Predicativity and parametric polymorphism of Brouwerian implication

A common objection to the definition of intuitionistic implication in the Proof Interpretation is that it is impredicative. I discuss the history of that objection, argue that in Brouwer's writings predicativity of implication is ensured through parametric polymorphism of functions on species, and compare this construal with the alternative approaches to predicative implication of Goodman, Dummett, Prawitz, and Martin-L\"of.Comment: Added further references (Pistone, Poincar\'e, Tabatabai, Van Atten

Semantic and Mathematical Foundations for Intuitionism

Thesis (Ph.D.) - Indiana University, Philosophy, 2013My dissertation concerns the proper foundation for the intuitionistic mathematics whose development began with L.E.J. Brouwer's work in the first half of the 20th Century. It is taken for granted by most philosophers, logicians, and mathematicians interested in foundational questions that intuitionistic mathematics presupposes a special, proof-conditional theory of meaning for mathematical statements. I challenge this commonplace. Classical mathematics is very successful as a coherent body of theories and a tool for practical application. Given this success, a view like Dummett's that attributes a systematic unintelligibility to the statements of classical mathematicians fails to save the relevant phenomena. Furthermore, Dummett's program assumes that his proposed semantics for mathematical language validates all and only the logical truths of intuitionistic logic. In fact, it validates some intuitionistically invalid principles, and given the lack of intuitionistic completeness proofs, there is little reason to think that every intuitionistic logical truth is valid according to his semantics. In light of the failure of Dummett's foundation for intuitionism, I propose and carry out a reexamination of Brouwer's own writings. Brouwer is frequently interpreted as a proto-Dummettian about his own mathematics. This is due to excessive emphasis on some of his more polemical writings and idiosyncratic philosophical views at the expense of his distinctively mathematical work. These polemical writings do not concern mathematical language, and their principal targets are Russell and Hilbert's foundational programs, not the semantic principle of bivalence. The failures of these foundational programs has diminished the importance of Brouwer's philosophical writings, but his work on reconstructing mathematics itself from intuitionistic principles continues to be worth studying. When one studies this work relieved of its philosophical burden, it becomes clear that an intuitionistic mathematician can make sense of her mathematical work and activity without relying on special philosophical or linguistic doctrines. Core intuitionistic results, especially the invalidity of the logical principle tertium non datur, can be demonstrated from basic mathematical principles; these principles, in turn, can be defended in ways akin to the basic axioms of other mathematical theories. I discuss three such principles: Brouwer's Continuity Principle, the Principle of Uniformity, and Constructive Church's Thesis

Subformula and separation properties in natural deduction via small Kripke models

Various natural deduction formulations of classical, minimal, intuitionist, and intermediate propositional and first-order logics are presented and investigated with respect to satisfaction of the separation and subformula properties. The technique employed is, for the most part, semantic, based on general versions of the Lindenbaum and Lindenbaum–Henkin constructions. Careful attention is paid (i) to which properties of theories result in the presence of which rules of inference, and (ii) to restrictions on the sets of formulas to which the rules may be employed, restrictions determined by the formulas occurring as premises and conclusion of the invalid inference for which a counterexample is to be constructed. We obtain an elegant formulation of classical propositional logic with the subformula property and a singularly inelegant formulation of classical first-order logic with the subformula property, the latter, unfortunately, not a product of the strategy otherwise used throughout the article. Along the way, we arrive at an optimal strengthening of the subformula results for classical first-order logic obtained as consequences of normalization theorems by Dag Prawitz and Gunnar Stalmarck

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

A Dempster-Shafer theory inspired logic.

Issues of formalising and interpreting epistemic uncertainty have always played a prominent role in Artificial Intelligence. The Dempster-Shafer (DS) theory of partial beliefs is one of the most-well known formalisms to address the partial knowledge. Similarly to the DS theory, which is a generalisation of the classical probability theory, fuzzy logic provides an alternative reasoning apparatus as compared to Boolean logic. Both theories are featured prominently within the Artificial Intelligence domain, but the unified framework accounting for all the aspects of imprecise knowledge is yet to be developed. Fuzzy logic apparatus is often used for reasoning based on vague information, and the beliefs are often processed with the aid of Boolean logic. The situation clearly calls for the development of a logic formalism targeted specifically for the needs of the theory of beliefs. Several frameworks exist based on interpreting epistemic uncertainty through an appropriately defined modal operator. There is an epistemic problem with this kind of frameworks: while addressing uncertain information, they also allow for non-constructive proofs, and in this sense the number of true statements within these frameworks is too large. In this work, it is argued that an inferential apparatus for the theory of beliefs should follow premises of Brouwer's intuitionism. A logic refuting tertium non daturìs constructed by defining a correspondence between the support functions representing beliefs in the DS theory and semantic models based on intuitionistic Kripke models with weighted nodes. Without addional constraints on the semantic models and without modal operators, the constructed logic is equivalent to the minimal intuitionistic logic. A number of possible constraints is considered resulting in additional axioms and making the proposed logic intermediate. Further analysis of the properties of the created framework shows that the approach preserves the Dempster-Shafer belief assignments and thus expresses modality through the belief assignments of the formulae within the developed logic

Constructive Criticism

Attempts to attain knowledge as certified true belief have failed to circumvent Hume’s injunction against induction. Theories must be viewed as unprovable, improbable, and undisprovable. The empirical basis is fallible, and yet the method of conjectures and refutations is untouched by Hume’s insights. The implications for statistical methodology is that the requisite severity of testing is achieved through the use of robust procedures, whose assumptions have not been shown to be substantially violated, to test predesignated range null hypotheses. Nonparametric range null hypothesis tests need to be developed to examine whether or not effect sizes or measures of association, as well as distributional assumptions underlying the tests themselves, meet satisficing criteria