Since, for Frege, semantic concepts are a function of epistemic concepts, and analytic expressions are knowable a priori, let me look briefly what it is that Frege takes a priori knowledge or justification to be.
I have already mentioned that he does not think of a priori justification as an immediate and intuitive rational apprehension of a necessary relationship of conceptual containment. “Intuitions” is how Kant explains the a priori knowledge that speakers have of, supposedly, “self-evident” conceptual
containment. For Frege, however, logical analysis is the only legitimate form of a priori justification. A priori justification is when we derive proofs from general laws, which do not need, or even allow, of proofs themselves (Juhl, C. and Loomis, E., 2010, p. 14).
The premises of logical proofs are axiomatic and never empirical. The sorts of proofs which will establish a synonymous relationship between the meanings of two terms are not the internal cognitive processes aiding clarification and comprehension of containment relationships, but are demonstrable processes performed by a speaker who understands the expression of it. Furthermore, it is required that the rules for proof, i.e. the rules for logic, are shared. For Frege it is impossible that the premises of a logical proof are informative if the proof is to end in an analytic statement/conclusion. Analytic propositions, when they are the conclusions to such proofs, are definitions or general logical laws and bear no information about the extra-linguistic world.
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If, however, it is impossible to give the proof without making use of truths which are not of a general logical nature, but belong to a sphere of some special science, then the proposition is a synthetic one.
(Frege, 1974, p. 4)
And
If, in carrying out this process, we come only on general logical laws and on definitions, then the truth is an analytic one, bearing in mind that we must take account also of all propositions upon which the admissibility of any of the definitions depends. (Frege, 1974, p. 4)
The corollary of this is that, for Frege, it is not possible to construct logical proofs, i.e. proofs which end in logically deductive conclusions, if the premises are understood as being in any way informative or bearing contingently true information. Why? For this seems contrary to what so many theorists believe is possible with deductive proof; surely one can take empirical premises and construct a proof from these and end up with an empirical conclusion? Frege denies this possibility. If this were to be the case, says Frege, the inferential pathways of a proof, i.e. the transference of truth from premises to
conclusion, would have to be able to do so for contingent truth. For Frege this is impossible since logical proof can only accommodate non-contingent expressions.
For a truth to be a posteriori, it must be impossible to construct a proof of it without including an appeal to facts, i.e., truths which cannot be proved and are not general, since they contain assertions about particular objects. But, if on the contrary, its proof can be derived exclusively from general laws, which themselves neither need or admit of proof, then the truth is a priori. (Frege, 1974, p. 4)
If it is impossible to have, as the premises of a logical proof, contingent expressions then it will not suffice for meaning to be that which is intuitively apprehended by a speaker. Because, for Frege, it seems to be contingently the case what that might be to the speaker. For Frege a priori justification is the demonstration of logical proof. This means that anything (i.e. the concluded expression) that is justified or known in this way is the product of logical proof. The only way in which it is possible to secure a consistent (objective) understanding of terms, i.e. their meanings, is to determine their meanings by definition. This has the desirable effect of all the premises and therefore the conclusion being composed of shared definitions for all the terms employed, which then enables a proof of a synonymous relationship (Coffa, 1991, p. 76).
Frege holds that empirical judgements are about what is “physically and psychologically actual” (Frege, 1974, p. 20). It is for this reason that Frege thinks it is wrong to say, as Kant does, that the foundations of
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mathematics presuppose a concept of space and time. Mathematics is infinite and the expressions thereof are ‘general’ in the same sense that purely logical expressions are. On the other hand, for Frege, it does not seem possible to have an a priori concept of space and time as infinite. The nature of space and time is therefore a matter of a posteriori judgement. And because it is a posteriori it does not admit of logical proof (Juhl, C. and Loomis, E., 2010, p. 14)
Frege goes on to say that there is no mention made by Kant of “any connexion with sensibility, which is, however, included in the notion of intuition in the Transcendental Aesthetic, and without which intuition cannot serve as the principle of our knowledge of synthetic a priori judgements” (Frege, 1974, p. 19).
This means that if Kant suggests that it is by intuition which we come to know the synthetic truth of arithmetical judgements, he is required to explain how such a faculty of apprehension works. It is not, to Frege, in any way obvious that a judgement which does not show a relationship of conceptual
containment, in Kant’s sense, can be known in a way which is not experiential. Bear in mind that for Frege the method of justification is a defining feature of the type of truth of an expression. Kant’s insistence that some expressions which are synthetic can be known a priori, and are therefore
necessary, makes no sense to Frege. Particularly in the absence of an account of the purported a priori intuition that might serve as the epistemic basis of some selected synthetic expressions.
The basis of arithmetic lies deeper, it seems, than that of any of the empirical sciences, and even that of geometry. The truths of arithmetic govern all that is numerable. This is the widest domain of all; for to it belongs not only the actual, not only the intuitable, but everything thinkable. Should not the laws of number, then, be connected very intimately with the laws of thought? (Frege, 1974, p. 21)
It seems that Frege, aside from thinking that Kant is not only wrong about the synthetic nature of mathematical sentences, has also failed to make mention of any way in which arithmetical judgements are known a posteriori, which they would need to be in order to be regarded synthetic. For Frege the
“general laws of and definitions” by which arithmetical judgements are justified describe only ways of thinking. And these laws and definitions are a priori and analytic (Juhl, C. and Loomis, E., 2010, p. 15).
