Hoop residuation algebras are the {, 1}-subreducts of hoops; they include Hilbert algebras and the {, 1}-reducts of MV-algebras (also known as Wajsberg algebras). The paper investigates the structure and cardinality of finitely generated free algebras in varieties of k-potent hoop residuation algebras. The assumption of k-potency guarantees local finiteness of the varieties considered. It is shown that the free algebra on n generators in any of these varieties can be represented as a union of n subalgebras, each of which (...) is a copy of the {, 1}-reduct of the same finite MV-algebra, i.e., of the same finite product of linearly ordered (simple) algebras. The cardinality of the product can be determined in principle, and an inclusion-exclusion type argument yields the cardinality of the free algebra. The methods are illustrated by applying them to various cases, both known (varieties generated by a finite linearly ordered Hilbert algebra) and new (residuation reducts of MV-algebras and of hoops). (shrink)

Łukasiewicz 3-valued logic Ł3 is often understood as the set of all valid formulas according to Łukasiewicz 3-valued matrices MŁ3. Following Wojcicki, in addition, we shall consider two alternative interpretations of Ł3: ‘truth-preserving’ Ł3a and ‘well-determined’ Ł3b defined by two different consequence relations on the 3-valued matrices MŁ3. The aim of this article is to provide a Routley–Meyer ternary semantics for each one of these three versions of Łukasiewicz 3-valued logic: Ł3, Ł3a and Ł3b.

We define the paraconsistent supra-logic Pσ by a type-shift from the booleans o of propositional logic Po to the supra-booleans σ of the propositional type logic P obtained as the propositional fragment of the transfinite type theory Q defined by Peter Andrews (North-Holland Studies in Logic 1965) as a classical foundation of mathematics. The supra-logic is in a sense a propositional logic only, but since there is an infinite number of supra-booleans and arithmetical operations are available for this and other (...) types, virtually anything can be specified. The supra-logic is a generalization of Lukasiewicz's three-valued logic, with the intermediate value duplicated many times and ordered such that none of the copies of this value imply other ones, but it differs from Lukasiewicz's many-valued logics as well as from logics based on bilattices. There are several automated theorem provers for classical higher order logic (finite type theory) and it should be possible to modify these to our needs. (shrink)

A reprint of the first edition together with three new chapters and an enlarged index. The new chapters include some previously published material and discuss Aristotle's modal logic of propositions, Lukasiewicz' new system of modal logic, and Aristotle's modal syllogistic. The author relates his interpretation of Aristotle's modal logic to the philosophical issues raised by modern modal logics. -- J. J. C.

Extending Lukasiewicz's approach of axiomatization to the modal syllogistic, McCall develops a system of fourteen axioms with decision procedure, in which exactly those necessity syllogisms recognized by Aristotle are provable. Primitives, besides those of propositional logic, are Necessity and the A and I statement forms. The approach thus contrasts with that of the "structuralists", who would analyze Aristotle's modal statements further in terms of contemporary logic systems. The seemingly insurmountable problems of the contingency syllogisms are circumvented by taking contingency (...) as an additional primitive requiring fourteen further axioms. In the resulting system, all of Aristotle's contingency syllogisms, as well as twenty-four which he overlooked, are provable. Although a "structuralist" analysis would seem indispensable for an ultimate understanding of the modal syllogistic, McCall's formalization is more faithful to Aristotle than any hitherto proposed.--J. B. B. (shrink)

In 1951 Lukasiewicz [[sic]] linked Aristotle's Prior Analytics with modern formal logic. This book attempts to analyze Aristotle's syllogistic theory in the light of Lukasiewcz's work and the whole tradition of classic interpretations of Aristotle's logic. The first of the book's five chapters shows that for Aristotle the syllogism is basically a relationship of terms couched in conditional form; a relationship of variables rather than concrete terms; and a relationship that sees S linked with P not by the copula (...) but by less metaphysical expressions. The classical S is P form is not Aristotelian at all but a much later renovation. The second chapter maintains that Aristotle distinguishes "relative necessity" from "absolute necessity." Aristotle himself ignores his own distinction at times and thus creates logical difficulties for himself. The oft-fought sea battle is fought again in this chapter. Chapter three deals with the controversial notions of "perfect" and "imperfect" syllogisms. The author asserts that by "perfect syllogism" Aristotle means self-evident syllogisms that serve as axioms for deducing other proofs. Chapters four and five deal with the figures of the syllogism and the syllogism as a deductive system respectively. Patzig concludes that Aristotle's syllogistic theory is purely formal in its structure and can only be properly understood if it is not tied to a metaphysics and converted into a philosophical logic. He denies that it is a universal logic, but rather sees it as a specialized theory of relations.--J. J. R. (shrink)

