logicandlanguage.net

This kind of sounds like what Lewis in Plurality calls a Lagadonian language, in which every object is the name of itself. That still seems to kind of limit the propositions to the merely actual, but if we accept one Lewisian doctrine, why not another?

Actually, that's what Berit said when I mentioned it to her.

Giving every real number a name merely gives us a set of names we cannot access as well as the numbers we could not access. This seems to exacerbate, not eliminate.

Is he saying there must be sentences expressing the proposition in an actual language? If not, then you can get around the cardinality objection if you assume that for any proposition you want to have, there is a possible language that has an expression for it. If there are uncountably many possible languages, then the cardinality objection fails. (Of course, it may still be true that there is no one language--even a merely possible one--which has sentences for every proposion.)

Hmm allowing possible languages seems like it might almost reduce his position to a sort of triviality, depending on what sorts of languages are taken to be possible. (I haven't actually read his paper yet.)

If we can have a language with names for every real number (as model theorists very often want to do) then why not have a language whose sentences have no logical structure to them, that directly express propositions? Maybe something like Wittgenstein's slab-language, but much bigger?

To me it doesn't necessarily sound that problematic that there will be no proposition corresponding to "x is a number" for the undescribable reals. Is there any particularly strong argument that there should be such a proposition, other than certain symmetries of the situation? If there's no procedure with which we can identify the particular real involved, then what is the actual content of the proposition?

Also, when I typed "Hmm" followed by three dots in the previous comment, it denied me for questionable content. I'm guessing that you've accidentally added the string with an m followed by some dots to your spam filter - this caused problems on my blog for a bit too.

Ah, that explains why it's denying me for questionable content as well. I'll have to have a word with it.

Hmm allowing possible languages seems like it might almost reduce his position to a sort of triviality, depending on what sorts of languages are taken to be possible. (I haven't actually read his paper yet.)

What do you mean by triviality in this instance, Kenny? If it just means that all the things we might want to say are propositions, then that seems like an advantage.

If we can have a language with names for every real number (as model theorists very often want to do) then why not have a language whose sentences have no logical structure to them, that directly express propositions? Maybe something like Wittgenstein's slab-language, but much bigger?

Well, in practice the language would be unlearnable, (I'm not saying fragments of it couldn't be learned, but there'd be an awful lot of vocab.) But I don't see the problem with saying such languages exist. The sentences of that language don't express the same propositions as sentences of ours though; on King's view propositions inherit their structure from underlying sentential structure (that is, not surface structure, but the input to semantic interpretation.)

To me it doesn't necessarily sound that problematic that there will be no proposition corresponding to "x is a number" for the undescribable reals.

But the problem isn't just for undescribable reals - it's for unnamable reals. If you use a description to pick out the object of your sentence the proposition has a different structure than if you used a name.

To me it doesn't necessarily sound that problematic that there will be no proposition corresponding to "x is a number" for the undescribable reals. Is there any particularly strong argument that there should be such a proposition, other than certain symmetries of the situation?

Yes. Lots of such propositions are true. There are many objects which satisfy the predicate "x is a real number", even though I don't have any way of replacing "x" with a name for the object.

If there's no procedure with which we can identify the particular real involved, then what is the actual content of the proposition?

It's a structure containing the number, and a property of the number: being real. The fact that we can't identify the object of a proposition doesn't seem to have anything to do with the content. I take it that there are lots of true propositions about objects no-one will ever identify.

I think the triviality I meant is that if we're allowed to stipulate languages whose sentences express whatever propositions we want, then the notion of a proposition would have to be antecedent to the notion of a sentence. But it sounds like his idea is that a proposition automatically inherits its structure from any sentence that expresses it, so it just isn't possible for any sentence in a language like that to express any of the propositions that we express in our language. The more I think about it, the more that sounds like a good consequence of his account. There might be a worry about what sorts of contents could be directly encoded in the lexicon though.

In our lexicon, we've got plenty of contents like entities, functions from entities to truth-values, functions from entities to entities, and the like. One might think that all of these should actually be replaced by functions from possible worlds to the things I mentioned above, so that a predicate doesn't have to pick out the same set of entities in every possible world. In the completely simple language I mentioned above, either there would be lots of unexpressed structure in the sentences (in which case the ends of the branches in the syntactic trees could have contents like those in our languages, except that we'd just have trouble getting from the surface structure to the tree) or else the trees would themselves be completely simple, so that the lexical items would have to express functions from possible worlds to truth-values directly. I guess this means that two sentences can have the same truth-conditions but express different propositions (one structured, and one unstructured), but perhaps that's not surprising.

