This paper considers proof-theoretic semantics for necessity within Dummett's and Prawitz's framework. Inspired by a system of Pfenning's and Davies's, the language of intuitionist logic is extended by a higher order operator which captures a notion of validity. A notion of relative necessary is defined in terms of it, which expresses a necessary connection between the assumptions and the conclusion of a deduction

Kürbis, Nils

English

This paper considers proof-theoretic semantics for necessity within Dummett’s and Prawitz’s framework. Inspired by a system of Pfenning’s and

Davies’s, the language of intuitionist logic is extended by a higher order

operator which captures a notion of validity. A notion of relative necessary

is defined in terms of it, which expresses a necessary connection between

the assumptions and the conclusion of a deduction

Kürbis, N

UCL Discovery

A Sketch of a Proof-Theoretic Semantics forNecessityNils KürbisPublished in Advances in Modal Logic 13. Booklet of Short Papers, edited by SaraNegri, Nicola Olivetti, Rineke Verbrugge (Helsinki 2020): 37-43https://www.helsinki.fi/sites/default/files/atoms/files/finalshortpapermain.pdfAbstractThis paper considers proof-theoretic semantics for necessity within Dum-mett’s and Prawitz’s framework. Inspired by a system of Pfenning’s andDavies’s, the language of intuitionist logic is extended by a higher orderoperator which captures a notion of validity. A notion of relative necessaryis defined in terms of it, which expresses a necessary connection betweenthe assumptions and the conclusion of a deduction.1 Proof-Theoretic SemanticsDummett and Prawitz do not consider how the meanings of modal operatorsmay be given by their theory of meaning for the logical constants. To investigatein outline how this may be done is the purpose of this short paper.According to proof-theoretic semantics, the rules governing a constant defineits meaning. Prior’s tonk shows that the rules cannot be arbitrary. Dummett andPrawitz impose the restriction that the introduction and elimination rules fora constant ∗ be in harmony, so that ∗E does not license the deduction of moreconsequences from A ∗ B than are justified by the grounds for deriving it asspecified by ∗I. (See (Dummett, 1993a), (Prawitz, 1987), (Prawitz, 1994), (Prawitz,2006), (Prawitz, 2007).) A necessary condition for harmony is that deductionscan be brought into normal form. A deduction is in normal form if it containsneither maximal formulas nor maximal segments. A maximal formula is one thatis the conclusion of an I-rule and major premise of an E-rule. A maximal segmentis a sequence of formulas all except the last of which are minor premises of ∨Eand the last one is major premise of an E-rule.11I am allowing myself a certain looseness in terminology, which, however, is quite commonin the literature. Strictly speaking, Dummett distinguishes intrinsic harmony, stability and totalharmony. Intrinsic harmony is captured by normalisation: the elimination rules of a constant arejustified relative to the introduction rules. Stability is harmony together with a suitable converse:the introduction rules are also justified relative to the elimination rules. Total harmony obtainsif the constant is conservative over the rest of the language. Dummett calls the permutativereduction steps to remove maximal segments ‘auxiliary reduction steps’. Sometimes, as in thecase of quantum disjunction, these cannot be carried out, which points to a defect in the rules for1Deductions in intuitionist logic I normalise (Prawitz, 1965, Ch 4):A∨I: A ∨ BBA ∨ BA ∨ B[A]iΠC[B] jΣC∨E: i, jC[A]iΠB⊃ I: iA ⊃ BA ⊃ B A⊃ E: B⊥⊥E: CA B∧I: A ∧ BA ∧ B∧E: AA ∧ BBThe constants occur only in conclusions of I-rules and major premises of E-rules.Thus the conditions for using an I-rule and the consequences of using an E-ruleare given independently of the constants.The rules of I exemplify Dummett’s notions of full-bloodedness and molecularityin the theory of meaning (see (Dummett, 1993b), (Dummett, 1993c)). A full-blooded theory of meaning characterises the knowledge of speakers in virtueof which they master a language in such a way that it exhibits how a speakerwho does not yet understand an expression could acquire a grasp of it. Amolecular theory of meaning does so piecemeal and specifies the meanings ofthe expressions of a language one