Once we have acknowledged that semantic representations play a crucial role in any theory of linguistic meaning and we are inclined to neglect possible ontological objections, I think we are in a position to appreciate the relevance of the questions I am trying to address.
Actually, classical first order logic, which, however expanded or modified, is at the base of standard modelizations of natural language semantics, was originally designed to deal with exactly the same ontology as RA, i.e. with the set of natural numbers. In particular, formal systems of arithmetic based on CL and incorpo-rating into their language non-logical symbols to define p.r. functions have been widely investigated and employed. Let F OA (First Order Arithmetic) be such a language where no restriction on formulæ instantiating its induction schema has been imposed13and let F OAt be the system obtained from F OA by adding to its alphabet the individual term t, which behaves exactly like in RAt, i.e. as a name of an unspecified number: then we can build a function f : L(RAt) 7→ L(F OAt), mapping equations of RAtinto formulæ of F OAt, such that, for any two equations α, β ∈ L(RAt), α `RAt β iff f (α) `F OAt f (β). However, the converse is not possi-ble, i.e. we cannot build an analogous mapping function g : L(F OAt) 7→ L(RAt) such that, for any two formulæ α, β ∈ L(F OAt), α `F OAt β iff g(α) `RAt g(β).
In particular, what would remain beyond the expressive power of RAt are, as I have already said, unbounded existential quantifiers and, when they are embedded within the scope of another operator, also unbounded universal ones.
13What I am going to say actually holds for much weaker formal systems imposing restrictions on formulæ instantiating the induction schema; in particular, systems where restrictions on the combinations of unbounded quantifiers appearing inside their formulæ have been well studied.
This point, however, is completely orthogonal to the one I am making.
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Now, if we could show that this extra expressive power is indeed needed once we employ logical formulæ of a certain formal system to build semantic representations of natural language sentences, then we would have shown ipso facto that RAt cannot do the job. This would be the case, in particular, if we were able to show one of the following things:
a) that at least some natural languages have verbs, nouns, adjectives, adverbs or prepositions which cannot be translated as p.r.u. functions;
b) that at least some natural languages have determiners which must be trans-lated as unbounded existential quantifiers;
c) that at least some natural languages have determiners which must be trans-lated as unbounded universal quantifiers and which are within the scope of another operator.
As for the first circumstance, I have no evidence that it holds for any language.
As for the other two, instead, I suspect that it would be pretty hard to empirically prove them, since it is a well-known fact that there is an abundance of covert arguments in natural language, and, besides, restrictors of DPs are known to be often incomplete and in need of being contextually constrained (seeKuroda(1982) and Sperber & Wilson (1986), among many others); so, it would be difficult to clearly show that there are cases like the ones just mentioned where the determiners in question truly come in an unbounded fashion.
Conversely, there is at least one way to suggest that a formalism like that of RAt best serves natural language semantics rather than its CL competitors:
to show that in natural languages there are elements corresponding to generic
variables, i.e. scopeless elements which are universal-alike and are interpreted as if they had wide scope over functional operators while they were scope-insensitive between themselves. This will be what I will argue in Ch. 7.
One could maintain that, even if I succeeded in showing that proper semantic representations for natural language sentences need to have at their disposal some-thing structurally analogous to generic variables, i.e. translating scopeless elements which are universal-alike, still this would not be, per se, an argument in favour of RA as the proper formal basis upon which to found natural language semantics, since scopeless elements of this kind could be, for instance, easily incorporated in a formal system like standard first-order or higher-order logic. However, I should reply that the really crucial way to argue against the application of RA to natural language is by showing that unbounded quantifiers (or, more generally, elements which are not defined as p.r.u. functions) are indeed present in natural languages:
RAt, where (bounded) quantifiers are only some special p.r.u. functions and, thus, are not primitives of the system, is indeed a very simple formal system, arguably minimally simple and certainly much simpler than any possible version of CL; thus, if no extra expressive power is needed, it should be preferred on the grounds of general epistemological principles. Moreover, if we had a system with unbounded universal quantifiers, there would be no need for generic variables, since all their inferential behaviour could be reproduced by wide scope universal quantifiers: the presence of those free variables in the system would thus require an independent motivation showing what has been gained at the cost of such an anti-economical move: the least we can say is that the task of providing such an independent motivation would hardly be a trivial one.
The hypothesis I am going to partly explore, hence, is that the outputs of J K 51
are terms of RAt.14
Of course, here I will give only a partial characterization of the interpretation function: this is to say that I will equate the term resulting from the application of the interpretation function to a compound linguistic expression to another term where the interpretation function, whenever present, is applied only to components of the previous expressions.15
Note, as well, that we should assume that something corresponding to indices appears at the level of the syntactic representation which feeds the semantic com-ponent (LF, for instance): I will assume that indices themselves do. The existence of something playing this role at some suitable level of the syntactic representa-tion is in line with most linguistic approaches to quantificarepresenta-tion, starting from the Quantifying-In device ofMontague(1970a,b,1973) to Lakoff’s Quantifier Lowering or the most famous Quantifier Raising by May.
The semantic theory I am going to sketch is a top-down semantics in the technical sense defined, for instance, inHodges(2012: §3): in other words, semantic derivation is taken to procede from the meaning of the largest expressions to that of their components, down to the semantic atoms.
14The assumption that what we need are terms of the system allows us to define the interpre-tation function in a much simpler way with respect to what we should do by assuming that the outputs were equations of the system, as in Goodstein(1957: Ch.3). Besides, the central intu-ition informing the theory of illocutionary operators (see §5.4below) suggests that this treatment should be preferred also for independent reasons.
I believe that this assumption is in line with Partee’s (2009) suggestion (ultimately based on Carstairs-McCarthy (1999)) of the viability of a semantic theory based on one single basic semantic type. But, again, Schönfinkel(1924) can be viewed as a forerunner.
15Note that this requirement does not imply that the semantic theory I am arguing for is a compositional one. In fact, I believe that, where fully developed, it would not be, like many others on the market. In particular, what I consider to be the crucial obstacle to a compositional theory of meaning, in agreement with nearly all scholars challenging compositionality, is the proper semantic treatment of intensionality.