Coarse-grained meanings and impossible worlds

To some extent, one can account for semantic phenomena without assigning meanings to words or sentences or thoughts. For instance, we might say that beliefs and other attitudes are relations to sentences, i.e. to strings of symbols. Roughly, to believe a sentence S is to be disposed to utter (or assent to) S (or some translation of S) under certain conditions. When people talk to each other, such dispositions may be transferred: after hearing me utter the sounds "it is raining", you acquire the disposition to utter those sounds yourself. Apart from communication, we can also account for things like synonymy and analyticity. Roughly, two sentences are synonymous if necessarily, anyone who stands in the belief relation to one of them also stands in the belief relation to the other. There is no compositional semantics in this picture, because there is no semantics at all. But there might be recursive rules for translating from one language to another.

What's missing in this picture are any relations between language and the non-linguistic world. Why does it matter to me whether you are disposed to utter the sounds "it is raining" rather than, say, "mitis laining", or "it isn't raining"? Why do we sanction the utterance of certain sounds and praise the utterance of others? How do people manage to get into a cinema or a plane by uttering certain sounds? What is the difference between a competent speaker of a language and someone who merely parrots everything they hear? A good answer to questions like these requires some link between linguistic expressions and non-linguistic matters.

At a minimum, we might draw a distinction between sentences that are true and sentences that are false. Equivalently, we might assign to every sentence one of two non-linguistic meanings, "the true" and "the false". We could then do a bit of compositional semantics with these meanings: we could give general rules relating the truth-values of certain complex sentences ("A or B") to the meanings of their parts. And we could say that there's a norm to utter only sentences that are true (or known to be true), which explains why it matters whether I say "it is raining" or "it isn't raining", and why it's not trivial to learn a language.

Of course, belief can hardly be just a relation to a truth-value, nor are truth-values the only things that are communicated from speaker to hearer. So here we might appeal once again to sentences: what is communicated is a truth-value under a particular sentential guise.

Even better, we might use more fine-grained entities, say, sets of possible worlds, as sentence meanings. Equivalently, we might distinguish sentences not only by their truth-value, but by their truth-conditions. If we're lucky, we could then also give a compositional semantics for some intensional constructions. Belief could be understood as a relation to a set of possible worlds under a sentential guise. Very roughly, to believe a set p under guise S is to (a) act in a way that would be reasonable if the actual situation were in p, and to (b) be disposed to utter (or assent to) S under certain conditions. Now we can explain why I can bring you to fetch an umbrella, or stay inside, by uttering the sounds "it is raining". And we can explain why it's useful to have linguistic conventions.

Perhaps at some point one or two people thought that one can account for all semantic phenomena in terms of truth-conditions, or sets of possible worlds. But, setting aside tendentious definitions of "semantic phenomena", it's obvious that this won't do, because sets of possible worlds are rather coarse-grained (as are, in a different way, Russellian propositions and other good candidates for semantic values). It makes a difference whether I say "17*21=357" or "if it's raining, then it's raining": the first might be helpful when the second is not. And there are plenty of linguistic constructions where the possible-worlds meaning of complex sentences is not determined by the possible-worlds meaning of their parts.

So why not go one step further and use very fine-grained meanings, so that we can account for all semantic phenomena in terms of meanings? Let S1 and S2 have different meaning whenever there is an embedding X (e.g. "Jones proved that ...") such that X(S1) is true and X(S2) is false, and whenever different communicative roles can be performed by uttering S1 and S2. This would mean that any two sentences whatsoever would have different semantic values. A simple and popular way to provide such fine-grained meanings is to define them as sets of possible or impossible worlds, where the "worlds" are simply sets of sentences (or some other entities from which such a set can be read off). For example, the semantic value of "it is raining" is the set of sets of sentences containing the sentence "it is raining". We can then say that belief is only a relation to a set of (possible or impossible) "worlds", without mentioning any sentential guise, and we can specify the meaning of every sentence, no matter how hyperintensional, entirely in terms of the meaning of its parts.

