Independence-friendly logic

Abstract: Game theory is a mathematical tool to study the behavior of independent agents in strategic interaction. Reasoning and communication have an essentially strategic aspect. Game theoretic is thus a suitable tool to illuminate the interactive aspects of logic and language.

Abstract: Many linguists and philosophers of language may have heard of informational independence, but most, not to say virtually all, of them consider it as a marginal feature of the semantics of natural languages. Yet in reality it is a widespread phenomenon in languages like English. In this paper, we shall develop an explicit unified formal treatment of all the different varieties of informational independence in linguistic semantics. This treatment amounts to a new type of logic, which is thereby opened for investigation. We shall also call attention to several actual linguistic phenomena which instantiate informational independence and provide evidence of its ubiquity. Last but not least, we shall show that the phenomenon of informational independence prompts several highly interesting methodological problems and suggestions.

Abstract: The syntax and semantics of quantifiers is of crucial significance in current linguistic theorizing for more than one reason. The last statement of his grammatical theories by the late Richard Montague (1 973) is modestly entitled \" The Proper Treatment of Quantification in Ordinary English \". In the authoritative statement of his \" Generative Semantics \" , George Lakoff (1971, especially pp. 238-267) uses as his first and foremost testing-ground the grammar of certain English quantifiers. In particular, they serve to illustrate, and show need of, his use of global constraints governing the derivation of English sentences. Evidence from the behavior of quantifiers (including numerical expressions 1) has likewise played a major role in recent discussions of such vital problems as the alleged meaningdpreservation of transformations , 2 co-reference, 3 the role of surface structure in semantical interpretation, and so on. In all these problems, the behavior of natural-language quantifiers is one of the main issues. Quantifiers have nevertheless entered the Methodenstreit of contemporary linguistics in another major way, too. (These two groups of problems are of course interrelated.) This is the idea that the structures studied iln the so-called quantification theory of symbolic logic-otherwise know as first-order logic, (lower) predicate calculus, or elementary logic-can serve and suffice 6 as semantical representations of English sentences. Views of this general type have been proposed by McCawley (1971)* and Lakoff (1972)' (among others). A related theory of \" Deep Structure as Logical Form \" has been put forward and defended by G. Harman (1972). Theories of this general type may be compared with the traditiomal idea that quantification 'theory can be viewed as an abstraction from the behavior

Abstract: In [3] Henkin made the observation that certain second-order existential formulas may be thought of as the Skolem normal forms of formulas of a language which is first-order in every respect except its incorporation of a form of partially-ordered quantification. One formulation of this sort of language is the closure of a first-order language under the formation rule that Qφ is a formula whenever φ is a formula and Q, which is to be thought of as a quantifier-prefix, is a system of partial order whose universe is a set of quantifiers. Although he introduced this idea in a discussion of infinitary logic, Henkin went on to discuss its application to finitary languages, and he concluded his discussion with a theorem of Ehrenfeucht that the incorporation of an extremely simple partially-ordered quantifier-prefix (the quantifiers ∀x, ∀y, ∃v, and ∃w, with the ordering {〈∀x, ∃v〉, 〈∀y, ∃w〉}) into any first-order language with identity gives a language capable of expressing the infinitary quantifier ∃zκ0x.