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Intuitionistic Logic (intuitionistic + logic)

Distribution by Scientific Domains

Information Science and Computing	50%
Humanities and Social Sciences	50%

Selected Abstracts

On open-set lattices and some of their applications in semantics

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, Issue 12 2003
Mouw-Ching Tjiok
In this article, we present the theory of Kripke semantics, along with the mathematical framework and applications of Kripke semantics. We take the Kripke-Sato approach to define the knowledge operator in relation to Hintikka's possible worlds model, which is an application of the semantics of intuitionistic logic and modal logic. The applications are interesting from the viewpoint of agent interactives and process interaction. We propose (i) an application of possible worlds semantics, which enables the evaluation of the truth value of a conditional sentence without explicitly defining the operator "," (implication), through clustering on the space of events (worlds) using the notion of neighborhood; and (ii) a semantical approach to treat discrete dynamic process using Kripke-Beth semantics. Starting from the topological approach, we define the measure-theoretical machinery, in particular, we adopt the methods developed in stochastic process,mainly the martingale,to our semantics; this involves some Boolean algebraic (BA) manipulations. The clustering on the space of events (worlds), using the notion of neighborhood, enables us to define an accessibility relation that is necessary for the evaluation of the conditional sentence. Our approach is by taking the neighborhood as an open set and looking at topological properties using metric space, in particular, the so-called ,-ball; then, we can perform the implication by computing Euclidean distance, whenever we introduce a certain enumerative scheme to transform the semantic objects into mathematical objects. Thus, this method provides an approach to quantify semantic notions. Combining with modal operators Ki operating on E set, it provides a more-computable way to recognize the "indistinguishability" in some applications, e.g., electronic catalogue. Because semantics used in this context is a local matter, we also propose the application of sheaf theory for passing local information to global information. By looking at Kripke interpretation as a function with values in an open-set lattice ,,U, which is formed by stepwise verification process, we obtain a topological space structure. Now, using the measure-theoretical approach by taking the Borel set and Borel function in defining measurable functions, this can be extended to treat the dynamical aspect of processes; from the stochastic process, considered as a family of random variables over a measure space (the probability space triple), we draw two strong parallels between Kripke semantics and stochastic process (mainly martingales): first, the strong affinity of Kripke-Beth path semantics and time path of the process; and second, the treatment of time as parametrization to the dynamic process using the technique of filtration, adapted process, and progressive process. The technique provides very effective manipulation of BA in the form of random variables and ,-subalgebra under the cover of measurable functions. This enables us to adopt the computational algorithms obtained for stochastic processes to path semantics. Besides, using the technique of measurable functions, we indeed obtain an intrinsic way to introduce the notion of time sequence. © 2003 Wiley Periodicals, Inc. [source]

Psychologism Revisited in Logic, Metaphysics, and Epistemology

METAPHILOSOPHY, Issue 3 2001
Dale Jacquette
Psychologism is a philosophical ideology that seeks to explain the principles of logic, metaphysics, and epistemology as psychological phenomena. Psychologism has been the storm center of concerted criticisms since the nineteenth century, and is thought by many to have been refuted once and for all by Kant, Frege, Husserl, and others. The project of accounting for objective philosophical or mathematical truths in terms of subjective psychological states has been largely discredited in mainstream analytic thought. Ironically, psychologism has resurfaced in unexpected guises in the form of intuitionistic logic and mathematics, cognitivism, and naturalized epistemology. I examine some of the principal objections to psychologism , distinguishing roughly between good and bad or philosophically acceptable versus unacceptable psychologism , and consider the extent to which a new wave of psychologism may be gaining prominence in contemporary philosophy, and the light its successes and failures may shed on the original concept and underlying perspective of classical psychologism. [source]

Intuitionistic notions of boundedness in ,

MLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 1 2009
Fred Richman
Abstract We consider notions of boundedness of subsets of the natural numbers , that occur when doing mathematics in the context of intuitionistic logic. We obtain a new characterization of the notion of a pseudobounded subset and we formulate the closely related notion of a detachably finite subset. We establish metric equivalents for a subset of , to be detachably finite and to satisfy the ascending chain condition. Following Ishihara, we spell out the relationship between detachable finiteness and sequential continuity. Most of the results do not require countable choice. (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Constructive Logic, Truth and Warranted Assertability

THE PHILOSOPHICAL QUARTERLY, Issue 205 2001
Greg Restall
Shapiro and Taschek have argued that simply using intuitionistic logic and its Heyting semantics, one can show that there are no gaps in warranted assertability. That is, given that a discourse is faithfully modelled using Heyting's semantics for the logical constants, then if a statement S is not warrantedly assertable, its negation ,S is. Tennant has argued for this conclusion on similar grounds. I show that these arguments fail, albeit in illuminating ways. An appeal to constructive logic does not commit one to this strong epistemological thesis, but appeals to semantics of intuitionistic logic none the less do give us certain conclusions about the connections between warranted assertability and truth. [source]