group of expressions at a time. A speaker neednot understand the constants of I in order to be informed about the conditions forthe application of their I- and E-rules. To understand the grounds for deriving aformula with ∗ as main operator, or to understand the consequences that followfrom it, a speaker only needs to grasp the meanings of some sentences, but notany sentences containing ∗. A speaker who does not already know the meaningsof the constants of intuitionist logic could acquire a grasp of their meanings bylearning the rules of inference of I. The rules are informative: the grounds andconsequences of a formula with ∗ as main operator are given without referenceto ∗. Its meaning is specified without presupposing that ∗ already has meaning.Contrast the rules of I with standard rules for  in S4:AI:AAE: Awhere in I all assumption on which A depends have the form B. Theconditions for applying I are not given independently of . Thus theypresuppose that  is meaningful. Hence they do not define its meaning. Put interms of speakers’ understanding, to be able to use I and to infer a formula ofthe connective from the meaning-theoretical perspective (Dummett, 1993a, 250, 289). Dummettobserves that normalisation implies that each logical constant is conservative over the rest of thelanguage (Dummett, 1993a, 250) and conjectures that ‘intrinsic harmony implies total harmony in acontext where stability prevails’ (Dummett, 1993a, 290). Dummett and Prawitz only count thosesegments as maximal that begin with the conclusion of an introduction rule. The more generalnotion used here is found in (Troestra and Schwichtenberg, 2000). It is required for philosophicalreasons. For more on these issues, see (Kürbis, 2019, Ch 2).2the form A, a speaker already needs to know how to use formulas of the formB in deductions, and so the speaker already needs to know the meaning of B.Thus a speaker could not acquire a grasp of the meaning of  by being taughtthose rules. As a definition of the meaning of , these rules are circular. TheI-rule for  presupposes that  already has meaning.2I propose that for the rules governing ∗ to define its meaning, they mustsatisfy a Principle of Molecularity: ∗must not occur in the premises and dischargedhypotheses of its I-rules, nor in any restrictions on their application, and ∗mustnot occur in the minor premises and discharged hypotheses of its E-rules, nor inany restrictions on their application. Generalising, there should be no sequenceof constants ∗1...∗n such that the rules for ∗i refer to ∗ j, i < j, and the rules for ∗nrefer to ∗1.A promising system of modal logic from the present perspective was for-malised by Pfenning and Davies (Pfenning and Davies, 2001). It is based onMartin-Löf’s account of judgements. They distinguish the judgment that aproposition is true from the judgement that a proposition is valid. ` is inter-preted as a hypothetical judgement. Validity is defined in terms of truth andhypothetical judgements, where ·marks an empty collection of hypotheses andΓ are hypotheses of the form ‘B true’: (1) If · ` A true, then A valid; (2) If A valid,then Γ ` A true.Their system has axioms for the two kinds of hypotheses and rules forimplication and necessity. Formulas assumed to be valid are to the left of thesemi-colon, those assumed to be true to its right:3hyp∆;Γ,A true,Γ′ ` A truehyp∗∆,B valid,∆′;Γ ` B true∆;Γ,A true ` B true⊃I∆;Γ ` A ⊃ B true∆;Γ ` A ⊃ B true ∆;Γ ` A true⊃E∆;Γ ` B true∆; · ` A trueI∆;Γ ` A true∆;Γ ` A true ∆,A valid;Γ ` C trueE∆;Γ ` C trueCall this system JM. It is a fragment of intuitionist S4. A normalisation theoremcan be proved for it. Its rules satisfy the Principle of Molecularity.2 Modal Logic with ValidityIn this section I reformulate, extend and generalise JM. The reformulation isthree-fold. (1) I use a system of natural deduction not in sequent calculus style.