Abstract
That there are analytic truths may challenge a principle of the homogeneity of truth. Unlike standard conceptions, in which analyticity is couched in terms of “truth in virtue of meanings”, Frege’s notions of analytic and a priori concern justification, respecting a principle of the homogeneity of truth. Where there is no justification these notions do not apply, Frege insists. Basic truths and axioms may be analytic (or a priori), though unprovable, which means there is a form of justification which is not (deductive) proof. This is also required for regarding singular factual propositions as a posteriori. A Fregean direction for explicating this wider notion of justification is suggested in terms of his notion of sense (Sinn)—modes in which what the axioms are about are given—and its general epistemological significance is sketched.
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Notes
Frege, FA. I emphasize "philosophical" to shunt here aside the formal achievements and the formal difficulties involved in FA, which are not our concern here.
The question was already discussed by Aristotle, who claimed that the foundations of a demonstration are not demonstrable—see Aristotle (1941) Posterior Analytics, 72b 10–18. He adds that one cannot ask a reason for everything, (Met. 1011a).
See for instance "Logic in Mathematics", in PW, 203–250, particularly 203–5. Frege, though, was well aware already in Begriffsschrift that there may be alternative (equivalent) systems such that an axiom in the one is provable in another. See Begriffsschrift, e.g. §13. Cf. also PW, p. 205.
In "Frege on Apriority" (Burge 2005, 356–387) Burge claims that these are definitions stating sufficient and necessary conditions for analyticity and apriority, and gives some reasons for thinking so (359). I remain unconvinced. The reasons given in the text for not thinking these formulations as stating necessary conditions—leaving open the possibility of regarding the axioms as analytic—seem to me to overweigh those given by Burge.
The general direction of the connection can be appreciated by realizing that there is an internal connection between mode of being given, and being a constituent of thought; these are Frege's two main characterizations of his notion of sense. A thought, however, is logically constituted by its inferential relations. This is explicit with regard to Frege's earlier notion of content (of a judgment) in Begriffsschrift, §3, but is also true, I believe, of his later notion of thought. Hence, modes of presentation, which are the constituents of thoughts, cannot be detached from this logical space of inferential relations. For a fuller account, cf. Bar-Elli (2001). For a somewhat different approach, couched in terms of the connection between "pragmatical justification" of a system, and the self-evidence of an axiom, see Burge (1998), §§III–IV.
The apriority of the axioms of geometry is presumed, for instance, in Dummett (1981), p. 464.
In his 1998 Burge says that Frege "neglects to formulate his notions of analyticity and apriority so as to either include or rule out the foundations of logic" (310; in Burge 2005, p. 322). In note 6 added there he agrees with Dummett that this was an "oversight" on Frege's part, easily emended so as to count the axioms analytic and a priori. In his "postscript to 'Frege on Apriority' " he withdraws this admission, and says that Frege intentionally followed suit with Kant in not regarding the axioms of logic as analytic, the reason being that "such laws are not subject to analysis" (Burge 2005, 388–389). I agree with Burge's recent view that this is not an oversight. Paying attention to the emphasized principle J (which Burge doesn't mention) is sufficient to show this conclusively. Burge, however, concludes that the axioms of logic are not analytic, and axioms of geometry are not a priori. This, as I argue in the text, seems to me unacceptable, and I suggest another way: proof, formal deductive demonstration, is not the only form of justification. The axioms can be justified, but not by proving them.
In his (1998, §IV) Burge, quite convincingly, suggests a distinction between a subjective notion of self-evidence, which he calls "obviousness", and an objective one, which he explicates in terms of an ideal mind. When using self evidence as characteristic of axioms, Frege refers to the latter, not the former.
Cf. Burge (1998), p. 336.
Cf. also Frege's Begriffsschrift.
For a fuller account, see Bar-Elli (2001).
It should be noted that the general idea of sense is evident not only in parts of FA, but already in Bs, §8, where Frege argues that understanding identity statements involve recognizing that names for the same content may differ in their Bestimmungsweise—the ways they determine it, which is somewhat akin to his later notion of ways of being given (Art des Gegebenseins), or sense, see Bar-Elli (2006).
