scholarly journals Negación dialéctica y lógica transitiva (I)

1983 ◽  
Vol 15 (43) ◽  
pp. 51-78
Author(s):  
Lorenzo Peña
Keyword(s):  
Mathematical Logic ◽  
Fuzzy Sets ◽  
Paraconsistent Logic ◽  
The World ◽  
Recent Developments ◽  
Law Of Excluded Middle ◽  
The Law ◽  
In Virtue Of ◽  
Excluded Middle

In this essay I bring up the issue of how to deal with dialectical views -especialIy with dialectic negation- from the standpoint of a transitive logic, which is a particular paraconsistent logic. After briefly tracing the development of the debate between dialectic thinkers and those who, hewing to entrenched logical theories, did out of hand reject any contradictorial proposal -up to recent developments of paraconsistent systems of mathematical logic- I canvass a variety of grounds shoring up the thesis of the contradictoriality of the world. Chief among them is fuzziness. The paper tries to show that fuzziness has nothing to do with uncertainty, and that accepting fuzzy sets and facts not only does not compel us to waive the law of excluded middle, but -on the base of reasonable presuppositions- entails recognition of that law's relevant instances -the ones that purportedly ought to be dropped as true sentences, should fuzziness be acknowledged. True enough, fuzziness plus excluded middle leads to contradiction, i.e. to negation inconsistency. But then fuzziness is -or can he viewed as being- negation inconsistency, since a fuzzy situation is one wherein something neither is nor fails to be the case: which -in virtue of involutivity of simple negation and De Morgan laws- means that something both ia and yet is not the case. [L.P.]

scholarly journals Exploring the Axiom of Excluded Middle and Axiom of Ontradiction in Fuzzy Sets

2019 ◽  
Vol 9 (1) ◽  
Keyword(s):  
Fuzzy Sets ◽  
Membership Functions ◽  
Law Of Excluded Middle ◽  
The Law ◽  
Excluded Middle

Fuzzy sets are considered as a fine extension of classical sets (crisp) in which the elements possess diverse degrees of membership functions. Zadeh is the initiator of fuzzy sets that predominantly deal with imprecision and vagueness. In this paper, the Law of Excluded middle and the Law of Contradiction were discussed in an exemplary mode. In addition to that the definitions of fuzzy sets, crisp sets and the various operations on them were presented in a consecutive manner.

The Mathematical Antinomies Resolved

2021 ◽  
pp. 245-276
Author(s):  
Ian Proops
Keyword(s):  
Singular Term ◽  
Transcendental Idealism ◽  
The Other ◽  
The World ◽  
Law Of Excluded Middle ◽  
The Law ◽  
Excluded Middle ◽  
Indirect Argument

This chapter identifies two lines of resolution in the mathematical antinomies, which lines, it argues, correspond to two traditional ways of attempting to generate counter-examples to the law of excluded middle. One line involves positing an instance of category clash, the other the suggestion that ‘the world’ is a non-referring singular term. The upshot, in either case, is that the thesis and antithesis are not contradictories but merely contraries (and both are false). The chapter criticizes, and then charitably reformulates, Kant’s indirect argument for Transcendental Idealism. It considers why Kant did not seek to resolve the antinomies by arguing that thesis or antithesis are nonsense. Also discussed are: reductio proofs in philosophy (and Kant’s attitude toward them, which is argued to be more sympathetic than is often supposed), regresses ad infinitum and ad indefinitum; the cosmological syllogism; the sceptical representation; the Lambert analogy, the indifferentists; and the comparison with Zeno.

Reconsidering Kant’s Rejection of Indirect Arguments in Transcendental Philosophy

2021 ◽  
pp. 1-19
Author(s):  
Marcel Buß
Keyword(s):  
Immanuel Kant ◽  
Explanatory Power ◽  
Transcendental Philosophy ◽  
Point Of View ◽  
Transcendental Arguments ◽  
Mathematical Proofs ◽  
Law Of Excluded Middle ◽  
The Law ◽  
Excluded Middle ◽  
Insight Into

Abstract Immanuel Kant states that indirect arguments are not suitable for the purposes of transcendental philosophy. If he is correct, this affects contemporary versions of transcendental arguments which are often used as an indirect refutation of scepticism. I discuss two reasons for Kant’s rejection of indirect arguments. Firstly, Kant argues that we are prone to misapply the law of excluded middle in philosophical contexts. Secondly, Kant points out that indirect arguments lack some explanatory power. They can show that something is true but they do not provide insight into why something is true. Using mathematical proofs as examples, I show that this is because indirect arguments are non-constructive. From a Kantian point of view, transcendental arguments need to be constructive in some way. In the last part of the paper, I briefly examine a comment made by P. F. Strawson. In my view, this comment also points toward a connection between transcendental and constructive reasoning.

