It is well known that no consistent normal modal logic contains (as theorems) both ◊A and ◊¬A (for any formula A). Here we observe that this claim can be strengthened to the following: for any formula A, either no consistent normal modal logic contains ◊A, or else no consistent normal modal logic contains ◊¬A

French, Rohan

Humberstone, Lloyd

Bulletin of the Section of Logic

English

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An Observation Concerning Porte’s Rule in Modal Logic

Repozytorium Uniwersytetu Łódzkiego (University of Lodz Repository)

Bulletin of the Section of LogicVolume 44:1/2 (2015), pp. 25–31Rohan French and Lloyd HumberstoneAN OBSERVATION CONCERNING PORTE’S RULE INMODAL LOGICAbstractIt is well known that no consistent normal modal logic contains (as theorems)both ♦A and ♦¬A (for any formula A). Here we observe that this claim can bestrengthened to the following: for any formula A, either no consistent normalmodal logic contains ♦A, or else no consistent normal modal logic contains ♦¬A.1. IntroductionWe work with the standard language of (propositional) monomodal logic,for convenience taken to have a countable stock of sentence letters, pi,constants (0-ary connectives) ⊤ and ⊥ among the Boolean connectives,alongside others together functionally complete (we use ¬, ∧, ∨, → and ↔below for the Boolean connectives commonly so notated), and for definite-ness, the 1-ary connective  (with ♦A defined as ¬¬A). A pure formula isone not containing any sentence letters (such as ⊥ ∨ ♦⊥). Here modallogics are treated as sets of formulas, and we are interested specificallyin normal modal logics, i.e. such sets as contain every truth-functionaltautology in the Boolean connectives as well as all formulas of the form(A→ B)→ (A→ B), and are closed under Modus Ponens, UniformSubstitution (of arbitrary formulas for sentence letters), and Necessitation(the rule taking us from A to A).Let us recall the status of Porte’s rule (from [11]):♦A ∧ ♦¬A⊥26 Rohan French and Lloyd Humberstonein this setting. Porte observed that (the set of theorems of) S5 was closedunder this rule – which appears somewhat differently notated as (R5) onp. 417 of [11] – and others (such as Fagin et al. [4]) have noted that the samegoes for all normal modal logics. Since the inconsistent logic is obviouslyclosed under the rule, it suffices to establish that all consistent normalmodal logics are, which can be done by noting that every such logic is asublogic of one of the two Post complete modal logics: that containingA ↔ A for all A (“the trivial system”), or that containing A for all A(“the Verum system”). Where S ⊕ X denotes the smallest normal modallogic extending S and containing all formulas A ∈ X and K is, as usual, thesmallest normal modal logic, then the latter system can alternatively bedescribed as K ⊕ {⊥}. Since no formula of the form ♦A ∧ ♦¬A belongsto either of these logics, no such formula belongs to any consistent normalmodal logic.Alternatively, one may draw the same conclusion semantically usingeither algebraic or model-theoretic semantics for modal logic. In the formercase (as in Makinson [8]) one considers normal modal expansions of the two-element Boolean algebra,1 noting that neither can validate any formulaof the form ♦A ∧ ♦¬A, and in the latter via a consideration (as in vanBenthem [2], Lemma 2.19, p. 31) of the special status of the two one-pointKripke frames, again neither of them validating a formula of the form inquestion.The consistent normal modal logics are thus closed under Porte’s rulebecause the rule is, as one says, vacuously admissible: no such logic containsany candidate premiss for an application of the rule. We can put thisanother way, rewriting the rule as a two-premiss rule, with premisses ♦Aand ♦¬A, and concentrating on K since the existence of some consistentnormal modal logic containing such and such formulas is equivalent to theconsistency of the normal extension of K by those formulas. The familiarpoint associated with Porte’s rule is then:For all formulas A, K ⊕ {♦A,♦¬A} is inconsistent. (*)By an elementary argument given in the following section, we show that aconsiderable strengthening of this point is available:1The notion of normality for a modal algebra used here is that defined in Blok andKo¨hler, p. 943.An Observation Concerning Porte’s Rule in Modal Logic 27For all formulas A, either K ⊕ {♦A} or K ⊕ {♦¬A}, is inconsistent.