Overview/motivation
The interpretation of logical quantifiers such as ‘there exists‘ and ‘for all‘ and the associated ontological implications of these interpretations is (apparently – or at least it was) an important topic in philosophy. I encountered these interpretations a few years ago when reading Haack’s ‘Philosophy of Logics’, but didn’t pay much attention. Quine is a central figure here – e.g. his famous (within philosophy, anyway) saying ‘to be is to be the value of a variable‘ concerns this issue.

I realised recently that I’ve been thinking about somewhat similar issues, albeit in a more ‘applied’ context, e.g. when talking about the interpretation of ‘for all’ in formulating ‘schematic’ model closure assumptions (see here). So, here a few notes on the topic. The obvious disclaimer applies – I am not a philosopher. I’m just hoping to get a few basic conceptual ideas straightened out in my head, so that I may better formalise some arguments useful in science and statistics. I am not aiming to ‘solve’ the general philosophical problems! Corrections or comments welcome.

The problem
Here is a brief sketch of the issue as it arises for the existential quantifier. The question is: how should we interpret statements of quantified logic of the form

We have (or there exists??), in fact, two options.

Objectual: There exists an object x such that it has property P.

Substitutional: There exists an instance of a statement having the general form P(x), obtained by substituting some name, term or expression etc for x, that is true.

In the former, the emphasis is placed on objects and their possession of properties, in the latter, the emphasis is placed on statement forms and the truth of particular statement instances.

In particular, in the latter, substitutional, case truth is a property of statements ‘as a whole’ and need not relate to ‘actual objects’ occurring in the sentence.

The classic example is that, on the objectual reading,

(S): Pegasus is a flying horse

can be taken to mean

(S via Obj.): “There exists an object (e.g. Pegasus) which is both a horse and can fly”

We would normally take this as false, since no such object ‘really exists’. On the other hand, on the substitutional reading we may take this to mean

(S via Subs.): There is a true statement of the form ‘x is a flying horse’ (e.g. Pegasus is a flying horse)

The justification for taking this as true is that, given our knowledge of mythology (certainly a real subject itself), we may take this to express a true statement without further commitment to (or even ‘attention to’) the existence of the ‘objects’ or ‘properties’ involved.

Thus the substitutional interpretation refers to the truth or falsity of resultant sentences/statements ‘as a whole’ (and the forms of such sentences/statements), while the objectual intepretation refers to the existence of objects with properties, and hence in a sense gives a more ‘granular’ interpretation.

Both seem to me to involve subtle issues of context, however – e.g. we can presumably only interpret the above statement instance as ‘true’ in the substitutional interpretation given the context of mythology.

Marcus and Kripke offered defenses of the substitutional interpretation while Quine advocated the objectual interpretation (hence ‘to be is to be the value of a variable’).

There is obviously much more to this topic – see e.g. Haack’s book, the SEP. For now, I note that I find myself reasonably sympathetic to the substitutional interpretation (or perhaps both interpretations, depending on the circumstances). This appears to be roughly consistent with what I was attempting to express here.

There also seems to be something here that depends on whether, given the ‘function’ P(x), we focus on the ‘domain of naming’ or on the ‘codomain of statements’. These issues hence also seem to connect with the issue of how to interpret (proper) names e.g. as ‘mere tags’ (Marcus), ‘rigid designators’ (Kripke), ‘definite descriptions’ (Russell) or as ‘predicates’ (Quine). The substitutional interpretation is generally allied with the view of proper names as ‘mere tags’ or as ‘rigid designators’, and I have become quite fond of (what I understand by) this idea. It would be too much to go into this in any detail at the moment, however.