I have been digging into Carnap's Autobiography (in the
Schilpp volume) paying attention to what he says (which is a
great deal) about the sources of his ideas, while trying to
progress the biographical parts of the Carnap/Grice
conversation I am working on with J.L. Speranza.
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I found a nice way of thinking about one of the issues which
I have recently been discussing on hist-analytic.org with
Speranza.
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In the Tractatus Wittgenstein presents a couple of ideas about the status of logical truths. He is working with Hume's fork which separates true propositions in to those which are logically true and those which are empirically true. He has some insights into the nature of propositions and of logical truth which allow this to be explained more fully than was possible for Hume.
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Two separate ideas contribute to a satisfactory explanation.
The first is that the relevant part of the meaning of a proposition can be captured by its truth conditions which tell you the truth value of the proposition in every possible situation. This idea suffices to distinguish between analytic and synthetic truths, the analytic truths being those which are true under all conditions, synthetic propositions are true under some and false under others.
This is Wittgenstein's insight that logical truths are tautologous.
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The second idea is that of a truth function.
It was well known that the classical propositional connectives are "truth functional", that the truth value of a complex proposition formed using them depends only on the truth value of the constituent propositions. Quantification is more complicated. A quantifier operates on a propositional function rather than a proposition, so cannot be truth functional in quite the same way as the connectives. Nevertheless in an appropriately augmented sense it too is "truth functional", and the effect is that any complex proposition formed from "atomic propositions" by
the propositional connectives or quantifiers is truth functional. It is not quite trivial to make this precise, but with some minor caveats this is what Wittgenstein does
in the Tractatus and this is what is done in a rather different way in the semantics of first order logic which we find (subsequently) at the beginnings of model theory.
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The second insight is more definite than the first, in that it takes us from the idea that a logical truth has tautologous truth conditions, to the idea that it is a tautologous truth function of its atomic propositions.
We have added the idea that the "possibilities" relative to which the truth conditions are defined are assignments of truth values to the constituent atomic propositions.
One step more and we get into trouble.
That step is to answer the question "but which assignments ot truth values are possible" and give the obvious answer:
well they are all logically possible.
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We have now moved from a truth conditional account of logical truth which can correctly account for that notion of logical truth which is complementary to the notion of empirical truth (i.e. the truths of reason in Hume's fork, by contrast with "matters of fact") to an essentially narrower conception which corresponds closely to that of first order validity (not yet defined at the time of the Tractatus).
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The problem then is for Wittgenstein that the conception of logical truth in the Tractatus fails to take account of "determinate exclusion" which is the possibility that two atomic propositions are not logically independent.
This according to Hacker was the primary consideration which undermined the Tractarian conception of the status of logic
(for Wittgenstein, in about 1930).
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So what has this to do with Carnap?
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Carnap gives several accounts of logical truth or L-truth (by which he almost always means analyticity).
The early ones in his syntactic phase don't concern us here, but in his later semantic phase we find in "Meaning and Necessity" that his semantic characterisation of L-truth
explicitly follows the Tractatus. He defines a state description as (effectively) an assignment of truth values to atomic propositions.
The semantics is then given as rules which tell us whether any proposition "holds" in any state description, i.e.which gives the truth conditions in terms of state descriptions (these are the "possibilities").
A proposition is L-true if it holds in every state description.
Carnap makes no provision here for restricting the possible state descriptions, though he is explicit in declaring L-truth to be an explication of Leibniz's "necessary truth" and Kant's "analytic truth", and goes on himself to define necessity in terms of L-truth.
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Though I see no aknowledgement of an error in "Meaning and Necessity" Carnap later offers another explication of analyticity in "Meaning Postulates". In this paper he again uses the term L-truth, but he now distinguishes two kinds of logical truth a narrow and a wide, identifies L-truth with the narrow conception (which has the same definition as was given in "Meaning and Necessity", ostensibly for the broad notion) identifies the broader conception with analyticity, and gives an explication using meaning postulates to restrict the states of affairs which count as possibilities in determining whether a proposition is tautologous.
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So far as I am aware this is the only place where he uses L-truth intending to explicate a narrow conception of logical truth.
When we come to his reformulation for the Schilpp volume he simply drops L-truth and introduces A-truth for analyticity, effectively removing the conflict with Quine over the scope of logical truth by giving him the concept.
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Meanwhile, seemingly by accident, Carnap has given, in "Meaning Postulates", a more precise characterisation of the narrow concept than I have seen elsewhere,
Carnap may be thought of here as defining analyticity in terms of tautological truth conditions, and logical truth in term of tautological truth functions. He has implicated that the logical connectives are the truth functional connectives (in the broad sense in which quantifiers count as truth functional). Nowhere does he explain this or offer it as a characterisation, (that I am aware of), it just falls out by accident.
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In fact the feature of his semantics out of which it falls is in other terms undesirable. for it prejudices the division of the semantics into evaluation rules (defining the truth conditions) and constraints on possibilities ("Meaning postulates"), forcing all but the truth functional aspects to be covered in the latter rather than the former.
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I should mention one other fly in the ointment, which I have glossed over and should be dealt with more carefully. That is the need for the semantics of quantification to know the domain of discourse. This makes quantification a truth function of the propositional function only if the domain of quantification is either fixed in advance or is somehow recoverable from the collection of true atomic propositions (e.g. if every value is known to have a name).
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RBJ
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