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.

scholarly journals Truth and Contradiction in Aristotle’s De Interpretatione 6-9

Phronesis ◽  
2010 ◽  
Vol 55 (1) ◽  
pp. 26-67 ◽  
Author(s):  
Russell E. Jones
Keyword(s):  
The Other ◽  
The Subject ◽  
Law Of Excluded Middle ◽  
The Law ◽  
De Interpretatione ◽  
Excluded Middle

AbstractIn De Interpretatione 6-9, Aristotle considers three logical principles: the principle of bivalence, the law of excluded middle, and the rule of contradictory pairs (according to which of any contradictory pair of statements, exactly one is true and the other false). Surprisingly, Aristotle accepts none of these without qualification. I offer a coherent interpretation of these chapters as a whole, while focusing special attention on two sorts of statements that are of particular interest to Aristotle: universal statements not made universally and future particular statements. With respect to the former, I argue that Aristotle takes them to be indeterminate and so to violate the rule of contradictory pairs. With respect to the latter, the subject of the much discussed ninth chapter, I argue that the rule of contradictory pairs, and not the principle of bivalence, is the focus of Aristotle’s refutation. Nevertheless, Aristotle rejects bivalence for future particular statements.

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.]

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 Third Antinomy Resolved

2021 ◽  
pp. 299-324
Author(s):  
Ian Proops
Keyword(s):  
Moral Responsibility ◽  
Transcendental Idealism ◽  
Human Freedom ◽  
The Other ◽  
Moral Argument ◽  
The Third ◽  
Moral Growth ◽  
Indirect Argument

This chapter examines the resolution of the third antinomy. Kant argues that the thesis and antithesis are (roughly speaking) sub-contraries rather than contradictories. However, the sense in which he maintains that the thesis and antithesis ‘can both be true’ is delicate. He holds that the truth of neither claim excludes the truth of the other; but this is compatible with necessary falsehood of the thesis, which affirms the existence of human freedom. Importantly, Kant does not take himself to show on theoretical grounds that freedom is even logically possible. The chapter also discusses: Kant’s conceptions of intelligible causality and of empirical and intelligible character; moral responsibility; moral growth; the rationality of blame; Kant’s criticisms of Leibniz’s compatibilism; the third antinomy as an indirect argument for Transcendental Idealism; and the first-Critique’s version of a moral argument for freedom. Kant emerges as a ‘soft determinist’ of a highly unusual stripe.

scholarly journals Breaking Up the Unity of the World: Peter Fitzpatrick’s Conception of Responsive Law

2020 ◽  
pp. 174387212097533
Author(s):  
Johan van der Walt
Keyword(s):  
Rule Of Law ◽  
The Other ◽  
Short Article ◽  
The Rule Of Law ◽  
Other Hand ◽  
The World ◽  
The Law ◽  
The One ◽  
The Ideal

This short article on Peter Fitzpatrick’s conception of “responsive law” analyzes the ambiguous temporality that Fitzpatrick discerned in modern law. On the one hand, law makes the claim of being fully present and therefore already and completely contained in itself. This aspect of law reflects the law’s claim to “immanence,” that is, its claim of always being able to rely strictly on its own operational terms without having to take recourse to any consideration not already contained within itself. It is this aspect of law that renders the ideal of the “rule of law” feasible. On the other hand, the law’s claim to doing justice to every unique and therefore every new case also demands that it takes leave of that which is already settled within it. This aspect of law can be called its “imminence.” The imminence of the law concerns the reality that law always finds itself on the threshold of that which has not yet been said and must still be said. The article shows how Fitzpatrick relied on Freud’s concept of the totem to explain the “wondrous” unity of its immanence and imminence.

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

Do Easterners and Westerners Treat Contradiction Differently?

2015 ◽  
Vol 15 (1-2) ◽  
pp. 45-63 ◽  
Author(s):  
Hugo Mercier ◽  
Jiehai Zhang ◽  
Yuping Qu ◽  
Peng Lu ◽  
Jean-Baptiste Van der Henst
Keyword(s):  
Cultural Differences ◽  
External Validity ◽  
Cross Cultural ◽  
The Other ◽  
Cross Cultural Differences ◽  
Middle Way ◽  
Points Of View ◽  
Law Of Excluded Middle ◽  
Excluded Middle ◽  
Influential Theory

Peng and Nisbett (1999) put forward an influential theory of the influence of culture on the resolution of contradiction. They suggested that Easterners deal with contradiction in a dialectical manner, trying to reconcile opposite points of view and seeking a middle-way. Westerners, by contrast, would follow the law of excluded middle, judging one side of the contradiction to be right and the other to be wrong. However, their work has already been questioned, both in terms of replicability and external validity. Here we test alternative interpretations of two of Peng and Nisbett’s experiments and conduct a new test of their theory in a third experiment. Overall, the Eastern (Chinese) and Western (French) participants behaved similarly, failing to exhibit the cross-cultural differences observed by Peng and Nisbett. Several interpretations of these failed replications and this failed new test are suggested. Together with previous failed replications, the present results raise questions about the breadth of Peng and Nisbett’s interpretation of cross-cultural differences in dealing with contradiction.

Sign in / Sign up

Export Citation Format

Share Document