Reconsidering Kant’s Rejection of Indirect Arguments in Transcendental Philosophy

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 Mathematical Antinomies Resolved

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.

Wright and Revisionism

Do considerations in the theory of meaning pose a challenge to classical logic, and in particular to the law of excluded middle? Michael Dummett suggested an affirmative answer to this question, and advocated a form of logical revisionism. In his 1981 study “Anti-Realism and Revisionism,” Crispin Wright developed a critique of Dummett’s case for logical revisionism, but in more recent work (e.g., his 1992 book Truth and Objectivity), Wright has advanced an argument in favour of logical revisionism. This chapter investigates the nature and limitations of anti-realist revisionism, and offers a critique of Wright’s arguments in favour of logical revisionism. It also develops an alternative proposal about how revisionism might proceed.

Negation and Opposition

The treatment of negation has long been linked to the treatment of opposition between propositions (or sentences) and between terms (or subsentential constituents). The primary types of opposition, usefully displayed on the post-Aristotelian Square of Opposition, are contradiction (two contradictories always differ in truth value) and contrariety (two contraries can both be false, but not both be true). The law of non-contradiction governs both oppositions, while the law of excluded middle applies only to contradictories. In principle, Aristotle’s semantic category of contradictory opposition lines up with the syntactic category of sentence (vs. constituent) negation, but in practice matters are more complicated, and while Klima’s diagnostics are helpful they are often not decisive. These complications are illustrated by the distribution of affixal negation, the phenomenon of logical double negation, the interaction of negation with quantifiers and modals, and the tendency for formal contradictory negation to be pragmatically strengthened to contrariety.

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

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.

Infinitesimals, Nations, and Persons

I compare three sorts of case in which philosophers have argued that we cannot assert the Law of Excluded Middle for statements of identity. Adherents of Smooth Infinitesimal Analysis deny that Excluded Middle holds for statements saying that an infinitesimal is identical with zero. Derek Parfit contended that, in certain sci-fi scenarios, the Law does not hold for some statements of personal identity. He also claimed that it fails for the statement ‘England in 1065 was the same nation as England in 1067’. I argue that none of these cases poses a serious threat to Excluded Middle. My analysis of the last example casts doubt on the principle of the Determinacy of Distinctness. While David Wiggins's ‘conceptualist realism’ provides a metaphysics which can dispense with that principle, it leaves no house-room for infinitesimals.

An enquiry into the ontological and logical foundations of sustainability: Toward a conceptual integration of the interface ‘Nature/Humanity’

Non-technical summary Implementing the logical and ontological principles (dualism, mechanicism, reductionism, law of excluded middle, etc.) of modernity has brought forth an unsustainable world. An overcoming of these principles is proposed by mesology (Umweltlehre, fûdoron), centring on the concept of trajection and the existential operator as (als, en tant que).

An enquiry into the ontological and logical foundations of sustainability: Toward a conceptual integration of the interface ‘Nature/Humanity’

Non-technical summary Implementing the logical and ontological principles (dualism, mechanicism, reductionism, law of excluded middle, etc.) of modernity has brought forth an unsustainable world. An overcoming of these principles is proposed by mesology (Umweltlehre, fûdoron), centring on the concept of trajection and the existential operator as (als, en tant que).