A reprint of the first edition together with three new chapters and an enlarged index. The new chapters include some previously published material and discuss Aristotle's modal logic of propositions, Lukasiewicz' new system of modal logic, and Aristotle's modal syllogistic. The author relates his interpretation of Aristotle's modal logic to the philosophical issues raised by modern modal logics. -- J. J. C.

In “A new proof of the completeness of the Lukasiewicz axioms” Chang proved that any totally ordered MV-algebra A was isomorphic to the segment \}\) of a totally ordered l-group with strong unit A *. This was done by the simple intuitive idea of putting denumerable copies of A on top of each other. Moreover, he also show that any such group G can be recovered from its segment since \^*}\), establishing an equivalence of categories. In “Interpretation of AF (...) C *-algebras in Lukasiewicz sentential calculus” Mundici extended this result to arbitrary MV-algebras and l-groups with strong unit. He takes the representation of A as a sub-direct product of chains A i, and observes that \ where \. Then he let A * be the l-subgroup generated by A inside \. He proves that this idea works, and establish an equivalence of categories in a rather elaborate way by means of his concept of good sequences and its complicated arithmetics. In this note, essentially self-contained except for Chang’s result, we give a simple proof of this equivalence taking advantage directly of the arithmetics of the the product l-group \, avoiding entirely the notion of good sequence. (shrink)

In Book Γ of the Metaphysics Aristotle states and attempts to prove what he calls the basic principle of the science of being as being: the law of non‐contradiction. In this paper I defend an interpretation of his proof, inspired by Elizabeth Anscombe's 1961 essay in ‘Three Philosophers’, though some of its features were remarked on by Lukasiewicz in 1910, according to which Aristotle is proving this principle only for substance predicates, and that it is to be understood as (...) the basic principle of metaphysics rather than of logic. It is the principle that every substance has an essential nature, and if it is of the essential nature of x to be F then it cannot be of the essential nature of x to be not‐F since then x's being F and x's being not‐F would be the same. He puts it by saying that in such a case F and not‐F would ‘signify the same’ for x. Since they do not, x cannot be both. (shrink)

Among the problems raised by admitting statements which are neither true nor false is the problem of how we are to cope with vague concepts. One method of dealing with such concepts has been suggested by Rosser/Turquette (1952), i.e. the employment of a many valued set theory. It is our intention in this paper to discuss the use of many valued logics, especially the set theoretical proposals of Zadeh and Brown, in dealing with this problem. Towards this end, we shall (...) pay close attention to the development of the concepts of fuzzy sets and fuzzy systems. In this regard, it will be argued that Zadeh had in mind a many valued logic where the connectives are interpreted according to Lukasiewicz and Tarski, whereas Brown's version may be based upon a Boolean-valued logic. Furthermore, we shall have occasion to distinguish between two sources of imprecision in specifying the membership of classes, i.e. conceptual vagueness and imprecision due to inexact measurement. Due to Suppes' work in this field, the latter sort of imprecision may be exemplified by providing a theory of the inexact measurement of subjective probability. Finally, we shall discuss some ways in which these two types of imprecision are intertwined. (shrink)

This detailed monograph deals with such problems as "The Unity of the Finite and Infinite," "Logic and the Concept of Function," "Mathematical Logic," "Formal and Dialectical Logic." The author mentions the work of Reichenbach and Lukasiewicz.--R. L. J.

Beginning with the traditional rules for syllogistic validity, Bird leads the reader without further ado through Aristotle's method of reduction, Keynes' treatment of the antilogism, Lukasiewicz's axiomatization, Ivo Thomas's extension of the last to negative terms, and on to the treatment of empty terms via informal Boolean algebra and the introduction of singular terms. Lukasiewicz's axiom system is investigated in some detail, and independence and consistency are proved. The problem of existential import is discussed quite thoroughly, though it (...) is doubtful that the last word is spoken here. Modalities come in for discussion, but their relation to syllogistic is hardly indicated. Not covered are quantificational analyses of syllogistic. The presentation is marred though not vitiated by various imprecisions in the metatheory. Symbolism is used from the start, and Venn diagrams are resorted to throughout.—J. B. B. (shrink)

This detailed monograph deals with such problems as "The Unity of the Finite and Infinite," "Logic and the Concept of Function," "Mathematical Logic," "Formal and Dialectical Logic." The author mentions the work of Reichenbach and Lukasiewicz.--R. L. J.

