We investigate the class of strongly distributive pregroups, a common abstraction of MV-algebras and Abelian l-groups which was introduced by E.Casari. The main result of the paper is a representation theorem which yields both Chang's representation of MV-algebras and Clifford's representation of Abelian l-groups as immediate corollaries.

The paper introduces a concept of logic applied to a formalization of the so-called inferences preserving degrees of truth. Semantical and syntactical characterizations of three kinds of logics preserving degrees of truth are provided. The other approach than in [3] and [9] to the problem of expressing that a sentence is less true than a sentence is presented.

The presence of premises expressing comparison is a problem for the Aristotelian theory of the dialectical method, first because there is no general theory of comparison in the Organon and secondly because along with propositions on the opposition and inflexion of the terms, comparative statements seem to fall outside the explicit description which Aristotle gives of possible premises. The purpose of this paper is to offer a synthetic theory of comparisons according to Aristotle’s Topics, in an attempt both to supply (...) the aforementioned absence and to highlight the importance of the second problem. There are three main types of premises on the more, the less and the similar: some comparing the degree of possession of a predicate, others comparing the plausibility of a predication and finally, others expressing an analogy. Once expounded the very marked differences that occur between these three classes of propositions, the paper analyses the various kinds of argumentation based on comparative premises and offers a formalization of its logical laws. The complexity of this theory and the volume of topoi involved in the comparisons show the magnitude of a problem that has not been sufficiently studied. (shrink)

A residuated lattice is said to be integrally closed if it satisfies the quasiequations \ and \, or equivalently, the equations \ and \. Every integral, cancellative, or divisible residuated lattice is integrally closed, and, conversely, every bounded integrally closed residuated lattice is integral. It is proved that the mapping \\backslash {\mathrm {e}}\) on any integrally closed residuated lattice is a homomorphism onto a lattice-ordered group. A Glivenko-style property is then established for varieties of integrally closed residuated lattices with respect (...) to varieties of lattice-ordered groups, showing in particular that integrally closed residuated lattices form the largest variety of residuated lattices admitting this property with respect to lattice-ordered groups. The Glivenko property is used to obtain a sequent calculus admitting cut-elimination for the variety of integrally closed residuated lattices and to establish the decidability, indeed PSPACE-completenes, of its equational theory. Finally, these results are related to previous work on BCI-algebras, semi-integral residuated pomonoids, and Casari’s comparative logic. (shrink)

Both W.D. Ross's and J. Brunschwig's editions of Aristotle's Topics contain the following passage: ἔτι εἰ τοῦ αὐτοῦ τινος τὸ μὲν μᾶλλον τὸ δὲ ἧττον τοιοῦτο· καὶ εἰ τὸ μὲν τοιούτου μᾶλλον τοιοῦτο, τὸ δὲ μὴ τοιούτου, δῆλον ὅτι τὸ πϱῶτον μᾶλλον τοιοῦτο. The passage is translated in the revised Oxford translation as follows: ‘Moreover, if in any character one thing exceeds and another falls short of the same standard; also, if the one exceeds something which possesses the character, while (...) the other exceeds something which does not, then clearly the first thing exhibits that character in a greater degree’. (shrink)

The presence of premises expressing comparison is a problem for the Aristotelian theory of the dialectical method, first because there is no general theory of comparison in the Organon and secondly because along with propositions on the opposition and inflexion of the terms, comparative statements seem to fall outside the explicit description which Aristotle gives of possible premises. The purpose of this paper is to offer a synthetic theory of comparisons according to Aristotle’s Topics , in an attempt both to (...) supply the aforementioned absence and to highlight the importance of the second problem. There are three main types of premises on the more, the less and the similar : some comparing the degree of possession of a predicate, others comparing the plausibility of a predication and finally, others expressing an analogy. Once expounded the very marked differences that occur between these three classes of propositions, the paper analyses the various kinds of argumentation based on comparative premises and offers a formalization of its logical laws. The complexity of this theory and the volume of topoi involved in the comparisons show the magnitude of a problem that has not been sufficiently studied. (shrink)

Motivated by aspects of reasoning in theories of physics, Robin Giles defined a characterization of infinite valued Łukasiewicz logic in terms of a game that combines Lorenzen-style dialogue rules for logical connectives with a scheme for betting on results of dispersive experiments for evaluating atomic propositions. We analyze this game and provide conditions on payoff functions that allow us to extract many-valued truth functions from dialogue rules of a quite general form. Besides finite and infinite valued Łukasiewicz logics, also Meyer (...) and Slaney’s Abelian logic and Cancellative Hoop Logic turn out to be characterizable in this manner. (shrink)

This article provides an intuitive semantic account of a new logic for comparisons (CL), in which atomic statements are assigned both a classical truth-value and a “how much” value or extension in the range [0, 1]. The truth-value of each comparison is determined by the extensions of its component sentences; the truth-value of each atomic depends on whether its extension matches a separate standard for its predicate; everything else is computed classically. CL is less radical than Casari’s comparative logics, in (...) that it does not allow for the formation of comparative statements out of truth-functional molecules. It is argued that CL provides a better analysis of predicate vagueness than classical logic, fuzzy logic or supervaluation theory. (shrink)

In this paper I provide some formal schemas for the analysis of vague predicates in terms of a set of semantic relations other than classical synonymy, including weak synonymy (as between "large" and "huge"), antonymy (as between "large" and "small"), relativity (as between "large" and "large for a dog"), and a kind of supervenience (as between "large" and "wide" or "long"). All of these relations are representable in the simple comparative logic CL, in accordance with the basic formula: the more (...) something is F, the more (or less) it is G. I use Carnapian meaning postulates to define these relations as constraints on interpretations of the formal language of CL. (shrink)

The reasoning process of analogy is characterized by a strict interdependence between a process of abstraction of a common feature and the transfer of an attribute of the Analogue to the Primary Subject. The first reasoning step is regarded as an abstraction of a generic characteristic that is relevant for the attribution of the predicate. The abstracted feature can be considered from a logic-semantic perspective as a functional genus, in the sense that it is contextually essential for the attribution of (...) the predicate, i.e. that is pragmatically fundamental (i.e. relevant) for the predica-tion, or rather the achievement of the communicative intention. While the transfer of the predicate from the Analogue to the analogical genus and from the genus to the Primary Subject is guaranteed by the maxims (or rules of inference) governing the genus-species relation, the connection between the genus and the predicate can be complex, characterized by various types of reasoning patterns. The relevance relation can hide implicit arguments, such as an implicit argument from classification , an evaluation based on values, consequences or rules, a causal relation, or an argument from practical reasoning. (shrink)