chapter 2: Fine's assertion that the two desiderata aren't compatible and response

  I've written much on this blog, and I think it's time for a bit of recycling. Here's a working out of Fine's assertion that the two proposals won't quite mesh.
  As far as the second part of chapter is concerned, our assignment is to show that with at least one semantic theory we can satisfy both desiderata. It's instructive to note that it seems that, as I think I'll have demonstrated in chapter one, we can satisfy both proposals if we're considering Carnap's system S₂. The present concern is to show that we can use an interpretive truth theory to do the same.
  The basic idea is laid out in Ludwig's "A Conservative Modal Semantics" (perhaps in the service of another goal), but it merits attention here as well. The straightforward case is for numerals and the numbers they name. First off, in the compositional meaning theory Ludwig's working in, a thesis of direct reference is maintained. He holds this because, in the theory, meanings are given by systematically interpretting object language sentences in a metalanguage. This is done with the help of a recursive method provided by the grammar of the object language (recursive interpretation procedures like those for 'and', 'or', 'there exists' and 'for all') and reference axioms (such as the referent of 'bob' is bob) and predicate axioms ('x je crvino' is true iff x is red, for instance). Essentially, we can say that reference is unmediated (direct) iff, in our particular semantic theory, we can give reference axioms for the object language terms. I guess this means that if there are singular referring terms in the object language and we want to use as a compositional meaning theory an interpretive truth theory, we must hold that reference is direct.

[It seems that there's no problem so long as the referents of the singular referring terms are abstract, but if we're trying to pick out concrete individuals, what happens then? It seems there are puzzles for this sort (any sort?) of compositional meaning theory when referents are concrete. Perhaps a challenge to this sort of view might come from one who held the view that reference to concreta is always mediated. How might the interpretive truth theory accomodate this? In the end it might not be quite so bad because once we acknowledge that reference to concreta must be mediated, then we're in the business of dealing with predicates when we pick things out. If we're in the business of dealing with predicates, it seems that (at least on the most promising view of concepts) we can handle any sort of modal claims about those things that we refer to -- or at least the ones that have some sort of bearing on the predicates that are involved in the act of reference. In any case, the meaning theory that we're using can give us answers presuming all the modal properties had by each thing in the extension of the predicate are part of the meaning of the predicate. Now when is unmediated reference most plausible? It seems it's in the case of demonstratives or indexicals. Ultimately, there may be a need to worry only about the reference axioms needed to refer to physical objects qua physical objects, the modally relevant properties being dictated in the act of reference to a physical object qua kind as in 'the table is red' making use of the predicate terms 'is a table' and 'is red'. This seems to explain why Ludwig was at such pains to account only for abstracta like numbers for which direct reference really seems to be a the only live option and physical objects qua physical objects.]

With direct reference on board, we guaranteed that we can satisfy Fine's the first proposal. First off, we can give an account of semantic uniformity in the context of a theory of meaning. So semantic uniformity needn't be quite so mysterious since we can give an axiomatic treatment of how meanings of sentences are achieved at the root from recursive axioms and reference and predicate base axioms. It seems reasonable to require semantic uniformity with both singular and predicate terms.
  Also, it seems that we can satisfy the proposal that singular terms occurring in the instances of quantified sentences have the right core content on the interpretive truth theory as a compositional meaning theory. Recall that the numerals are singular terms which refer to the numbers directly because of the reference axioms. It also seems that the reference axioms are such that they "induce" certain relations of those things that are referred to. For example, the first few axioms of the theory are 'ref('0') = {x: x ≠ x}', 'ref('1') = sucessor(0)', 'ref('2') = successor(successor(0))'. We define reference just in terms of the referent of '0' and successor. If we define the relation '>' which is such that it is appropriate in sentences like 'a > b' where 'a' and 'b' are stand-ins for numerals, and 'a > b' is true iff (1) the number named by 'a' is the successor of the number named by 'b' or there is 'c' such that 'a > c' and 'c > b'. We see that sentences with numerals and '>' are true just in virtue of the meaning. If we assume that to know the meanings of the terms in sentences like 'a > b', is to have the concepts associated with the constituent terms of those sentences, then it seems like these sentences will have the right conceptual content. This content is had by the relations borne by those things referred to in virute of the reference axioms that serve to pick them out.