Abstract
Kit Fine has replied to my criticism of a technical objection he had given to the version of Millianism that I advocate. Fine evidently objects to my use of classical existential instantiation (EI) in an object-theoretic rendering of his meta-proof. Fine’s reply appears to involve both an egregious misreading of my criticism and a significant logical error. I argue that my rendering is unimpeachable, that the issue over my use of classical EI is a red herring, and that Fine’s original argument commits the straw-man fallacy. I argue further that contrary to Fine’s gratuitous attribution, what Kripke’s Pierre lacks and a typical bilingual has is not knowledge (“possession”) of a “manifest-making” (in fact, spectacularly false) premise, but the capacity to recognize London when it is differently designated. Fine’s argument refutes a preposterous theory no one advocates while leaving standard Millianism unscathed. The failure of his argument threatens to render Fine’s central notion of “coordination” redundant or empty.
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Notes
Fine (2007, 2014) and Salmon (2012, pp. 407–441).
I take this opportunity to correct some of the misformulations in “Recurrence”: On p. 412, line 4, ‘dissents from’ should read ‘assents to’. On p. 422, lines 13 from top, 8 from bottom, and also 5–4 from bottom, the recurring clause ‘each is self-consistent’ should read ‘they are consistent’. On p. 423, line 6, ‘each of the two propositions is itself’ should read ‘the two propositions are’. On p. 423, lines 7–8, ‘are then also perfectly consistent with each other’ should read ‘in that case do not entail any contradiction’.
Standard compositionality holds that mere recurrence of an expression within a sentence (as in ‘Bachelors socialize with other bachelors’, as opposed to ‘Unmarried men socialize with other bachelors’) does not itself contribute to the sentence content.
Kripke (2007, pp. 1002–1036)
SR, pp. 82. The word ‘legitimize’ is my term for the relation between whatever condition or state of affairs it is, the satisfaction of which would justify the deduction in question, and the deduction itself. See “Recurrence,” p. 414.
SR, p. 137n4. An inference of a conclusion χ from premises ϕ1, ϕ2, …, ϕ n is a deduction (“proof”) of χ from (“as a consequence of”) those premises. This definition is adapted from Church (1983, pp. 198–199).
The “complete, purely qualitative description” of London is supposed to be simply ‘pretty capital’—this on the ground that the hypothesized legitimizing knowledge together with the propositions that London is pretty and that London is a capital allegedly entail that any pretty capital is purely-qualitatively indiscernible from London. It should be noted that even if something is a pretty capital iff it has London’s entire purely qualitative profile, it does not follow that the phrase ‘pretty capital’ entails that profile. It is quite bad enough, however, if standard Millianism commits Cousteau to the thesis that all pretty capitals are purely-qualitatively indiscernible. See note 8 below.
On the other hand, as characterized, ⊨ m ‘(Fa & Ga) → ∃x(Fx & Gx)’. The theorems of manifest logic are exactly those of classical logic. Like Cousteau, Pierre knows by logic alone that if London is pretty and London is a capital, then something is a pretty capital. Also like Cousteau, Pierre believes that London (“Londres”) is pretty and London is a capital. Unlike Cousteau, and like the tortoise, Pierre does not infer that something is a pretty capital.
Fine’s definition of manifest consequence in SR, pp. 48–49, is incorrect. See “Recurrence,” p. 418n17 where I suggested a possible repair. Weiss (2014) explores my proposal and alternative characterizations in “A Closer Look at Manifest Consequence,” Journal of Philosophical Logic, doi: 10.1007/s10992-013-9269-3. I am indebted to Weiss for discussion. The characterization provided here is based both on my proposal and Weiss’s favored definition. The alternative definition in “Rejoinder” (first page), although imprecise, may be adequate for present purposes.
One sure-fire way to construct a proof-theoretically valid manifest-logical deduction for an argument ⌜ϕ1. ϕ2. … ϕ n ∴ χ⌝ is to proceed in two stages: first construct a proof-theoretically valid classical deduction for an argument ⌜ϕ′1. ϕ′2. … ϕ′ n ∴ χ′⌝ whose premises lack free recurrence of any constant or variable and from which ⌜ϕ1. ϕ2. … ϕ n ∴ χ⌝ is obtainable by proper substitutions; then perform those very substitutions throughout the classical deduction.
The following suffices as a third and manifestly-validating premise: ‘If anything is pretty then any capital is pretty’. It is bad enough if standard Millianism commits Cousteau to something such as this as a third premise. Fine’s tacit assumption of (ii) makes I out to be a good deal worse than this.
