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Seem I can't use latex commands on here. Latex works on math.stackexchange, maybe it would be a good idea here as well if linguistics.stackexchange is intended to be a place for questions about formal semantics.


I am basing my question on chapters 'Montague Grammar' and 'Intensional Type Theory' of Gamut 1991.

Basically, I wonder about the interpretation of (higher order) variables in intensional type theory, and the effect on meaning postulates in MG.

Gamut presents 'Meaning Postulate 2', essentially saying that a relation between individuals and second-order properties representing an extensional transitive verb must be equivalent to a relation between individuals, as follows:

MP2 (v1) Exist S, For all x, For all X, Box, Delta(x, X) <--> Caret_Down X(Caret_Up lambda-y Caret_down S(x, y))

Here, S is a variable of type <s, <e, <e, t>>> (two-place FO relation), X is a variable of type <s, <<s, <e, t>>, t>> (SO property), and Delta is a constant symbol representing any extensional TV.

Here's the alternative version I have in mind:

MP2 (v2) Exist S, For all x, For all X, Box Delta(x, Caret_up X) <--> X(Caret_Up lambda-y Caret_down S(x, y))

S and Delta as above, but variable X is now of type <<s, <e, t>>, t>, i.e. a set of first-order properties, instead of a second-order property.

I hope despite the lack of typesetting the difference became clear:

The original MP2 (in the GAMUT version) demands the existence of a relation S s.t. for all individuals x and for all second-order properties X, if Delta holds between x and the SO property X, then the extension of the SO property X behaves in a certain way.

Version 2 instead requires the existence of a relation S s.t. for all individuals x and for all sets of first-order properties X, if Delta holds between x and the intension of X, then the set of FO properties X behaves in a certain way.

Even shorter, and a bit imprecise: (v1) quantifies over the intension of terms, (v2) over the extension of terms.

By our definition of the semantics of ITT we can show the equivalence of interpretation of "Intension of alpha", "I(alpha)" (where I is the interpretation function of constants) and "Caret_Up(alpha)", so leaving quantification aside, (v1) and (v2) could be taken to be equivalent.

However, by our interpretation of quantifiers, "For all X" will range over (semantic) objects, not syntactic objects (i.e. not term symbols, but sets and functions over our domain), so I am not sure if I am right to think the two are equivalent, i.e. whether models of (v1) and (v2) are equivalent w.r.t. any expression of L_ITT.

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The point that Gamut are making here is that (i) verbs like seek express relations between individuals and second-order properties (not relations between individuals), (ii) verbs like kiss express relations between individuals, and (iii) we want to be to treat verbs uniformly, but also in a way that is able to handle the contrast between seek-type verbs on the one hand and kiss-like verbs on the other hand.

The solution is to treat all verbs as expressing relations between individuals and second-order properties and then, for a certain subset of verbs (the kiss-like ones), define a meaning postulate that says that those verbs also express relations between individuals. (This is an instance of what in Montague Grammar is sometimes called "generalizing to the worst case". The same occurs when we assume that John has some really high type, akin to every boy, rather than just type e.)

So the way to do that is to say, if delta is the relation that is normally expressed by kiss, i.e. delta is a relation between individuals and second-order properties, then kiss also expresses a relation S between individuals.

The reason why X has to range over second-order properties (and not first-order properties) is simply because X is the second argument of delta, and delta is the relation between individuals and second-order properties normally expressed by a verb (any verb, in Montague Grammar).

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  • Thank you very much for reading and answering my question. Please note, I understand the motivation of the MPs, as well as the general treatment of TVs as intensional objects in MG, then "cutting down" the extensional ones via MP2. However, I'm not sure if your last paragraph answers my question: I understand why the usual phrasing of MP2 quantifies over SO properties, but I am trying to derive whether my alternative, with quantification over (objects interpreting) "terms", which are then "made intensional" via ^- operator, yields an equivalent axiom to MP2 or not. (1/2) – Bert Zangle Jun 24 '15 at 19:00
  • I am basically looking for a model that is a counter example, or a proof (sketch) that the two are equivalent. (2/2) – Bert Zangle Jun 24 '15 at 19:00
  • "The reason why X has to range over second-order properties (and not first-order properties) ..." Probably difficult to see my presentation of the formula, but I suggest X ranges over sets of FO properties (i.e. "terms"), not FO properties themselves. – Bert Zangle Jun 24 '15 at 19:04
  • I see -- sorry, I misunderstood. Proving that your formulation is equivalent would involve proving that whenever delta(x, X) is true (where X has the type from MP2v1), then X = Caret_Up Y for some Y (with the type from MP2v2). Whether this is true or not, I'm not sure, but it's not in general true that X = Caret_Up Caret_Down X (see p. 128 of Gamut), so it can't be that Y = Caret_Down X. – Brian Buccola Jun 25 '15 at 0:50
  • Thanks again. That's the track I am on as well. I can't shake off the feeling that a complete answer depends on what the quant. range over, i.e. whether we assume full semantics or Henkin semantics, but I can't quite put my finger on it yet. By the way, I'd vote up your answer if I could, but I lack the necessary rep. to do so. – Bert Zangle Jun 25 '15 at 14:30
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I decided to quickly write down an answer to my own question, on the off chance that someone else will come to LSE with a similar question at some point in the future.

MP2 v1 (the original formulation) and MP2 v2 (my alternative suggested above) are not identical - see the following (informal) argument by counter-example:

Note that MP2 v2 quantifies over variable X of type corresponding to terms, as opposed to quantification over variable X of type of intensions of terms like in the original MP2.

(a) The intension of a term, say, "caret_up [lambda X. exist x. P(x) and caret_down X(x)]" (for some constant symbol P), is not generally a constant function, since the extension of a term (i.e. the set of properties denoting the term in w) can vary depending on the world of interpretation.

(b) In IL, the intension of a variable is necessarily a constant function, since for any X of VAR_L, the interpretation of X in w only depends on variable assignment g, which is independent of the world of interpretation. So, for any X of VAR_L, the intension of X, or equivalently: caret_up X, is a constant function.

As we see from (b) then, MP2 v2 is a weaker statement than MP2 v1, since it only expresses the equivalence of relation Delta holding between x (of type (e)) and a constant function of type (s, ((s, (e, t)), t)) with relation Delta* holding between two objects of type (e).

As per argument (a), not every statement we want to fall under MP2 is of this form however, i.e. the intension of a (quantified) term is not generally a constant function.

So for any term s.t. the intension of the term is not a constant function, MP2 v2 will not apply, i.e. in models of MP2 v2, "Delta(x, caret_up [lambda X. exist x. P(x) and caret_down X(x)]" is not generally equivalent to "[lambda X. exist x. P(x) and caret_down X(x)](caret_up [lambda y. Delta*(x, y)]), while in models of MP2 v1, the equivalence holds in general (see Thm. 1, p. 177, Gamut II).

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