## Abstract

On one important notion of irrelevance, evidence that is irrelevant in an inquiry may rationally be discarded, and attempts to obtain evidence amount to a waste of resources if they are directed at irrelevant evidence. The familiar Bayesian criterion of irrelevance, whatever its merits, is not adequate with respect to this notion. I show that a modification of the criterion due to Ken Gemes, though a significant improvement, still has highly implausible consequences. To make progress, I argue, we need to adopt a hyperintensional conception of content. I go on to formulate a better, hyperintensional criterion of irrelevance, drawing heavily on the framework of the truthmaker conception of propositions as recently developed by Kit Fine.

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## Notes

- 1.
As Cohen has emphasized (cf. 1994, p. 171f), there are also important applications of the notion of relevance which cannot readily be represented as concerning a relation of evidence to hypothesis. In this paper, I restrict attention to applications which can naturally be so understood. As I mention below, however, I suspect that the tools I shall employ in accounting for these instances of relevance can also fruitfully be applied in a much greater range of cases.

- 2.
Pr(H) denotes the (prior) probability of the hypothesis H, and Pr(H|E) denotes the (posterior) probability of H given the evidence E. The standard provided by (BI), and any of its refinements to be considered below, is thus relative to a suitable prior probability distribution. Following common practice in the debate, we won’t worry here about the exact nature of the probabilities in question. The notion of irrelevance is often further relativized to a body of background information K and is then taken to be characterized by the condition that Pr(H|E\(\wedge \)K) = Pr(H|K). I have left out reference to K throughout. Doing so facilitates comparison with Gemes’ (2007) account to be considered below, and the difference is of no import for our purposes.

- 3.
This is not to say that (BI) may not amount to an adequate explication of

*a*useful notion of irrelevance. The claim is that the notion sketched above is not adequately explicated by (BI). - 4.
- 5.
See (Gemes 2007, p. 165). Gemes’ formulation has ‘content part’ in place of ‘part’, which is Gemes’ term for his explication of the notion of a part of a content. For our purposes, it is better to view as separate the specific account of that notion and the proposed revision of (BI). The term ‘part’, here and throughout, is to be understood as ‘proper or improper part’.

- 6.
I use the terms ‘content’ and ‘proposition’ interchangeably. I have found it convenient to speak of evidence and hypotheses as themselves propositions rather than sentences expressing these propositions.

- 7.
I use the phrases ‘content part’ and ‘partial content’ interchangeably. The former is favoured by Gemes, the latter by Fine.

- 8.
The obvious prior probability of \(\hbox {H}_1\) is 1/4, which is also the probability of \(\hbox {H}_1\) given \(\hbox {E}_1\), since \(\hbox {E}_1\) is compatible with four equally probable outcomes of the tosses, exactly one of which makes \(\hbox {H}_1\) true.

- 9.
Cf. (Gemes 2007, p. 165f). Note that to deal with this particular example, a more modest deviation from (BI) than is embodied in (GI) would have been sufficient. For as we saw, (GI) overdetermines, as it were, the result that \(\hbox {E}_1\) is relevant to \(\hbox {H}_1\) in that we have both that part of the evidence is probabilistically relevant to the hypothesis as a whole, and that the evidence as a whole is probabilistically relevant to part of the hypothesis. But this feature is specific to the current example. The next example in the main text plausibly can only be captured by allowing as sufficient for relevance the probabilistic relevance of the evidence for part of the hypothesis. For an example of the ‘converse’ sort, consider any hypothesis H and the corresponding evidence that Bill said H was false and Bob said H was true, where Bill and Bob are generally reliable sources equally likely to be wrong (or lying) with respect to H or any part of H.

- 10.
The quoted phrase, and in particular the parenthetical qualification, is the bit which requires further clarification. Firstly, the vocabulary of \(\gamma \) in question is supposed to be non-logical vocabulary, otherwise the fact that \(P \wedge Q\) is a stronger consequence of \(P \wedge Q\) than \(P \vee Q\) would not prevent the latter from being a part of \(P \wedge Q\), as Gemes clearly intends it to do (cf. Gemes 1994, p. 603). Secondly, from the official definition offered in (Gemes 1994, p. 605), we may extract that a piece of non-logical vocabulary is said to occur essentially in a sentence just in case there is no logically equivalent sentence in which it does not occur. The motivation for the restriction to \(\alpha \)’s essential vocabulary is to ensure that logically equivalent sentences stand in the same parthood relations (cf. Gemes 1994, p. 604f). To see the point, note that without the restriction,

*P*is part of \(P \wedge Q\), but the logically equivalent \(P \wedge (Q \vee \lnot Q)\) is not, since \(P \wedge Q\) itself is a stronger logical consequence of \(P \wedge Q\) which contains all the vocabulary in \(P \wedge (Q \vee \lnot Q)\). - 11.
We can construct a similar case pertaining to parts of the evidence. Consider some hypothesis H with prior probability 1/2 and two generally reliable sources Bill and Bob who are equally likely to be wrong (or lying) about H, and let the evidence be that (Bob or Bill said H is true) and (Bob or Bill said H is false). Let it be given that both Bob and Bill said either that H is true or that H is false, so that the evidence may be represented as \((P \vee Q) \wedge (\lnot P \vee \lnot Q)\). The evidence as a whole is intuitively, but not probabilistically, relevant for the hypothesis, whereas the conjuncts are probabilistically relevant. For the same reasons as before, however, the conjuncts do not qualify as parts of the evidence on Gemes’ account.