(2) As any formula in JM is followed by either ‘valid’ or ‘true’, I drop the latterand simply write ‘A’. This has a philosophical point: it accords with an accountof logical inference as relating propositions, not judgements. (3) I do not treatvalidity as a judgement either, but as a sentential operator. The generalisationconsists in the observation that validity is a relation between the assumptions2Prawitz proves a normalisation theorem for intuitionist S4 and S5 (Prawitz, 1965, Ch 6). Othersuch systems of intuitionist S4 are formalised by Biermann and de Paiva (Biermann and de Paiva,2000) and von Plato (von Plato, 2005). Thus normalisation is not a sufficient condition for rules todefine meaning.3The restriction on Γ of clause (2) of the definition is not explained further. The point may wellbe to avoid circularity. It is effectively lifted in the axiom hyp∗.3and the conclusion of a deduction. The extension consists in formulating rulesof inference for a higher level operator ` for this generalised notion of validity.The rules for  appeal to it. HM extends I by ` and .Formulas of level 0 are those of I extended by . Formulas of level 1 areall formulas B1...Bn ` A, where B1...Bn,A are formulas of level 0, for 0 ≤ n.B1...Bn ` A can be derived if there is a deduction of A from B1...Bn. Applying anelimination rule for `, this is what we should get back. We may not know howA was derived from B1...Bn, but as we know that there is such a deduction, theinference of A from B1...Bn is valid. ` has the following rules:[B1]i1 ...[Bn]inΠAVI : i1...inB1...Bn ` AΣB1...Bn ` AΞ1B1 ...ΞnBnVE : Awhere B1...Bn, 0 ≤ n, are representatives of all the open assumption classes of Πin any order (as the Bs must be of level 0, there are no open assumptions of level1). Vacuous discharge is allowed: a representative to the left of `may belong toan empty assumption class of Π; this corresponds to Thinning.VI and VE are generalisations of Pfenning’s and Davies’s definition ofvalidity cast into rules of a system of natural deduction. Next we generalise theI- and E-rules for necessity.  is treated as a multi-grade constant which hasone formula to its right and 0 to finite n formulas on its left. I abbreviate B1...Bnby Γ and write Γ ` A instead of B1...Bn ` A and [Γ]i instead of [B1]i1 ...[Bn]in . Therules for  are:[Γ]iΠAI : iΓAΣΓA[Γ ` A]iΞCE : iCwhere in I, all open assumptions of level 0 of Π are in Γ (any other openassumptions are of level 1 and have the form ∆ ` B). Vacuous discharge isallowed. In E, C is a 0-level formula. I propose to read  as relative necessity.It expresses that there is a necessary connection between the formulas in Γ andA, or necessarily, A given Γ.4 When Γ is empty, we get the usual unary necessityoperator: it behaves as in intuitionist S4.Maximal formulas of the form Γ ` A are removed by the following reductionstep:[B1]i1 ...[Bn]inΠA i1...inB1...Bn ` AΞ1B1 ...ΞnBnAΣ Ξ1[B1] ...Ξn[Bn]ΠAΣ4For a few more thoughts on this modal notion, see (Kürbis, 2015). It should be noted that onthis reading, > is necessary relative to everything, while everything is necessary relative to ⊥: thenotion of relative necessity proposed here is not a relevant relative necessity.4The restrictions on VI and I require that all open formulas or all open 0-level formulas are discharged above their premises, and hence there can be noapplication of these rules in Π below B1...Bn, except where an assumption class[B]i is empty. So the transformation cannot lead to any violations of restrictionson rules in Π. Any applications of those rules also remain correct in Σ, asthe reduction procedure does not introduce new open assumptions into thededuction. For essentially the same reason, Prawitz’s reduction procedures formaximal formulas and segments continue to work for the constants HM shareswith I.Removing maximal formulas ΓA is slightly more original:[Γ]iΠAiΓA[Γ ` A] jΣΓAΞC jCΘ Σ[Γ]ΠAΞCΘΣ are the deductions of the formulas in Γ. A maximal formula of type ΓA canonly occur in the context on the left, unless Γ ` A is discharged vacuously byE, in which case its removal is trivial. The only thing one can do with Γ ` Ais to apply VE to it. Due to the restriction on C in E and the formation rulesfor the language of HM, such a formula cannot be assumed and immediatelydischarged by a rule. Due to the restrictions on VI and I, there can be noapplications of these rules below the Γs in Π (unless in the case of vacuousdischarge, which is trivial): hence concluding the Γs with the deductions in Σcannot lead to violations of rules inΠ. Due to the restrictions on VI, there can beno application of that rule below A in Ξ, as there is at least the open assumptionΓ ` A that prevents such an application. If there is an application ofI in Ξ, thenall open assumptions of the deductions in Σ are of the form ∆ ` B, and hencethey remain correct after the