This article studies the mathematical properties of two systems that model Aristotle's original syllogistic and the relationship obtaining between them. These systems are Corcoran's natural deduction syllogistic and ?ukasiewicz's axiomatization of the syllogistic. We show that by translating the former into a first-order theory, which we call T RD, we can establish a precise relationship between the two systems. We prove within the framework of first-order logic a number of logical properties about T RD that bear upon the same properties (...) of the natural deduction counterpart ? that is, Corcoran's system. Moreover, the first-order logic framework that we work with allows us to understand how complicated the semantics of the syllogistic is in providing us with examples of bizarre, unexpected interpretations of the syllogistic rules. Finally, we provide a first attempt at finding the structure of that semantics, reducing the search to the characterization of the class of models of T RD. (shrink)

In the paper, Jan Łukasiewicz’s program of the logicization of philosophy is presented and discussed. Łukasiewicz, known mostly for his invention of trivalent logic as well as his achievements in propositional calculus and metalogic, had always been concerned with the methodological condition of philosophy. He finally found “the measure of exactness” in mathematical logic. According to him, only the use of logical tools may provide philosophical investigations with an appropriate level of exactness. He expressed his views most firmly and directly (...) in the paper, “A call for the method of Philosophy”. Łukasiewicz proposed giving philosophical theories the form of axiomatic systems by indicating the primitive terms of their language, selecting suitable axioms, and explicitly determining the applied rules of inference. All the theses of these systems should be consequences of the accepted axioms and confronted with the data of experience and the results of science. Łukasiewicz’s program is presented together with its inspirations and prospects. (shrink)

Natural syllogisms are expressed in terms of classes and properties of the real world. They exploit a categorisation present in semantic memory that provides a class inclusion structure. they are enthymematic and typically occur within a dialogue. Their form is identical to a formal syllogism once the minor premise is made explicit. It is claimed that reasoners routinely execute natural_syllogisms in an effortless manner based on ecthesis, which is primed by the class inclusion structure kept in long term memory.

Resumen En esta contribución ofrecemos una interpretación del pons asinorum que se basa en una lógica de términos contemporánea. Esto nos permite revitalizar la idea del pons asinorum para generar el -políticamente correcto- pons scholasticorum, una versión terminística del pons asinorum que conecta la inventio medii con el dictum de omni et nullo.In this contribution we offer an interpretation of the pons asinorum by using a contemporary term logic. This interpretation allows us to revitalize the concept of the pons asinorum (...) in order to produce the -politically correct- pons scholasticorum, a terministic version of the pons asinorum that relates the inventio medii with the dictum de omni et nullo. (shrink)

The solution of the problem of the future random events truth is considered in Vasil’ev’s logic. N. A. Vasil’ev graded the logic according to two levels—the level of facts, i.e. time fixed events, and the level of notions or rules, governing these facts. The mathematical construction previously suggested for imaginary Vasil’ev’s logic, extends to the early variant of his logic—a logic of notions. In the paper, we investigate the meaning of problematic and uncertain assertions introduced by Vasil’ev. As a result, (...) we developed a model of Vasil’ev’s logic of facts that resolves also the truth problem of future random events. The imaginary logic has also been extended to the level of notions, and the law of the excluded eighth is gotten in it. The correspondence between Vasil’ev’s terms “some” and “all” and modern quantifiers is discussed. (shrink)

Neste artigo é defendida a tese de que a noção aristotélica de perfeição silogística não é completamente arbitrária e reflete características lógicas, apesar de alguns aspectos não lógicos. Em consonância com a sugestão de alguns intérpretes de que a validade dos silogismos em primeira figura se fundamenta no dictum de omni et nullo, será apontado como essa fundamentação se desdobra em procedimentos dedutivos encontrados nos textos de Aristóteles. Tomando definições ou explicitações das proposições categóricas como parâmetro, a perfeição ou imperfeição (...) de um silogismo, independentemente das relações de prova que se adote entre as figuras silogísticas, não se mostra arbitrária. (shrink)

Constructive logic with Nelson negation is an extension of the intuitionistic logic with a special type of negation expressing some features of constructive falsity and refutation by counterexample. In this paper we generalize this logic weakening maximally the underlying intuitionistic negation. The resulting system, called subminimal logic with Nelson negation, is studied by means of a kind of algebras called generalized N-lattices. We show that generalized N-lattices admit representation formalizing the intuitive idea of refutation by means of counterexamples giving in (...) this way a counterexample semantics of the logic in question and some of its natural extensions. Among the extensions which are near to the intuitionistic logic are the minimal logic with Nelson negation which is an extension of the Johansson's minimal logic with Nelson negation and its in a sense dual version — the co-minimal logic with Nelson negation. Among the extensions near to the classical logic are the well known 3-valued logic of Lukasiewicz, two 12-valued logics and one 48-valued logic. Standard questions for all these logics — decidability, Kripke-style semantics, complete axiomatizability, conservativeness are studied. At the end of the paper extensions based on a new connective of self-dual conjunction and an analog of the Lukasiewicz middle value ½ have also been considered. (shrink)

Humans have used arguments for defending or refuting statements long before the creation of logic as a specialized discipline. This can be interpreted as the fact that an intuitive notion of “logical consequence” or a psychic disposition to articulate reasoning according to this pattern is present in common sense, and logic simply aims at describing and codifying the features of this spontaneous capacity of human reason. It is well known, however, that several arguments easily accepted by common sense are actually (...) “logical fallacies”, and this indicates that logic is not just a descriptive, but also a prescriptive or normative enterprise, in which the notion of logical consequence is defined in a precise way and then certain rules are established in order to maintain the discourse in keeping with this notion. Yet in the justification of the correctness and adequacy of these rules commonsense reasoning must necessarily be used, and in such a way its foundational role is recognized. Moreover, it remains also true that several branches and forms of logic have been elaborated precisely in order to reflect the structural features of correct argument used in different fields of human reasoning and yet insufficiently mirrored by the most familiar logical formalisms. (shrink)

The purpose of this paper is to contribute to the natural logic program which invents logics in natural language. This study presents two logics: a logical system called d R containing transitive verbs and a more expressive logical system R containing both transitive verbs and intersective adjectives. The paper offers three different set-theoretic semantics which are equivalent for the logics.

This paper examines how the work of Frege was known and received in Poland in the period 1910–1935 (with one exception concerning the later work of Suszko). The main thesis is that Frege's reception in Poland was perhaps faster and deeper than in other countries, except England, due to works of Russell and Jourdain. The works of Łukasiewicz, Leśniewski and Czeżowski are described.

(1993). Alexander of aphrodisias and others on a controversial demonstration in aristotle’s modal syllogistic. History and Philosophy of Logic: Vol. 14, No. 2, pp. 201-214.

The aim of this paper is to provide a dynamic interpretation of Kant’s logical hylomorphism. Firstly, various types of the logical hylomorphism will be illustrated. Secondly, I propose to reevaluate Kant’s constitutivity thesis about logic. Finally, I focus on the design of logical norms as specific kinds of artefacts.