The Machinery of International Law and Democratic Backsliding

2021 ◽  
pp. 136-146
Author(s):  
Tom Ginsburg
Keyword(s):  
Political Participation ◽  
International Law ◽  
Term Limits ◽  
Constitutional Court ◽  
Constitutional Amendments ◽  
The World ◽  
International Rights ◽  
Recent Developments ◽  
The Law ◽  
The Right

This chapter focuses on the abuse of international rights to political participation so as to facilitate a leader's remaining in office beyond the constitutionally mandated term. This involves not only the abuse of the interpretation of rights, but also the abuse of the doctrine of unconstitutional constitutional amendments, which has spread around the world in recent years. How does this happen and what, if anything, can international law do about it? After introducing a motivating case — the famous decision of the Colombian Constitutional Court in the second re-election decision, in which courts stood for the protection of democracy — the chapter examines recent 'bad' cases in which rights and constitutional amendments are abused to extend leaders' terms. It surveys recent developments in the law of term limits, and briefly proposes a normative interpretation of the right to political participation which ought to be consistent with the emerging doctrine. The chapter suggests that there is an emerging consensus, at least in some regions of the world, that there are limits in states' ability to modify term limits unconditionally.

On Weakening the Law of Excluded Middle

Dialogue ◽  
1966 ◽  
Vol 5 (2) ◽  
pp. 232-236
Author(s):  
Douglas Odegard
Keyword(s):  
Law Of Excluded Middle ◽  
The Law ◽  
Excluded Middle

Let us use ‘false’ and ‘not true’ (and cognates) in such a way that the latter expression covers the broader territory of the two; in other words, a statement's falsity implies its non-truth but not vice versa. For example, ‘John is ill’ cannot be false without being nontrue; but it can be non-true without being false, since it may not be true when ‘John is not ill’ is also not true, a situation we could describe by saying ‘It is neither the case that John is ill nor the case that John is not ill.’

The Law of Excluded Middle

Mind ◽  
1978 ◽  
Vol LXXXVII (2) ◽  
pp. 161-180 ◽  
Author(s):  
NEIL COOPER
Keyword(s):  
Law Of Excluded Middle ◽  
The Law ◽  
Excluded Middle

Frege's theorem in a constructive setting

1999 ◽  
Vol 64 (2) ◽  
pp. 486-488 ◽  
Author(s):  
John L. Bell
Keyword(s):  
Set Theory ◽  
Natural Numbers ◽  
The Family ◽  
Power Set ◽  
Law Of Excluded Middle ◽  
The Law ◽  
Image Position ◽  
Excluded Middle ◽  
Intuitionistic Set Theory

By Frege's Theorem is meant the result, implicit in Frege's Grundlagen, that, for any set E, if there exists a map υ from the power set of E to E satisfying the conditionthen E has a subset which is the domain of a model of Peano's axioms for the natural numbers. (This result is proved explicitly, using classical reasoning, in Section 3 of [1].) My purpose in this note is to strengthen this result in two directions: first, the premise will be weakened so as to require only that the map υ be defined on the family of (Kuratowski) finite subsets of the set E, and secondly, the argument will be constructive, i.e., will involve no use of the law of excluded middle. To be precise, we will prove, in constructive (or intuitionistic) set theory, the followingTheorem. Let υ be a map with domain a family of subsets of a set E to E satisfying the following conditions:(i) ø ϵdom(υ)(ii)∀U ϵdom(υ)∀x ϵ E − UU ∪ x ϵdom(υ)(iii)∀UV ϵdom(5) υ(U) = υ(V) ⇔ U ≈ V.Then we can define a subset N of E which is the domain of a model of Peano's axioms.

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