Category theory and set theory as theories about complementary types of universals

Instead of the half-century old foundational feud between set theory and category theory, this paper argues that they are theories about two different complementary types of universals. The set-theoretic antinomies forced naïve set theory to be reformulated using some iterative notion of a set so that a set would always have higher type or rank than its members. Then the universal u_{F}={x|F(x)} for a property F() could never be self-predicative in the sense of u_{F}∈u_{F}. But the mathematical theory of categories, dating from the mid-twentieth century, includes a theory of always-self-predicative universals--which can be seen as forming the "other bookend" to the never-self-predicative universals of set theory. The self-predicative universals of category theory show that the problem in the antinomies was not self-predication per se, but negated self-predication. They also provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato's Theory of Forms as well as for the idea of a "concrete universal" in Hegel and similar ideas of paradigmatic exemplars in ordinary thought

F.Smarandache

Florentin Smarandache is Professor Asociado de Matemática to the University of New Mexico in EE. Publicó more than 75 books and 100 artícuculos and notes of mathematician, physicist, philosophy, psychology, Literature, rebus. In mathematical his investigation it is in theory of numbers, not- Euclidean geometry, synthetic geometry, algebraic structures, statistic, neutrosofica logic and joint (generalization of blurred intuicionistica logic and joint respectively), neutrosofica probability (generalization of vague and classic probability). Also, few contributions to nuclear and physical of particles, information of fusion (it sees the theory of Dezert-Smarandache of reasonable reasoning paradojisto and, 2002), philosophy (it sees neutrosofía, a generaralización of dialectic), psychology (leya of sensations and stimuli), sociology (paradoxical sociologicales), linguistic (paradoxical semantic and tautología) etc. In Literature the movement established in 1980 Paradoxismo, based of the use excess of contradictions, antinomies, paradoxes in creatio

The Principle Of Excluded Middle Then And Now: Aristotle And Principia Mathematica

The prevailing truth-functional logic of the twentieth century, it is argued, is incapable of expressing the subtlety and richness of Aristotle's Principle of Excluded Middle, and hence cannot but misinterpret it. Furthermore, the manner in which truth-functional logic expresses its own Principle of Excluded Middle is less than satisfactory in its application to mathematics. Finally, there are glimpses of the "realism" which is the metaphysics demanded by twentieth century logic, with the remarkable consequent that Classical logic is a particularly inept instrument to analyze those philosophies which stand opposed to the "realism" it demands

Can contradictions be asserted?

In a universal logic containing naive semantics the semantic antinomies will be provable. Although being provable they are not assertible because of some pragmatic constraints on assertion I will argue for. Furthermore, since it is not acceptable that the thesis of dialethism is a dialethia itself, what it would be according to naive semantics and the prefered logical systems of dialethism, a corresponding restriction on proof theory is necessary

Semantyczna teoria prawdy a antynomie semantyczne [Semantic Theory of Truth vs. Semantic Antinomies]

The paper presents Alfred Tarski’s debate with the semantic antinomies: the basic Liar Paradox, and its more sophisticated versions, which are currently discussed in philosophy: Strengthen Liar Paradox, Cyclical Liar Paradox, Contingent Liar Paradox, Correct Liar Paradox, Card Paradox, Yablo’s Paradox and a few others. Since Tarski, himself did not addressed these paradoxes—neither in his famous work published in 1933, nor in later papers in which he developed the Semantic Theory of Truth—therefore, We try to defend his concept of truth against these antinomies. It is demonstrated that Tarskian theory of truth is resistant to the paradoxes and it is still the best solution to avoid the antinomies and remain within a classical logic, that is, accepting the laws of noncontradiction, excluded middle, and the principle of bivalence. Thus, the goal of the paper is double—firstly, to show that none of the versions of the Liar Paradox’s is a serious threat to Tarski’s concept of truth, and secondly, that Semantic Theory of Truth allows to remain within classical logic, and at the same time, avoid antinomies—which makes it the most attractive among classical theories of truth