Abstract
§§3–4 of the Begriffsschrift present Frege’s objections to a dominant if murky nineteenth century semantic picture. I sketch a minimalist variant of the pre-Fregean picture which escapes Frege’s criticisms by positing a thin notion of semantic content which then interacts with a multiplicity of kinds of truth to account for phenomena such as modality. After exploring several ways in which we can understand the existence of multiple truth properties, I discuss the roles of pointwise and setwise truth properties in modal logic. I argue that thinking of supertruth and determinate truth as setwise truth properties allows an understanding of supervaluationist approaches to vagueness which escapes both Williamson’s objections to and a needless metalinguistic orientation of traditional supervaluationism.
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Notes
- 1.
See Williamson (1994), 112, for this objection. The objection is an application to the domain of vagueness of the Gödel-McKenzie-Dugundji objection discussed in Section 6 below.
- 2.
I don’t particularly recommend this way of thinking about truth. If assertion is taken to be subject to an epistemically-specified regulative norm (such as ‘assert only that which you know’), truth construed as the internal accusative for assertion threatens to push one toward a substructural logic. If, of course, the regulative norm for assertion is truth-based (‘assert only that which is true’), then truth cannot be analyzed or illuminated by being characterized as assertion’s internal accusative.
- 3.
Quantity, quality, relation, and mood, in the classical terminology. See, for example, Lotze (1884) for a clear articulation of these categories, although Lotze is, not surprisingly, much closer to Frege than the nineteenth century norm in his inclination to identify a force-free content underlying these logical differentia.
- 4.
Thus, one suspects, Venn’s failure in his review of the Begriffsschrift to comprehend the radicality of Frege’s logical moves:
The merits he claims as novel for his own method are common to every symbolic method. … He calls attention to the fact that, on his scheme, the distinction, so important in grammar and on the predication-view of ordinary logic, between subject and predicate loses all its significance, that hypothetical and disjunctive propositions become equivalent to categorical, and so on; all these being points which must have forced themselves upon the attention of those who have studied this development of Logic. (Venn (1880), 297)
- 5.
So, perhaps, answers are ‘truth values’ for interrogatives and satisfactions for imperatives.
- 6.
Of course, eventually an adequate account must yield that a sentence \(\phi\) is true\(_{\Box}\) just in case it is necessary that \(\phi\) is true\(_{@}\). The present thought is, however, that this result will be a downstream consequence of an independent understanding of true\(_{\Box}\), rather than the very definition of that would-be VF.
- 7.
See the end of Section 9 below for more on this point.
- 8.
This constraint follows from the superclassicality of \(\vDash_X\) if we take ‘truth preservation’ to block the move from truth\(_X\) to falsehood \(_X\). In the case of truth\(_\square\), the constraint can be thought of as a realization of the D axiom.
- 9.
Hence the result extends slightly further, blocking a truth-functional analysis for logics S5-Alt \(_2\) and weaker.
- 10.
See both Szabo (2000) and Dever (2006) for some of the shortcomings of the standard answers, and Dever (2003) for some problems with the novel answer that Szabo defends.
- 11.
More or less; for refinements, see the discussion in Dever (2006).
- 12.
This version of Geach’s point is taken from Szabo (2000).
- 13.
We also know that, once we leave the extensional realm, there is no compositional treatment of truth\(_{@}\) values either, and hence that these values cannot by themselves reveal our ontological and conceptual commitments.
- 14.
The applicability of Lindenbaum’s lemma follows from the regularity of truth\(_\square\). Superclassicality suffices to guarantee that at most one of \(\phi\), \(\neg\phi\) can consistently \(_\square\) be added to a consistent \(_\square\) set.
- 15.
The right-to-left direction of both conditionals requires the assumption that if \(\phi\) is neither true\(_\square\) nor false \(_\square\), then \(\neg\phi\) also is neither true\(_\square\) nor false \(_\square\). This assumption is guaranteed by the regularity of truth\(_\square\).
- 16.
I set aside here worries about, e.g., pronouns under modal subordination, and other facets of the generally Meinongian tendencies of natural language reference.
- 17.
The same emergence of molecular claims occurs elsewhere when speech act specifiers are added to a language. The negation-containing command ‘Don’t do that!’ can be read only as a command not to perform some action. However, once a specifier ‘I command’ is added, we can distinguish between ’I command you not to do that’ and ‘I don’t command you to do that.’
- 18.
A Google search for ’necessarily possibly’, for example, turns up only two uses that are not part of philosophical discussions.
