It is instructive, though not historically accurate, to see the development of contemporary semantic theories as motivated by the deficiencies that are uncovered when one tries to take the FOPC example further as a model for how to do natural language semantics. For example, the technique of associating set theoretic denotations directly with syntactic units is clear and straightforward for the artificial FOPC example. But when a similar programme is attempted for a natural language like English, whose syntax is vastly more complicated, the statement of the interpretation clauses becomes in practice extremely baroque and unwieldy, especially so when sentences that are semantically but not syntactically ambiguous are considered [Coo83]. For this reason, in most semantic theories, and in all computer implementations, the interpretation of sentences is given indirectly. A syntactically disambiguated sentence is first translated into an expression of some artificial logical language, where this expression in its turn is given an interpretation by rules analogous to the interpretation rules of FOPC. This process factors out the two sources of complexity whose product makes direct interpretation cumbersome: reducing syntactic variation to a set of common semantic constructs; and building the appropriate set-theoretical objects to serve as interpretations.
The first large scale semantic description of this type was developed by]. Montague made a further departure from the model provided by FOPC in using a more powerful logic (intensional logic) as an intermediate representation language. All later approaches to semantics follow Montague in using more powerful logical languages: while FOPC captures an important range of inferences (involving, among others, words like every, and some as in the example above), the range of valid inference patterns in natural languages is far wider. Some of the constructs that motivate the use of richer logics are sentences involving concepts like necessity or possibility and propositional attitude verbs like believe or know, as well as the inference patterns associated with other English quantifying expressions like most or more than half, which cannot be fully captured within FOPC .
For Montague, and others working in frameworks descended from that tradition (among others, Partee, e.g., [Par86], Krifka, e.g., [Kri89], and Groenendijk and Stokhof, e.g., [GS84,GS91a]) the intermediate logical language was merely a matter of convenience which could in principle always be dispensed with provided the principle of compositionality was observed. (I.e., The meaning of a sentence is a function of the meanings of its constituents, attributed to Frege, [Fre92]). For other approaches, (e.g., Discourse Representation Theory, [Kam81]) an intermediate level of representation is a necessary component of the theory, justified on psychological grounds, or in terms of the necessity for explicit reference to representations in order to capture the meanings of, for example, pronouns or other referentially dependent items, elliptical sentences or sentences ascribing mental states (beliefs, hopes, intentions). In the case of computational implementations, of course, the issue of the dispensability of representations does not arise: for practical purposes, some kind of meaning representation is a sine qua non for any kind of computing.
Discourse Representation Theory (DRT) [Kam81,KR93], as the name implies, has taken the notion of an intermediate representation as an indispensable theoretical construct, and, as also implied, sees the main unit of description as being a discourse rather than sentences in isolation. One of the things that makes a sequence of sentences constitute a discourse is their connectivity with each other, as expressed through the use of pronouns and ellipsis or similar devices. This connectivity is mediated through the intermediate representation, however, and cannot be expressed without it.
Dynamic semantics takes the view that the standard truth-conditional view of sentence meaning deriving from the paradigm of FOPC does not do sufficient justice to the fact that uttering a sentence changes the context it was uttered in. Deriving inspiration in part from work on the semantics of programming languages, dynamic semantic theories have developed several variations on the idea that the meaning of a sentence is to be equated with the changes it makes to a context.
Update semantics (e.g., [Vel85,vEdV92]) approaches have been developed to model the effect of asserting a sequence of sentences in a particular context. In general, the order of such a sequence has its own significance. A sequence like:
Someone's at the door. Perhaps it's John. It's Mary!
is coherent, but not all permutations of it would be:
Someone's at the door. It's Mary. Perhaps it's John.
Recent strands of this work make connections with the artificial intelligence literature on truth maintenance and belief revision .
Dynamic predicate logic [GS91a,GS90] extends the interpretation clauses for FOPC (or richer logics) by allowing assignments of denotations to subexpressions to carry over from one sentence to its successors in a sequence. This means that dependencies that are difficult to capture in FOPC or other non-dynamic logics, such as that between someone and it in:
Someone's at the door. It's Mary.
can be correctly modeled, without sacrificing any of the other advantages that traditional logics offer.
One of the assumptions of most semantic theories descended from Montague is that information is total, in the sense that in every situation, a proposition is either true or it is not. This enables propositions to be identified with the set of situations (or possible worlds) in which they are true. This has many technical conveniences, but is descriptively incorrect, for it means that any proposition conjoined with a tautology (a logical truth) will remain the same proposition according to the technical definition. But this is clearly wrong: all cats are cats is a tautology, but The computer crashed, and The computer crashed and all cats are cats are clearly different propositions (reporting the first is not the same as reporting the second, for example).Situation theory [BP83 has attempted to rework the whole logical foundation underlying the more traditional semantic theories in order to arrive at a satisfactory formulation of the notion of a partial state of the world or situation, and in turn, a more satisfactory notion of proposition. This reformulation has also attempted to generalize the logical underpinnings away from previously accepted restrictions (for example, restrictions prohibiting sets containing themselves, and other apparently paradoxical notions) in order to be able to explore the ability of language to refer to itself in ways that have previously resisted a coherent formal description