X 80
Completeness Theorem/deductive/Quantifier Logic/Quine:
(B) A scheme fulfilled by each model is provable.
Theorem (B) can be proven for many proof methods. If we imagine such a method, then (II) follows from (B).
(II) If a scheme is fulfilled by every model, then e is true for all insertions of propositions.
X 83
Proof Procedure/Evidence Method/Quine: some complete ones do not necessarily refer to schemata, but can also be applied directly to the sentences
X 84
that emerge from the scheme by insertion.
Such methods produce true sentences directly from other true sentences. Then we can leave aside schemata and validity and define logical truth as the proposition produced by these proof procedures.
1. VsQuine: this usually triggers a protest: the property "to be provable by a certain method of proof" is uninteresting in itself. It is only interesting because of the completeness theorem, which allows to equate provability with logical truth.
2. VsQuine: if one defines logical truth indirectly by reference to a suitable method of proof, one deprives the completeness theorem of its basis. It becomes empty.
QuineVsVs: the danger does not exist at all: the principle of completeness in the formulation (B) does not depend on how we define logical truth, because it is not mentioned at all! Part of its meaning, however, is that it shows that we can define logical truth by merely describing the method of proof, without losing anything of what makes logical truth interesting in the first place.
X 100
Fake Theory/quantities/classes/relation/Quine: is masked pure logic. Mathematics: begins when we accept the element relationship "ε" as a real predicate and accept classes as values of the quantified variables. Then we leave the realm of complete proof procedure. Logic: quantifier logic is complete. Mathematics: is incomplete.
>
Logical Truth/Quine.
X 119
Intuitionism/Quine: gained buoyancy through Goedel's incompleteness evidence.
XIII 157
Predicate Logic/completeness/Goedel/Quine: Goedel proved its completeness in 1930.