Representing constructive and non-constructive claims in formal semantics

Question

I came across Montague's treatment of "John seeks a unicorn". He distinguishes de re and de dicto interpretations of it on the basis that the de dicto meaning does not entail the existence of any unicorn; in other words, John is seeking some set of properties that answers to the conventional description of a unicorn, but this property may not have any instances.

I'm interested in a slightly different example: "Jack seeks a woman". Here there is no doubt that women do in fact exist, but it seems ambiguous as to whether Jack is seeking a particular woman, such as his wife, or any woman, i.e. he just desires some female company. Attempting to represent this sentence in classical predicate logic does not support this distinction.

As I see it there are several options:

1. We could assimilate it to Montague's unicorn and say on the second interpretation it means Jack seeks some set of properties common to women.

2. We could go 'modal' and say that on the first interpretation, Jack seeks a set of possible worlds in which he is with a particular woman, while on the second he seeks a set of possible worlds in which he is with any woman.

3. We could perhaps express the distinction using intuitionistic logic, as the difference between (∃x)(Woman(x) ∧ Seeks(jack,x)) and ¬¬(∃x)(Woman(x) ∧ Seeks(jack,x)).

4. Maybe we could say there is no semantic ambiguity at all and the difference in the readings is just a matter of pragmatics.

I would appreciate any illumination that theoretical linguistics might shed on it. Thanks.

Answer 1

I'd say the answer lies in the co-reference between the object being predicated the existence and the verbal-object status over.

Recall the formalization of the two readings:

de dicto: seek'(j, ^λQ∃x[unicorn'(x) ∧ Q{x}])
de re: ∃x[unicorn'(x) ∧ [seek'(j, ^λP[P{x}])]]

In the second reading, the x that is existentially quantified over is coreferential between the unicorn' and the seek' predication, with the two predicates being conjoined by conjunction. So whenever there is such an x which is a unicorn, this must be the same x that John is seeking. If it is true that there is some unicorn (possibly even several ones) but this none of these unicorns is the one that John is seeking, then the sentence becomes false, because what the existential quantifier ranges over is the conjunction of both the extensional unicorn predication and the reference in the intensional seeking construction.

To make it less formal, the de dicto reading can be reworded as "John seeks something which is a unicorn", whereas the de re reading can be reworded as "There is a unicorn which John seeks", where it is clear that what John seeks must refer to the same entity that is claimed existence about.

Applied to your example, if it is clear that there is some woman, this doesn't entail the de re formalization - because if none of these women is what John is looking for, then the coreference between the (extensional) woman'(x)and the intensional seek'predication is incorrect. So as soon as you choose the second reading, you automatically imply that among the women that exist, one of them is what John is looking for. If you want to exclude that possibility, i.e. leave open the option that there is some woman but not that kind of woman that John seeks, you would have to go with the de dicto reading, since in that case, as soon as one of the conjuncts in the intensional expression becomes false, the existence of something which is both a woman and seeked by John is negated.

So my answer is: Whether among the women that definitely do exist there is one which John seeks is the same distinction as between whether any women exist at all, due to the coreference between the variables being predicated over. The the dicto reading is underspecified w.r.t. existence and doesn't exclude the possibility of there always (extensionally) being women, whereas the de re formalization commits you to the women claimed existence over being that woman that John seeks, if the sentence is to become true.
If you choose the weaker statement, in which the existence of a woman doesn't entail the existence of the thing John is seeking, then this is only represented by the de dicto reading.

I'm not sure whether one could tweak the representations in such a way that one would get a formulation of precisely the case you described, where the existence of unicorns is extensionally true but the coreference with the entity involved in the intensional predication is not. At the moment I can't think of one.

Answer 2

Perhaps it is pragmatic, in a sense, and has to do with who is responsible for identifying what Jack seeks as a woman. Consider for instance "Jack seeks a woman, although he may be mistaken about her sex," where it is I the speaker who identifies her as being a woman. When a sentence is assigned a representation that makes explicit various conversational participants (or others), that is a "performative analysis".