You're right - I hadn't paid attention to the distinction between describing a real and naming it.

Now, if you want there to be a proposition for any object X and property P that asserts that X has P, then I guess it makes sense to resort to possible languages of the sort the model theorists would use (or at least, possible extensions of English). Certainly, it seems that for any object we can identify, we can stipulate a name for that object, and we can probably talk about possible names for objects even that we can't identify or describe. It's less clear whether we can stipulate lexical items to stand for arbitrary properties, but I don't see a principled reason why not. Thus, for any object and any property, whether or not we actually have words for the object and the property, we can stipulate a name for the object and a predicate for the property and string them together as an ordinary sentence, which will (on King's account, it seems) express the proposition that that object has that property. (At least, if there are any sentences at all in our language whose trees are that simple - maybe every tree has more structure than this, so that all propositions end up being more complicated, but still, we should be able to do what it sounds like you want to do.)

I love the picture of the surfer on the website--nice job Gillian!

A few quick thoughts:

First, I agree with Kenny that it isn't clear that there is any strong argument for the claim that for every real there is a proposition to the effect that it is a number or whatever. Gillian says there is: lots of such propositions are true. But that seems to me to beg the question. What is the argument that there are many such true propositions? Here we need to distinguish two claims: 1) there are lots of reals out there possessing the property of being a number--call these "real facts". They are just mathematical facts--mathematical objects possessing properties. For every real, there is the fact consisting of it possessing the property of being a number, just like for every human there is the fact of her possessing the properety of being an animal; 2) for each real, there is a proposition to the effect that it is a number, and this proposition is made true by the real fact in question. I agree with 1 and I am not sure that there is any strong argument for 2. I think it is easy to illegitimately move from the (to me) obvious existence of the real facts (1) to the existence of propositions that those facts make true (2). Hence it is easy for it to seem as though it is obvious that there are all these true propositions. But I don't think it is at all obvious.

Second, there are expressions designating the reals (I hesitate to call them names--in effect, I think they are definite description-like), namely infinite decimal expansions. So there is a language whose sentences express propositions about all reals: English combined with infinite decimal expansions. Despite the doubts expressed above, this would seem to give us propositions about the reals.

Third, a number of you are right that sentences of English won't express the same propositions as sentences of the slab language on my view. I think this is a good consequences of the account, as some of you do.

Fourth (cryptically), I don't think that propositions exist in virtue of merely possible languages.

Finally, this is a very timely exchange for me because I am just finishing a book on my view of propositions (which has evolved a bit from the view in the paper you are discussing). I hope it will clarify all the questions you raise. It will be out with OUP late this year or early next.

Hi Jeff,

It's nice to have you on the site. We're reading your paper "Tense, Modality and Semantic Values" at the Wash U Logic and Language reading group next week!

Your comment helped me to get a bit clearer on what I think are two different background pictures, one of which I was assuming, and one of which I think you and some of the other commenters are working with.

On one picture - the one I'm familiar with - true propositions and facts are just the same thing. Hence sentences:

(1) It's true that John loves Mary

and

(2) It's a fact that John loves Mary

both imply that the proposition that John loves Mary is a true proposition, from which it follows that the proposition exists.

One nice thing about this view is that it allows for a very straightforward explanation of our use of "that" in expressions like "it's a fact that..."/"the fact that John loves Mary implies that..." Just like "Sam believes that...", "Kate said that..." and "it is necessary that...", these expressions are used to make claims about propositions: that they are necessary, that they were said, that they are believed, and that they are facts.

But on this picture, the consequence of your account of the structure of propositions in "Structured Propositions and Complex Predicates" (according to which there are no propopsitions unless there are sentences which express them) is radical. If there is no sentence which expresses the proposition that p, then the proposition that p does not exist, and and so there is no fact that p. What's radical about the position is that it makes what facts there are depend on what languages there are. Some people might like this, I just don't. It seems right to me to say that the facts exist independently of our ability to express them.

On the alternative picture, the one you describe above, facts and true propositions are distinct kinds of thing, and so the consequence of your view has much less radical implications: there might not be any propositions which no sentence expresses, but that doesn't have any implications for what facts there are.

Call the first picture the binary picture and the second the ternary picture.

I'm not wholly opposed to the ternary picture (I haven't thought about these issues very much) but I do wonder:

- how come we say "it's a fact that..." if facts are never propositions?

