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fully vindicated is not the final word on its overall correctness. It is the notion that terms have “unique’
and “absolute” logically primitive terms which forms part of Frege’s case for realism in mathematics (that mathematical terms refer to mind-independent entities). This is, however, not for discussion here, but is in part what Carnap reacts against in his language relative account of analytic truth.
I think it is worth ending this gloss of Frege’s account of analytic truth with a quote I take to capture the sentiments of Frege’s overall project. It will then also be easier to understand why and how Carnap retains some and rejects some of what Frege proposed.
The aim of proof is, in fact, not merely to place the truth of a proposition beyond all doubt, but also to afford us insight into the dependence of truths upon one another. After we have convinced ourselves that a boulder is immovable, by trying unsuccessfully to move it, there remains the further question, what is it that supports it so securely? The further we pursue these enquiries, the fewer become the primitive truths to which we reduce everything; and this simplification is in itself a goal worth pursuing. But there may even be justification for further hope: if, by examining the simplest cases, we can bring to light what mankind has there done by instinct, and can extract from such procedures what is universally valid in them, may we not thus arrive at general methods for forming concepts and establishing principles which will be applicable also in more complicated cases? (Frege, 1974, p. 2)
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account of truth and meaning which he believes demonstrates their objectivity. For Frege, there are objective facts about the truth and meaning of analytic sentences. He argues that analytic sentences are logically reducible, by way of logical proofs, to logical primitives. It is these primitives that are objective in the sense that a valid logical proof for any concept will yield the same definitions for anyone.
Proof is now demanded of many things that formerly passed as self-evident… Negative and irrational numbers, which had long since been admitted into science, have had to submit to a closer scrutiny of their credentials. In all directions these same ideals can be seen to work – rigours of proof, precise delimitation of extent of validity, and as a means to this, sharp definitions of concepts. (Frege, 1974, p. 1)
In what follows, I explain that, even though Carnap was greatly influenced by Frege’s logicism (Coffa, 1991), he forwards an account of analytic truth as logical truth (here Carnap is still aligned to Frege), but argues that such truth can only be necessary truth if true relative to constructed language systems. By
‘constructed’ I mean that they are made up, or “freely invented” (Hale, B. and Wright, C., 2003), by a linguistic community. In this sense, Carnap’s account of analytic truth subscribes to a type of
‘formalism’, because he takes the rules of logic to be constructed linguistic rules or conventions (Coffa, 1991; Bohnert 1963).
Carnap’s account of analytic truth is based on the following two underlying semantic suppositions: 1.
Not all accounts of truth and therefore meaning are ‘cashed out’ in terms of verification (here he eventually departs from early logical positivist theories about meaning), 2. When the meaning of a sentence is not ‘cashed out’ in terms of verification, the truth of that sentence is logical truth and the meaning of such a sentence has no knowable objective grounds, and is therefore taken to be non- factual and constructed. This ‘non-factualism’ is epistemological rather than metaphysical, insofar as we have no knowledge of the supposed objective and foundational facts which some accounts of truth and meaning presuppose (Coffa, 1991, p. 307). ‘Objective’, henceforth, will be used in the Fregean sense, as explained in section 2.
In section 3.2 below I show how Carnap’s position is not a metaphysical one, but rather an epistemic one (Coffa, 1991, p. 317). Understanding these two points goes a long way to understanding Carnap’s notion of analytic truth. He resists taking a metaphysical stance on the factuality of meaning and concludes that analytic truth is logical truth, as determined by constructed logical systems.
It is near impossible to render Carnap’s account of analytic truth without also giving some explanation of his epistemology; at least his account of a priori knowledge. Like Frege’s, Carnap’s conception of analytic
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truth is closely tied in with his conception of how we might know things non-empirically. Like for Frege, Carnap insists that all claims to truth must be justifiable or provable. And like for Frege, justification is always either empirical or logical. This means, respectively, justification is always either a posteriori or a priori. But against Frege (and Russell) Carnap argues that logical proof includes mathematical proof, only if logic and mathematics are accepted to be without knowable objective foundations (Coffa, 1991, pp.
307 - 309).
I show, particularly in section 3.2, that for Carnap a sentence is analytic partly because of the way in which it is justified. For Kant, Frege and Carnap some sentences are justified a priori. But, unlike for Kant, Frege and Carnap hold that the a priori justification and knowledge of analytic truths is
demonstrable logical proof. Demonstrable logical proof is proof which is executed according to accepted rules of logic or rules of deductive inference. Plainly then Carnap’s account of analytic truth is
determined by what he thinks logical truth is. So, some of what I do in what follows is to explain what it is that Carnap thinks about logic and necessity. To say, for instance, that Carnap posits a “non-factualist and semantic conventionalist account” (Coffa, 1991, pp. 306 - 326) for the sentences of logic is to say the same about analytically true sentences, even when they are not expressing directly, for instance, the laws or rules of logic.
I give a very brief précis of Carnap’s account of a priori knowledge. The précis should be enough to show in what way Carnap’s account of a priori knowledge is central to his explication of analytic truth. A more comprehensive investigation of a priori knowledge follows in Part 2. There I discuss the relationship between logical truth and a priori knowledge and also under what conditions a priori knowledge might be non-inferential. We will see that in some particular cases, also for theorists such as Carnap, it is possible to have non-inferential a priori knowledge (Hale, B. and Wright, C., 2003).17
In part Carnap’s account of analytic truth is a reaction to the Platonism underpinning Frege’s, for whom the objectivity of meaning was dependent on thought being objective. And for this to be the case thought had to exist independently of the construction of language. Language creation had to match or track such an independent realm of objective thought.
Frege believed that he needed to postulate the third realm in order to safeguard the objectivity of thoughts and their accessibility to different individuals; but, as Wittgenstein taught us, these things are
17 I offer a detailed account and defence of non-inferential a priori knowledge is Part 2, section 4.
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sufficiently secured by the fact that the use of language is common practice in which its many speakers have learned to engage. (Dummett, 2006, p. 10)
Carnap, with Wittgenstein,18 rejects this or any other metaphysical stance as a requirement for semantic theory. Analytic sentences are logical truths and logic cannot be confirmed to be objective in the sense that Frege thinks it is. It is well known that Carnap’s interest in logical truth is expressed across two phases: the syntactic and the semantic (Bohnert, 1963, p. 411). My précis of Carnap’s account of analytic truth is focussed entirely on his semantic phase (i.e. post-Tarski), which is also his later phase. The reason, simply, is that it is his treatment of definitions, not simply as grammatical forms, but also as conventions determining ‘rich’ meaning, which is of interest to this project. It is where sentences seem to bear ‘rich’ content that drawing the distinction between factual and logical truth becomes, to me, even more important to a campaign set out to reject the synthetic a priori.