[hist-analytic] The "Analytic A Posteriori"
Danny Frederick
danny.frederick at tiscali.co.uk
Thu Sep 3 07:34:12 EDT 2009
Hi Steve,
I thought it might be worth expanding on something I said last time. You
claimed that even if the designators are not rigid 'Nec a=b' follows from
a=b by first order logic by substitution of predicates. This claim is an
expression of what is sometimes known as 'the Frege Argument.' The argument
is supposed to show that any referentially transparent context in which
logical equivalents (sentences or predicates) are intersubstitutable must be
truth-functional. The argument is used in Quine's 'Word and Object' (197-98)
to 'obliterate modal distinctions' and in Davidson's 'The Logical Form of
Action Sentences' ('Essays on Actions and Events,' 1982, 115-18) to show
that all facts are identical. The argument is invalid, as I will now
explain.
Here is a version of the argument.
(1) The context 'K()' permits the substitution salva veritate of
co-designative singular terms. [Assumption]
(2) The context 'K()' permits the substitution salva veritate of
logically equivalent sentences. [Assumption]
(3) The letters 'p' and 'q' represent any arbitrary sentences
with the same truth value. [Assumption]
(4) 'The x such that [(x=1 and p) v (x=0 & ~p)] = 1' is logically
equivalent to 'p.'
(5) 'The x such that [(x=1 and q) v (x=0 & ~q)] = 1' is logically
equivalent to 'q.'
(6) 'The x such that [(x=1 and p) v (x=0 & ~p)]' has the same
reference as 'The x such that [(x=1 and q) v (x=0 & ~q)].' [From (3)]
(7) 'K(p)' has the same truth value as 'K(The x such that [(x=1
and p) v (x=0 & ~p)] = 1).' [From (2) and (4)]
(8) 'K(The x such that [(x=1 and p) v (x=0 & ~p)] = 1)' has the
same truth value as 'K(The x such that [(x=1 and q) v (x=0 & ~q)] = 1).'
[From (1) and (6)]
(9) 'K(p)' has the same truth value as 'K(The x such that [(x=1
and q) v (x=0 & ~q)] = 1).' [From (7) and (8)]
(10) 'K(The x such that [(x=1 and q) v (x=0 & ~q)] = 1)' has the same
truth value as 'K(q)' [From (2) and (5)]
(11) 'K(p)' has the same truth value as 'K(q).' [From (9) and (10)]
We can now discharge the three assumptions to get the required conclusion:
since 'p' and 'q' represent any arbitrary sentences with the same truth
value, then any context of which (1) and (2) hold must be truth-functional.
The fault in the argument concerns the inference of (8) from (1) and (6).
This involves a confusion about singular terms and definite descriptions, or
rigid and non-rigid designators. In any extensional context, both rigid and
non-rigid designators are substitutable salva veritate. But in modal
contexts only rigid designators (singular terms proper) are
inter-substitutable salva veritate. This may also be so in some other
intensional contexts, such as talk about facts or propositions.
Thus, the argument is valid only if (1) is interpreted so that definite
descriptions are singular terms; but that restricts the argument to
extensional contexts, which makes the conclusion unsurprising. The
surprising conclusion requires that (1) be interpreted as concerning
singular terms proper; but then the argument is invalid. The argument shows
no more than that extensional contexts are extensional. It sucks.
Danny
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