Proceedings of the 15th Meeting on the Mathematics of Language

Abstract

The volume contains eleven regular papers and two invited papers. It also includes an abstract of a third invited talk. The regular papers were selected from a total of 23 submissions, using the EasyChair conference management system. The conference benefited from the financial support of the British Logic Colloquium (http://www. blc-logic.org) and of AYLIEN (http://aylien.com), which we gratefully acknowledge.Last but not least, we would like to express our sincere gratitude to all the reviewers for MOL 2017 and to all the people who helped with the local organization. Partee (1986) claimed without proof that the function BE is the only homomorphism that makes the Partee triangle commute. This paper shows that this claim is incorrect unless "homomorphism" is understood as "complete homomorphism." It also shows that BE and A are the inverses of each other on certain natural assumptions. IntroductionIn a famous and influential paper, Partee (1986) discussed type-shifting operators for NP interpretations, including lift, ident and BE: lift = λxλP. P (x), ident = λxλy. [y = x], BE = λPλx. P(λy. [y = x]).She pointed out that these operators satisfy the equality BE • lift = ident, so the following diagram, now often referred to as the Partee triangle, commutes. Partee declared that BE is "natural" because of the following two "facts." Fact 1. BE is a homomorphism from e, t , t to e, t viewed as Boolean structures, i.e., BE(P 1 P 2 ) = BE(P 1 ) BE(P 2 ), BE(P 1 P 2 ) = BE(P 1 ) BE(P 2 ), BE(¬P 1 ) = ¬BE(P 1 ).Fact 2. BE is the unique homomorphism that makes the diagram commute.While Fact 1 is immediate, Fact 2 is not obvious. Partee (1986) nevertheless did not give a proof of Fact 2, but only a note saying, "Thanks to Johan van Benthem for the fact, which he knows how to prove but I don't." Meanwhile, van Benthem (1986) referred to Partee's work and stated Fact 2 on p. 68, but gave no proof either. Despite this quite obscure exposition, because of the classic status of Partee's and van Benthem's work, I suspect that many linguists take Fact 2 for granted while unable to explain it. Not only is this unfortunate, but it is actually as expected, because Fact 2 turns out to be not quite correct unless "homomorphism" is read as "complete homomorphism." The main purpose of this paper is to rectify this detrimental situation.Van Benthem (1986) took the domain of entities to be finite, writing, "Our general feeling is that natural language requires the use of finite models only" (p. 7). Fact 2 is indeed correct on this assumption. However, natural language has predicates like natural number whose extensions are obviously infinite. Also, if we take the domain of portions of matter in the sense of Link (1983) to be a nonatomic join-semilattice, then the domain of entities will surely be infinite, whether countable or uncountable. It is a fact that a single sentence of natural language, albeit only finitely long, can talk about an infinite number of entities, as exemplified in (1).(1) a. Every natural number is odd or even...