If p is a singular proposition that satisfies hypothesis (i)–(ii) and in which an individual recurs, then some variant of p satisfies all of (i)–(iii). Otherwise, Cousteau would need in addition to p a separate legitimizing proposition, p′, in order to recognize and capitalize upon the recurrence. In that case, it would not be knowledge of p that in itself legitimizes Cousteau’s deducing that London is a pretty capital, contradicting (i). The threat of infinite regress is avoided by taking I to be free of individual recurrence. Similarly, according to (i)–(ii) Cousteau cannot legitimately exploit any recurrence of London among {that London is pretty, that London is a capital, I}, in order to deduce that London is a pretty capital.
The meta-proof Fine originally offered in SR (p. 137n4) concerns the notion—presumably proof-theoretic—of a reasoner’s being justified in deducing (“inferring”) a conclusion from premises. In “Rejoinder” he recasts his meta-proof as one concerning instead the notion of manifest consequence. (He says, second page, n4, “I also talk explicitly of manifest consequence rather than justified inference.” He means this in contrast to his original meta-proof, which is the subject of my criticism in “Recurrence.”) The reformulation of his meta-proof as one that does not (or might not) specifically concern proof theory unnecessarily renders potentially curious an otherwise obviously appropriate focus on an object-theoretic deduction.
Fine does not specifically object to it but line 10 is also obtained by standard EI.
Fine refereed an early draft of “Recurrence.” In “Rejoinder” (second-third pages), he characterizes a criticism from the unpublished early draft. In the draft I had taken for granted that in SR Fine had attributed to standard Millianism the more natural hypothesis that the alleged legitimizing information I somehow correlates the distinct occurrences of London in Cousteau’s separate beliefs. The unpublished draft had challenged Fine’s apparent inference from ‘{I, [F&P]x, [G& ~ P]x} ⊢ ⊥’ to ‘{I, ∃x[F&P]x, ∃x[G& ~ P]x} ⊢ ⊥’. Contrary to Fine’s characterization, the early draft had not in effect questioned the corresponding inference invoking ‘⊨ m ’ in lieu of ‘⊢’. Fine pointed out in his referee’s comments that he had taken the legitimizing information I to be a manifestly-validating premise. I was grateful to have Fine’s comments and revised “Recurrence” accordingly. See notes 13, 16, and 17 below and “Recurrence,” p. 424n22.
Fine says in “Rejoinder” (third page, n5) that he explicitly assumed in SR that standard Millianism hypothesizes that I legitimizes Cousteau’s deduction by being a third and manifestly-validating premise. Fine also strongly suggests (second page) that his assumption should have been clear from his having argued (SR, p. 81) that Cousteau’s deduction cannot be legitimized by depicting it as taking a detour through semantic ascent. The assumption was nowhere stated or clearly implied in SR. Perhaps it is entailed by what Fine intended in saying of the standard Millian that
he must work with a conception of propositional knowledge that is closed under manifest rather than classical consequence. Given that a thinker knows the proposition that x Fs and also knows the proposition that x Gs, he does not necessarily know the proposition x both Fs and Gs, no matter how logically competent he may be (SR, pp. 80–81).
Neither belief nor knowledge is closed under either manifest or classical consequence. Given his peculiar circumstances, Pierre cannot justifiably deduce that London is a pretty capital from his two beliefs concerning London. It does not follow that one cannot do so even under normal circumstances. On the contrary, Cousteau presumably does exactly this. Manifest logic might be useful in determining what someone in a Pierre-type predicament is not justified in deducing—subject to significant limitations. (Suppose Pierre correctly translates the Italian ‘Londra’ into French as ‘Londres’ and comes to accept ‘Londra è una capitale’.) It does not apply in this way to a normal speaker/thinker not in a Pierre-type predicament. Cf. “Recurrence,” p. 426n24.
In “Rejoinder” Fine apparently continues to maintain that standard Millianism must hypothesize (ii) to account for Cousteau’s justification in deducing that London is a pretty capital. He writes (second page), “if the inference is to be justified, then it should be justified … on the basis of the content of some premises, and if it is justified on the basis of the content of some premises, then it must be manifestly valid … Salmon is surely right in thinking that the Millian’s best—and indeed, his only—defense is to deny that I need play this manifest-making role. Perhaps Salmon means to say no more than this and is perfectly happy to admit that my argument is valid under the assumption” of hypothesis (ii). I admitted this in “Recurrence”; I also argued that the alleged near-reductio as presented in SR is invalid, and by contrast that the intended argument tacitly assumes that standard Millianism hypothesizes (ii), and is thereby valid but disproves a ludicrous straw-man theory. (See note 11.) “Rejoinder” proceeds on the erroneous assumption that I declared the intended argument invalid.
