Semantic Types: Two Is Better Than Too Many

Thomason [32] urged a different project in which linguistics—or at least studies of syntax and semantics—would be developed as a branch of mathematics (“Montague Grammar”), without focusing on properties of human languages/procedures that are “merely psychologically universal.” But as Chomsky remarks [8, pp. 29–30], if the envisioned enterprise is to be evaluated in terms of the interesting theorems that have emerged, it hasn’t been a great success; and one wouldn’t expect to find mathematicians (e.g., David Hilbert) describing physicists as being unduly concerned with the “merely physical” properties of the universe. Similarly, insisting on a “general semantics” that covers Slangs and also sundry invented languages that meet certain stipulated conditions (see [24]) may be like insisting on a “general biology” that is not limited to living things but also covers logically possible animals like unicorns and dragons. Such a project might lead to describing actual animals in ways that are less than ideal for purposes of actual biology.

And they are usually arbitrary. The meanings expressed with /bæŋk/ could be expressed, as in many languages, with lexical items that have distinct pronunciations. The polysemous word ‘window’ seems to have a meaning that supports related “subsenses,” which can be used to talk about certain openings in walls or framed panes of glass that fill such openings. But even if polysemy is open-ended, the number of subsenses is presumably finite for each speaker.

For example, if π-μ pairs are generated in structure-dependent ways involving transformations (but no context-sensitive operations of inversion), that is relevant; see, e.g., [4‐6].

One can define a task of describing certain facts (e.g., those concerning apparent motions of celestial bodies from a certain vantage point) without regard to other facts (e.g., those concerning the motions of terrestrial pendula and balls rolling down inclined planes, or correlations between tides and phases of the moon). But whatever the value of such tasks, they shouldn’t be confused with the goal of explaining natural phenomena. History suggests that this goal is hindered by trying to define the relevant explananda in advance, but that when studying Slangs, it’s easy to slide into behavioristic stipulations that restrict attention to data that is accessible in certain ways.

Compare ‘easy/eager to please’ and ‘persuaded/expected John to leave’; see [6, 7]. Note that ‘I persuaded him that he should leave’ is fine, unlike ‘I persuaded that he should leave’. But ‘I expected that he would leave’ is fine, unlike ‘I expected him that he should leave’.

Some of this section is drawn, with slight modifications, from [27] and [28].

Hence, Predecessor(2, 3) is a truth value, even if ‘Predecessor(3)’ denotes a number. Likewise, Prime(2) is a truth value, even if ‘Prime(2)’ does not denote a truth value but instead has a Tarskian satisfaction condition. In this sense, expressions of type <e, t> are relational, even if they also count as monadic; they indicate mappings from entities to truth values, highlighted here with boldface. In this sense, λx.Predecessor(x) and λx.Prime(x) are on a par with regard to arity/adicity. If only for simplicity, I ignore Frege’s [16] talk of Functions/Concepts being unsaturated and use lambda expressions to talk about denotable functions as in [12].

Note that the function λD′.λD.ANCESTRAL-OF(D, D′) is like λD.TRANSITIVE(D) in being second-order, but also like λy.λx.Predecessor(x, y) in being dyadic. By contrast, the function λD.ANCESTRAL(D) maps λy.λx.Predecessor(x, y) to λy.λx.Precedes(x, y).

The index is not posited as an expression of type <t, et>; but neither is the displaced element in [which₁ [t₁ ran quickly]_<t>]_<et>. Heim and Kratzer posit a rule according which: if a sentence S contains a trace with index i and combines with a copy of i, the result is an expression of type <e, t>; and relative to any assignment A, i^S indicates a function that maps each entity e to T iff S denotes T relative to the minimally different assignment A* that assigns e to i.

For these purposes, let’s not worry about the indefinite descriptions. Suppose that ‘sald’ would be of type <e, <e, <e, <et>>>> and not the Level Five type <et, <et, <et, <et, t>>>>. For these purposes, let’s also ignore adverbial modification and the need for an event variable.

See [28] for further discussion, a treatment of quantificational determiners as plural monadic predicates, and the puzzles presented by “conservativity” if we say that words like ‘every’ and ‘most’ express second-order relations exhibited by first-order monadic predicates.

My thanks to the participants, and especially to Naoya Fujikawa, for helpful discussions both during and after the conference.