Section 3 of Glanzberg’s paper gives an overview of the ways in explicit and common-or-garden-contextual restrictions on quantifiers work (as background to a discussion in later sections about how “background domains” are fixed). This section isn’t intended to be more than just a quick set of reminders about familiar stuff, so we can be equally speedy here.
Going along with Glanzberg for the moment, suppose the “background domain” in a context is M (the absolutist says there is ultimately one fixed background domain containing everything: the contextualist says that M can vary from context to context). More or less explicit restrictions on quantifiers plus common-or-garden-contextual restrictions carve out from this background a subdomain D (so that Every A is B is interpreted as true just when the As in D are B, and so on). How?
Explicit restrictions are relatively unproblematic. But how is the contextual carving done? There are cases and cases. For example, there is carving by ‘anaphora on predicates from the context’, as in
1. Susan found most books which Bill needs, but few were important,
where ‘few’ is naturally heard as restricted to the books that Susan found and Bill needs. Then second, there is ‘accommodation’, where we rely perhaps on some Gricean mechanisms to read quantifiers so that claims made are sensible contributions to the conversational context. For example, as we are about to leave for the airport, I reassuringly say that, yes, I’m sure,
2. Everything is packed
when maybe some salient things (the passports, say) are in plain view in my hand and my keys are jingling in my pocket. Here, my claim is heard, and is intended to be heard, as generalizing over those things that it was appropriate to pack, or some such.
There’s a third, rather different way, in which context can constrain domain selection, that isn’t a matter of domain restriction but rather a matter of how, when an object which is already featuring prominently enough as a focal topic of discourse at a particular point, “we will expect contextually set quantifier domains to include it”. (Though I guess that this point has to be handled with a bit of care. The taxi for the airport arrives very early: we comment on it. “But,” I say, “we might as well leave in the taxi now. Everything is packed” The quantifier of course doesn’t now include the taxi, even though it is the current topic of discourse.)
Well, so far, let’s suppose, so good. But how are these reminders about common-or-garden contextual settings of domains going to help us with understanding what is going on in fixing ‘background’ domains? The story continues …
10 thoughts on “Absolute Generality 9: Restricting quantifiers”
…unless there’s more than one unicorn in your garden. If there are two unicorns in your garden, then the first premise is certainly true. The second premise has two possible interpretations (and, correspondingly, two possible formalizations). If we formalize it in the traditional Russellian way, then it says “There is a unique unicorn in your garden, and it is white,” — then it is FALSE.
If we formalize it in such a way that it is within the scope of the existential quantification of the first premise, then we’d be saying (in effect) “There is a white unicorn in my garden”.
I don’t suppose either of these formalizations appeal to you — but I can’t find any meaning to attach to the sentence ‘The unicorn is white,’ which cannot be formalized by one of these two.
Having said that, you emphasise that your original point was about inference. In which case, it seems that the second premise contains further quantificational information, viz. that the unicorn is unique. Perhaps this is your point? That the first (indefinite) quantification narrows the domain to permit the second (definite) quantification? In which case, I stand by my original claim that they should be formalized together since the first is a proper part of the second — I think one naturall hears a suppressed ‘in the garden’, i.e. ‘The unicorn in the garden is white’, which formalizes in such a way that the first premise is a straight-forward consequence.
>> when you talk about ‘the unicorn in question’, you presuppose that there is some particular unicorn in question
Is that true? My original point was about inference.
There is a unicorn in my garden
The unicorn is white
Ergo, something white is in my garden
Now you may argue that the use of the word ‘the’ is unacceptable or whatever. But it seems a perfectly valid inference to me. After all, if (a) there really is a unicorn in my garden and (b) the unicorn really is white then (c) surely there is something white in my garden.
Truth-value and content are different things. If we consider ‘There is at least one cat on the mat. The cat is fat’, I don’t think that the latter sentence has a determinate truth-value.
This is what I meant by calling the use of the definite article “unacceptable”. Here is an extract from the OED’s definition of ‘the’: “Marking an object as before mentioned or already known, or contextually particularized (e.g. ‘We keep a dog. We are all fond of the dog’).” Thus when you talk about ‘the unicorn in question’, you presuppose that there is some particular unicorn in question — which there may not be when one uses the indefinite article.
As for the business with proper names, that’s a whole other kettle of fish/can of worms.
>>This strikes me as an unacceptable use of the definite article since we have not offered anything to define the individual in question.
That I find baffling. In what sense is the use unacceptable? What counts as acceptable use? How do we ‘define’ an individual?
>>Ultimately, though, it seems that the question hinges on whether distinct propositions should be formalizable as distinct formulae in an extensional language. If that’s a fixed point for you, then I think we might have to agree that one man’s ponens is another man’s tollens.
There you have it. But there are many arguments for formalising separately (given a formalisation possible at all).
For example in
Ex (Fx & Gx)
it makes sense to ask whether the whole thing is true or false. It makes NO sense to ask whether Fx is true, or Gx. Whereas, if it is true that there is a unicorn in my garden, then we can reasonably ask whether it (or the unicorn, or that unicorn, or the unicorn in question &c) is white or not.
Also what about
There is a unicorn in my garden called ‘Fred’.
Fred is white.
seems a perfectly acceptable use of a proper name. Indeed, in fiction and in history, all proper names are introduced to us this way.
