The Notion of Logical Consequence in the Logic of Inexact Predicates

Cleave, John P.

doi:10.1002/malq.19740201903

Cited by 38 publications

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“…After some discussion, he proposes that the old truth values be understood in terms of "degrees of error in deviation from the truth". This is not very far from the idea of degrees of truth 11 . Moreover, since the intersection of all the sets [a) is the set [1) = {1}, it is clear that the theorems 12 of L α are the same as those of L α .…”

Section: Scott's Ideamentioning

confidence: 90%

“…10 He explains his proposal in the concrete case of 5-valued logic in [46], and of 6-valued logic in [47], but the idea is clearly the general one. 11 Another close interpretation is found in the framework of Rényi-Ulam games with lies, which interprets Lukasiewicz's truth values as measuring the distance from each value to the condition of falsifying too many answers (which means falsity); see [41]. 12 For semantically defined logics there is a tendency to speak of tautologies rather than theorems.…”

Section: Scott's Ideamentioning

confidence: 98%

“…A close proposal is that in [52], where it is suggested that a more refined notion of many-valued logic might be the family of consequences A, { i : i ∈ I} . According to (11), the sets D i play the role of "logical values". Therefore, when using these more general structures of degrees of truth, it is reasonable to say that the logical values correspond to the degrees of truth.…”

Section: Generalizingmentioning

confidence: 99%

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Taking Degrees of Truth Seriously

Font

2009

Stud Logica

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This is a contribution to the discussion on the role of truth degrees in manyvalued logics from the perspective of abstract algebraic logic. It starts with some thoughts on the so-called Suszko's Thesis (that every logic is two-valued) and on the conception of semantics that underlies it, which includes the truth-preserving notion of consequence. The alternative usage of truth values in order to define logics that preserve degrees of truth is presented and discussed. Some recent works studying these in the particular cases of Lukasiewicz's many-valued logics and of logics associated with varieties of residuated lattices are also presented. Finally the extension of this paradigm to other, more general situations is discussed, highlighting the need for philosophical or applied motivations in the selection of the truth degrees, due both to the interpretation of the idea of truth degree and to some mathematical difficulties.

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Section: Scott's Ideamentioning

confidence: 90%

Section: Scott's Ideamentioning

confidence: 98%

Section: Generalizingmentioning

confidence: 99%

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Taking Degrees of Truth Seriously

Font

2009

Stud Logica

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“…Also, GBL is regarded as a logic with strong negation. sequent calculi for the →-free parts of Nelson's logics N − and N + 6) presented in [1], and C be Cleave's logic [3] 7) . We can conclude the following fact 8) :…”

Section: Correspondence Between DC and Gj −mentioning

confidence: 99%

“…A constructive logic N − , which is a variant of logics with strong negation, was proposed by Almukdad and Nelson [1], who showed a relationship between constructive logics N − and N + , and Cleave's three-valued logic [3]. Logics such as N − , N and N + have been studied by many logicians and computer scientists (e. g. [9,11,12,14,15]).…”

Section: Introductionmentioning

confidence: 99%

A note on dual‐intuitionistic logic

Kamide

2003

Mathematical Logic Qtrly

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Quasi‐Boolean Algebras, Empirical Continuity and Three‐Valued Logic J. P. Cleave in Bristol (Great Britain)

Cleave¹

1976

Mathematical Logic Qtrly

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Z&chr. f. matb, Lol?i& und (hundlagcn d. M d h . Bd.82, S. 481-600 (1976) QUASI-B0OI;EAN ALGEBRAS, EMPIRICAL CONTINUITY AND THREE-VALUED LOGIC by J. P. CLEAVE in Bristol (Great Britain) rlnl Klg. 1We introduce QBA's by means of KORNER'S theory of inexact classes ([7] and [2]). Let A be a fixed domain of objects. By identifying classes with characteristic functions, a subclass of A is usually conceived as a function q~: A 4 (0, 1}, an element x of A being "in" the class if y(x) = 1 and excluded from the class if y(x) = 0. For the study of inexactness a third entity n can be introduced and an inexact class (or, rather, its characteristic function) conceived as a function q~: A --f (0, n, 1); x is in the class if ~( x ) = 1, excluded from the class if ~( x ) = 0, and is a neutral c m e if ~( x ) = 11. The subclasses of A form a Boolean algebra, C ( A ) , under the operat,ions of union, intersection and complementation : these operations have also been defined for inexact classes ([2], [7]) and their definitions can be rephrased in the following way. Let K, = ((0, n, 1); 0, 1, W, n, ') where the functions W, n, ' are defined by the tables shown in Fig. 1. Then the operations of union W, intersection n, and complementation -, on (characteristic functions of) inexact classes 9, p are defined byfor all a E A . (In [2] we used the integers -1 , 0 , + 1 instead of 0, n, 1 ; f , u , t are used in [7].) As a technical convenience we introduce here the universal class, 1, and the empty class 0 defined by l(a) = 1 and O(a) = 0 for all a E

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The Notion of Logical Consequence in the Logic of Inexact Predicates

Cited by 38 publications

References 0 publications

Taking Degrees of Truth Seriously

Taking Degrees of Truth Seriously

A note on dual‐intuitionistic logic

Quasi‐Boolean Algebras, Empirical Continuity and Three‐Valued Logic J. P. Cleave in Bristol (Great Britain)

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