How does Frege’s defence of the a priori justification of arithmetic, and what this says about the analytic truth of arithmetical sentences, bear on his account of analytic truth in general?
It is easier to detect synonymy in an arithmetical expression than in an expression in natural language. If Frege claims that there is a relationship of synonymy between the terms in some natural language expressions in the same manner as there is between the numbers and symbols of arithmetical
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expressions, then what does this amount to and how do we recognise synonymy in natural language?
After all, in maths we do sums, but it is not so clear what it is that we do in natural language to permit the diagnosis of an expression as analytically true. To begin, logical statements, which are written using symbols for logical constants and variables are the laws of thought, for Frege. The laws of thought are the laws of arithmetic. Arithmetic is based on logic. Let us say logical equivalence is expressed like this:
P ↔ Q :: (P > Q) & (Q > P) P ↔ Q :: (P & Q) V ( ~ P & ~ Q)
This means that saying that there is a materially bi-conditional relationship between the sentences ‘P’
and ‘Q’ is the exactly the same as saying that ‘if P then Q’ as well as ‘if Q then P’ are true. Also, to say that there is a materially bi-conditional relationship between ‘P’ and ‘Q’ is exactly the same as saying that either it is the case that ‘P’ and ‘Q’ are true or that neither are true. But what cannot be the case with a material bi-conditional is that only one (either ‘P’ or ‘Q’) is true. Now it is the meaning of ‘exactly the same’ which brings us closer to what is meant by logical equivalence. Frege suggests that we
understand ‘=’ in arithmetic as signifying the logical equivalence of what appears on either side of it. And it is logical equivalence, as well as the justification of a claim to logical equivalence, which is, at bottom, what characterises analytically true expressions. Why?
Firstly, the method of justification: It is not possible, according to Frege, to observe, a posteriori, a claim to necessity, because this would entail that the necessity is a relationship between parts of the extra- linguistic world. There is evidently no manner in which necessary claims can be justified a posteriori. It is for this reason, i.e. that we justify the sentences of arithmetic a priori, that we take them to be analytic.
And secondly, logical equivalence: An analytic sentence is one which is logically reducible to a tautology.
To be reducible to a tautology means replacing synonyms for synonyms, and the success of this is dependent on the logical equivalence of two terms standing in a synonymous relationship with each other (Dummett, 1981, p. 341). Dummett explains that for Frege sense is something perhaps richer (in meaning and associated thought) and therefore less rigid than merely the technical definitional meaning of a term which is capable of being intensionally isomorphic with another term. I have already alluded to this above in Miller’s (1998) explication of sense. For Frege, analysed (note the priority of
justification) terms and expressions yield analytic definitions (Dummett, 1981, pp. 255 - 260). Frege divides terms or expressions into those which are analysable and those which are non-analysable. Non- analysable terms are “logically primitive”. And logical analysis is incomplete…
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...until we have reached a definiens framed wholly in terms of expressions that may be claimed as indefinable in an absolute sense. (Dummett, 1981, pp. 256 - 257)
For an expression to be a tautology the meanings of the analysed terms must be in such a form as to indicate no possible variation of meaning and thought to anyone who understands the statement. For this to be the case terms are, let us say, sanitised – all rich and variable meaning content or associated thought is ‘cut away’. But what happens when an analysed term ends up, after analysis, not bearing much resemblance to its more complex forms? In other words, it bears no resemblance to what we originally understood when using it. What happens when we have reduced the meaning of a term and in doing so it loses the very meaning we take it to have in the first place? For Frege this is not a problem; it is in fact the aim of logical analysis and exhibits the exact character he takes analytic truth to have. An analytic sentence is not one which must convey the ordinarily associated sense of the sentence.
In any case, Frege’s position is clear: logical analysis does not require that the definitions given strictly preserve the senses of the defined expressions. (Dummett, 1981, p. 259)
That a logical analysis will not necessarily preserve the sense of a defined expression is explained, as I have above, by the fact that sense is not what is logically reducible, but is what is grasped when an expression is understood. For Frege, an unanalysed definiendum is a theorem. Such a theorem is then only fully analysed when the definiens is explicated in primitive terms. And this fully analysed definiens yields what Frege calls an analytic axiom. An analytic axiom will look like a logically primitive definition.
The parts of that definition which are equivalent to each other are intensions. And the expression is analytic because the intensions are isomorphic with each other; they logically resemble each other.
Frege holds (see above quote) that analytic expressions, or logically axiomatic propositions are absolutes or universal (Dummett, 1981, pp. 256 - 257). This means that, for Frege, there is only one correct way in which an expression can be analysed. It is for this reason that Frege says that meaning is objective. The implication is, of course, that there are objective facts about the primitive or axiomatic meanings of terms. If there were not it would be impossible to say of the analysis of expressions that they render absolute axioms. There are some problems which arise with what “absolute” is and whether or not it is possible to show that an analysis in fact renders an “absolute” axiom. What is certain though, according to Dummett, is that even though analysis might possibly not issue “absolute” axioms, this does not preclude that some terms, or all terms, do not have a “unique” analysis into its constituent terms (Dummett, 1981, p. 258). So, for Dummett, the fact that Frege’s account of analytic truth has not been
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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)