transformation. For similar reasons, applicationsof these rules in Θ also remain correct.All reduction steps reduce the complexity of the deduction: a maximalsegment is shortened, a maximal formula of higher degree than those that maybe introduced by the reduction procedure removed. A standard induction overthe complexity of deductions establishes the normalisation theorem for HM.3 ConclusionHM is a natural system of modal logic with higher order rules. It fulfils necessaryconditions for a proof-theoretic account of the meaning of . Deductionsnormalise. Its rules are harmonious and satisfy the molecularity principle. Themeaning of  is given in terms of the meaning of `, the meaning of which isgiven in terms of inferences in I.HM generalises JM in introducing a more general notion of validity andallowing validities to occur as conclusions of rules. But it remains close to JM,in that the restrictions on VI and the rules for  are directly lifted from JM. Anatural question is how the restrictions on VI could be loosened to allow further5ways of deriving Γ ` A. The restriction on VI blocks a derivation of a versionof cut: If (1) Γ ` A and (2) ∆,A ` C, then (3) Γ,∆ ` C. It is possible to concludeA from (1) by assuming all formulas in Γ, and then to conclude C from (2) byassuming all formulas in ∆, but the restriction on VI prevents the conclusion of(3), as besides the 0-level formulas in Γ and ∆, the conclusion C depends on theundischarged first level formulas Γ ` A and ∆,A ` C.Došen proposes systems of higher order sequents for intuitionist and classicalS4 and S5 (see (Došen, 1985), (Došen, 1986)), in which, he explains, A means‘A is assumed as a theorem’. This sounds similar to Pfenning’s and Davies’saccount of modality. Došen’s system implements a stricter distinction of levelsof formulas and rules than HM. To the left and right of Došen’s turnstile of level2, there must be formulas of level 1, not of level 1 or 0. Thus transposed into asystem of natural deduction, Došen’s rules for , which are of level 2, wouldrequire premises and conclusions of level 1. These rules are derivable usingpresent the rules if VI may also be applied when all assumptions on whichits premise depends are of level 1, i.e. of form ∆ ` C. Furthermore, with therestriction on VI so loosened that amongst the assumptions on which its premisedepends there may be formulas of level 1, the version of cut mentioned in theprevious paragraph becomes derivable. Modifying HM is an avenue for furtherresearch.ReferencesBiermann, G. M. and V. C. V. de Paiva (2000). On an intuitionistic modal logic.Studia Logica 65, 383–416.Došen, K. (1985). Sequent-systems for modal logic. The Journal of SymbolicLogic 50(1), 149–168.Došen, K. (1986). Higher-level sequent systems for intuitionistic modal logic.Publications der L’Institut Mathématique 38(53), 3–12.Dummett, M. (1993a). The Logical Basis of Metaphysics. Cambridge, Mass.:Harvard University Press.Dummett, M. (1993b). What is a theory of meaning? (I). In The Seas of Language,pp. 1–33. Oxford: Clarendon.Dummett, M. (1993c). What is a theory of meaning? (II). In The Seas of Language,pp. 34–93. Oxford: Clarendon.Kürbis, N. (2015). Proof-theoretic semantics, a problem with negation andprospects for modality. The Journal of Philosophical Logic 44, 713–727.Kürbis, N. (2019). Proof and Falsity. A Logical Investigation. Cambridge UniversityPress.Pfenning, F. and R. Davies (2001). A judgemental reconstruction of modal logic.Mathematical Structures in Computer Science 11, 511–540.Prawitz, D. (1965). Natural Deduction. Stockholm, Göteborg, Uppsala: Almqvistand Wiksell.Prawitz, D. (1987). Dummett on a theory of meaning and its impact on logic.In B. Taylor (Ed.), Michael Dummett: Contributions to Philosophy, pp. 117–165.Dordrecht: Nijhoff.Prawitz, D. (1994). Meaning theory and anti-realism. In B. McGuiness (Ed.), ThePhilosophy of Michael Dummett, pp. 79–89. Dordrecht: Kluwer.6Prawitz, D. (2006). Meaning approached via proofs. Synthese 148, 507–524.Prawitz, D. (2007). Pragmatist and verificationist theories of meaning. InR. Auxier and L. Hahn (Eds.), The Philosophy of Michael Dummett, pp. 455–481.Chicago: Open Court.Troestra, A. and H. Schwichtenberg (2000). Basic Proof Theory (2 ed.). CambridgeUniversity Press.von Plato, J. (2005). Normal derivability in modal logic. Mathematical LogicQuarterly 51(6), 632–638.7

