Handbook of the History of Logic: - Fordham University Faculty

400 Gyula Klima
stead, representing the subject term by the quantifiable restricted variable ‘x.Mx’,
we get a formula that need not contain a conjunction: ‘(∃x.Mx)(Ax.)’. 22
In fact, since the syntax of this formula is much closer to the original in this
regard than formulae of standard quantification theory, it should be clear that
just as changing the determiner in the English sentence (using ‘every’, ‘the’, ‘this’,
‘no’, or, switching to the plural, ‘two’, ‘five’, ‘most’, ‘twenty percent of’, in the
place of the indefinite article, etc.) does not introduce new conjunctives into the
sentence, so the addition of the corresponding (non-standard, numerical, and even
“pleonotetic” quantifiers) to this sort of formula need not introduce different logical
connectives into the syntax of this formula, as it does in the case of standard
quantificational formulae (as when we have to switch from conjunction to implication
when replacing the existential quantifier with the universal quantifier — but
we cannot do that with the pleonotetic quantifiers). 23
Thus, using restricted variables to represent common terms in personal supposition
clearly has the advantage of providing a better “match” with the syntax of
natural languages than the formulae of standard quantification theory can provide.
However, with the appropriate semantic interpretation restricted variables can do
even more.
If restricted variables are used to represent common terms in their referring
function, then the supposition of these terms can best be interpreted as the valueassignment
of such variables. For example, in accordance with the doctrine of
personal supposition, ‘Every man is an animal’ was analyzed by the medievals in
terms of the conjunction ‘This man is an animal and that man is an animal ...’,
where the demonstrative pronouns pick out all individuals falling under ‘man’. But
if we represent this sentence as ‘(∀x.Mx)(Ax.)’, then the restricted variable in this
formula, ‘x.Mx’, does exactly the same thing, namely, it takes its values from the
extension of its matrix. Thus, we can justifiably define a supposition function for
this variable analogously to the value assignment function of ordinary variables
of standard quantification theory, with the only difference that whereas ordinary
variables range over the entire universe of discourse, restricted variables range only
over the extension of their matrix.
However, there can obviously be cases when the extension of the matrix of a
restricted variable is empty, namely, when the common term represented by the
variable is true of nothing. In such a case we may assign the variable some artificial
value, whatever that may be, of which no simple predication is true. 24 This move at
22 This is the approach to the reconstruction of certain features of medieval supposition theory
I first presented in my Ars Artium. But there are a number of other, basically equivalent
approaches in the literature e.g. by G. Englebretsen, D. P. Henry, A. Orenstein and T. Parsons.
23 For a discussion of this observation see essay III of my Ars Artium. The impossibility of
representing pleonotetic quantifiers in standard quantification theory was first proven (for ‘most’
interpreted as ‘more than half the’) in J. Barwise and R. Cooper, “Generalized Quantifiers and
Natural Language”, Linguistics and Philosophy, 4(1981), pp.159-219, pp.214-215. (C13), setting
off a whole cottage industry of generalized quantification theory in the eighties.
24 Again, I took this approach in Essay II of my Ars Artium as well as in “Existence and
Reference in Medieval Logic”, in: A. Hieke — E. Morscher (eds.): New Essays in Free Logic,
Kluwer Academic Publishers, 2001, pp. 197-226.