Intuitionism and the Modal Logic of Vagueness

Intuitionistic logic provides an elegant solution to the Sorites Paradox. Its acceptance has been hampered by two factors. First, the lack of an accepted semantics for languages containing vague terms has led even philosophers sympathetic to intuitionism to complain that no explanation has been given of why intuitionistic logic is the correct logic for such languages. Second, switching from classical to intuitionistic logic, while it may help with the Sorites, does not appear to offer any advantages when dealing with the so-called paradoxes of higher-order vagueness. We offer a proposal that makes strides on both issues. We argue that the intuitionist’s characteristic rejection of any third alethic value alongside true and false is best elaborated by taking the normal modal system S4M to be the sentential logic of the operator ‘it is clearly the case that’. S4M opens the way to an account of higher-order vagueness which avoids the paradoxes that have been thought to infect the notion. S4M is one of the modal counterparts of the intuitionistic sentential calculus and we use this fact to explain why IPC is the correct sentential logic to use when reasoning with vague statements. We also show that our key results go through in an intuitionistic version of S4M. Finally, we deploy our analysis to reply to Timothy Williamson’s objections to intuitionistic treatments of vagueness

An Objection to Naturalism and Atheism from Logic

I proffer a success argument for classical logical consequence. I articulate in what sense that notion of consequence should be regarded as the privileged notion for metaphysical inquiry aimed at uncovering the fundamental nature of the world. Classical logic breeds necessitism. I use necessitism to produce problems for both ontological naturalism and atheism

Introducing choice sequences into mathematical ontology

Tese de mestrado, Filosofia (Epistemologia e Metafísica), Universidade de Lisboa, Faculdade de Letras, 2012A ideia de objectos matemáticos que estão em permanente desenvolvimento no tempo foi pela primeira vez avançada por L.E.J. Brouwer. Na matemática intuicionista estes objectos são concebidos como sequência infinitas de números naturais que em qualquer estágio do seu crescimento têm apenas um número finito de valores, além disso, tais valores podem ser livremente escolhidos, no sentido em que a sua produção não necessita de ser determinada por nenhuma regra matemática definida. Tais objectos são denominados de sequências de escolha. O presente trabalho tem como objectivo fornecer uma resposta à sequinte questão: são as sequências de escolha legítimos objectos matemáticos? A resposta que iremos propor e à qual iremos argumentar favoravelmente é a seguinte: tais objectos não podem ser considerados objectos matemáticos legítimos. Com esta tese em vista, iremos discutir as propriedades intrínsecas das sequências de escolha relativamente à maneira como são incorporadas no contexto matemático e as suas implicações. Seguindo esta metodologia pretendemos atingir um cabal entendimento filosófico das consequências em que incorremos ao aceitarmos sequências de escolha como objectos da ontologia matemática e das razões que temos para não as aceitarmos como tal.Abstract: The idea of mathematical objects which are in a permanent state of growth in time was by the first time defended by L.E.J. Brouwer. In intuitionistic mathematics these objects are conceived as infinite sequences of natural numbers that at any stage of growth have only finitely many values defined. Additionally, these values may be freely chosen, in the sense that their generation has not to follow any determinate mathematical rule. These objects are called choice sequences. The present work aims at providing the answer to the following question: are choice sequences legitimate mathematical objects? The answer that we will propose and argue for is a negative one: that they cannot be considered legitimate mathematical objects. In order to do this we will discuss the intrinsic features of choice sequences concerning the way they are incorporated into a mathematical framework and their implications. Following this methodology we expect to achieve a good philosophical understanding of the consequences of accepting choice sequences into our mathematical ontology and of the reasons that we have not to accept them as such

Infinity

This essay surveys the different types of infinity that occur in pure and applied mathematics, with emphasis on: 1. the contrast between potential infinity and actual infinity; 2. Cantor's distinction between transfinite sets and absolute infinity; 3. the constructivist view of infinite quantifiers and the meaning of constructive proof; 4. the concept of feasibility and the philosophical problems surrounding feasible arithmetic; 5. Zeno's paradoxes and modern paradoxes of physical infinity involving supertasks

Nature of Philosophy

The aim of this paper is to examine the nature, scope and importance of philosophy in the light of its relation to other disciplines. This work pays its focus on the various fundamental problems of philosophy, relating to Ethics, Metaphysics, Epistemology Logic, and its association with scientific realism. It will also highlight the various facets of these problems and the role of philosophers to point out the various issues relating to human issues. It is widely agreed that philosophy as a multi-dimensional subject that shows affinity to others branches of philosophy like, Philosophy of Science, Humanities, Physics and Mathematics, but this paper also seeks, a philosophical nature towards the universal problems of nature. It evaluates the contribution and sacrifices of the great sages of philosophers to promote the clarity and progress in the field of philosophy