Willard Van Orman Quine

208 dagfinn føllesdal
points out that not only can one have open sentences ‘Fx’ and ‘Gx’
such that
(∃ x)(⊓ Fx.Gx.∼ ⊓Gx)
but one must require that there are open sentences fulfilling
(x)(⊓ Fx.Gx.∼ ⊓Gx).
“An appropriate choice of ‘Fx’ is easy: ‘x = x’. And an appropriate
choice of ‘Gx’is‘x = x.p’ where in place of ‘p’ any statement is chosen
which is true but not necessarily true” (TGMI 81).
collapse of modal distinctions
In Word and Object, Quine draws a further disastrous consequence
of the requirement on open sentences at which he arrived in From a
Logical Point of View (requirement (EQUIV) above), viz., the consequence
that every true sentence is necessarily true (WO 197–8).
Let ‘p’ stand for any true sentence, let x be any object in our purified
universe of discourse, and let w = x. Then
(8) (y)(p.y = w.≡. y = x)
(9) (y)(y = w.≡. y = x)
Introducing ‘p. 1○=w’ for ‘F 1○’ and ‘ 1○=w’ for ‘G 1○’ in (EQUIV),
one gets
(y)(p.y = w.≡. y = x). (y)(y = w.≡. y = x) ⊃
(10)
⊓ (y) (p.y = w.≡. y = w),
which together with (8) and (9) implies
(11) ⊓ (y)(p.y = w.≡. y = w).
But the quantification in (11) implies in particular ‘p.w = w .≡. w =
w’, which in turn implies ‘p’, so from (11) we conclude ‘⊓ p’.
Since in this proof nothing is assumed about the objects over
which we quantify, restricting the values to intensional objects does
not prevent this collapse of modal distinctions. So, unless quantification
into modal contexts can be interpreted without assuming
(EQUIV), the prospects for a quantified modal logic are very gloomy
indeed.
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