In this paper we address some central problems of combination of logics through the study of a very simple but highly informative case, the combination of the logics of disjunction and conjunction. At first it seems that it would be very easy to combine such logics, but the following problem arises: if we combine these logics in a straightforward way, distributivity holds. On the other hand, distributivity does not arise if we use the usual notion of extension between consequence relations. (...) A detailed discussion about this phenomenon, as well as some possible solutions for it, are given. (shrink)

ABSTRACT: The 3rd BCE Stoic logician "Chrysippus says that the number of conjunctions constructible from ten propositions exceeds one million. Hipparchus refuted this, demonstrating that the affirmative encompasses 103,049 conjunctions and the negative 310,952." After laying dormant for over 2000 years, the numbers in this Plutarch passage were recently identified as the 10th (and a derivative of the 11th) Schröder number, and F. Acerbi showed how the 2nd BCE astronomer Hipparchus could have calculated them. What remained unexplained is why Hipparchus’ (...) logic differed from Stoic logic, and consequently, whether Hipparchus actually refuted Chrysippus. This paper closes these explanatory gaps. (1) I reconstruct Hipparchus’ notions of conjunction and negation, and show how they differ from Stoic logic. (2) Based on evidence from Stoic logic, I reconstruct Chrysippus’ calculations, thereby (a) showing that Chrysippus’ claim of over a million conjunctions was correct; and (b) shedding new light on Stoic logic and – possibly – on 3rd century BCE combinatorics. (3) Using evidence about the developments in logic from the 3rd to the 2nd centuries, including the amalgamation of Peripatetic and Stoic theories, I explain why Hipparchus, in his calculations, used the logical notions he did, and why he may have thought they were Stoic. (shrink)

In this paper we study the relations between the fragment L of classical logic having just conjunction and disjunction and the variety D of distributive lattices, within the context of Algebraic Logic. We prove that these relations cannot be fully expressed either with the tools of Blok and Pigozzi's theory of algebraizable logics or with the use of reduced matrices for L. However, these relations can be naturally formulated when we introduce a new notion of model of a sequent calculus. (...) When applied to a certain natural calculus for L, the resulting models are equivalent to a class of abstract logics (in the sense of Brown and Suszko) which we call distributive. Among other results, we prove that D is exactly the class of the algebraic reducts of the reduced models of L, that there is an embedding of the theories of L into the theories of the equational consequence (in the sense of Blok and Pigozzi) relative to D, and that for any algebra A of type (2,2) there is an isomorphism between the D-congruences of A and the models of L over A. In the second part of this paper (which will be published separately) we will also apply some results to give proofs with a logical flavour for several new or well-known lattice-theoretical properties. (shrink)

I argue that the conjunctive distribution of permissibility over or, which is a puzzling feature of free-choice permission is just one instance of a more general class of conjunctive occurrences of the word, and that these conjunctive uses are more directly explicable by the consideration that or is a descendant of oper than by reference to the disjunctive occurrences which logicalist prejudices may tempt us to regard as semantically more fundamental. I offer an account of how the disjunctive uses of (...) or may have come about through an intermediate discourse-adverbial use of or, drawing a parallel with but, which, etymologically, is disjunctive rather than conjunctive and whose conjunctive uses seem to represent just such a discourse-adverbial application. (shrink)

This is a comprehensive study of the English word 'or', and the logical operators variously proposed to present its meaning. Although there are indisputably disjunctive uses of or in English, it is a mistake to suppose that logical disjunction represents its core meaning. 'Or' is descended from the Anglo-Saxon word meaning second, a form which survives in such expressions as "every other day." Its disjunctive uses arise through metalinguistic applications of an intermediate adverbial meaning which is conjunctive rather than disjunctive (...) in character. These conjunctive uses have puzzled philosophers and logicians, and have been discussed extensively under such headings as "free choice permission." This study examines the textbook myths that have clouded our understanding of how or and other "logical" vocabulary comes to have something approaching its logical meaning in natural languages. It considers the various historical conceptions of disjunction and its place in logic from the Stoics to the present day. (shrink)

In the first part of this article, we show that, contrary to the Gricean tradition, inter-clausal and is not semantically equivalent to logical conjunction and, contrary to temporal approaches such as Bar-Levand Palacas 1980, it is not temporallyloaded. We then explore a commonsenseidea – namely that while sentence juxtaposition might be interpreted either as discourse coordination or subordination, and indicates coordination. SDRT already includes notions of coordinating and subordinating discourse relations (cf. Lascarides and Asher 1993, Asher 1993), and the meaning (...) of and is related to this distinction. Similar distinctions that play a crucial role in anaphora resolution have also appeared in AI – cf. Scha and Polanyi 1988, or Webber 1991. However, this discourse-structure-based distinction has not been well defined yet, and our approach provides independent motivation for it. This paper argues that the semantics of and includes a notion of coordination expressed as the requirement of a Coordinated Discourse Topic (CDT).CDT characterizes aclass of discourse relations, among which are Narration and Result.Once the basic semanticcontribution of and is isolated, effects related to its presence such as changes in temporal structure, blocking of a Discourse Relation, or conditional meanings are shown to follow from the defeasiblearchitecture set up by SDRT. (shrink)