Whereas Fine’s intended reductio of standard Millianism is logically valid, there is a significant logical problem with SR as a whole. According to his underlying reductionism, what Fine actually means by some of his statements ostensibly conflicting with tenets of standard Millianism are things that standard Millianism in fact embraces. A central thesis of SR might be encapsulated thus: According to standard Millianism the English sentence ‘Cicero = Cicero’ expresses the uncoordinated singular identity proposition about Cicero that he is him; whereas (in contrast to ‘Cicero = Tully’) ‘Cicero = Cicero’ in fact expresses the corresponding positively coordinated singular proposition. Yet by Fine’s lights, standard Millianism holds that ‘Cicero = Cicero’ “expresses the positively coordinated identity singular proposition” about Cicero that he is him, in Fine’s misleading sense of the phrase (on which it means nothing more than that it is a manifest theorem of English semantics that the sentence expresses the singular identity proposition in which Cicero occupies both the subject and object positions). Fine’s reductionism apparently renders SR committed to both the claims that standard Millianism entails p (for a particular p), and that standard Millianism does not entail p. See “Recurrence,” pp. 439–440.
Standard EI is manifest-logically legitimate iff: whenever (a) ϕγ results from uniform substitution of free occurrences of the variable γ for the free occurrences of the variable α throughout ϕα; (b) there is a classically legitimate deduction of χ from Γ ∪ {ϕγ} that does not exploit any free recurrences of a constant or a variable that come from the term’s free recurrence among the elements of Γ ∪ {⌜∃αϕα⌝} (though it may exploit the free recurrence, if any, of γ in ϕγ); and (c) γ does not occur free in ⌜∃αϕα⌝, in χ, or in any element of Γ (and whatever further restrictions the apparatus requires of classical EI), then Γ ∪ {⌜∃αϕα⌝} ⊨ m χ. Consider ϕγ as the result of applying EI to ⌜∃αϕα⌝ in accordance with standard restrictions in a purported manifest-logical deduction of χ from Γ ∪ {⌜∃αϕα⌝}. Assume (a)–(c). In that case there are Γ′, ϕ′α, ϕ′γ, and χ′ such that: (1) Γ′ ∪ {ϕ′γ} ⊢ χ′, where (2) Γ, ⌜∃αϕα⌝, and χ result from uniform substitution of free occurrences for the free-variable occurrences throughout Γ′, ⌜∃αϕ′α⌝, and χ′, respectively, and ϕ′γ results from uniform substitution of free occurrences of γ for the free occurrences of α throughout ϕ′α; (3) there is no free singular-term recurrence among the elements of Γ′ ∪ {⌜∃αϕ′α⌝} (though there may be free recurrence of γ in ϕ′γ); and (4) γ does not occur free in ⌜∃αϕ′α⌝, in χ′, or in any element of Γ′ (etc.). It follows by classical EI (and soundness) that (5) Γ′ ∪ {⌜∃αϕ′α⌝} ⊨ χ′. By (2), (3), and (5), Γ ∪ {⌜∃αϕα⌝} ⊨ m χ. (See note 7 above.)
Standard EI does not preserve truth in every model. It is truth-preserving only in a weaker sense: For any model and any assignment s to variables of values taken from the model’s universe, if the antecedent line ⌜∃αϕα⌝ is true in that model under that assignment, then the inferred line ϕγ is true in that model under at least one assignment s' that agrees with s with regard to the free variables of ⌜∃αϕα⌝. The so-called inference is more an “assumption without loss of generality,” even more an assumption with intentional non-specificity (and not a posit of a philosophically peculiar object). To accommodate this the deductive apparatus imposes severe restrictions on γ. By contrast, the classically valid argument ‘R(aa) ∴ ∃xR(xx)’ is not manifestly valid. Fine accepts as manifestly valid the EG inference from ‘Fx’ to ‘∃xFx’ (“Rejoinder,” first page). The converse move, though deductively legitimate only in the weaker sense (provided the safeguards are respected), is equally “manifest.”
The following corollary of ∃2E m sanctions line 11a:
∃1E m :If Γ ∪ {ϕγ} ⊨ m χ and no individual constant or variable occurs free in χ, then Γ ∪ {⌜∃αϕα⌝} ⊨ m χ.
Let: Γ be {lines 1 and 10}; ⌜∃αϕα⌝ be line 9; ϕγ be line 11a; and χ be ‘⊥’. (There is no free singular-term recurrence among lines 1, 9, 10, and 11a.)