See the Preface to Bs; FA §3; PW, p. 3; ibid. 147.
See for instance FA §26, BL xvii/15, PW 149/137. Cf. Bar-Elli (1996), Chap. 2, particularly pp. 36–47.
See for instance, "Logic", in PW, p. 3; "17 Key Sentences…", PW, 175; cf. also Bar-Elli (2001).
I talk of "objects" and "things an axiom is about" rather loosely here. They include not only objects in the narrow sense, but also functions and concepts. Axioms (whether analytic or only a priori) are usually general judgments, and as such are strictly about concepts. Only rarely (as in e.g. axiom V of BL) about objects. However, in a wider sense we can say that the axioms of arithmetic are about numbers, and those of geometry—about points and lines. This looser way of talk does not affect the main point made here.
Recall the crucial passage from "Logic" (PW 3) cited above: "… But if there are any truths recognized by us at all, this cannot be the only form that justification takes …" Cf. also FA §26. Already in Bs (§13) Frege spoke of the axioms (the basic truths) as comprising the "kernel" of the whole system. He emphasize this again with regard to mathematics e.g. in "Logic in Mathematics", PW p. 204–5. With regard to geometry, he insists on the truth of the axioms, and on holding either Euclidean or non-Euclidean geometry to be true—"No man can serve two masters" ("On Euclidean Geometry", PW, p. 169).
I thus think that Burge, for instance, in grounding Frege's belief in the truth of Euclidean geometry just in its longevity, underrates the force of this argument (cf. Burge 1998, p. 327).
This, with a comparison to his later view in "On Sense and Reference" is elaborated in Bar-Elli (2006).
Scholars have debated on whether this way of being given is basically Kantian spatial intuition, and whether Frege can be regarded as a Kantian in this respect. I shall not discuss this here. Cf. Dummett (1981), pp. 463–470.
Bar-Elli (2001).
Cf. Frege's discussion of "snow is white" in FA, §26. For an elaborated discussion, cf. Bar-Elli (1996), 35–43.
This may have escaped Boghossian. He distinguishes an epistemic notion of analyticity from a metaphysical one, and says of the former: "According to Frege, a statement's analyticity (in my epistemological sense) is to be explained by the fact that it is transformable into a logical truth by the substitution of synonyms for synonyms." He talks of this as a "semantical condition". However, the relevant kind of definitions (what Frege calls "analytical definitions") are not statements of synonymy or sameness of conventional meaning, but explications of modes of being given. Therefore, "semantical" in Boghossian's formulation also seems inappropriate. I elaborate on it in my (2009), Chap. 7. See also n. 30.
This has been argued in detail in Bar-Elli (1985).
Reality, in the Tractatus is the totality of facts, or facts in logical space, and facts are “existing states of affairs”. All these raise notorious questions we shall not deal with.
This has been argued in detail in Bar-Elli (2009), Chap. 7 ("Definition and Analysis in Frege"), where it is shown that the course of argument of Frege's FA expounds a sophisticated view of the nature of philosophical analysis, in which analytical definitions (to be distinguished from "constructive" definitions) are part of a philosophical explication of ways in which the things concerned are given to us. This explication respects three fundamental principles:
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The principle of about—propositions are about things referred to by their terms.
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The context principle—one of whose implications is that securing a sense for a proposition guarantees meaning for its terms.
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The principle of the explanatory value of sense—Explicating the sense of a term is explicating a way its reference is presented, which explains and justifies propositions about it.
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The point may be implied by (or at least connected to) Kripke's insistence to use "analytic" as narrower than "necessary". An analytic truth, he insists, is not only necessary, i.e. true in all possible worlds, but is so by virtue of its meaning (see Kripke 1972, p. 39 and note 63 on p. 122). "True by virtue of meaning" remain operative (and unclear).
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Acknowledgments
I have been benefited, in this research, from a grant of The Israel Academy of Science.
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Bar-Elli, G. Analyticity and Justification in Frege. Erkenn 73, 165–184 (2010). https://doi.org/10.1007/s10670-010-9248-9
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DOI: https://doi.org/10.1007/s10670-010-9248-9