This paper embeds the core part of Discourse Representation Theory in the classical theory of types plus a few simple axioms that allow the theory to express key facts about variables and assignments on the object level of the logic. It is shown how the embedding can be used to combine core analyses of natural language phenomena in Discourse Representation Theory with analyses that can be obtained in Montague Semantics.

This paper developes a relational---as opposed to a functional---theory of types. The theory is based on Hilbert and Bernays' eta operator plus the identity symbol, from which Church's lambda and the other usual operators are then defined. The logic is intended for use in the semantics of natural language.

It is now a quarter of a century ago that Wolfgang Stegmfiller wrote his monograph 'Das Wahrheitsproblem und die Idee der Semantik' (1957) which dealt with Tarski's and Carnap's foundational work in the field of semantics. While this book is about the definition of the basic semantical concepts in artificial formal languages there is an article written a year earlier (1956) in which Stegmfiller addresses himself specifically to the relation between logic and natural language. Here he gives a logical analysis (...) of the standard structural expressions in language that are still of primary concern for current semantics: quantifiers, pronouns, articles, etc. The motives for such an analysis at that time were mainly philosophical: the aim was to expose the misconceptions and pitfalls of traditional philosophy arising from the disregard of various systematic semantic ambiguities in everyday language. Or as Stegmfiller puts it: Ober sie [i.e. einige nicht triviale F~ille von Vagheit in der A11tagssprache] Klarheit zu gewinnen, ist schon deshalb yon auBerordentlicher Bedeutung, weil Unkenntnis fiber sie zu schwersten philosophischen Verirrungen ffihren kann, namlich entweder der Unterlassung von berechtigten Fragestellungen, oder, was weit h~ufiger vorgekommen ist, der Formulierung yon falsch gestellten Fragen, denen gegenfiber man dann nur die Wahl hat, entweder fiberhaupt keine oder nur sinnlose Antworten zu geben. So logic was to regiment language. Sentences involving the copula and the above-mentioned structure words are assigned one or more unambiguous formal representations in an already interpreted formal language, usually the first order predicate calculus. The natural language expressions thereby receive a precise meaning, since the semantics of the formal language has been specified in advance, as is always assumed. This procedure has, of course, always been common practice, witness the typical syntactical argot of the mathematicians. It carries, however, an obvious methodological presupposition: it is the idea that the logic of our choice to which the formal representations belong is in some sense an adequate framework to express our thoughts. (shrink)

We revisit a typological puzzle due to Horn (Doctoral Dissertation, UCLA, 1972) regarding the lexicalization of logical operators: in instantiations of the traditional square of opposition across categories and languages, the O corner, corresponding to ‘nand’ (= not and), ‘nevery’ (= not every), etc., is never lexicalized. We discuss Horn’s proposal, which involves the interaction of two economy conditions, one that relies on scalar implicatures and one that relies on markedness. We observe that in order to express markedness and to (...) account for a bigger typological puzzle, namely the absence of lexicalizations of ‘XOR’ (= exclusive or), ‘all-or-none’, and many other imaginable logical operators, one must restrict the basic lexicalizable elements to a small set of primitives. We suggest that an ordering based perspective, following Keenan and Faltz (Boolean semantics for natural language, 1985), makes the stipulated primitives that we arrive at more natural. We also propose a modification to Horn’s proposal, based on recent work on implicatures, in which only the implicature condition is operative and in which markedness is part of the definition of the alternatives for scalar implicatures rather than an independent condition. (shrink)

The exclusive component of unembedded disjunctions is standardly derived as a conversational implicature by assuming that or forms a lexical scale with and. It is well known, however, that this assumption does not suffice to determine the required scalar competitors of disjunctions with more than two atomic disjuncts (McCawley, Everything that linguists have always wanted to know about logic* (But were ashamed to ask). Chicago University Press, Chicago, 1993, p. 324; Simons, “Or”: Issues in the semantics and pragmatics of disjunction. (...) Ph.D. thesis, Cornell University, Ithaca, NY, 1998). To solve this, Sauerland (Linguist Philos 27(3): 367–391, 2004) assumes that or forms a lexical scale with two otherwise unattested silent connectives ( ${\mathbb{L}}$ and ${\mathbb{R}}$ ) that retrieve the left and right terms of a disjunction. A number of recent works have proposed an Alternative Semantics for indefinites and disjunction to account for their interaction with modals and other propositional operators (Kratzer and Shimoyama, In: Otsu Y (ed) The Proceedings of the Third Tokyo Conference on Psycholinguistics. Hituzi Syobo, Tokyo, pp. 1–25, 2002; Aloni, In: Weisgerber M (ed) Proceedings of the Conference “SuB7—Sinn und Bedeutung”. Arbeitspapier Nr. 114. Konstanz, pp. 28–37, 2003; Simons, Nat Lang Semantics 13: 271–316, 2005; Alonso-Ovalle, Disjunction in alternative semantics. Ph.D. thesis, University of Massachusetts, Amherst, MA, 2006). We note that the McCawley–Simons problem does not arise in an Alternative Semantics, if we assume that the set of pragmatic competitors to a disjunction is the closure under intersection of the set of propositions that it denotes. An adaptation of the strengthening mechanism presented in Fox (In: Sauerland U, Stateva P (eds) Presupposition and implicature in compositional semantics. MacMillan, Palgrave, pp. 71–120, 2007) allows for the derivation of the exclusive component of disjunctions with more than two atomic disjuncts without having to rely on the ${\mathbb{L}}$ and ${\mathbb{R}}$ operators. The proposal extends to the case of disjunctions with logically dependent disjuncts. (shrink)