- 12.
- 13.
An anonymous referee has suggested to me that it might after all appear counter-intuitive that one can turn anything into a partial truth just by tacking on a logically idle conjunct. I concede that our unreflective judgement with respect to this principle—especially in this particular, somewhat leading wording—may be negative or at least sceptical. However, part of the intuitive resistance to the principle appears to vanish already when it is reformulated in a more explicit and neutral fashion. The point is that for any proposition

*P*, there is some logical consequence*R*of*P*such that \(P \wedge R\) is (at least) partially true. More importantly, it seems to me that any remaining intuitive uneasiness with respect to the principle disappears once one reflects that: (i) it seems intuitively very plausible to say that anything can be turned into a partial truth by tacking on a*true*conjunct; (ii) anything has some true logical consequences, and (iii) even given this principle, if one wants to turn a proposition one does not know to be true into a proposition one knows to be partially true, then one needs to ‘invest’ some known truth. That is, one needs to add a conjunct which one independently knows to be true. - 14.
The fullest published exposition by Fine of the truthmaker conception of content and the notion of partial content is given in Fine (2015) in the context of a discussion of Angell’s logic of analytic entailment. A more general presentation and discussion of the framework is contained in the as yet unpublished manuscripts Fine (msa) and Fine (msb).

- 15.
While this may make it sound as though the views make incompatible claims about the same kind of thing, viz. propositions, it is not necessary for my purposes that we think of the views in this way. We may instead take them to concern different concepts of propositions, suited to different theoretical purposes.

- 16.
Strictly speaking, Fine distinguishes a number of different conceptions of verification. I am here concerned with what Fine calls exact verification, which is the basic notion of verification in terms of which he defines other, looser conceptions. Cf. (Fine 2015, pp. 7f, 20f), and (Fine, msa, p. 35f). Fine sometimes describes his notion of exact verification as embodying a constraint of holistic relevance in the sense that for a state to exactly verify a proposition, it must be wholly relevant to the proposition, and so must not have any part that is irrelevant to the proposition (cf. e.g. (Fine, msa, p. 1)). This may invite the worry that some sort of untoward circularity is involved in using Fine’s framework to describe and study relations of relevance. However, the appeal to a notion of relevance is confined solely to Fine’s informal commentary on his theory, and not part of the theory itself. It might perhaps still be claimed that to the extent that exact verification imposes relevance constraints, an analysis of relevance within the truthmaker framework is in that sense not fully reductive; I would be content to concede that much.

- 17.
This is a slight exaggeration, since other treatments of negation are possible within truthmaker semantics that do not require a separate specification of falsifiers, appealing instead to modal connections on the states. These approaches to negation will not be considered here.

- 18.
To see this, it suffices to note that none of these propositions—the disjuncts of \(\hbox {H}_3\) and their conjuncts—are even logical consequences of \(\hbox {H}_3\). On Gemes’ view, content parts are by definition a special kind of logical consequence. On Fine’s view, this is clear from the fact that content parts are conjuncts of the propositions they are part of.

- 19.
There is no corresponding motivation to also take into consideration mere ways for the evidence to hold, or mere parts of such ways. Indeed, for any hypothesis H with \(0 < \hbox {Pr(H) } < 1\) and arbitrary \(P,\, (\hbox {H} \vee \lnot \hbox {H}) \wedge P = (\hbox {H} \wedge P) \vee (\lnot \hbox {H} \wedge P)\) would otherwise turn out relevant to H on the strength of the probabilistic relevance of H to H. This would seem a bad result. Surely, amassing evidence of this sort by procuring arbitrary information

*P*would amount to an objectionable waste of resources in an inquiry into H. Note that the ‘converse’ claim of relevance, that H is relevant to \((\hbox {H} \vee \lnot \hbox {H}) \wedge P\), which I endorse, is not subject to same objection, since it does not yield a recipe for producing lots of irrelevant-seeming evidence. It does, of course, yield a recipe for producing lots of somewhat strange hypotheses to which the evidence at hand is classified as relevant. But although we may at times start with a piece of evidence, and then ask what hypotheses the evidence might sensibly lead us to inquire into, we would not expect a criterion of evidential irrelevance on its own to provide the answer to this question. This point also serves, firstly, to highlight that on my approach—in contrast to the Bayesian and to Gemes’ account—irrelevance is*not symmetric*, and secondly, to indicate how this may be justified in terms of the constraints by which I have introduced my target notion of irrelevance. Thanks to an anonymous referee for raising the issue of symmetry here. - 20.
Cf. (Fine, msa, p. 16); Fine says that

