Logicality, Double-Line Rules, and Modalities

Abstract

This paper deals with the question of the logicality of modal logics from a proof-theoretic perspective. It is argued that if Dosen's analysis of logical constants as punctuation marks is embraced, it is possible to show that all the modalities in the cube of normal modal logics are indeed logical constants. It will be proved that the display calculus for each displayable modality admits a purely structural presentation based on double-line rules which, following Dosen's analysis, allows us to claim that the corresponding modal operators are logical constants.

Proof-theoretic pluralism

“…More precisely, a derivability relation is a binary relation between multisets of fomulas that is reflexive and transitive -i.e., a set of sequents that is closed under the rules Id and Cut. 16 As a consequence, the putative proof-theoretic definition of a logical operator • is successful if and only if the rules governing its behaviour are conservativei.e., they don't alter the derivabilty relation for the •-free language -and they satisfy uniqueness -i.e., if • is a notational variant of • we must be able to show that • and • are interderivable. The need for conservativity springs from the assumption that the •-free derivability relation already provides 'all universally valid deducibility-statement not involving [•]' [3, p. 132].…”

“…We have chosen Belnap's definition because it is well known and it allows us to make our point without having to dwell on many formal details; cf [7]. for an argument in favour of Belnap-like global harmony 16. This is not the place to enter into the interesting discussion concerning whether there is a set of invariant features at the core of the concept of following from (and hence of the derivability relation).…”

Hopeful Monsters: A Note on Multiple Conclusions

“…A suitable criterion of definitional success-occasionally emerging, albeit in different forms, in the literature, particularly when logics are presented by means of sequent calculi-is invertibility [1,10,30]. I shall not go into a detailed account of why invertibility is a formal property the obtaining of which is indicative of harmony; an explicit endorsement and analysis of invertibility as a criterion of harmony can be found in [15,16]. In the present context it is enough to notice that it suffices to rule out pathological connectives like tonk.…”

“…The threat of circularity is, at least in this case, mitigated. 16 Somewhere in the same ballpark lies the worry that multiple-conclusion derivations go against the proof-theoretic desideratum of separability [36]. Technically, this is the requirement that the (schematic formulation of the) defining rules for the connectives do not mention any logical connectives but the one they define.…”

“…Incidentally, this suggests that the efforts to show that multiple conclusions are not part of the vernacular practice are pointless. If appealing to them in proof-theoretic arguments is somehow circular, then the circularity would not be fixed by there being multiple-conclusion derivations in 'nature' 16. For a brief discussion of this kind of inter-conectedness of the connectives see[7] 17.…”