A Sketch of a Proof-Theoretic Semantics for Necessity

PhilPapers

A Sketch of a Proof-Theoretic Semantics forNecessityNils Ku¨rbisUniversity of  Lo´dz´AbstractThis paper considers proof-theoretic semantics for necessity within Dummett’s andPrawitz’s framework. Inspired by a system of Pfenning’s and Davies’s, the languageof intuitionist logic is extended by a higher order operator which captures a notionof validity. A notion of relative necessary is defined in terms of it, which expresses anecessary connection between the assumptions and the conclusion of a deduction.Keywords: proof-theoretic semantics, modal logic, necessity, higher-order rules.1 Proof-Theoretic SemanticsDummett and Prawitz do not consider how the meanings of modal operatorsmay be given by their theory of meaning for the logical constants. To investigatein outline how this may be done is the purpose of this short paper.According to proof-theoretic semantics, the rules governing a constant de-fine its meaning. Prior’s tonk shows that the rules cannot be arbitrary. Dum-mett and Prawitz impose the restriction that the introduction and eliminationrules for a constant ∗ be in harmony, so that ∗E does not license the deductionof more consequences from A ∗B than are justified by the grounds for derivingit as specified by ∗I. (See [4], [11], [12], [13], [14].) A necessary condition forharmony is that deductions can be brought into normal form. A deduction isin normal form if it contains neither maximal formulas nor maximal segments.A maximal formula is one that is the conclusion of an I-rule and major premiseof an E-rule. A maximal segment is a sequence of formulas all except the lastof which are minor premises of ∨E and the last one is major premise of anE-rule. 11 I am allowing myself a certain looseness in terminology, which, however, is quite commonin the literature. Strictly speaking, Dummett distinguishes intrinsic harmony, stability andtotal harmony. Intrinsic harmony is captured by normalisation: the elimination rules of aconstant are justified relative to the introduction rules. Stability is harmony together witha suitable converse: the introduction rules are also justified relative to the elimination rules.Total harmony obtains if the constant is conservative over the rest of the language. Dummettcalls the permutative reduction steps to remove maximal segments ‘auxiliary reduction steps’.Sometimes, as in the case of quantum disjunction, these cannot be carried out, which pointsto a defect in the rules for the connective from the meaning-theoretical perspective [4, 250,2 A Sketch of a Proof-Theoretic Semantics for NecessityDeductions in intuitionist logic I normalise [10, Ch 4]:A∨I:A ∨BBA ∨BA ∨B[A]iΠC[B]jΣC∨E: i,jC[A]iΠB⊃ I: iA ⊃ BA ⊃ B A⊃ E:B⊥⊥E:CA B∧I:A ∧BA ∧B∧E:AA ∧BBThe constants occur only in conclusions of I-rules and major premises of E-rules. Thus the conditions for using an I-rule and the consequences of usingan E-rule are given independently of the constants.The rules of I exemplify Dummett’s notions of full-bloodedness and molec-ularity in the theory of meaning [5], [6]. A full-blooded theory of meaningcharacterises the knowledge of speakers in virtue of which they master a lan-guage in such a way that it exhibits how a speaker who does not yet understandan expression could acquire a grasp of it. A molecular theory of meaning doesso piecemeal and specifies the meanings of the expressions of a language onegroup of expressions at a time. A speaker need not understand the constantsof I in order to be informed about the conditions for the application of their I-and E-rules. To understand the grounds for deriving a formula with ∗ as mainoperator, or to understand the consequences that follow from it, a speaker onlyneeds to grasp the meanings of some sentences, but not any sentences contain-ing ∗. A speaker who does not already know the meanings of the constants ofintuitionist logic could acquire a grasp of their meanings by learning the rulesof inference of I. The rules are informative: the grounds and consequences of aformula with ∗ as main operator are given without reference to ∗. Its meaningis specified without presupposing that ∗ already has meaning.Contrast the