General reflections on "General Semantics": this is a fully intensional, "generalize-to-the-worst-case" system. It appears that what inspired Lewis to generalize to the worst case was intensional adjectives like "alleged." He writes that an adjective like [[alleged]] is a function from adjectives to adjectives--but
"It is best to foresake extensions and Carnapian intensions...most adjectives do not have extensions. What is the set of things to which `alleged' applies?...[there is none.]
In general, an adjective takes a common noun to make a new, compound common noun; and the intension of the new common noun depends on the intension of the original common noun in a manner determined by the meaning of the adjective...
More generally, let us say that an appropriate intension for a [D_ node]...is any n-place function from \pi intensions to \tau intensions.
...We will call these intensions for derived categories compositional intensions." (27-28)
The important thing here is the distinction between compositional intensions and Carnapian intensions; I suspect the distinction corresponds to the Prof.-MacFarlane-recommended Dummettian distinction between Assertoric and Ingredient senses. I have a fair idea at this point of what the compositional intension is. But what is a Carnapian Intension?--
"We call the truth-value of a sentence the extension of that sentence; we call the thing named by a name the extension of that name; we call the set of things to which a common noun applies the extension of that common noun. The extension of something in one of these categories depends on its meaning and, in general on other things as well: on facts about the world, on the time, place, speaker and surrounding discourse of the utterance, etc. It is the meaning which determines how the extension depends upon the combination of other relevant factors. What sort of things determine how something depends on something else? Functions, of course...We have now found something to do at least part of what a meaning for a sentence, name, or common noun does: a function which yields as output an appropriate extension when given as input a package of the various factors on which the extension may depend. We will call such an input package of relevant factors an index; and we will call any function from indices to appropriate extensions for a sentence, name, or common noun an intension.
Thus an appropriate intension for a sentence is any function from indices to truth-values; an appropriate intension for a name is any function from indices to things; an appropriate intension for a common noun is any function from indices to sets. The plan to construe intensions as extension-determining functions originated with Carnap (1947 and 1963). Accordingly, let us call such functions Carnapian intensions. But whereas Carnap's extension-determining functions take as their arguments models or state-descriptions representing possible worlds, I will adopt the suggestion (Montague, 1968; Scott, 1970) of letting the arguments be packages of miscellaneous factors relevant to determining extensions."
...That was a mouthful. But the idea should be clear now. The distinction between Carnapian intension and compositional intension appears here to come from the assumption that an intensional adjective like `alleged' is still an adjective (aka a `common noun.') It is a common noun with no Carnapian intension because there is no set of alleged things. (Not even one that is enumerable with the help of free variables for w, t, and the like...and the problem is obviously not due to an insufficiently rich index.) Yet `alleged' obviously has a compositional intension because it combines in a systematic way with another predicate g, such that the meaning (compositional or Carnapian intension) of [[alleged g]] depends systematically on the compositional intensions of [[g]] and [[alleged]].
...What to say here? It seems like the problem comes in part from taking the term ``adjective" at face-value. [[Alleged]] and [[Communist]] are of different semantic types: one is in D_et, et, and one is in D_et. They have the same output types--both [[g]] and [[alleged g]] are of the same type--but that means they are a type match only in their ranges.
It seems right to say that for Lewis, anything with a Carnapian intension has a compositional intension, but not vice-versa (`alleged' is a case in point.) But with our emendation, we could just as easily say that the Carnapian intension of [[alleged]] is a function from indices to a function from adjectives to adjectives. In fact, that is the kind of semantic entry we give it in H & K, Ch 13:
[[alleged]]^w, t = \lambda f \in D_et . \lambda x \in D . \forall w' \in W compatible with what is alleged in w at t, f(x) = 1.
Here is Lewis defending the ``generalize to the worst case" strategy: basically, it preserves the underlying simplicity of FA.
"I promised simplicity; I deliver functions from functions from functions to functions to functions from functions to functions. ... Yet I think no apology is called for. Intensions are complicated constructs, but the principles of their construction are extremely simple.
...In some cases, it would be possible to find simpler intensions, but at exorbitant cost: we would have to give up the uniform function-and-arguments form for semantic projection rules. We have noted already that some adjectives are extensional, though mot are not. The extensional adjectives could be given sets as extensions and functions from indices to sets as Carnapian intensions...'Grunts,' for instance, is an extensional verb phrase; its extension at an index i is the set of things that grunt at the world and the time given by the world coordinate and the time coordinate of the index i.
There is no harm in noting that extensional adjectives and verb phrases have Carnapian intensions as well as compositional intensions. However, it is the compositional intension that should be used to determine the intension of an extensional-adjective-plus-common-noun or extensional-verb-phrase-plus-name combination. If we used Carnapian intensions, we would have a miscellany of semantic projection rules rather than the uniform function-and-arguments rule....Moreover, we would sacrifice generality: non-extensional adjectives and verb phrases would have to be treated separately fro the extensional ones, or not at all. This loss of generality would be serious in the case of adjectives.
I am a little confused by what is going on here. The point is surely that we need intensions to be the arguments taken in FA, because while e.g. Ext(renate) = Ext(cordate), their intensions differ, and it is that difference that is responsible for the nonequivalence of e.g. [[alleged renate]] and [[alleged cordate]]. But it seems that if we assign [[renate]] and [[cordate]] mere Carnapian intensions, we are ok--since their Carnapian intensions differ. Lewis must be thinking that they don't differ (they don't differ if you fix the w variable at w = actual world...)
Hmm. I'm not sure this is giving me the ingredient/assertoric sense distinction I need.
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