See notes 11–13 above. In the unpublished draft that Fine refereed, I offered a deduction based on his argument as it was presented in SR, which specified (i) but not (ii). In “Rejoinder” (third page), Fine writes, “To my astonishment, I discovered upon reading the published paper that [Salmon] insisted upon presenting the same problematic reconstruction of the argument … and then accused me of fallaciously arguing” to line 11b. The deduction in the unpublished draft and the deduction in “Recurrence” (p. 421) are significantly different. Most significant, in the former deduction line 11 is obtained through application of FI to line 9; in the “Recurrence” deduction, which took account of Fine’s unstated assumption of (ii), line 9 is instantiated to ‘y’ and line 11 depends on line 1. As a direct result, the former deduction is highly problematic, the latter entirely unproblematic; indeed, the former is classically invalid, the latter manifest-logically valid. Fine evidently did not in fact read the relevant portions of “Recurrence,” especially pp. 423–426 as well as the annotation for line 11. (I thank Daniel Kwon for pointing this out.)
The apologist’s comment is ambiguous because the distinction was not clearly drawn between a proof-theoretic requirement on an unspecified premise for a particular deduction to be valid, and an additional premise to the effect that the unspecified premise satisfies the requirement. However, the apologist seemed to concede that Fine’s attempted reductio would indeed fail if the unspecified premise I had to be sufficient to validate the sub-deduction from it together with lines 10 and 11a either to line 11b or to ‘y is both pretty and P’ for the object-theoretic deduction best corresponding to Fine’s meta-proof to be manifest-logically valid, as I must do for the deduction provided in “Recurrence” to be proof-theoretically valid. (The apologist presents the deduction employing FI as the apologist’s own reconstruction, based on his/her more careful consideration of Fine’s meta-proof. See the preceding note.)
Here is a sketch of a proof: Assume (ii) and suppose that {‘London is both pretty and P’, ‘London is both a capital and Q’, I} ⊨ m ‘London is at once pretty, P, a capital, and Q’, where the predicate-variables ‘P’ and ‘Q’, neither of which occurs free in I, range over purely qualitative properties, there is at most one occurrence of a name of London in I, and by (iii), there is no free singular-term recurrence in I. Let I z be the result of substituting a free occurrence of the variable ‘z’ for the occurrence of a name of London, if any, in I (I z may be I itself), and let Γ be the alternative premise set {‘x is both pretty and P’, ‘y is both a capital and Q’, I z } in which there is no free singular-term recurrence. There are three cases to consider: Either (1) Γ ⊨ ‘x is at once pretty, P, a capital, and Q’; or (2) Γ ⊨ ‘y is at once pretty, P, a capital, and Q’; or (3) Γ ⊨ ‘z is at once pretty, P, a capital, and Q’. In case (1) (by the deduction theorem), I entails I F . In case (2), I entails \( I^{\prime}_{F} \). In case (3), I entails (\( I^{\prime\prime}_{F} \)) that London has all the purely qualitative properties both of anything pretty and of any capital. Each of I F , \( I^{\prime}_{F} \), and \( I^{\prime\prime}_{F} \) separately entails I′. For example, to deduce \( I^{\prime}_{F} \) from I F , assume I F and suppose that x is a capital and that y is both pretty and P. Assume for a reductio that unlike y, x is not P. In that case x is non-P. Then by I F , y is also non-P. Hence, y is not P. But y is P. Therefore, x is like y in being P.
References
Church, A. (1983). Entry on ‘logistic system’. In D. D. Runes (Ed.), The standard dictionary of philosophy (pp. 198–199). New York: Philosophical Library.
Fine, K. (2007). Semantic relationism. Oxford: Blackwell.
Fine, K. (2014) Recurrence: A rejoinder. Philosophical Studies. doi:10.1007/s11098-013-0189-4.
Kripke, S. (2007) A Puzzle about Belief. Reprinted in On Sense and Direct Reference, pp. 1002–1036 by M. Davidson, ed., 2007, Boston: McGraw-Hill.
Salmon, N. (2012). Recurrence. Philosophical Studies, 159, 407–441.
Weiss, M. (2014) A closer look at manifest consequence. Journal of Philosophical Logic. doi:10.1007/s10992-013-9269-3.
Acknowledgements
I owe thanks to Philip Atkins, Daniel Kwon, Teresa Robertson, Max Weiss, and the Santa Barbarians for discussion; and to an anonymous referee and apologist for Kit Fine for providing comments that, although hostile, helpfully fill some critical gaps in Fine’s published positions. I am especially grateful to the late Donald Kalish, who was a brilliant thinker and a generous and superb teacher, and who is responsible for most of my knowledge about existential instantiation.