Abstract
In this paper, we present a new semantic framework designed to capture a distinctly cognitive or epistemic notion of meaning akin to Fregean senses. Traditional Carnapian intensions are too coarse-grained for this purpose: they fail to draw semantic distinctions between sentences that, from a Fregean perspective, differ in meaning. This has led some philosophers to introduce more fine-grained hyperintensions that allow us to draw semantic distinctions among co-intensional sentences. But the hyperintensional strategy has a flip-side: it risks drawing semantic distinctions between sentences that, from a Fregean perspective, do not differ in meaning. This is what we call the ‘new problem’ of hyperintensionality to distinguish it from the ‘old problem’ that faced the intensional theory. We show that our semantic framework offers a joint solution to both these problems by virtue of satisfying a version of Frege’s so-called ‘equipollence principle’ for sense individuation. Frege’s principle, we argue, not only captures the semantic intuitions that give rise to the old and the new problem of hyperintensionality, but also points the way to an independently motivated solution to both problems.
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Notes
Throughout we shall understand capital letters (\(A,B,\dots \)) as placeholders for interpreted sentences (‘snow is white’, ‘Joe is hungry’,\(\dots \)).
The pairs (5) and (6) are extracted from Frege (1984, pp. 390–406). While some might find the particular examples in (4)–(6) contentious, they are in our opinion well-suited to motivate the idea that Fregean content should not be arbitrarily fine-grained. See also Chalmers (2011), Jago (2014), Bjerring and Schwarz (2017), and Nolan (2018) for similar examples.
See King (2016) for an overview of structuralist approaches to semantic content.
Note that Frege’s equipollence principle does not make content individuation depend on the stock of empirical information that an agent has available. Obviously, what is “immediately recognizable as equivalent” strictly speaking depends on one’s empirical information. For example, an astronomer might immediately recognize that “Hesperus is Phosphorus” is equivalent to “Hesperus is Hesperus.” But since Frege’s equipollence principle quantifies over all agents, regardless of their empirical background, content individuation remains insensitive to empirical information.
For example, the sequence \((A_0,A_1,A_2) = (p \wedge q, p, p \vee r)\) can be seen as a two-step derivation from \(p \wedge q\) to \(p \vee r\) in a proof system that includes conjunction elimination and disjunction introduction. Strictly speaking, the number of inferential steps needed to derive \(A_n\) from \(A_0\) need not coincide with the number n, since there may be inference rules with more than one premise. For example, the sequence \((B_0, B_1, B_2) = (p \rightarrow q, p, q)\) is a one-step derivation in a proof theory that includes modus ponens. But this is immaterial for the general point that the relation of being ‘immediately recognizable as equivalent’ cannot be transitive.
Of course, it is a moot point exactly what it takes for an inference rule to be “trivial” or “obvious”. For now we shall rely on an intuitive grasp of these notions, but later on, we will offer a formally precise account.
To be sure, nothing of importance hinges on the exact choice of terminology. The notion of “indistinguishability” strikes us as a particularly natural way of replacing “sameness” with an intransitive counterpart, but other intuitive notions such as “cognitive synonymy” or “likeness in meaning” may do the job just as well.
See Russell (2017) for additional background on defeasibility of a priori reasoning.
A detailed account of the relation between \({\mathcal {L}}\) and English would appeal to a translation of tokens in English to types in \({\mathcal {L}}\); see Jago (2014, §5) for such an account.
Accordingly, whether a contradiction can be derived from w in more than n steps is irrelevant for whether w is n-consistent. Thanks to an anonymous reviewer for asking us to clarify this point.
For details, see for instance Elliott (2017).
For those interested in the formal details, here is a brief proof of the upper subset relation involving conjunction: suppose \(w \in |A \wedge B|^n\). By (Epistemic n-intension) and (Verification), \(A \wedge B \in w^n\). Given that \({\mathcal {S}}\) contains conjunction elimination, \(w^n \vdash _{{\mathcal {S}}}^1 A\) and \(w^n \vdash _{{\mathcal {S}}}^{1} B\). Hence, \(w \vdash _{{\mathcal {S}}}^{n+1} A\) and \(w \vdash _{{\mathcal {S}}}^{n+1} B\). By (n-expansion), \(A \in w^{n+1}\) and \(B \in w^{n+1}\). By (Epistemic n-intension) and (Verification), the epistemic \(n+1\)-intension of A is true at w, and similarly for B. Thus, \(w \in |A|^{n+1}\) and \(w \in |B|^{n+1}\), which means that \(w \in |A|^{n+1} \cap |B|^{n+1}\). The other subset relations can be established in similar ways.
Of course, these similarities between epistemic n-intensions and primary intensions presuppose that \({\mathcal {S}}\) contains enough rules to allow us to infer the class of sentences that count as a priori in Chalmers’ framework.
Again, depending on how we exactly specify the logical fragment of \({\mathcal {S}}\), further steps might be needed to infer \(A \leftrightarrow (A \wedge A)\). For example, if we adopt a Lemmon-style natural deduction system, it would take five proof steps to infer \(A \leftrightarrow (A \wedge A)\), because we would need additional steps for making and discharging assumptions. But such details are unimportant for the issue at hand.
For further discussion of this result, see Bjerring (2013) and Bjerring and Skipper (forthcoming).
Bjerring and Skipper (forthcoming, §4).
For a survey of questions and problems in this direction, see Szabó (2017).
See Carnap and Bar-Hillel (1952) for an early exposition of the intensional account of information.
For further considerations in this direction, see Bjerring and Schwarz (2017, §§5–6). See also Chalmers (2011), Bjerring (2010), and Jago (2009) for discussions of how proof-theoretic structure might help to construct a ‘non-ideal epistemic space’ that is sensitive to the limited cognitive resources of non-ideal epistemic agents.
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Acknowledgements
Earlier versions of this paper were presented at Massey University, National University of Singapore, Umeå University, and University of Auckland. We thank the audience on those occasions for valuable feedback. A special thanks to two anonymous reviewers for Synthese, and to Andreas Stokke, Ben Blumson, John Matthewson, Lars Bo Gundersen, Weng Hong Tang, and Wolfgang Schwarz for very helpful comments and criticism on earlier versions of this paper.
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Skipper, M., Bjerring, J.C. Hyperintensional semantics: a Fregean approach. Synthese 197, 3535–3558 (2020). https://doi.org/10.1007/s11229-018-01900-4
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DOI: https://doi.org/10.1007/s11229-018-01900-4