rules of I with standard rules for 2 in S4:A2I:2A2A2E:Awhere in 2I all assumption on which A depends have the form 2B. The condi-tions for applying 2I are not given independently of 2. Thus they presupposethat 2 is meaningful. Hence they do not define its meaning. Put in terms of289]. Dummett observes that normalisation implies that each logical constant is conservativeover the rest of the language [4, 250] and conjectures that ‘intrinsic harmony implies totalharmony in a context where stability prevails’ [4, 290]. Dummett and Prawitz only countthose segments as maximal that begin with the conclusion of an introduction rule. The moregeneral notion used here is found in [15]. It is required for philosophical reasons. For moreon these issues, see [8, Ch 2].Ku¨rbis 3speakers’ understanding, to be able to use 2I and to infer a formula of theform 2A, a speaker already needs to know how to use formulas of the form 2Bin deductions, and so the speaker already needs to know the meaning of 2B.Thus a speaker could not acquire a grasp of the meaning of 2 by being taughtthose rules. As a definition of the meaning of 2, these rules are circular. TheI-rule for 2 presupposes that 2 already has meaning. 2I propose that for the rules governing ∗ to define its meaning, they mustsatisfy a Principle of Molecularity : ∗ must not occur in the premises anddischarged hypotheses of its I-rules, nor in any restrictions on their application,and ∗ must not occur in the minor premises and discharged hypotheses of itsE-rules, nor in any restrictions on their application. Generalising, there shouldbe no sequence of constants ∗1...∗n such that the rules for ∗i refer to ∗j , i < j,and the rules for ∗n refer to ∗1.A promising system of modal logic from the present perspective was for-malised by Pfenning and Davies [9]. It is based on Martin-Lo¨f’s account ofjudgements. They distinguish the judgment that a proposition is true fromthe judgement that a proposition is valid. ` is interpreted as a hypotheticaljudgement. Validity is defined in terms of truth and hypothetical judgements,where · marks an empty collection of hypotheses and Γ are hypotheses of theform ‘B true’: (1) If · ` A true, then A valid; (2) If A valid, then Γ ` A true.Their system has axioms for the two kinds of hypotheses and rules forimplication and necessity. Formulas assumed to be valid are to the left of thesemi-colon, those assumed to be true to its right: 3hyp∆; Γ, A true,Γ′ ` A true hyp∗∆, B valid,∆′; Γ ` B true∆; Γ, A true ` B true⊃I∆; Γ ` A ⊃ B true∆; Γ ` A ⊃ B true ∆; Γ ` A true⊃E∆; Γ ` B true∆; · ` A true2I∆; Γ ` 2A true∆; Γ ` 2A true ∆, A valid; Γ ` C true2E∆; Γ ` C trueCall this system JM. It is a fragment of intuitionist S4. A normalisationtheorem can be proved for it. Its rules satisfy the Principle of Molecularity.2 Modal Logic with ValidityIn this section I reformulate, extend and generalise JM. The reformulationis three-fold. (1) I use a system of natural deduction not in sequent calculusstyle. (2) As any formula in JM is followed by either ‘valid’ or ‘true’, I dropthe latter and simply write ‘A’. This has a philosophical point: it accords withan account of logical inference as relating propositions, not judgements. (3) I2 Prawitz proves a normalisation theorem for intuitionist S4 and S5 [10, Ch 6]. Other suchsystems of intuitionist S4 are formalised by Biermann and de Paiva [1] and von Plato [16].Thus normalisation is not a sufficient condition for rules to define meaning.3 The restriction on Γ of clause (2) of the definition is not explained further. The point maywell be to avoid circularity. It is effectively lifted in the axiom hyp∗.4 A Sketch of a Proof-Theoretic Semantics for Necessitydo not treat validity as a judgement either, but as a sentential operator. Thegeneralisation consists in the observation that validity is a relation betweenthe assumptions and the conclusion of a deduction. The extension consists informulating rules of inference for a higher level operator ` for this generalisednotion of validity. The rules for 2 appeal to it. HM extends I by ` and 2.Formulas of level 0 are those of I extended by 2. Formulas of level 1 areall formulas B1...Bn ` A, where B1...Bn, A are formulas of level 0, for 0 ≤ n.B1...Bn ` A can be derived if there is a deduction of A from B1...Bn. Applyingan elimination rule for `, this is what we should get back. We may not knowhow A was derived from B1...Bn, but as we know that there is such a deduction,the inference of A from B1...Bn is valid. ` has the following rules:[B1]i1 ...