The aim of this paper is to reconstruct Brouwer’s justification for the intuitionistic revision of logic and mathematics. It is attempted to show that pivotal premisses of his argument are supplied by his philosophy. To this end, the basic tenets of his philosophical doctrine are discussed: the concepts of mind, causal attention, intuition of two-ity and his repudiation of realism.The restriction of intuitionistically allowable objects to spreads and species is traced back to Brouwer’s concept of intuition that is a defining (...) feature of his notion of mind. On the other hand, it is argued that his objections to some laws of classical logic result from the rejection of the rule of double negation elimination, which in turn follows from both, the claim that rules of logic should preserve evidence for assertions rather than truth, and too restrictive a concept of evidence. (shrink)

Infinity and negation are in various relations and interdependencies one to another. The analysis of negation and infinity aims to better understanding them. Semantical, syntactical, and pragmatic issues will be considered.

Strictly speaking, intuitionistic logic is not a modal logic. There are, after all, no modal operators in the language. It is a subsystem of classical logic, not [like modal logic] an extension of it. But... (thus Fitting, p. 437, trying to justify inclusion of a large chapter on intuitionist logic in a book that is largely about modal logics).

Locatedness is one of the fundamental notions in constructive mathematics. The existence of a positivity predicate on a locale, i.e. the locale being overt, or open, has proved to be fundamental in constructive locale theory. We show that the two notions are intimately connected.Bishop defines a metric space to be compact if it is complete and totally bounded. A subset of a totally bounded set is again totally bounded iff it is located. So a closed subset of a Bishop compact (...) set is Bishop compact iff it is located. We translate this result to formal topology. ‘Bishop compact’ is translated as compact and overt. We propose a definition of locatedness on subspaces of a formal topology, and prove that a closed subspace of a compact regular formal space is located iff it is overt. Moreover, a Bishop-closed subset of a complete metric space is Bishop compact — that is, totally bounded and complete — iff its localic completion is compact overt.Finally, we show by elementary methods that the points of the Vietoris locale of a compact regular locale are precisely its compact overt sublocales.We work constructively, predicatively and avoid the use of the axiom of countable choice. (shrink)

[The paper is in Polish, an English abstract is given only for information.] This article is intended for philosophers and logicians as a short partial introduction to category theory and its peculiar connection with logic. First, we consider CT itself. We give a brief insight into its history, introduce some basic definitions and present examples. In the second part, we focus on categorical topos semantics for propositional logic. We give some properties of logic in toposes, which, in general, is an (...) intuitionistic logic. We next present two families of toposes whose tautologies are identical with those of classical propositional logic. The relatively extensive bibliography is given in order to support further studies. (shrink)

This paper studies an economy whose agents perceive their consumption possibilities subjectively, and whose preferences are defined on what they subjectively experience, rather than on those alternatives that are objectively present. The model of agents' perceptions is based on intuitionistic logic. Roughly, this means that agents reason constructively: a solution to a problem exists only if there is a construction by which the problem can be solved. The theorems that can be proved determine how an agent perceives a set of (...) alternatives. A dual model relates perceived alternatives to a shared language, which the agents use in trading. So perceptions relate objective alternatives to an agent's subjective view of them, and reporting dually relates an agent's subjective world to a shared language. It turns out that an appropriately modified notion of competitive equilibrium always exists. However, in contrast with standard results in economic theory, competitive equilibrium need not be efficient. (shrink)