But now we've come full circle. The alleged semantic values in the final account are really just the sentences themselves, decorated with set-theoretic distractions. The difference between a sentence S and the set of sets of sentences containing S is much like the difference between S and the unit set of S: there is a trivial 1-1 map between the two kinds of thing. Any theory involving the one can trivially be converted into a theory involving the other. For example, have a look at the compositional semantics with possible and impossible "worlds". In ordinary possible world semantics, the meaning of "it is not raining" is the complement of the meaning of "it is raining": a possible world is in the semantic value of the first sentence iff it is not in the value of the second. Not so if we have impossible "worlds" where it is both raining and not raining. In the impossible-worlds framework, we don't need to look at the meanings of the parts of a sentence at all. We know from the start that the meaning of "it is not raining" is precisely the set of sets of sentences containing the sentence "it is not raining". Compositional semantics has become utterly trivial.

Semantics with very fine-grained (hyperintensional) semantic values is really the same thing as semantics without semantic values. As before, we thereby lose every contact to the non-linguistic world. Elementary set theory is enough to know the hyperintensional impossible-worlds content of every French sentence. But this won't help you in the streets of Paris. If you only know that "il pleut" means the set of sets of sentences containing "il pleut", you will have no idea whether to pack an umbrella or sunscreen when people utter those words. You won't actually know what the sentence means.

The lesson is that we shouldn't be over-ambitious. Let's have coarse-grained semantic values that provide us with non-trivial language-world relations. And let's concede that not all semantic phenomena can be accounted for on this level. Sometimes, for example, the meaning of a complex sentence will be determined not only by the meaning of its parts, but also by the sound and shape of its parts. This is not a problem. Those non-semantic features are after all not a secret to competent speakers. If you know a language, you will trivially know the hyperintensional impossible-worlds meanings in addition to the possible-worlds meanings, so we shouldn't be surprised if the possible-worlds meaning of some complex constructions also makes use of impossible-worlds meanings.

So we do need hyperintensional, impossible-worlds meanings. I'd rather we call them by their right name: they are simply the sentences themselves, individuated as strings of sounds and shapes. But we should not let these things replace our coarse-grained meanings.

Comments

# on 30 June 2011, 19:04

Interesting stuff, but it may be worth pointing out (maybe too obvious) that sounds and shapes won't do it alone - otherwise 'if every woman likes a man then every woman likes a man' which has one trivial meaning, one non trivial meaning but the same shapes and sounds will be problematic. Same with 'they saw her duck but not her duck'.

# on 07 July 2011, 16:51

Adam 's obvious problem has an obvious solution: Simply take hyper-intensional, impossible-worlds meanings to be ordered pairs consisting of sentences (strings of types of sounds and shapes) and their syntactic analyses. This revised version of wo 's semantics correctly predicts that 'if every woman likes a man then every woman likes a man' is semantically associated with at least two different meanings. However it may be doubted whether this revision captures the difference in cognitive siginifinace between the trivial and the nontrivial reading of the sentence.

# on 08 July 2011, 06:13

'In ordinary possible world semantics, the meaning of "it is not raining" is the complement of the meaning of "it is raining": a possible world is in the semantic value of the first sentence iff it is not in the value of the second. Not so if we have impossible "worlds" where it is both raining and not raining. In the impossible-worlds framework, we don't need to look at the meanings of the parts of a sentence at all. We know from the start that the meaning of "it is not raining" is precisely the set of sets of sentences containing the sentence "it is not raining".'

I'm a bit unclear about how you take these two things to be related. The failure of the first thing (about negation and complements) in the impossible worlds framework doesn't seem to be what is behind the second thing (about "not having to look"). Doesn't the second thing hold equally of the possible worlds framework, if possible worlds are taken to be maximal consistent sets of sentences? If that's right, then do you have some other conception of possible worlds which is in the background here?