The set (X, J) of fuzzy subsetsf:XJ of a setX can be equipped with a structure of -valued ukasiewicz-Moisil algebra, where is the order type of the totally ordered setJ. Conversely, every ukasiewicz-Moisil algebra — and in particular every Post algebra — is isomorphic to a subalgebra of an algebra of the form (X, J), whereJ has an order type . The first result of this paper is a characterization of those -valued ukasiewicz-Moisil algebras which are isomorphic to an algebra (...) of the form (X, J) (Theorem 1). Then we prove that (X, J) is a Post algebra if and only if the setJ is dually well-ordered (Theorem 2) and we give a characterization of those -valued Post algebras with are isomorphic to an algebra of the form (X, J) (Theorem 3 and Proposition 2). (shrink)

J.L. Schellenberg offers a path to a new kind of religious outlook. Reflection on our early stage in the evolutionary process leads to skepticism about religion, but also offers a new answer to the problem of faith and reason, and the possibility of a new, evolutionary form of religion.

The paper discusses interpretations of Aristotle’s modal notions by modern commentators (J. Hintikka, J. Lukasiewiez, M. Mignucci). It is shown that the semantics of modal notions which the above mentioned authors attribute to Aristotle is based on the algebraic idea of multiplier.

What happens when someone acts? A familiar answer goes like this. There is something that the agent wants, and there is an action that he believes conducive to its attainment. His desire for the end, and his belief in the action as a means, justify taking the action, and they jointly cause an intention to take it, which in turn causes the corresponding movements of the agent's body. I think that the standard story is flawed in several respects. The flaw (...) that will concern me in this paper is that the story fails to include an agent-or, more precisely, fails to cast the agent in his proper role. (shrink)

Providing an original and systematic treatment of foundational issues in philosophy of religion, J. L. Schellenberg's new book addresses the structure of..

Abstract I argue that meaning in life is importantly influenced by bioloical ties. More specifically, I maintain that knowing one's relatives and especially one's parents provides a kind of self-knowledge that is of irreplaceable value in the life-task of identity formation. These claims lead me to the conclusion that it is immoral to create children with the intention that they be alienated from their bioloical relatives?for example, by donor conception.

Supervaluationism is often described as the most popular semantic treatment of indeterminacy. There’s little consensus, however, about how to fill out the bare-bones idea to include a characterization of logical consequence. The paper explores one methodology for choosing between the logics: pick a logic thatnorms beliefas classical consequence is standardly thought to do. The main focus of the paper considers a variant of standard supervaluational, on which we can characterizedegrees of determinacy. It applies the methodology above to focus ondegree logic. (...) This is developed first in a basic, single-premise case; and then extended to the multipremise case, and to allowdegreesof consequence. The metatheoretic properties of degree logic are set out. On the positive side, the logic is supraclassical—all classical valid sequents are degree logic valid. Strikingly, metarules such as cut and conjunction introduction fail. (shrink)

Hailed by the Bulletin of the American Mathematical Society as "undoubtedly a major addition to the literature of mathematical logic," this volume examines the essential topics and theorems of mathematical reasoning. No background in logic is assumed, and the examples are chosen from a variety of mathematical fields. Starting with an introduction to symbolic logic, the first eight chapters develop logic through the restricted predicate calculus. Topics include the statement calculus, the use of names, an axiomatic treatment of the statement (...) calculus, descriptions, and equality. Succeeding chapters explore abstract set theory—with examinations of class membership as well as relations and functions—cardinal and ordinal arithmetic, and the axiom of choice. An invaluable reference book for all mathematicians, this text is suitable for advanced undergraduates and graduate students. Numerous exercises make it particularly appropriate for classroom use. (shrink)

Ultimism and the aims of human immaturity -- Faith without details, or how to practice skeptical religion -- Simple faith and the complexities of tradition -- The structure of faith justification -- How skeptical faith is true to reason -- Anselm's idea -- Leibniz's ambition -- Paley's wonder -- Pascal's wager -- Kant's postulate -- James's will -- Faith is positively justified : the many modes of religious vision.