theory INC#∞# in infinitary set theoretical language is considered. External induction principle in nonstandard intuitionistic arithmetic were derived. Non trivial application in number theory is considered.The Goldbach-Euler theorem is obtained without any references to Catalan conjecture. Main results are: (i) number ee is transcendental; (ii) the both numbers e + π and e − π are irrational.

Intuitionism Reconsidered

2009 ◽  
Author(s):  
Roy Cook
Keyword(s):  
Intuitionistic Logic ◽  
Algebraic Structure ◽  
Subject Matter ◽  
Logical Consequence ◽  
Consequence Relation ◽  
Mathematical Structures

The debate between intuitionists and classical logicians is fought on two fronts. First, there is the battle over subject matter—the disputants disagree regarding which mathematical structures are legitimate domains of inquiry. Second, there is the battle over logic—they disagree over which algebraic structure correctly codifies logical consequence. In this article the emphasis is on the latter issue—it focuses on what the correct (formal) account of correct inference might look like, and, given such an account, how we should understand disagreements regarding the extension of the logical consequence relation. In the next two sections of the article, two typical sorts of arguments for intuitionistic logic are examined. The article then examines exactly what is at stake when one provides a logic as an account of logical consequence.

The consistency of classical set theory relative to a set theory with intu1tionistic logic

1973 ◽  
Vol 38 (2) ◽  
pp. 315-319 ◽  
Author(s):  
Harvey Friedman
Keyword(s):  
Set Theory ◽  
Classical Theory ◽  
Intuitionistic Logic ◽  
Classical Logic ◽  
Predicate Calculus ◽  
Basic Tool ◽  
Axiom Scheme ◽  
Power Set ◽  
Image Position

Let ZF be the usual Zermelo-Fraenkel set theory formulated without identity, and with the collection axiom scheme. Let ZF−-extensionality be obtained from ZF by using intuitionistic logic instead of classical logic, and dropping the axiom of extensionality. We give a syntactic transformation of ZF into ZF−-extensionality.Let CPC, HPC respectively be classical, intuitionistic predicate calculus without identity, whose only homological symbol is ∈. We use the ~ ~-translation, a basic tool in the metatheory of intuitionistic systems, which is defined byThe two fundamental lemmas about this ~ ~ -translation we will use areFor proofs, see Kleene [3, Lemma 43a, Theorem 60d].This - would provide the desired syntactic transformation at least for ZF into ZF− with extensionality, if A− were provable in ZF− for each axiom A of ZF. Unfortunately, the ~ ~-translations of extensionality and power set appear not to be provable in ZF−. We therefore form an auxiliary classical theory S which has no extensionality and has a weakened power set axiom, and show in §2 that the ~ ~-translation of each axiom of Sis provable in ZF−-extensionality. §1 is devoted to the translation of ZF in S.

scholarly journals Set Theory INC# ∞# Based on Infinitary Intuitionistic Logic with Restricted Modus Ponens Rule (Part.II) Hyper Inductive Definitions

2021 ◽  
pp. 90-112
Author(s):  
Jaykov Foukzon
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Intuitionistic Logic ◽  
Theoretical Language ◽  
Modus Ponens ◽  
Inductive Definitions ◽  
Induction Principle ◽  
Intuitionistic Set Theory ◽  
Intuitionistic Arithmetic

In this paper intuitionistic set theory INC# ∞# in infinitary set theoretical language is considered. External induction principle in nonstandard intuitionistic arithmetic were derived. Non trivial application in number theory is considered.The Goldbach-Euler theorem is obtained without anyreferences to Catalan conjecture.

Higher‐order Logic Reconsidered

2009 ◽  
Author(s):  
Ignacio Jané
Keyword(s):  
Set Theory ◽  
Logical Consequence ◽  
Higher Order ◽  
Second Order ◽  
Order Logic ◽  
Higher Order Logic ◽  
Consequence Relation ◽  
Fourth Section ◽  
Second Order Logic

This article discusses canonical (i.e., full, or standard) second-order consequence and argues against it being a case of logical consequence. The discussion is divided into three parts. The first part comprises the first three sections. After stating the problem in Section 1, Sections 2 and 3 examine the role that the consequence relation is expected to play in axiomatic theories. This leads to put forward two requirements on logical consequence, which are called “formality” and “noninterference.” It is this last requirement that canonical second-order consequence violates, as the article sets out to substantiate. The fourth section argues that canonical second-order logic is inadequate for axiomatizing set theory, on the grounds that it codes a significant amount of set-theoretical content.

scholarly journals Local External/Internal Symmetry of Smooth Manifolds and Lack of Tovariance in Physics

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1429
Author(s):  
Torsten Asselmeyer-Maluga ◽  
Jerzy Król
Keyword(s):  
Set Theory ◽  
Intuitionistic Logic ◽  
Category Theory ◽  
Internal Symmetry ◽  
Smooth Manifolds ◽  
Natural Numbers ◽  
Local Invariance ◽  
Geometric Morphisms

Category theory allows one to treat logic and set theory as internal to certain categories. What is internal to SET is 2-valued logic with classical Zermelo–Fraenkel set theory, while for general toposes it is typically intuitionistic logic and set theory. We extend symmetries of smooth manifolds with atlases defined in Set towards atlases with some of their local maps in a topos T . In the case of the Basel topos and R 4 , the local invariance with respect to the corresponding atlases implies exotic smoothness on R 4 . The smoothness structures do not refer directly to Casson handless or handle decompositions, which may be potentially useful for describing the so far merely putative exotic R 4 underlying an exotic S 4 (should it exist). The tovariance principle claims that (physical) theories should be invariant with respect to the choice of topos with natural numbers object and geometric morphisms changing the toposes. We show that the local T -invariance breaks tovariance even in the weaker sense.

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