One:
The compiled list is not necessarily possibly incomplete. looks like a misprint to me; the other:
Well, doesn’t one necessarily possibly mean the other? If you’re going to disengage, you may have to pull people out of their houses?
is suggestive in its apparent use of ‘necessarily’ as an indicator of discourse structure, rather than sentential content.
- 19.
Also suggestive, returning briefly to the thought that quantificational specification might be seen as a mode of truth, is the historical preference for a logic of quantification which avoids quantificational iteration, although there are important distinctions between this case and the modal case.
- 20.
For a more detailed argument that disquotational principles should hold with respect to the truth property under evaluation, rather than uniformly with respect to truth\(_{@}\),see Asher, Dever, Pappas (2005)
- 21.
In fact, both hold in a weaker logic requiring transitivity plus the reflexivity of all worlds which are accessed by some world.
- 22.
Earlier truth\(_\square\) was related to the evaluation point space universally, so that truth\(_\square\) was truth\(_W\) for all evaluation points \(w\). However, setwise modes needn’t receive this universal interpretation. If truth\(_Y\) is commensurate with truth\(_X\), then it can be given setwise truth conditions relative to the larger evaluation point space generated by truth\(_X\).
- 23.
Williamson (2000), among others, argues that the regulative norm of assertion is based on knowledge, rather than truth, and Sutton (2005) extends the claim to belief as well. The point will play no significant role in the current discussion, but I am inclined to think that a norm of truth suffices, and that the apparent role of knowledge in the norms is a result of the fact that the norm is a rule to be followed, and to follow a rule is to adopt a course of action which one knows to be in keeping with the rule.
- 24.
Can we retrench by pointing out the role of false \(_\square\)? I don’t think so; such a retrenchment still fails to account for the appropriateness, and perhaps even the obligatoriness, of believing and otherwise cognitively integrating claims which are not true\(_\square\). But there are messy questions about the role and conceptual coherence of truth value gaps here; the considerations raised by Dummett (1959) will be relevant.
- 25.
DET requires that the logic of \(\mathcal{D}\) obey both 4 and E.
- 26.
See Asher et al. (2005) for full presentation of all three arguments.
- 27.
Depending on how truth-value gaps are treated, it may be necessary to restrict the evaluation point quantifier of the local consequence relation to complete points. See Asher et al. (2005) for more on the logical issues here.
- 28.
Whether they are, in addition, actualizable will depend more or less on whether epistemicism about vagueness is true.
- 29.
Maybe – not everyone feels the draw of this. A disunity approach accounts also for the split in intuitions here.
- 30.
If the epistemic modal is interpreted in terms of belief instead of knowledge, the argument is more properly an omnidogmatic, rather than omniscience, one. Nothing I say here will depend on which interpretation is taken.
- 31.
I use here the simpler of the update systems developed in Veltman (1996), ignoring the complications added by the introduction of expectation patterns in the less simple system.
- 32.
Following the most common practice in the update semantics literature, I implement the epistemic modals only when they have scope over non-modal sentences, disallowing iterated modal operators. Note the interaction between this practice and the Iteration Hypothesis set out above.
- 33.
This logical consequence relation is \(\Vdash_2\), the second of the three consequence relations Veltman (1996) considers for update semantics.
- 34.
Thus the conclusion follows from the premises just in case any information state updated with the premises is a fixed point of the update function of the conclusion.
- 35.
UTC consequence is easily seen to be transitive. Suppose \(\phi\Vdash\psi\). Then:
1. \(\sigma[\phi]=\sigma[\phi][\psi]\)
Suppose also that \(\psi\Vdash\theta\). Then (taking one particular instance):
2. \(\sigma[\phi][\psi]=\sigma[\phi][\psi][\theta]\)
Chaining together the two equalities, we obtain \(\sigma[\phi]=\sigma[\phi][\psi][\theta]\). But applying the first identity, we then obtain \(\sigma[\phi]=\sigma[\phi][\theta]\), so \(\phi\Vdash\theta\).
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Acknowledgment
I am grateful to Nicholas Asher, Anthony Gillies, and Rob Stainton for comments on an earlier draft of this paper. Portions of this material were presented to audiences at the University of Michigan and Princeton University, and benefited from comments and questions received there.
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Dever, J. (2009). The Disunity of Truth. In: Stainton, R.J., Viger, C. (eds) Compositionality, Context and Semantic Values. Studies in Linguistics and Philosophy, vol 85. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-8310-5_7
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