Was sollen wir tun? Was dürfen wir glauben? - bei DuEPublico ...

36 DOLBY
proposition expressed by ‘ϕ’. This would be perfectly feasible. The problem is simply that
knowledge of the satisfaction conditions for a sentence involving such an expression would
only suffice for understanding if one knew what it is for a proposition to be the conjunction of
two other propositions. That is, one would need to know that:
∀x∀P∀Q (x = the conjunction of [P] and [Q] ↔ x = [P & Q])
In the notion of the conjunction of two propositions the satisfaction conditions would
therefore be employing a further concept which must be explained in terms of non-nominal
generalisation. Similar notions will need to be introduced for any other extension of the
language giving rise to new intensional contexts.
5. Implications
One consequence of the explicit or implicit appearance of non-nominal quantification in the
satisfaction conditions is that the statement of truth-conditions for a non-nominal
quantification will involve greater quantification than the sentence with which we started.
Consider the standard interpretation of nominal quantification: this tells us that an
existential quantification is true if and only if some object in the domain satisfies a certain
condition specified by the open sentence. This condition will be specified with one fewer
quantifiers than were contained in the original quantification. For instance, ‘Bill kicked
someone’ will be true if and only if some object in the domain satisfied the condition specified
by ‘Bill kicked x’, and this condition is met by someone if Bill kicked them. The end result is
therefore quantified to precisely the same degree as the original sentence.
The situation with objectual non-nominal quantification is different: for the interpretation
effectively transforms the non-nominal quantifier (things are thus and so) in the object
language into an objectual quantifier (some proposition) in the metalanguage, but is then
forced to introduce additional implicit non-nominal quantification into the condition to be
met by the values of the variable. If this implicit quantification is made explicit then the
interpretation involves more quantifiers than the original quantification.
Quantificational logic holds out the prospect of articulating the relationship between the truth
of a quantification and the satisfaction of more basic conditions, of relating a complex
sentence to its parts. Complexity may, of course, remain after the analysis: we may translate
the predicate ‘is married’ as ‘Fx’ in which case one could argue that the axiom for the
satisfaction of the predicate, and therefore the resulting analysis, will be implicitly quantified,
since to be married is to be married to someone. But implicit quantification of this sort is
something that we can in principle make explicit by translating ‘is married’ as ‘∃yRxy’
instead. And while it may not be possible or even make sense to give an exhaustive analysis of
a language into its elements, we can make explicit any given implicit quantification, and it
might be fruitful to do so. However, the non-nominal quantification implicit in the objectual
interpretation’s use of the terms ‘exemplifies’ and ‘true’, in contrast, could not be made
explicit in this way since these terms are introduced by the interpretation itself. Any attempt
to use the objectual interpretation to make this explicit would merely generate further
occurrences of the very same implicitly quantified concepts.
Of course, how troublesome any of this is will depend on what we take the function of an
interpretation of quantification to be. If our interest is simply in providing a workable
semantics for the purposes of displaying relationships of implication between sentences of a
language with certain expressive capacities, then there is no objection to giving an objectual
interpretation of non-nominal quantification. However, the interpretation of quantification is
generally taken to have greater significance than this. It is often argued, for instance, that the
standard substitutional interpretation is inappropriate for the interpretation of a