Abstract
Without directly addressing the Demarcation Problem for logic—the problem of distinguishing logical vocabulary from others—we focus on distinctive aspects of logical vocabulary in pursuit of a second goal in the philosophy of logic, namely, proposing criteria for the justification of logical rules. Our preferred approach has three components. Two of these are effectively Belnap’s, but with a twist. We agree with Belnap’s response to Prior’s challenge to inferentialist characterisations of the meanings of logical constants. Belnap argued that for a logical constant to exist, its rules must be conservative over a previously given consequence relation and guarantee the uniqueness of the constant. The twist is that we require logical vocabulary to be provably conservative, not over a previously given formal consequence relation, but over a previously given meaningful vocabulary of a language. Uniqueness is also a feature defined in terms of provability: if two syntactically distinguished expressions are governed by the same rules, then formulas with them as main operators are interderivable. Belnap’s criteria are not only those for the existence of a logical constant, but more: they are what distinguishes logical vocabulary from all other expressions. It is the defining mark of a logical constant that it is provably conservative over the fragment of the language which excludes it and that its rules guarantee its uniqueness. The third component is the topic neutrality of logic.
The provable conservativeness of logic over a previously given vocabulary of a language is motivated, in part, by appeal to molecuralism in the theory of meaning. Molecularity is a feature endorsed by the theory of meaning as a whole, so it does not distinguish logical constants from other expressions. The same is true for conservativeness. We argue that molecularity, and hence conservativeness, are implicit presuppositions of speakers’ use of language: the addition of new vocabulary to a language is presupposed to be conservative. But this presupposition is one that we should be able reflectively to endorse (such as when attempting a rational reconstruction of the use of the vocabulary). Thus, where conservativeness is found to fail, this defect should be remedied and the use of the vocabulary corrected (as in classical negation) or excised altogether (as in pejoratives). We go on to characterise a notion of topic neutrality, which we argue applies to logical vocabulary. We then note that reflective endorsement of the conservativeness of a topic neutral vocabulary requires a proof that the vocabulary is conservative relative to any base vocabulary. Thus we require that logical vocabulary be demonstrably conservative. Allying this requirement, distinctive of logic, to a general consideration about the commitments of assertion yields a mode of justification for logical constants akin to some conceptions of Harmony, i.e., to the idea that the consequences of assertion of a logical complex need to be warranted by the grounds and ought to be the strongest consequences warranted by the grounds. Intuitionistic logic acquires a somewhat familiar justification but emanating from a new motivation.
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Notes
- 1.
See, for instance his ‘What is a theory of meaning II’ in Truth and meaning, edited by G. Evans and McDowell, 1976, reprinted as, pp. 38–42 in the latter.
- 2.
By ‘meaning’ we shall mean semantic content, acknowledging that there may be other aspects to meaning more broadly construed such as Fregean colour or tone.
- 3.
- 4.
- 5.
So including language entry and exit rules in inferential role.
- 6.
Requiring the vocabulary to be a proper part conveniently excludes various trivial cases in the definitions below. But nothing hangs on it and the definitions are easily adjusted to accommodate them.
- 7.
See, e.g., (Steinberger, 2011).
- 8.
It is worth stressing that our notion of conservativeness is a meaning theoretical notion. As such it does not coincide with notions of conservativeness formulated by Dummett, which are epistemic notions. For a plausible way of spelling out such an epistemic notion of conservativeness in the spirit of Dummett, see Cozzo, 2002. It is also worth stressing the neutrality of our notion of conservativeness: its definition does not depend on any specific meaning determining feature. Dummett would favour a notion of verification conditions as that in terms of which meanings are determined, or sometimes falsification conditions. Other authors prefer truth conditions or consequences of assertions. They can all accept our notion of conservativeness. If some notions of conservativeness are not acceptable to some theorists of meaning, say, an epistemic notion of conservativeness to a truth conditional theorist of meaning, this again shows that these notions of conservativeness are not ours.Prawitz (1987) argues that Gödel’s Incompleteness Theorems show that arithmetic, though molecular, is not conservative relative to provability. It’s worth noting that the construal of molecularity that Prawitz has in mind is what he calls Dummett’s second notion of molecularity, which amounts to no more than compositionality: ‘the specification of [the meaning determining] feature of a compound sentence should be determined in terms of this feature for the constituents and the way they are put together’ (1987: 129). The notion of conservative extension used here is allied to a stronger notion of molecularity, namely, one in which the meaning of a sentence is given by reference to, at most, a fragment of the language. The bearing of Gödel’s results is, of course, an interesting topic and may call for a distinctively non-formal characterisation of canonical proof—for argument to this effect see Dummett (1973) and Weiss (1997)—but investigation of this issue cannot be on the current agenda.
- 9.
See below for more discussion and motivation.
- 10.