[Bn]inΠAV I : i1...inB1...Bn ` AΣB1...Bn ` AΞ1B1 ...ΞnBnV E :Awhere B1...Bn, 0 ≤ n, are representatives of all the open assumption classes ofΠ in any order (as the Bs must be of level 0, there are no open assumptionsof level 1). Vacuous discharge is allowed: a representative to the left of ` maybelong to an empty assumption class of Π; this corresponds to Thinning.V I and V E are generalisations of Pfenning’s and Davies’s definition ofvalidity cast into rules of a system of natural deduction. Next we generalisethe I- and E-rules for necessity. 2 is treated as a multi-grade constant whichhas one formula to its right and 0 to finite n formulas on its left. I abbreviateB1...Bn by Γ and write Γ ` A instead of B1...Bn ` A and [Γ]i instead of[B1]i1 ...[Bn]in . The rules for 2 are:[Γ]iΠA2I : iΓ2AΣΓ2A[Γ ` A]iΞC2E : iCwhere in 2I, all open assumptions of level 0 of Π are in Γ (any other openassumptions are of level 1 and have the form ∆ ` B). Vacuous discharge isallowed. In 2E, C is a 0-level formula. I propose to read 2 as relative necessity.It expresses that there is a necessary connection between the formulas in Γ andA, or necessarily, A given Γ. 4 When Γ is empty, we get the usual unarynecessity operator: it behaves as in intuitionist S4.Maximal formulas of the form Γ ` A are removed by the following reductionstep:4 For a few more thoughts on this modal notion, see [7]. It should be noted that on thisreading, > is necessary relative to everything, while everything is necessary relative to ⊥:the notion of relative necessity proposed here is not a relevant relative necessity.Ku¨rbis 5[B1]i1 ...[Bn]inΠAi1...inB1...Bn ` AΞ1B1 ...ΞnBnAΣ;Ξ1[B1] ...Ξn[Bn]ΠAΣThe restrictions on V I and 2I require that all open formulas or all open 0-level formulas are discharged above their premises, and hence there can be noapplication of these rules in Π below B1...Bn, except where an assumptionclass [B]i is empty. So the transformation cannot lead to any violations ofrestrictions on rules in Π. Any applications of those rules also remain correctin Σ, as the reduction procedure does not introduce new open assumptions intothe deduction. For essentially the same reason, Prawitz’s reduction proceduresfor maximal formulas and segments continue to work for the constants HMshares with I.Removing maximal formulas Γ2A is slightly more original:[Γ]iΠAiΓ2A[Γ ` A]jΣΓAΞCjCΘ;Σ[Γ]ΠAΞCΘΣ are the deductions of the formulas in Γ. A maximal formula of type Γ2A canonly occur in the context on the left, unless Γ ` A is discharged vacuously by2E, in which case its removal is trivial. The only thing one can do with Γ ` Ais to apply V E to it. Due to the restriction on C in 2E and the formation rulesfor the language of HM, such a formula cannot be assumed and immediatelydischarged by a rule. Due to the restrictions on V I and 2I, there can be noapplications of these rules below the Γs in Π (unless in the case of vacuousdischarge, which is trivial): hence concluding the Γs with the deductions in Σcannot lead to violations of rules in Π. Due to the restrictions on V I, therecan be no application of that rule below A in Ξ, as there is at least the openassumption Γ ` A that prevents such an application. If there is an applicationof 2I in Ξ, then all open assumptions of the deductions in Σ are of the form∆ ` B, and hence they remain correct after the transformation. For similarreasons, applications of these rules in Θ also remain correct.All reduction steps reduce the complexity of the deduction: a maximalsegment is shortened, a maximal formula of higher degree than those that maybe introduced by the reduction procedure removed. A standard induction overthe complexity of deductions establishes the normalisation theorem for HM.6 A Sketch of a Proof-Theoretic Semantics for Necessity3 ConclusionHM is a natural system of modal logic with higher order rules. It fulfils neces-sary