Substitutional Quantification

What's the difference between substitutional and objectual quantification? I'll use the old-fashioned round brackets for objectual quantifiers and square brackets for substitutional quantifiers. The standard interpretations are

OB) (x)A is true under an interpretation I iff for some new constant t, A(x/t) is true under all interpretations I' that differ from I at most in what they assign to t.

SUB) [x]A is true under an interpretation I iff for all constants t, A(x/t) is true under I.

Assume that predication (and the truth functors) is interpreted in one of the usual ways, for instance by ruling that Ft is true under I iff I(t) is in I(F). Then if (x)A is true under any interpretation, [x]A is also true under that interpretation. The converse holds iff every interpretation assigns every object in the domain to some constant.

Sometimes, substitutional quantification is defended by the claim that unnamed objects should be banned from logic. For example, Reinhard Kleinknecht writes in his 1998 article "Referentielle und substitutionelle Logik" (my translation):

An argument against unnamed objects is that one can't properly talk about such objects. At best, their existence can be derived, they are an ontological fiction, not a factum. [p.204]

To me, that sounds quite bizarre, but suppose we are convinced, and consequently rule out unnamed objects. Is that a reason to use substitutional quantifiers? Not at all. As I just noted, once we have ruled out unnamed objects, the two kinds of quantification coincide. One might find (SUB) slightly simpler, but there really is no longer any substantial difference. (The difference is more like the difference between recursion on truth and recursion on satisfaction in the interpretion of objectual quantification.)

In particular, the meta-logical properties of predicate logic with substitutional quantification all carry over to predicate logic with objectual quantification plus the ban on unnamed objects: The resulting logic is not compact and neither positively nor negatively decidable (in other words, it is incomplete and undecidable). The reason, in case you forgot, is that the set of sentences At for all constants t now semantically entails (x)Ax (and [x]Ax), but no finite subset does.

So given the ban on unnamed objects, there is no difference between substitutional and objectual quantification. What if we lift that ban? Then the two quantifications really differ: While (x)A says that all objects in the domain satisfy A, [x]A only says that all named objects do. But what is such a strange quantifier good for? What's so special about the named objects? Why not introduce another quantifier { }, so that {x}A is true iff all unnamed objects satisfy A, or, for that matter, all objects thought about by a logician born on a Thursday?

All along, I've assumed a classical interpretation of predication. I've argued that on this account the substitutional quantifier either coincides with the objectual one or it is philosophically uninteresting. Things change if we adopt a deviant semantics for (at least some) predications. And that's where substitutional quantifiers have usually been employed (back in the 1960s, when they were hip).

To take an example from Quine, we might interpret sentences like '{x:Fx} = {x:Gx}' as notational variations of '(x)(Fx iff Gx)'. Then we could also allow quantifications like '(Ey)(y = {x:Gx})'. What does this mean? It doesn't mean that some object in the domain is identical with the object denoted by '{x:Gx}', for there is no such object. Here comes the substitutional interpretation: '(Ey)(y = {x:Gx})', or rather '[Ey](y = {x:Gx})', is true iff 't = {x:Gx}' is true for some pseudo-constant t of the form '{x:Fx}'. In other words, it is true iff for some formula F, '(x)(F iff Gx)' is true.

To take an example from Marcus, we might interpret sentences like 'Pegasus is a winged horse' in some way such they are true even though we don't want to have Pegasus in our domain of objects. For instance, we might want to say that 'Pegasus is a winged horse' is true iff the sentence 'Pegasus is a winged horse' occurs in some book about Greek mythology. Then we could allow a quantification '[Ex](x is a winged horse)' and interpret it as true iff for some pseudo-name t, 't is a winged horse' is true, that is, occurs in the book.

In both cases, the really interesting thing is not the substitutional interpretion of the quantifiers but the deviant interpretation of the respective predications.

Some have argued that some or all quantifiers in ordinary language should be interpreted substitutionally. In most cases, the latter view gets the truth conditions wrong: 'most stars are unnamed' is true even though it is certainly not the case that most of the named stars are unnamed. So the only viable option is that while ordinary quantification is sort of objectual, we still use, or at least could use, sort of substitutional quantifiers in special kinds of discourse. (In fact, I don't think ordinary quantifiers are very much like objectual quantifiers, because those are tailor-made for extensional languages.) I have no good reason against that. I agree with van Inwagen though that if those alleged quantifications are really to be interpreted as substitutional quantifications than one can't at the same time deny that they are in some sense really just ordinary quantifications over linguistic entities.

Oh well, I really don't have anything new to say on this subject.


# on 21 February 2006, 14:30


is there anyone who can just in short to explain me what does it mean susbtitutional and objectual quantification. I have found that in the work of Quine,and not having that discovered I can not go on. Please, I am really in need. Is there anyone who can make a difference between those ones. Just let me give an example.
Thank You in advance.
Please, the answer send on my e-mail address.

# on 21 February 2006, 14:40

very briefly: the difference here concerns the *interpretation* of the quantifier. On the objectual interpretation, "(Ex)Fx" is true iff some object has the property expressed by "Fx"; whereas on the substitutional interpretation, "(Ex)Fx" is true iff there is some name "a" such that "Fa" is true.

