Wolfgang Schwarz

First bunch of question: properties and semantic values

This is a follow-up to yesterday's entry.

Andy Egan argues that functions from worlds and times to sets of things are ideally suited as semantic values of predicates, even better than mere sets of things.

I agree, and so would Lewis. In fact, Lewis would say that functions from worlds and times are still too simple to do the job of semantic values. There are more intensional operators in our language than temporal and modal operators. Among others, there are also spatial operators and precision operators ("strictly speaking"). So our semantic values for predicates should be functions from a world, a time, a place, a precision standard and various other 'index coordinates' to sets of objects. This is more or less what Lewis assigns to common nouns in "General Semantics" (see in particular §III). Other predicates like "is green" that do not belong to any basic syntactic category get assigned more complicated semantic values: functions from functions from indices to things to functions from indices to truth values. In later papers, Lewis argues that we may need several of the world and time coordinates and, more importantly, a further mapping that accounts for context-dependence (and to deliver the kind of truth-conditions needed in his theory of linguistic conventions). Thus for predicates, we get something like a function from centered worlds to functions from functions from possibly several worlds, times, places, precision standards, etc. to functions from such worlds, times etc. to truth values. (Alternatively, if we go for the 'moderate external strategy' (Plurality) and reserve "semantic value" for 'simple, but variable semantic values' ("Index, Context and Content"), we can say that the semantic value of a predicate in a given context is the value of the function just mentioned for that context.)

So if Egan is right that "what properties do (among other things, but first and foremost) is provide semantic values for predicates", it seems that Lewis would reject his proposal to identify properties with functions from worlds (and times) to sets as much too simple. Even more would he reject the alternative proposal to identify these properties just with sets of possibilia.

Egan adds in a footnote that Lewis also took the main role of properties to serve as semantic values for predicates. That would be odd. Let's see. Egan points at "New Work", where Lewis writes:

It is properties [previously defined as sets of possibilia] that we need [...] to provide an adequate supply of semantic values for linguistic expressions. Consider such sentences as these:

1. Red resembles orange more than it resembles blue.
2. Red is a colour.
3. Humility is a virtue.
4. Redness is a sign of ripeness.

Prima facie, these sentences contain names that cannot be taken to denote particular, individual things. What is the semantic role of these words? If we are to do compositional semantics in the way that is best developed, we need entities to assign as semantic values to these words [...]. Perhaps sometimes we might find paraphrases that will absolve us from the need to subject the original sentence to semantic analysis. [...] But even if such paraphrases sometimes exist -- even if they \emph{always} exist, which seems unlikely -- they work piecemeal and frustrate any systematic approach to semantics" (p.16f.).

As far as I can tell, Lewis is talking about names here, not predicates. He argues that we need sets of possibilia as referents of "Redness" and "humility", not as semantic values of "--- is red" and "--- is humble". So no contradiction there between Lewis' theory of properties and his semantics.

Nevertheless, some questions remain.

First, if we are to do compositional semantics in the way that is best developed, the semantic values we should assign to singular terms are not referents. We need intensions: functions from indices to things, or better, functions from contexts to functions from inices to things, or maybe more complicated constructions (see "General Semantics", §VIII, Plurality, fn.31, p.41f.). So how could Lewis think that for property names, referents will do?

Second, why the fuzz about compositional semantics anyway? Why doesn't Lewis say that because humility is a virtue and redness a sign of ripeness, there are such things as humility and redness? (How could they be virtues or signs if they don't even exist?) This would also fit his remark about paraphrases. We know that not all apparent singular terms and quantifiers carry genuine ontological commitment. Some are just misleading idioms, easily paraphrased away. But this, Lewis notes, doesn't seem to be a good strategy for our talk about properties. All this has little or nothing to do with formal semantics.

Third, how are Lewis' properties related to his semantic values for predicates? Presumably Lewis would not mind calling the semantic values of predicates 'properties', as Egan does: "[I]t is wrong to speak of the role associated with the word 'property', as if it were fully and uncontroversially settled. [...] The question worth asking is: which entities, if any, among those we should believe in, can occupy which versions of the property role?" (Plurality, 55). Nevertheless, it seems a little odd that properties, things like redness, play no role in the semantics of "this is red".

There is a more general question here: How are Lewis' philosophical analyses of modal or temporal predication in terms of counterparts and temporal parts related to his semantic analyses? It is often said that in Lewis' semantics, modal operators are interpreted as quantifiers over worlds. But this is not what we see in his semantics (e.g. in "General Semantics", "Index, Context and Content" and Plurality 40-50). Nor do we see any mention of temporal parts or counterparts there. I'm not sure, but I guess Lewis regarded philosophical analysis as a different project than the development of a general compositional grammar for English. All truths are made true by the distribution of fundamental properties -- but that does not mean that all English sentences, deep down under their surface grammar, only involve predicates for fundamental properties (and no names at all).

More questions later, I hope.