Thursday, 1 October 2015

Rudolf Carnap's Modal Semantics

Intensions and Extensions

Rudolf Carnap believed that an expression “meant” what it means because it possess both an extension and an intension (or a reference to a class and to a sense). He believed that this approach would effectively bring the meanings of expressions back down to earth (away from abstract propositions). Extensions are, after all, collections of concrete objects and intensions are the means to get at those extensions (or at the objects within those extensions). That means that concrete objects (or collections of such objects) are important when it comes to extracting the meanings from expressions.

In the traditional picture, it's abstract objects that fundamentally determine the meanings of expressions. Of course Carnap still allowed the existence and use of abstract intensions (if they are in fact abstract). That means that expressions have an extension for the parts and then the whole; and also an intension for the parts and then the whole. (In this, he basically updates Frege.)

How did Carnap define the term ‘intension’? Are intensions essentially concepts or predicates/properties?

Again, intensions, whatever they are, help determine extensions.

Carnap on Necessity

Carnap saw modal logic in terms of semantics. That effectively means that modal properties don't belong to objects qua objects: they belong to statements qua statements. Statements have modal properties, not objects.

Traditionalists would say:

                  “A is necessarily B.”

That seems to be a reference to things; not to sentences or words. Thus Carnap offers this alternative:

                  “The statement ‘A is B’ is a necessary one.”

What's necessary in the above isn't the properties of things: it's the properties of sentences. Necessity is only a property of sentences (e.g., in conceptually analytic statements).

Thus “A is B” implicitly (or elliptically) refers to statements which generate a further necessary statement. That means that necessity is either stipulated and/or conventional. That is, it's a property of the sentences and concepts we use (i.e., de dicto necessity): not the things we refer to with those sentences (i.e., de re necessity).

Thus, according to Carnap, modal logic is a branch of semantics. That means that necessity is generated (as it were) by meanings or concepts which are themselves properties of the sentences that assert certain necessities. If necessity were a property of things and their properties, then it would be ontology (rather than semantics) which deals with such properties.

Carnap on Analyticity

Carnap accepted analyticity. He saw analyticity in terms of what he called “individual concepts”.

Thus we can explain necessity by way of analyticity. That is, in

                    a = b

both a and b “are concepts of the same individual”; not variables for concrete objects. Thus perhaps we should write:

              [Ca] = [Cb]

Thus if a and b are concepts of/for the same individual, we can create, from this, an analytic statement. That is, in the often-used example

             “All bachelors are unmarried men.”

the words “bachelors” and “unmarried men” both refer or denote different concepts of the same set of individuals (i.e., they have the same extension).

Thus, according to Carnap, modal contexts were really disguised quotational contexts. That is 

  i) “Bachelors are necessarily unmarried men.”

is transformed into:
          ii) “The statement 'Bachelors are unmarried men' is analytic.”

Thus i) is an example of de re necessity. It's a statement about concrete objects: bachelors and unmarried men. And ii), on the other hand, is an example of de dicto necessity: it's a statement about the concepts BACHELOR and UNMARRIED MAN.
In terms of necessity, it's not necessary that the concrete objects unmarried men are bachelors. In terms of stipulational necessity, it is necessary that the concept UNMARRIED MAN implies the concept BACHELOR.
Alternatively, we can used second-order logic to get the above points across:

           (c) (UcBc)

However, that conditional (in which 'c' means concept) doesn't really capture necessity. Thus we can have a biconditional instead:

           (c) (Uc Bc)
Carnap, Rudolph. (1947) Meaning and Necessity

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