*P**exactly entails**Q*when I say that*P*is a way for*Q*to hold. - 21.
Note that (W) implies that whenever \(0 < \hbox { Pr}(P) < 1,\, P\) is relevant to \(P \vee \lnot P\), since

*P*is then probabilistically relevant to*P*. This is in marked contrast to (BI), and to (GI) on Gemes’ account of content parts, on which nothing can be relevant to a logical truth. Since \(\hbox {Pr}(P \vee \lnot P)\) is always 1, \(P \vee \lnot P\) makes for a somewhat peculiar choice of a hypothesis to investigate, so it is not obvious what significance to attach to our result. However, if we wish to allow for rational inquiry into a hypothesis that is a logical truth like \(P \vee \lnot P\), then the result seems very plausible to me. For*P*then*is*evidence that bears on \(P \vee \lnot P\) in a way in which it does not bear on arbitrary \(Q \vee \lnot Q\), and it seems to me a feature, not a bug, of the present proposal, that it enables us to capture this fact. - 22.
It is plausible that (P) may be strengthened in the analogous way, so that consequently E’s probabilistic relevance to a way for \(\hbox {H}^\prime \) to hold, where \(\hbox {H}^\prime \) is part of a way for H to hold, is also sufficient for E’s relevance to H. Fortunately, this is already implied by (WP). For in this case, H may be written \(((P \vee Q) \wedge R) \vee S\), where E is probabilistically relevant to

*P*. We can then show that \(P \wedge R\) is a way for H to hold, and thus*P*part of a way for H to hold, using that \((P \vee Q) \wedge R = (P \wedge R) \vee (Q \wedge R)\), and thus \(\hbox {H} = ((P \wedge R) \vee (Q \wedge R)) \vee S = (P \wedge R) \vee ((Q \wedge R) \vee S)\). The identities used here are implicit in the soundness results of (Fine 2015, Sects. 6, 9). - 23.
Note, though, that the fact that

*s*is part of a verifier of*P*does not rule out that*s*is also part of a*falsifier*of*P*. Indeed, there are various possible scenarios in which it would be natural to say that*s*goes more of the way towards making*P*false than it goes towards making*P*true. It bears emphasis, then, that on my use of the phrase, that*s*goes some way towards making*P*true does not imply that*on balance*,*s*goes further along the way to*P*’s truth than to its falsity. (Thanks here to an anonymous referee). - 24.
We can construct a proposition

*R*of which*P*is part and which is a way for*Q*to hold as follows. Let the set of verifiers of*R*be the set of verifiers of*Q*that have a part which verifies*P*. Let the set of falsifiers of*R*be \((P^- \cup Q^-)^\circ \). It is then straightforward to show that*R*is a proposition, and that it relates to*P*and*Q*in the desired way. - 25.
In this respect, the present paper would seem to follow something of a trend. Hyperintensional accounts have in recent years been proposed for many philosophically central concepts, such as essence, ground, conditionals, subject matter, and, closest to our present concerns, confirmation—cf. here esp. Yablo (2015). It is striking that in many cases, considerations of relevance play an important role in motivating the claim to hyperintensionality. It would be very interesting to explore the connections between these debates and the arguments I have here advanced in detail. A particularly tight connection may obtain to

*ground*, for which Fine has offered a semantics within the same truthmaker framework we have employed here (cf. Fine (2012a, b)). Indeed, our notion of a cd-part coincides with (a non-factive version of) Fine’s notion of a weak partial ground, and our notion of a way for a proposition to hold coincides with (a non-factive version of) Fine’s notion of a weak full ground.

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*Metaphysical grounding*(pp. 37–80). Cambridge: Cambridge University Press.Fine, K. (2012b). The pure logic of ground.

*The Review of Symbolic Logic*,*5*(1), 1–25.Fine, K. (2013). A note on partial content.

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*Journal of Philosophical Logic Online First*, 1–28.Fine, K. (msa) A theory of truth-conditional content I, unpublished manuscript

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*Journal of Philosophical Logic*,*23*, 596–620.Gemes, K. (1997). A new theory of content II: Model theory and some alternatives.

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## Acknowledgments

The research for this paper was funded by the Deutsche Forschungsgemeinschaft (Grant KR 4516/1-1), and I gratefully acknowledge the support. An earlier version of this paper was presented at a research colloquium at the University of Hamburg. I thank the members of the audience for their comments and criticisms. Special thanks are due to Sebastian Krug and to my fellow *phlox* members Michael Clark, Yannic Kappes, Martin Lipman, Giovanni Merlo, Stefan Roski, Benjamin Schnieder, and Nathan Wildman. I have also greatly benefitted from a number of conversations with Kit Fine on the general topic of relevance and truthmaker semantics. Finally, I would like to thank an anonymous referee for this journal for some very helpful comments and suggestions which have resulted in a number of improvements.

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Krämer, S. A hyperintensional criterion of irrelevance.
*Synthese* **194, **2917–2930 (2017). https://doi.org/10.1007/s11229-016-1078-0

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### Keywords

- Relevance
- Hyperintensionality
- Partial content
- Truthmaker semantics