V 54 ff
Loewenheim/reference/PutnamVsTradition: Loewenheim tries to fix the intension und extension of single expressions via the determination of the truth values for whole sentences.
V 56f
PutnamVsOperationalism: e.g. (1) "E and a cat is on the mat." If we re-interpret this with cherries and trees, all truth values remain unchanged.
Cat* to mat*:
a) some cats on some mats and some cherries on some trees,
b) ditto, but no cherry on a tree,
c) none of these cases.
Definition cat*: x is a cat* iff. a) and x = cherry, or b) and x = cat or c) and x = cherry. Definition mat*: x = mat* iff. a) and x = tree or b) and x = mat or c) and x = quark.
Ad c) Here all respective sentences become false ((s) "cat* to mat*" is the more comprehensive (disjunctive) statement and therefore true in all worlds a) or b)).
Putnam: cat will be enhanced to cat* by reinterpretation. Then there might be infinitely many reinterpretations of predicates that will always attribute the right truth value. Then we might even hold "impression" constant as the only expression. The reference will be undetermined because of the truth conditions for whole sentences (>
Gavagai).
V 58
We can even reinterpret "sees" (as sees*) so that the sentence "Otto sees a cat" and "Otto sees* a cat" have the same truth values in every world.
V 61
Which properties are intrinsic or extrinsic is relative to the decision, which predicates we use as basic concepts, cat or cat*. Properties are not in themselves extrinsic/intrinsic.
V 286ff
Loewenheim/Putnam: theorem: S be a language with predicates F1, F2, ...Fk. I be an interpretation in the sense that each predicate S gets an intension. Then, there will be a second interpretation J that is not concordant with I but will make the same sentences true in every possible world that are made true by I. Proof: W1, W2, ... all be possible worlds in a well-ordering, Ui be the set of possible individuals existing in world Wi. Ri be the set, forming the extension of the predicate Fi in the possible world Wj. The structure [Uj;Rij(i=1,2...k)] is the "intended Model" of S in world Wj relative to I (i.e. Uj is the domain of S in world Wj, and Rij is (with i = 1, 2, ...k) the extension of the predicate Fi in Wj). J be the interpretation of S which attributes to predicate Fi (i=1, 2, ...k) the following intension: the function fi(W), which has the value Pj(Rij) in every possible world Wj. In other words: the extension of Fi in every world Wj under interpretation J is defined as such, that it is Pj(Rij). Because [Uj;Pj(Rij)(i=1,2...k)] is a model for the same set of sentences as [Uj;Rij(i=1,2...k)] (because of the isomorphism), in every possible world the same sentences are true under J as under I. J is distinguished from I in every world, in which at least one predicate has got a non-trivial extension.
V 66
Loewenheim/intention/meaning/Putnam: this is no solution, because to have intentions presupposes the ability to refer to things.
Intention/mind State: is ambiguous: "pure": is e.g. pain, "impure": means e.g. whether I know that snow is white does not depend on me like pain (>
twin earth). Non-bracketed belief presupposes that there really is water (twin earth). Intentions are no mental events that evoke the reference.
V 70
Reference/Loewenheim/PutnamVsField: a rule like "x prefers to y iff. x is in relation R to y" does not help: even when we know that it is true, could relation R be any kind of a relation (while Field assumes that it is physical).
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I (d) 102ff
E.g. the sentence: (1) ~(ER)(R is 1:1. The domain is R < N. The range of R is S). Problem: when we replace S by the set of real numbers (in our favourite set theory), then (1) will be a theorem. In the following our set theory will say that a certain set ("S") is not countable. Then S must in all models of our set theory (e.g. Zermelo-Fraenkel, ZF) be non-countable.
Loewenheim: his sentence now tells us, that there is no theory with only uncountable models. This is a contradiction. But this is not the real antinomy. Solution: (1) "tells us" that S is non-countable only, if the quantifier (ER) is interpreted in such a way that is goes over all relations of N x S.
I (d) 103
But if we choose a countable model for the language of our set theory, then "(ER)" will not go over all relations but only over the relations in the model. Then (1) tells us only, that S is uncountable in a relative sense of uncountable.
"Finite"/"Infinite" are then relative within an axiomatic set theory. Problem: "unintended" models, that should be uncountable will "in reality" be countable.
Skolem shows, that the whole use of our language (i.e. theoretical and operational conditions) will not determine the "uniquely intended interpretation". Solution: platonism: postulates "magical reference". Realism: offers no solution.
I (d) 105
In the end the sentences of set theory have no fixed truth value.
I (d) 116
Solution: thesis: we have to define interpretation in another way than by models.
