Page 390 - Encyclopedia of Philosophy of Language

Formal Semantics
Figure 5. Valuation space construction of/~cfta/ and /dj
result in the empty valuation space, as (/<//u/~<//)n /~c/=0. ThisisillustratedinFig.5,wherethearea covered by vertical lines is / ~ c/.
Under this construal of the notion of sequential discourse, we are practically back at Strawson'sGBC (Fig. 1), but with the extra provision of sentences (and other linguistic objects) as possible reference objects. It follows from this analysis that natural language
being interpretable as 'belongs to NL(D) ,' and thus providing a functional instrument for correcting any discourse whose sequential reconstruction (that is, with full post hoc suppletion of all implicit pre- suppositions) is inconsistent.
This may provoke fears of a resurrection of the 'liar paradox.' Yet this fear is unfounded. Following the medieval solution to this paradox in terms of vacatio, that is, lack of reference (Bottin 1976; Seuren 1987), we say that the sentence This proposition isfalse fails to deliver a reference object for the subject term this proposition, so that the sentence fails to deliver a truth- value. This solution implies an infraction of PET, but so does the whole analysis.
A generalized trivalent logic is still useful in |hat it can express the logical properties of NOT. If NOT('</') (in the example of Fig. 5) is interpreted, after ~crf, as
^d,thatis,astheintersectionof/~c/and/^d/ (=/~clsince/^d/ =U-/c/),itisvalued'1,'yethasno eflFect on the incrementation of D: it does not restrict further the valuation space of D and thus violates condition (c) of the sequentially criterion. But now the metalinguistic character of NOTis ignored and i(~</) is true but not informative.
2
From this point of view, TGC' and Toe differ
= (UAuUB)). Thus, U[A+BA+c.+Dc) = U(BA+c.+Dd = /C/, as it makes no difference whether the pre- supposition is implicit or explicit. Since it makes a D radically false as soon as a radically false sentence is added, it allows for the rule that a minimally false D is made true simply by negating all its false sentences, without any sentence having to be rejected as belong- ing to NL(D), whereas a radically false D is made true by negatingall its minimally false sentences and eliminating all sentences valued '3.' Such a trivalent
negation is ambiguous between ~ and NOT, the latter now
suppositions.
4. The Structural Source of Presuppositions
The structural source of three of the four types of presupposition that were distinguished at the outset of Sect. 1 can be identified uniformly: it lies in the lexical meaning conditions of the main predicate of the sentence (clause) in question. The lexical conditions of a predicate P" over individual objects are the con- ditions that must be satisfied for any object, or n-tuple of objects, to be truthfully predicated by means of P°. Thus, for the unary predicate bald the conditions must be specified under which any object can truthfully be called 'bald/ Or for the binary predicate wash it must be specified under what conditions it can truthfully be said of any pair of objects <i,j> that 'i washes j.' Analogously for predicates whose terms refer to things other than individual objects, such as sets of objects, or facts, or imbedded propositions.
In the light of presupposition theory, one can now, following Fillmore (1971), make a distinction between two kinds of lexical conditions, which we shall call the 'preconditions' and the 'satisfaction conditions.' The criterion distinguishing the two is that when any pre- conditionisnotfulfilledthesentenceisradicallyfalse, whereasfailureofasatisfaction condition resultsin minimal falsity. Fulfillment of all conditions results in truth.
The following notation makes the distinction for- mally clear. Let the extension of a predicate P° be characterized by the function symbol a. Then an n-ary predicate P" over individuals will have the following schema for the specification of its lexical conditions:
2
<T(P")=«i',i ,...,i°>:...(preconditions)...|... (37)
(satisfaction conditions)...}
as follows. In TGC', as can be seen from Fig. 2, AB
U A =(UAnUB)u/~A/u/~B/. Thus, if A has no presuppositions, UAAB = (Un/A/)u(U-/A/)u (/A/- /B/)=U.Consequently,/~BA/=/A/- /B/,but
~(AABA)=—,BA, even though /BA/ = /AABA/. A fully sequential discourse will thus never be radically false in TGC', even when A is minimally false and all
or: 'the extension of P is the set of all n-tuples of 2n
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the remaining sentences are therefore radically false. But a discourse [BA+CB+DC+...] (that is, with the initial presupposition kept implicit) will be radically
2
false just in case A is not true. In TGC, on the other
hand, the subuniverses for conjunction (and dis- AB AB
junction) run parallel: U A =(UAnUB) (and U v
logic requires that the notion of presupposition be 2
limited to single increment units. In TGC, con- junctions (discourses) have subuniverses, but no pre-
individuals<i',i ,...,i >suchthat... (preconditions) ... and ... (satisfaction conditions) ...' The pre-