- what are the primary bearers of truth on this picture? I'm not just inclined to say that the facts are what they are, independently of our ability to express them. I'm also inclined to say that there may be truths that we can't express. Or that none of our sentences express. Obviously I'm can't be talking about sentential truth. I thought I was talking about propositional truth, i.e. that I was saying that there are truths (true propositions) that we can't express. Perhaps the ternary picture has to deny this, but perhaps it could hold that:

- truth is (in the first place) a property of facts.

- we say a propositon is true if it expresses a true fact.

- we say a sentence is true if it expresses a true proposition.

Anyway, I really liked the paper and I think the existence of the two different background pictures explains some of the differences in reaction to this consequence of the view.

Thanks for the welcome and the kind words Gillian!

(Btw, let me know about any thoughts you or others have about 'Tense, Modality and Semantic Values'.)

Your comment is right on target and really helps clear things up.

A couple things before answering your two questions:

1. Right, I think that the fact of my being human is different from the proposition that I am human and that the former makes the latter true. (I hate to say this but it turns out that propositions are themselves facts on my view. However, the fact of my being human is a different fact from the fact that I claim is the proposition that I am human)

2. There still is a further question about the English locution 'the fact that p'. It is a substantive claim that these expressions designate the things I am calling 'facts' (and not e.g. propositions). So it may be that the English word 'fact' does not apply to (all) the things I call 'facts' (in my technical sense)

Now, to your questions

"-- how come we say "it's a fact that" if facts are never propositions?"

There are at least two possibilities (there are more): 1) the that clause here designates a proposition and some propositions satisfy the English predicate 'is a fact'. Of course, as I said above, that doesn't mean that propositons are the things I am calling 'facts'; 2) that-clauses sometimes designate what I am calling 'facts'.

Some evidence for the second view is given by examples like the following:

That the bear was angry caused him to attack Alan.

The fact that the bear was angry caused him to attack Alan.

*The proposition that the bear was angry caused him to attack Alan.

One way to explain this is that the that-clause in the first sentence ( and 'the fact that te bear was angry' in the second) designates a fact and not a proposition. Note that these sentences also provide prima facie evidence that 'the fact that p' and 'the proposition that p' don't designate the same thing. But I am not saying I actually think all this (though I a tempted to every other day).

Your other question:

"- what are the primary bearers of truth on this picture?"

I think propositions are. I would deny that there are truths that we can't express (except in the sense that e.g. there are sentences too long to ever utter and etc.). When we say, as you are inclined to, that there may be truths we can't express (and intend something more robust than that there are sentences too long to utter), I'm gonna say we are trying to give voice to the claim that there are facts that we can't express truths about.

Oops, forgot to sign in. Don't accept that other comment I made. Will just repeat it here

Hmm, you want there to be propositions involving reals we can't express in language? Does this even make sense on any non-platonic view of math? If so what would it mean?

Thanks Jeff, I have a much better understanding of where you're coming from now.

So what's wrong with the suggestion that a language could infinite strings? Then you'll have a numeral for every real.

And why isn't English such a language? When we write something like:

0.16538761...

who's to say we're writing a 13-character word, as opposed to abbreviating an omega-character one?

[Oops. I accidenally posted this on the wrong page originally, but it belongs here.]

Logicnazi:

You can show that there are more reals than (finitely long) sentences by diagonalization.

I'm curious as to what you consider a "non-Platonic view of math"? Does Cantor's Theorem make sense on a "non-Platonic view of math"? If not, then there's an awful lot of math that a non-Platonist will have to throw out.

I've read with interest your posts. Most interesting is the proposed distinction sentences/propositions/facts. Let me add a couple of ideas (and excuse my bad English).

It is really not clear that there are more reals than can be defined by means of finite phrases. I'll point out two reasons:

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1. It is only so for platonists, not for definitionists (or predicativists), who require for an object to possess mathematical existence that it be definable. Cantor's theorem does not make sense for most definitionists since they do not admit the existence of arbitrary sets.

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2. I'm not quite sure the definitions of reals or the propositions we can express about them (by means of finite sentences) form a countable set. Richard paradox can be employed to argue against that possibility. Of course, the 'signifiers' (in Saussurian terms) of those definitions and propositions are enumerable, since they constitute a subset of the enumerable set of all finite letter-strings. But perhaps the signifieds cannot be bijected to their signifiers (I'm perfectly aware of how odd this can sound).

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The argument is (roughly) that if there were an enumeration of all definitions of reals, there would be a definable enumeration of them (by reference to some length-alphabetical order) and then there would be a definition not in the enumeration, obtained by Richardian diagonalization.

This is customarily taken as a paradox and dismissed by appealing to the fuzziness of the concepts involved but I'm not sure this is the right way out.