# on 21 February 2006, 22:58

and one can add: Quine's standard argument to endorse the objectual q. version is a "weak" and straightforward one: We do have domains where we do not have names for all objects in the domain.
Frankly, what wo explained is totally OK as always - but I think there might be a metaphysical issue of some weight behind this distinction.
BTW: What is the idea behinds Kripke's "Is there a problem about substitutional quantification?" and BTW BTW: Is it planned to have a "Collected Papers" for Saul Kripke - I have no knowledge of the paper mentioned because it is so hard to get.


# on 22 February 2006, 15:32

hey, I agree that some deep metaphysical (or even metametaphysical) at least appear to be lurking here. Re the Kripke paper, is the Evans & McDowell volume where it first appeared that hard to get?

# on 10 April 2006, 22:43

Can someone explain why de re quantification requires the assumption of transworld identity?

and on another note, can you explain to me why 'it is not the case that' is truth functional, but 'it is necessary that' is not?

# on 23 May 2006, 22:20


many, many excuses upfront, the one thing I just don't want to do is hijack your blog (so please delete this entry if it is not fitting): I am looking for some kind of "HELP! I need advise"-section for serious questions in analytic philosophy (and the usenet is not the place where I can find that).

Can somebody please point out to me where the semantics of Fodor sits vis a vis the model- theoretic and the proof-theoretic approach?
I am not asking for help on a term paper of something, I just want to understand this.
Any help & hint is much appreciated

# on 24 May 2006, 16:17

hi M, this comments thread seems to have degenerated into a general help thread anyway, so your question is not off topic.

Though I'm afraid I don't really understand it. You're talking about Fodor's theory of intentionality for the language of thought? And the model-theoretic versus proof-theoretic approach to what -- to characterizing logical validity? Fodor's theory of intentionality, if I recall correctly, combines a causal/counterfactual semantics for primitive concepts with a Tarskian compositional semantics ("truth theory") for complex expressions. Formal semantics like Tarski's are often called model theories. From this perspective, one could say that the truth theory provides the semantics, and the causal/counterfactual theory the meta-semantics that explains how the primitive terms got their semantic values. One could also model a semantics on proof theoretic considerations, but that would lead to a kind of conceptual role semantics which Fodor wouldn't want at all. As I said, I don't really know what you mean.

# on 25 May 2006, 11:05


Thanks for this - I think I have all the distinctions I need on board now. I was confused by the oberservation that the semantics looks like model theory (as you said), but JF often comes up with proof theory (see quotes below). This can be seen as answers to two different questions. Not sure however if JF would like to opt for a pt-semantics for logical constants and how this meets with the mt-approach.

"It would not be unreasonable to describe Classical Cognitive Science as an extended attempt to apply the methods of proof theory to the modeling of thought (and similarly, of whatever other mental processes are plausibly viewed as involving inferences; preeminently learning and perception.)" Fodor/Pylyshyn, 1988

"The basic idea in cognitive science is the idea of proof theory, that is, that you can simulate semantic relations - in particular, semantic relations among thoughts - by syntactical process."
Fodor interview in Baumgartner&Payr(1995). Speaking minds


# on 29 May 2006, 00:47

ah, I'm not sure Fodor is talking about the semantics of thoughts here at all. He believes (that cognitive science presupposes) that cognitive processes are best understood as operations on mentalese symbols, and that these operations are independent of the meaning those symbols might carry. This is compatible with the view that the meaning of the symbols is determined by a combination of causal/counterfactual relations for atoms and Tarskian rules for complex expressions.

So when Fodor here speaks of "proof theory", I think he only means "purely syntactic (and algorithmic) symbol manipulation", not something like conceptual role semantics.

Though you're probably right that Fodor accepts some kind of CRS for logical vocabulary. I don't remember his exact position here, but it seems unlikely that some mentalese symbol stands in a relation of asymmetric counterfactual depence to, say, negation.

# on 30 August 2007, 04:23


I've been knocking something around, and I wouldn't mind your opinion, if you've got the time.

It seems that if we are careful about the interpretations, both should be read subjunctively. When we speak of "substitution instances", what we really mean is, "were these names substituted into 'F . .', one such instance would be true". And with objectual quantification assigning the (objects in the domain as) values to variables is similary counterfactual. No actual assignment is done. Not surprising, since for large domains it is not clear that it could be done without using a metaquantifier.

The difficulty for the objectual account is that if you read the assignment condition subjunctively, the rest of the account must be read subjuntively as well - and that leads to some very odd semantics (incoherent, even).

The way to resolve the problem (if there is a problem), is to use a hybrid interpretation - something like this. For some Domain D and some list of variables x1,x2 . ..

"(Ex) Fx" - means -

"If it were the case that the members of D were assigned as values to the variables x1, x2, . .., and were those variables substituted in the expression 'F . . .', at least one such instance would be true."

or something like that.

This is obviously a substitutional account, but one that does not require that all objects in the domain have names. In fact there need be no names at all. Adjusting Quine then, to be is to be able to be the value of a bound variable - to be nameable, in other words - and who would object to that?!

It is a bit unpalatable to read ordinary quantified expressions as subjuntive conditionals, but if the "standard" account are committed to this anyway. . . .

# on 10 February 2011, 10:25

Hello, do you agree with this:
[x](Ey). x=y ; not(x)[Ey].x=y?
Best Regards,F.A.

Add a comment

Please leave these fields blank (spam trap):

No HTML please.
You can edit this comment until 30 minutes after posting.