Abstract
In this Chapter, I will explicitly specify the formal semantic framework that I argued for in the previous Chapter based on the reduced language of propositional logic. I will then provide precise descriptions within this framework for various aspects of lexical meaning, in particular for the precisification properties and restrictions that were informally outlined in Chapters 3 and 4. As before, the semantic formalism presented here will not be directly based on natural language, nor will it establish a formal language that corresponds completely to the syntactic and semantic richness of natural languages. In order to avoid unnecessary technical complexity, we will restrict ourselves to a language that, although richer than LA, still has a relatively simple syntax and classical semantics: the language of first-order predicate logic (with identity). I don’t see any fundamental problems that would stand in the way of extending the precisification logic approach to fully developed intensional logics or λ-categorial languages (see Cresswell (1973)), to name two examples.
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References
The extension to the intensional level of precisification logic in a possible worlds semantics is done in Pinkal (1983a). — For a necessary modification of the strictly extensional approach, see 4.4 above (p. 105ff.) and section 3 of this Chapter (p. 233).
At best, they could be compared to “bound” personal pronouns used in sentence anaphora.
For the connectives equivalences are given as (v), (vi), and (vii) in 5:(8), p. 123f..
Occasionally, “satisfaction” as the “assignment-specific truth” of open expressions is distinguished from the model-dependent truth of sentences. This distinction is not intended here: all of these semantic concepts are defined for sentences.
Actually, these are theorem schemata or “metatheorems”, since they are expressed with the aid of the metavariables α and β3. They become theorems when a and β are instantiated with any wff. — The theorems of propositional logic cited in 5.1, which I will occasionally make use of in the following, should also be read correctly as metatheorems. I think that this point does not create any intuitive or formal difficulties, and thus I will leave out an explicit correction.
With the exception of idiomatic expressions — of which there can be only a finite number, whereas the number of sentences of Lp (and, of course, the number of sentences of natural language) is potentially infinite.
The result is trivial for the connectives. The proof for the quantifiers is somewhat more difficult, since it must make use of values that are dependent on assignments.
For example, one alternative renders an intuitionistic logic, while another leads to supervaluations.
Cf. (5), p. 204; the analogy between quantifiers and precisification operators can be read off from (19).
In the SV version without precisification operators, the problem does not turn up in this form, since the base interpretation does not have any special formal status.
For a survey of systems of modal logic, see Hughes/Cresswell (1968).
Standard modal logic, in which the individual domain is fixed, and “free logic”, in which the individual domain may vary from world to world, differ in their recognition or refutation, respectively, of the Barcan formula (cf. Hughes/Cresswell 1968, p. 173ff.).
The technique of meaning postulates is originally due to Carnap (1947) and constitutes an integral part of the framework of Montague Grammar.
I will discuss these matters at length in Ch. 7.
This is especially evident in the case of point indefinite predicates: (40)(ii) can be suspended by reference to the ideal meaning of the expression (“actually”, no one is exactly 1.80m tall) — which is, according to Grice (1975), an important characteristic of conversational rules.
A remark on the notion of dimensionality: I have intentionally used the term “non-one-dimensional” instead of “multi-dimensional”. Concrete findings on semantic dimensionality are the subject matter of context theory (see 3.4 above); the terminology of precisification semantics can and should remain neutral in this respect. In the special case of the structural property specified by (41)(i), it can be concluded that no more than one dimension of evaluation is relevant; for our example heavy, this dimension is that of weight. A corresponding conclusion about the kind of context dependence that holds for a predicate that satisfies (41)(ii) is not as clear. The number of dimensions that are relevant for the predicate’s contextual precisification, or whether just a “global impression” is the determining factor in certain contexts (instead of clearly distinguished dimensions), are all issues that are left open by the postulate, and they can remain open. A typical example is the predicate large. The only conclusion that can clearly be drawn from (41)(ii) about the specific nature of a predicate’s context dependence is that it does not result from a strictly linear dimension of evaluation.
The asymmetric case of point indefiniteness does not appear here, since definitely positive cases of application can always be translated into definitely negative cases by exchanging the arguments, and vice versa.
Examples of the predicates at issue here are institutional concepts for which a limited definite domain is given ostensively, any extension being highly uncertain (cf. the examples democracy and nation in 4.4, or the naturalism example from Erdmann 1910).
The notion of lexical fields was introduced by Trier (1931) and elaborated on by Weisgerber (e.g. in 1962). A number of precise definitions have been proposed for this informal concept of fields. For a general presentation of lexical field theory, see Lyons (1977; Ch. 8).
Körner actually uses set-theoretic notation, thus he speaks of “inclusion” and “exclusion” rather than “hyponymy” and “incompatibility”.
In all of the schemata under (47), ∀x and o ∀ xa are equivalent; cf. note 27.
In the polarity of antonym pairs, cf. Seuren (1978), Bierwisch (1989).
Quantification over predicates would be a more elegant approach, but it is not available here since I have restricted myself to first-order predicate logic for reasons of simplicity.
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Pinkal, M. (1995). Precisification Semantics. In: Pinkal, M. (eds) Logic and Lexicon. Studies in Linguistics and Philosophy, vol 56. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8445-6_7
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