There seems to be some tension here between this conception of the content of assertions, in general, and Brandom’s logical expressivism. For, the assertion A seems to express as a claim the inferential commitment from GA to CA. But that was logic’s peculiar role.
- 11.
This chimes with some of Dummett’s complaints about holism, namely, that ‘[f]or the holist, we ought not to strive to command a clear view of the workings of our language, because there is no clear view to be had’ (1993a: 241). Here we think we have a clear conception of the content of A, in terms of the inferential connection it forges. But its very introduction changes the content of the terms of that inference and so demolishes our handle on its substantive content.
- 12.
See (Weiss, 2007: 609–14).
- 13.
Cf. The Generality Constraint (Evans, 1982: p. 75).
- 14.
Brandom, following Sellars, is sceptical about whether there could be an autonomous practice purely of observational reporting, arguing that observational reporting is only possible in the context of inferring too. We’re not here concerned with this level of nuance and will treat reporting and inferring as asserting—the latter depending on semantic relations between assertions.
- 15.
Brandom also develops a conception of logic which makes it topic neutral—‘I will argue that logical vocabulary both can be algorithmically elaborated from and is explicative of practices that are PV-necessary for the autonomous deployment of any vocabulary at all’ (our emphasis; 2008: 28)—but arrives at such a view from his expressivist understanding of logic, namely, that the role of logic is in explicitating inferential practice. The issue is somewhat tangential to current concerns; but our sense is that it would be better to understand logic in terms of its topic neutrality (treated as distinctive). Weiss thinks logical expressivism falters because its expressive role depends on a prior role of compounding contents; and the notion of explicitating is radically unclear (2010: 247–62).
- 16.
We might agree with Brandom (see chapter 5 of his 1994) that truth need not be metalinguistic and is logical; though our reasons for holding this would be to do with its neutrality rather than its expressive role.
- 17.
Namely, that discourse involving ascription of propositional attitudes depends semantically on discourse involving behavioural ascriptions.
- 18.
Hel** ourselves to a suitable mode of generalising over sentences: not straightforward because substitutional quantification would be limited by the resources of a language, whereas we are concerned with all possible vocabularies. Being in a position feasibly to obtain a warrant is to have a warrant. It is a further question whether agents are thereby also in a position to know that they have a warrant. We do not assume a transparent notion of warrant. On the other hand, transparency would appear to follow if it is the case that if an agent can feasibly achieve a purely epistemic goal, then they ought to be able to reflect on this and come to know it. Thus transparency would follow from the additional assumption that agents can always reflect on what they can feasibly do. If we reject transparency of warrant, it is this latter thought we should reject.
- 19.
Testimony might seem to constitute a counter-example and, though it involves deferring a commitment to another, it does involve the commitment to provide one’s testimonial warrant. So, rather than being a counter-example, this kind of case falls into a mixed bag of what Dummett calls indirect warrants for assertion.
- 20.
See Dummett’s discussion of the relation of demonstrations to canonical proofs, the former being a means, recognizably, to furnish the latter (2000: 269–74).
- 21.
Grasp of the rule is manifestable. In gras** the syntax of the language, a speaker has a manifestable capacity to recognise any sentence of the language; and in gras** the introduction rule for ⊥ a speaker recognises that a warrant to assert ⊥ is a warrant to assert any member of the set of sentences, which, as just noted, she grasps via grasp of the syntax of the language.
- 22.
See (Steinberger, 2011: §5).
- 23.
As we mentioned above, p. 466ff.
- 24.
For further discussion of the philosophical reasons of requiring permutation conversions see (Kürbis, 2019: Ch 2).
- 25.
This manoeuvre effectively establishes the derivability of intuitionist disjunction elimination if quantum disjunction is coupled with intuitionist implication.
- 26.
One might object straightaway that the modal operators are not uniquely characterised by their rules of inference. But there is no justified worry here. Some notions of modality are metaphysical notions, so it is not surprising if formal rules do not determine their meanings completely. Maybe we also need to know something about the natures of things or their essences. (As argue, e.g., (Fine, 1994, 1995) and (Hale, 2013). For a review of the latter, see Kürbis (2015).). For purely logical notions of modality, of which one would desire that they belong amongst the logical constants, this merely shows that a suitable modification of standard proof systems should be chosen for their formalisation in which they are uniquely characterised. Dosen’s systems provide an example (Došen, 1985). See also Kürbis (2020).
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Weiss, B., Kürbis, N. (2024). Molecularity in the Theory of Meaning and the Topic Neutrality of Logic. In: Piccolomini d'Aragona, A. (eds) Perspectives on Deduction: Contemporary Studies in the Philosophy, History and Formal Theories of Deduction. Synthese Library, vol 481. Springer, Cham. https://doi.org/10.1007/978-3-031-51406-7_9
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