conditions for a proof-theoretic account of the meaning of 2. Deductionsnormalise. Its rules are harmonious and satisfy the molecularity principle. Themeaning of 2 is given in terms of the meaning of `, the meaning of which isgiven in terms of inferences in I.HM generalises JM in introducing a more general notion of validity andallowing validities to occur as conclusions of rules. But it remains close to JM,in that the restrictions on V I and the rules for 2 are directly lifted from JM.A natural question is how the restrictions on V I could be loosened to allowfurther ways of deriving Γ ` A. The restriction on V I blocks a derivation of aversion of cut: If (1) Γ ` A and (2) ∆, A ` C, then (3) Γ,∆ ` C. It is possibleto conclude A from (1) by assuming all formulas in Γ, and then to conclude Cfrom (2) by assuming all formulas in ∆, but the restriction on V I prevents theconclusion of (3), as besides the 0-level formulas in Γ and ∆, the conclusion Cdepends on the undischarged first level formulas Γ ` A and ∆, A ` C.Dosˇen proposes systems of higher order sequents for intuitionist and classi-cal S4 and S5 ([2], [3]), in which, he explains, 2A means ‘A is assumed as atheorem’. This sounds similar to Pfenning’s and Davies’s account of modality.Dosˇen’s system implements a stricter distinction of levels of formulas and rulesthan HM. To the left and right of Dosˇen’s turnstile of level 2, there must beformulas of level 1, not of level 1 or 0. Thus transposed into a system of naturaldeduction, Dosˇen’s rules for 2, which are of level 2, would require premises andconclusions of level 1. These rules are derivable using present the rules if V Imay also be applied when all assumptions on which its premise depends are oflevel 1, i.e. of form ∆ ` C. Furthermore, with the restriction on V I so loos-ened that amongst the assumptions on which its premise depends there maybe formulas of level 1, the version of cut mentioned in the previous paragraphbecomes derivable. Modifying HM is an avenue for further research.References[1] Biermann, G. M. and V. C. V. de Paiva, On an intuitionistic modal logic, Studia Logica65 (2000), pp. 383–416.[2] Dosˇen, K., Sequent-systems for modal logic, The Journal of Symbolic Logic 50 (1985),pp. 149–168.[3] Dosˇen, K., Higher-level sequent systems for intuitionistic modal logic, Publications derL’Institut Mathe´matique 38 (1986), pp. 3–12.[4] Dummett, M., “The Logical Basis of Metaphysics,” Cambridge, Mass.: HarvardUniversity Press, 1993.[5] Dummett, M., What is a theory of meaning? (I), in: The Seas of Language, Oxford:Clarendon, 1993 pp. 1–33.[6] Dummett, M., What is a theory of meaning? (II), in: The Seas of Language, Oxford:Clarendon, 1993 pp. 34–93.[7] Ku¨rbis, N., Proof-theoretic semantics, a problem with negation and prospects formodality, The Journal of Philosophical Logic 44 (2015), pp. 713–727.Ku¨rbis 7[8] Ku¨rbis, N., “Proof and Falsity. A Logical Investigation,” Cambridge University Press,2019.[9] Pfenning, F. and R. Davies, A judgemental reconstruction of modal logic, MathematicalStructures in Computer Science 11 (2001), pp. 511–540.[10] Prawitz, D., “Natural Deduction,” Stockholm, Go¨teborg, Uppsala: Almqvist and Wiksell,1965.[11] Prawitz, D., Dummett on a theory of meaning and its impact on logic, in: B. Taylor,editor, Michael Dummett: Contributions to Philosophy, Dordrecht: Nijhoff, 1987 pp.117–165.[12] Prawitz, D., Meaning theory and anti-realism, in: B. McGuiness, editor, The Philosophyof Michael Dummett, Dordrecht: Kluwer, 1994 pp. 79–89.[13] Prawitz, D., Meaning approached via proofs, Synthese 148 (2006), pp. 507–524.[14] Prawitz, D., Pragmatist and verificationist theories of meaning, in: R. Auxier andL. Hahn, editors, The Philosophy of Michael Dummett, Chicago: Open Court, 2007pp. 455–481.[15] Troestra, A. and H. Schwichtenberg, “Basic Proof Theory,” Cambridge University Press,2000, 2 edition.[16] von Plato, J., Normal derivability in modal logic, Mathematical Logic Quarterly 51(2005), pp. 632–638.

Sketch of a Proof-Theoretic Semantics for Necessity

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Sketch of a Proof-Theoretic Semantics for Necessity

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