# on 08 July 2011, 08:38

Adam: I'm not quite sure I see the problem you see there. But if there is one, then yes, I'd go for something like celarent's solution.

Tristan: you're right that my worries apply just as much to possible-worlds accounts in which worlds are taken to be maximal consistent sets of English sentences. I should have been more explicit about this. I think of worlds as maximally specific properties or states of affairs.

# on 11 July 2011, 16:00

'The lesson is that we shouldn't be over-ambitious. Let's have coarse-grained semantic values that provide us with non-trivial language-world relations. And let's concede that not all semantic phenomena can be accounted for on this level. Sometimes, for example, the meaning of a complex sentence will be determined not only by the meaning of its parts, but also by the sound and shape of its parts.'

I'm not quite sure what the lesson is supposed to be. As I understand it the lesson entails two things: (a) Each (declarative) sentence (type) S is equipped with a pair consisting of a coarse-grained semantic value (a set of possible worlds) and a fine-grained semantic value (the sentence (type) itself); and (b) there are embeddings X and X* taking CP-complements such that 'C(complementizer) + S' refers to the coarse-grained semantic value associated with S, if 'C + S' is preceded by X and 'C + S' refers to the fine-grained semantic value associated with S, if 'C + S' is preceded by X*.

If this is part of the lesson, then I'd like to know what values the variables 'X' and 'X*' are intended to take. Some early versions of two-dimensionalism (such as the one outlined in Graeme Forbes' 'Languages of Possibility') were inclined to assume that 'X' and 'X*' range over modal constructions like "It is necessary" and attitude verbs like 'believe' respectively. However if wo's semantics accords with early two-dimensionalism in this regard, it faces well known problems arising from 'mixed contexts', i.e. contexts, where attitude and modal constructions cooccur.

Here's one: Consider the argument "It is necessary that 1=1; celarent believes that 1=1; so, there is something necessary celarent believes." This argument is valid. However (assuming it entails (a) and (b) as well as early two-dimensionalist doctrines concerning the values of 'X' and 'X*') wo's semantics implies that it is invalid, since according to it, the occurrences of 'that 1=1' to be found in the premises are not coreferential.

# on 12 July 2011, 11:23

Hi celarent,

good point. I don't think "x believes that S" should be analysed as stating a relation between x and some sort of entity denoted by "S". Quantification into belief contexts is therefore a bit tricky. I think it works somewhat like the quantifier in: "philosophers talk about all kinds of bizarre things like zombies and round squares and empty boxes full of statues". I don't have a clear view on what is going on here. Perhaps the quantifiers are substitutional in some sense.

I'm not familiar with the discussions you mention. What is the scenario in which the premises of your argument would be true according to Forbes and the conclusion false?

# on 15 July 2011, 01:16

Hi wo,

so your (non-relational) account of the semantics of 'believes' and its relatives entails that believing is not a relation and therefore that-clauses (or CPs more generally) never function as singular terms referring to entities of some sort (I call them 'propositions'). To be sure, an interesting (Priorian) case can be made for such a view: A singular term t designates x only if for each singular term t' having the same designatum as t , ...t... is true iff ...t'... is true, and the biconditional is falsified by the pair consisting of 'Pierre hopes that France will win the world cup' and 'Pierre hopes the proposition that France will win the world cup'; so, that-clauses do not function as singular terms referring to propositions. However interesting as this argument is, its first premise is false as instances of apposition show (I think this point is due to Horwich): While Aloysius Bertrand is the author of 'Gaspard de la nuit', the sentence 'The French writer Aloysius Bertrand wrote poems' is not equivalent to 'The French writer the author of 'Gaspard de la nuit' wrote poems’, since the former is true and the latter truth-valueless because ill-formed.

