Abstract
Fine (2007) argues that Frege’s puzzle and its relatives demonstrate a need for a basic reorientation of the field of semantics. According to this reorientation, the domain of semantic facts would be closed not under the classical consequence relation but only under a stronger relation Fine calls “manifest consequence.” I examine Fine’s informally sketched analyses of manifest consequence, showing that each can be amended to determine a class of strong consequence relations. A best candidate relation emerges from each of the two classes, and I prove that the two candidates extensionally coincide. The resulting consequence relation is of independent interest, for it might be held to constitute a cogent standard of reasoning that proceeds under a deficient grasp on the identity of objects.
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Notes
Fine ([2], henceforth SR), 3, 35.
SR, 43.
SR 5, 39–40.
SR, 43.
Frege [3].
SR, 62.
SR, 49–50.
SR, 48.
SR, 48.
SR, 55ff, 136n14.
See Goldfarb [5] for an illuminating treatment of this point.
At least, not in ZF. In NBG, they would still be second-class citizens.
Kaplan [7].
In his later [6], Kaplan develops in more detail an analogy of a proposition with what he calls a “valuated formula”, which is, roughly speaking, a formula together with an assignment of values to its free variables. The latter notion is more or less familiar from logic, as a formula with parameters drawn from a given domain; see for example Krivine ([9], 63). But on Kaplan’s approach, as on the approach to be taken in what follows, one and the same expression results from assigning a as the value of ‘x’ and ‘y’ in the formulas ‘Fx’ and ‘Fy’.
Of course, this presupposes some procedure to ensure that the new terms do not clash with the existing apparatus—for example, that they do not overlap with the set of variables. Cf. Kaplan ([6], 274).
As Kaplan puts it: “Assigning me to ‘x’ yields a valuation of \([(y)y \text {is unmarried }\supset x \text { is unmarried}]\) which is not true in the domain of bachelors” ([6], 250).
The evaluation of contents in the earlier ([7], circulated in the 1970s) takes a smaller step to the same result, despite its possibilistic reading of the quantifiers. On this alternative approach, a non-descriptive linguistic term may receive as its content a ‘fallback value’, \(\dagger \), which is excluded from the domain of quantification and from the extensions of predicates. Now, suppose that the term a receives the value \(\dagger \). Then, at a world and time where the extension of the predicate F coincides with the domain of quantification, the Kaplanian contents of \(\ulcorner \neg Fa\urcorner \) and \(\ulcorner \exists x\neg Fx\urcorner \) are true and false respectively. The evaluation of Kaplanian contents resembles in this respect a special case of evaluation of \({\mathcal L}_{\mathcal U}\), in which the domain of the structure includes all but exactly one of the elements of \({\mathcal U}\).
This is a relettering of the statement at SR, 48.
SR, 135n11.
SR, 135n11.
I’m indebted to Nathan Salmon for discussion here.
According to the U.S. Department of Justice [18], “in 2010, 7.0 % of households in the United States…had at least one member age 12 or older who experienced one or more types of identity theft victimization.”
This stipulation can be weakened in various ways. As a first pass, the problem actually arises only in case \(\mathrm {Card}({\mathcal U}- \textrm {Ran}(\sigma \frown \tau ))<\mathrm {Card}(|\sigma \frown \tau |)\). This condition holds provided that, for example, \({\mathcal U}\)is the domain of a model of set theory and \(\mathrm {Ran}(\sigma \frown \tau )\) exists as a set in that model.
For the sake of informal discussion, M-consequence may here be taken to be any consequence relation I over \({\mathcal L}_{\mathcal U}\) such that \(M(K\cap L)\subseteq I\subseteq ML\).
SR, 41. The arguments of this passage may rely partly on earlier remarks about an “intuitive notion of meaning” with respect to which trivial and nontrivial statements of identity would differ (35).
In Fine’s initial presentation (SR 5) he does not sharply distinguish strict coreference from coordination; but the official exposition (SR 54ff) draws the distinction carefully.
SR, 54ff.
SR, 52.
SR, 67.
SR, 73.
For an example, see again U.S.D.O.J., op. cit.
SR, 136n14.
SR, 108ff.
Maybe not all historians of philosophy are fated to benefit from Proposition 12.
In particular, the underlying relation must invalidate statements of distinctness. Otherwise, the proposition \(\ulcorner a=b\urcorner \) K-entails the absurdity under the scheme which does not coordinate the two occurrences; yet we need to explain how the truth of identity could be an open question.
There seems to be room here for improvements of Proposition 16. Confusion about hypotheses is a fact of life, but we might still want to find consequences. We could ensure the soundness of our inferences by insisting that their soundness be manifest. But is there a better, or even optimal strategy for reasoning from possibly confused hypotheses in such a way as to quarantine the confusions, wherever they happen to be? Note that it will not suffice to insist that the conclusion contain none of the objects that had been confused, since inferentially exploited objects can be laundered out through quantification.
I take it that this is the source of immunity of Finean proxies to the fantastical “Bruce” objection (SR, 36).
I’m indebted here to the comments of an anonymous reviewer.
Cf. SR, 57.
More precisely, (2) amounts to the claim that the correspondence derives from those features of the occurrence of the name which are independent of the sentential context of the occurrence. Supposing that, roughly speaking, the occurrence of a name in a sentence, considered independently of the sentence, is just the name, then, (2) may be understood as the claim that the correspondence of the entity to the occurrence derives from the properties of the name which occurs.
To derive the property (1), it suffices to apply to a coordinated proposition the following theorem: for any set S and any equivalence relation E over S, there is a function f such \(\forall x\in S(f(x)=f(y)\leftrightarrow Exy)\). To derive (4), note that full understanding presupposes grasp of the extension of the coordination scheme, and that the formula just given is manifestly applicable.
SR, 35.
With respect to original Sinn, it seems clear that Frege accepted (3) and (4); if, taking a stand on a controversial interpretive issue, we grant that Frege did think of his doctrine of Sinn as part of a semantic theory, then he might also accept (1). However, Frege maintained that (1) excludes (2): see for example Frege [4], in Austin, trans., ([4], pp. x and 71), and the letter to Russell from 29 June 1902, in McGuinness et al., (1980, 135ff). For representative discussion of these issues, see Potter and Ricketts [14].
See SR, 57ff for Fine’s own account of the contrasts.
This suggestion is due to an anonymous reviewer.
Cf. SR, 124–125.
I hope that Kaplan would not be offended by this allusion to his [8].
I wish to thank Roger Clarke, Sid Grewal, Yannig Luthra, Nathan Salmon, and Ori Simchen for discussion. Thanks are due as well to the two anonymous reviewers for this journal, whose generosity enriched the paper throughout. I should note here my primary motive for the explorations of this paper, which is curiosity.
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Weiss, M. A Closer Look at Manifest Consequence. J Philos Logic 43, 471–498 (2014). https://doi.org/10.1007/s10992-013-9269-3
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DOI: https://doi.org/10.1007/s10992-013-9269-3