with itself. Clearly, this alone does nothing to show that logical truths are somehow insubstantial in any metaphysical, epistemologically explanatory sense (see the end of Section 1). For instance, it is compatible with the hypothesis that there are truths of second-order logic which characterize the necessary structure of reality in profound ways and can never be known by any mind. A fortiori, nothing has been done to show that Frege-analytic truths are insubstantial.15
To make the point vivid, call a sentence “Einstein-analytic” just in case it is synonymous with a truth once uttered by Einstein. Trivially, every truth once uttered by Einstein is Einstein-analytic. That does nothing to show that truths once uttered by Einstein are in any sense insubstantial; a fortiori, nothing has been done to show that Einstein-analytic truths are somehow insubstantial. Of course, if we had independent reason to regard all logical truths as somehow insubstantial, that would presumably give us reason to regard all Frege-analytic truths as insubstantial in some related way, but the mere definition of “Frege-analytic” provides no such reason. Quine devoted some of his most powerful early work to arguing that logical truths are not analytic in a less trivial sense (Quine 1936).
To explain why “All furze is furze” is a logical truth while “All furze is gorse” is not, use was made of Tarski’s standard model-theoretic account of logical consequence as truth-preservation under all interpretations which preserve logical form, and in particular of logical truth as truth under all such interpretations (Tarski 1983b). It lends no support to any conception of logical truths as somehow insubstantial. The truth of a sentence under all interpretations which preserve its logical form in no way make its truth under its intended interpretation insubstantial.16 To use a style of argument from Section 2, consider this simple logical truth (with “if” read as the material conditional):
(7) If Barbara is a barrister, Barbara is a barrister
Its compositional semantic evaluation proceeds in parallel to that for the non-logical analytic truth (3) and the synthetic truth (4); each is true because it is a material conditional with a true antecedent and a true consequent. All three are true in the same way. From the perspective of compositional semantics, logical truths are true in the same way as other truths.
In one good sense, sentences of the form “P if and only if actually P” are logical truths, and therefore Frege-analytic, because true in every model (Davies and Humberstone 1980, Kaplan 1989). Nevertheless, they can express contingent truths on the same reading; it is not necessary for me to be my actual height. Although we could add a modal qualification to the definition of logical truth in order to exclude such examples, by requiring logical truths to be true at every world in every model, this mixing together of the modal dimension with the world dimension is bad taxonomy; perspicuous basic notions keep such different dimensions separate. Thus Frege-analyticity, like modal-analyticity, violates Kripke’s constraint that analyticity implies necessity. In this respect Frege-analyticity too may diverge from the traditional conception.
The mathematical rigor, elegance, and fertility of model-theoretic definitions of logical consequence depend on their freedom from modal and epistemological accretions. As a result, such definitions provide no automatic guarantee that logical truths express necessary or a priori propositions. This is no criticism. As a theoretical discipline, logic only recently attained maturity. Tarski’s model-theoretic notion of logical consequence has turned out to be a key theoretical notion. To reject it on the basis of preconceived extraneous constraints would subvert the autonomy of logic as a discipline. Pretheoretic conceptions of logical consequence are in any case too confused to provide much guidance on subtle issues.17 Still, those who do have a non-standard account of logical truth can feed it into the definition of “Frege-analytic” if they like.
“All furze is furze,” unlike many logical truths, is obvious. That does not justify the idea that it imposes no constraint on the world, rather than one which, by logic, we easily know to be met (Wittgenstein, Tractatus Logico-Philosophicus, 4.461–4.4661 and 6.1–613). What case does the constraint exclude? That not all furze is furze, of course. To complain that “Not all furze is furze” does not express a genuine case is to argue in a circle. For it is to assume that a genuine constraint must exclude some logically consistent case. Since substantiality was being understood to consist in imposing a genuine constraint, that is tantamount to assuming that no logical truth is substantial, the very point at issue. Concentration on obvious logical truths obscures this circularity.
We may hope, given an epistemology for logical truths, to extend it to an epistemology for Frege-analytic truths. That task will not be trivial, for cognitive differences may arise between synonymous expressions, even for those who understand them. For example, Kripke (1979) has argued persuasively that a competent speaker of English can understand the synonymous expressions “furze” and “gorse” in the normal way without being in a position to know that they refer to the same thing. Such a speaker will assent to the logical truth “All furze is furze” while refusing assent to the Frege-analytic truth “All furze is gorse.” Similarly, on standard theories of direct reference, coreferential proper names such as “Hesperus” and “Phosphorus” are synonymous, so an astronomically ignorant competent speaker may assent to the logical truth “If Hesperus is bright then Hesperus is bright” while refusing assent to the Frege-analytic truth “If Hesperus is bright then Phosphorus is bright.”
The epistemological consequences of such examples are contested. According to some direct reference theorists, the proposition that if Hesperus is bright then Phosphorus is bright is the proposition that if Hesperus is bright then Hesperus is bright, so whoever knows that if Hesperus is bright then Hesperus is bright ipso facto knows that if Hesperus is bright then Phosphorus is bright.18 However, even granted that view of propositional attitude ascriptions, that speaker is in no position to know that if Hesperus is bright then Phosphorus is bright under the guise of the sentence “If Hesperus is bright then Phosphorus is bright,” but only under the guise of the sentence “If Hesperus is bright then Hesperus is bright.” In a sense the speaker cannot express their knowledge by using the merely Frege-analytic sentence, even though it expresses the content of that knowledge: if they do use the sentence, their utterance will not be causally connected to their knowledge state in the right way. In elliptical terms, the speaker knows “If Hesperus is bright then Hesperus is bright” without being in a position to know “If Hesperus is bright then Phosphorus is bright”; they know the logically true sentence without being in a position to know the merely Frege-analytically true sentence.
If propositions are individuated in that coarse-grained direct reference way, what matters for progress in philosophy is less which propositions we know than which sentential guises we know them under. Suppose, just for the sake of argument, that some form of physicalism is true, and pain is in fact identical with π, where “π” is a name whose reference is fixed by a neuroscientific description. According to a hard-line direct reference theory, “pain” and “π” are synonymous. The hypothesis “Pain is π” becomes a focus of philosophical controversy. On some direct reference theories, everyone knew all along that pain is π, because they knew all along that pain is pain and the proposition that pain is π just is the proposition that pain is pain. If that view is correct, it just shows that such attitude ascriptions constitute the wrong level of description for understanding philosophical activity. What matters is that although everyone knew the proposition under the guise of the logical truth “Pain is pain,” they did not know or even believe it under the guise of the merely Frege-analytic truth “Pain is π.” In elliptical terms, they knew “Pain is pain” but not “Pain is π.” Perhaps such physicalist theories are false, but we can hardly expect philosophy to be a discipline in which there are no informative identities; the moral of the example stands. The need for such finer-grained descriptions of propositional attitudes is even more urgent if propositions as the objects of knowledge and belief are identified with sets of possible worlds, for then all necessary truths are identical with the set of all possible worlds: anyone who knows one necessary truth knows them all (Lewis 1996, Stalnaker 1999: 241–73). Thus a coarse-grained account of attitude ascriptions does not trivialize the problem of extending an epistemology for logical truths to an