As far as quantification into belief contexts is concerned, I don't think non-relationalists about 'believes' should opt for substitutional quantification. In giving natural language quantifiers a substitutional interpretation one must be extremely careful in characterizing its substitution class in order to see to it that one's predictions about truth-conditions and logical consequence-relations come out true. In particular one must ensure that the members of the substitution class are appropriately rigid with respect to those parameters of circumstances of evaluation that can be shifted by operators interacting with the quantifiers. For instance, if the substitution class of the quantifier associated with the determiner 'a(n)' is specified in such a way that it includes the (modally) non-rigid 'the teacher of Alexander', the following sentence comes out valid, which clearly it isn't: 'If (necessarily, there is an x such that x = the teacher of Alexander), then there is an x such that(x = the teacher of Alexander and necessarily, x = the teacher of Alexander).' The same subtleties arise in connection with treating propositional quantifiers substitutionally. Consider the apparently invalid argument: ‘Aloysius believed that Charles is a hypocritical reader. Aloysius believes everything he once did. So, Aloysius believes that Charles is a hypocritical reader.’ If the propositional quantifier occurring in the second premise gets a substitutional interpretation, one must ensure that the sentence ‘Charles is a hypocritical reader’ doesn’t belong to the quantifier’s substitution class, for otherwise the argument comes out valid. (My qualms about treating natural language quantifiers substitutionally of course just repeat what Mark Richard has said on this matter).

Rather than giving the quantifier in your example sentence a substitutional treatment, one could treat it as an objectual quantifier, whose domain contains theories (sets of interpreted sentences) about zombies etc. (where a theory is a theory about zombies, if its members amount to an implicit definition of the predicate ‘is a zombie’). So one could analyze your example sentence as ‘philosophers talk about all (objectual quantification) kinds of bizarre theories like theories about zombies and theories about round squares and theories about empty boxes full of statues’. Maybe this idea can provide a basis for developing an account of quantification into attitude contexts being compatible with non-relationalism about ‘believes’ and a non-substitutional treatment of propositional quantifiers. A first shot inspired by modal fictionalism might be: ‘There is something p such that x believes that p’ can be anaysed as ‘According to the (false) relational theory about ‘believes’ there is a proposition p such that x stands in the relation of believing to p’.

Turning finally to your question concerning my argument: This argument comes out invalid, if the conjunction of relationalism about ‘believes’, (a), (b), and early two-dimensionalist doctrines about the values of ‘X’ and ‘X*’ is true (so your non-relational account is not affected by my objection). The invalidity of the argument given the truth of the conjunction simply follows from the fact that, given the conjunction, the argument exemplifies a trivially invalid argument scheme, namely ‘Ft; t*Rt**; so, $x(Fx&t*Rx)’ (t, t*, t** are metavariables ranging over terms and $ is the existential quantifier). The argument exemplifies this scheme, according to the conjunction, because the conjunction entails that the occurrences of ‘that 1=1’ appearing in the argument’s premises aren’t coreferential and so must be translated into distinct terms. Thus my argument’s fallacy is more or less of the same nature as the one being involved in the argument ‘Aloysius scratched his back; Charles scratched his back; so, there is something they both scratched.’

# on 15 July 2011, 06:53

Hey celarent,

doesn't every one-premise argument instantiate the trivially invalid schema 'p, therefore q'? For a particular argument to be invalid, I would have thought that there have to be conceivable cases in which the premises are true and the conclusion false. I agree that on your rendering of its logical form, the argument in question is not logically valid, i.e. not valid merely in virtue of its logical form. But why does it have to be *logically* valid?

I'm not convinced by the Richardian points against the substitutional interpretation. I think the allegedly invalid argument "Aloysius believed that Charles is a hypocritical reader. Aloysius believes everything he once did. So, Aloysius believes that Charles is a hypocritical reader." definitely has a reading on which it is valid. But you're right that the full story is more complicated, for example because there is also an invalid reading. Unlike the "early two-dimensionalists" you mention, I don't claim to have a semantics of attitude reports.

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