Witti-Buch2 2001.qxd - Austrian Ludwig Wittgenstein Society

Is Logic Syntax of Language?
already part of the system. Thus the claim that any particular system of syntactical rules
explains all the logical or mathematical truths is contradicted since the necessary
consistency proof can never be part of that system. 16
3. Wittgenstein's Evasion
Among the factors that could exempt a theory of logical syntax from these objections are
(3.1) a rejection of the inference from a contradiction to any arbitrary proposition, (3.2) a
rejection of the idea that rules can be inconsistent, (3.3) a rejection of the idea of a hidden
contradiction, (3.4) a rejection of the idea that logical syntax has to be complete. 17
3.1 Wittgenstein's endorsement of the first factor is found in his Lectures on the
Foundations of Mathematics: "One may say: 'From a contradiction everything would
follow.' The reply to that is: Well then, don't draw any conclusions from a contradiction;
make that a rule." (p. 209) In fact, EFQ and the intuitionist absurdity rule are valid only
on a model-theoretical definition of logical consequence and validity, to which neither
Carnap nor Wittgenstein subscribes.
3.2 Second, on Wittgenstein's view axioms just are the permitted patterns or moves
within a system-they are not statements at all. But a contradiction consists of a
statement conjoined with its negation, so if axioms are not statements they cannot
contradict. The same holds for conventions.
3.3 Third, in logical syntax a sentence is part of a system only if the sentence has been
constructed according to the rules of the system. So a contradiction cannot lurk deep
within a system, for to be a sentence of the system at all entails having been derived. 18
3.4 Fourth, Gödel's criticism that logical syntax has to have recourse to more
powerful mathematics than it captures in order to prove its consistency, and is in this
sense incomplete, would not bother either Wittgenstein or Carnap. Carnap, for one,
states quite plainly that "everything mathematical can be formalized, but mathematics
cannot be exhausted by one formal system." (LSL, p. 222) Gödel takes it for granted
that an adequate foundation of mathematics-an epistemological reduction- would issue
in "a full exhibition of the syntactical nature of mathematics." 19 There appears to be a
standoff here between the logical analysis offered by Carnap and Wittgenstein, and the
epistemological grounding that Gödel pursued.
Gödel's view here is that an adequate logico-mathematical theory has to be
semantically complete, that is where "all valid formulas or all valid inference patterns (in
the underlying language) can be derived in it as theorems." (Hintikka 1998, p. 65) 20
However, if one's notion of consequence is relativized to a linguistic framework or
language game (see 1.3 and 1.4 above), then no one framework will include all valid
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