“Proper names may acquire intelligibility either through our being introduced in some way to their bearers, or by being incorporated into a story – “Once upon a time someone lived in Rome who was called “Julius Caesar”, and this Julius Caesar conquered Gaul, crossed the Rubicon, and so forth”. We believe that Julius Caesar crossed the Rubicon in the sense that we believe the relevant part of this story; there is no one to whom we are related as believing the whole story of him, but we identify the subject of the later part of the story, ultimately, as simply the “someone” with whom the story begins. … there is no one of whom I say that he crossed the Rubicon; but I do say that someone crossed the Rubicon, and I have previously said (among much else) that this same someone was called “Julius Caesar”. “
Why take separable formalization (as it were) to be a necessary condition of expressing distinct propositions? I prefer the contra-positive conclusion that distinct propositions can’t always be formalized separately.
One major advantage of this is that I have absolutely no problem representing the inference you mention: ExGx is a straight-forward consequence of Ex[Fx&Gx].
The ambiguity that I am finding (which you aren’t) lies in the fact that if we take ‘A cat sat on the mat’ to be synonymous with ‘At least one cat sat on a mat’, then the second sentence doesn’t get its reference: At least one cat sat on a mat. The cat is fat. Which cat is fat?
This strikes me as an unacceptable use of the definite article since we have not offered anything to define the individual in question.
Ultimately, though, it seems that the question hinges on whether distinct propositions should be formalizable as distinct formulae in an extensional language. If that’s a fixed point for you, then I think we might have to agree that one man’s ponens is another man’s tollens.
>>The propositions are distinct,
OK then you clearly cannot formalise in the way you stated. That was the point.
>>but the phrase ‘the soldier’ lacks reference outside of the context provided by the previous sentence. The sense of ‘the soldier’ in vacuo is insufficient to determine reference. That’s the sense in which they are “linked”.
>>Lastly, in your example where S1 says something about ‘a soldier’ and S2 says something about ‘the soldier’, the difficulty lies in the ambiguity of what S1 says.
It is pretty clear to me that the two sentences taken together are entirely UNambiguous. The meaning of the two sentences is such as to have a certain logical implication. I.e. S1 says that a soldier was returning home, and S2 says that the soldier had travelled far, and if what S1 says is true, i.e. if a soldier was returning home, and if what S2 says is true, i.e. if the soldier had travelled far, then it logically follows that some person who was returning home had travelled far. No ambiguity at all.
>>If there is some particular soldier of whom he is speaking, then that is the soldier to whom S2 refers. If he is not speaking of any particular soldier, then S2’s utterance lacks reference. The two formalizations you mention seem to reflect this ambiguity.
No they don’t, because neither captures the conventional meaning of the two sentences, read in sequence, namely the meaning which includes the logical implication just mentioned. Suppose I say
There is a unicorn in my garden. It is white.
Anyone can understand this, and everyone understands that if both sentences are true, i.e. if there is a unicorn in my garden, and if it is true that it is white, it logically follows that something in my garden is white. You need to formalise these separate sentences in a way that preserves the implication, which is difficult.
I didn’t intend to provide a translation scheme for the schematic “Some A is B. The A is C.” Rather, I was concerning myself with the particular example of “A soldier was returning home. The soldier had travelled far.” A different example — e.g. one with the notoriously tricky ‘that’ clauses — will require a different strategy.
Which brings me back to my point about the word ‘The’. There is no royal road between grammatical form and logical form.
The propositions are distinct, but the phrase ‘the soldier’ lacks reference outside of the context provided by the previous sentence. The sense of ‘the soldier’ in vacuo is insufficient to determine reference. That’s the sense in which they are “linked”.
Lastly, in your example where S1 says something about ‘a soldier’ and S2 says something about ‘the soldier’, the difficulty lies in the ambiguity of what S1 says. If there is some particular soldier of whom he is speaking, then that is the soldier to whom S2 refers. If he is not speaking of any particular soldier, then S2’s utterance lacks reference. The two formalizations you mention seem to reflect this ambiguity.
>>The propositions expressed by the two sentences are linked.
In what sense, then, are they 2 propositions? Geach considers the ‘bound variable’ solution in Mental Acts and rejects it. He imagines a story in which the pronoun ‘he’ occurs in many successive sentences.
S1 says that a soldier was returning home.
S2 says that the soldier had travelled far.
How do you get the quantifier outside the ‘that’ clause. E.g.
S1 says that Ex, x was a soldier and x was returning home.
is no good for obvious reasons. Nor is
Ex S1 says x was a soldier and x was returning home.
because it commits us to the existence of an x that S1 is talking about.
The propositions expressed by the two sentences are linked. They should be formalized together since the phrase “The soldier” is behaving like a bound variable within the scope of an existential quantifier represented by “A soldier”. Hence, the sentences ought to be formalized as Ex[Rx&Fx].
The schematic version is misleading because ‘The’ can play different logical roles. In the soldier example it is playing the part of a demonstrative pronoun (and hence is synonymous with ‘That’ in the example); in other contexts it plays the role of a quantifier.
Even the ‘contextual setting’ seems a bit problematic to me. Consider:
Some A is B. The A is C
We agree that the two separate sentences express two distinct propositions, which can be rendered formally, separate of one another. We agree that the two sentences imply ‘some B is C’. But (in order to make the inference formally valid) we can’t resolve the second proposition in any ‘Russellian’ way, i.e. as saying that every A is C. For example
A soldier was returning home. The soldier had travelled far.
Clearly you can’t read the second sentence as saying that every soldier had travelled far. Alternatively you could suppose that there is some sort of ‘domain restriction’ going on. But that is very difficult to make explicit, moreover there are all kinds of counter-examples to consider. Suppose we restrict the background domain to the single soldier. Then the second sentence is read as ‘every soldier [in the domain] had travelled far’. But intuitively the first sentence doesn’t presuppose such a narrow domain. We talk of ‘a soldier’ as though one soldier among many. Moreover we could easily continue the story to say ‘another soldier was returning home with him. And so on.