(**)Thus, reverting to Porte’s conjunctive form, the inconsistency of a nor-mal modal logic S containing any formula ♦A∧♦¬A can always be ‘blamedon’ one of the conjuncts and is never due to any interaction between them(or to any interaction between them and whatever else is in S other thanwhat is already in K).2. Justifying the Stronger ClaimThe present section is devoted to establishing the observation (**) above.It is a well known fact that the logic KD ( = K ⊕ ♦⊤) contains, for everypure formula B, either B itself or else ¬B; we will put this by saying thatKD decides every pure formula. (See the first part of the proof of Theorem1 in [15].)With a view to showing that for any formula A, either K ⊕ {♦A} orK ⊕ {♦¬A} is inconsistent, fix on one such A and note that each of theselogics extends KD. Let B be any pure substitution instance of A and notethat(1) ♦B ∈ K⊕ {♦A} (2) ♦¬B ∈ K⊕ {♦¬A}.Now, since B is a pure formula and KD decides each such formula we haveB ∈ KD or ¬B ∈ KD. In the first case, therefore, B ∈ KD (by normality)while in the second case we have ¬B ∈ KD. Since each of K ⊕ {♦A},K ⊕ {♦¬A} ⊇ KD, we therefore have either ¬B ∈ K ⊕ {♦A}, showingK ⊕ {♦A} to be inconsistent in view of (1), or else B ∈ K ⊕ {♦¬A},showing K⊕{♦¬A} to be inconsistent in view of (2). Thus one or other ofthese extensions of K is inconsistent, as (**) claims.3. Enriching the LanguageIt is worth asking what becomes of (*) and (**) in the setting of bimodallogic, i.e., when there are two primitive non-Boolean connectives instead ofone. Consider the case of tense logic, in which each of the two operators –we shall write them as  and −1 here – is 1-ary, and satisfies the normalityconditions given in connection with  in Section 1, as well as the ‘Lemmon28 Rohan French and Lloyd Humberstonebridging axioms’, all formulas of the form ♦−1A→ A and ♦−1A→ A.(Here ♦−1 abbreviates ¬−1¬; , ♦, −1 and ♦−1 are often written, ina notation due to A. N. Prior, as G, F , H and P respectively.) The basictense logic Kt is smallest set of formulas in this language satisfying theseconditions, and we can ask about the fate of (*) and (**) for Kt ratherthan monomodal K. Naturally one could focus on ♦−1 rather than ♦ inpursuing this question but in view of the symmetrical treatment of the twooperators in Kt there is nothing to be gained by doing so. When discussingextensions of a logic S in this language, S⊕X denotes the smallest extensionof S containing all formulas in X and in which each of ,−1 satisfies thenormality conditions.As is well known, frames for (i.e., validating every formula in) Kt canbe presented with a single accessibility relation, taken as interpreting ,with −1 interpreted by its converse. Such frames reveal a contrast withthe monomodal case:Proposition 3.1. The claim resulting from (∗∗) by putting “Kt” for “K”is false.Proof: We need to supply a formula A for which each of K ⊕ {♦A},K ⊕ {♦¬A}, is consistent. Take A as −1♦−1⊤. Thus we can write ¬Aas ♦−1−1⊥. To see that Kt ⊕ {♦A} is consistent, note that the one-element reflexive frame – or more explicitly〈{0}, {〈0, 0〉}〉– is a framefor this logic. In the case of Kt ⊕ {♦¬A} consider, instead, the frame〈{0, 1}, {〈0, 1〉, 〈1, 1〉}〉.The choice of A in the proof just given refutes the tense-logical ana-logue of (**) but not that of (*), since from ♦A, i.e., ♦−1♦−1⊤, by oneof the bridging axioms (and Modus Ponens) we get ♦−1⊤, from which Ne-cessitation (w.r.t. −1 and then, after that, ) delivers −1♦⊤, whichis equivalent to A, making the envisaged extension by both {♦A,♦¬A}inconsistent. But must the same happen, as it does in the monomodal case,for all possible choices of A? That is:Question. Is (*) correct for Kt in place of K?A remark spanning pages 712 and 713 in Segerberg [15] (and echoed in thebottom paragraph of p. 135 of [6]) suggests that the answer to this questionmay be negative, but we have no definite information.An Observation Concerning Porte’s Rule in Modal Logic 29A simpler extension of the language of monomodal logic would involvethe addition of a non-Boolean sentential constant. As has recently beenobserved ([5], [6]) adding such a constant κ makes for considerable changesto standard metatheoretical results familiar from the straight monomodalcase, and indeed the multimodal-with-all-operators-normal case. We canuse our constant by considering the smallest modal logic in this language inwhich  obeys the normality conditions and which also contains, in orderto contain a candidate premiss for Porte’s rule, the axiom ♦κ ∧ ♦¬κ. Ifthe notion of normality is extended in a natural way to other than 1-aryconnectives (see Schotch and Jennings [14], p. 271, where the definitiongiven is credited to unpublished work of Goldblatt), this would be countedas a bimodal logic in which  but not κ was normal. (The latter wouldrequire, by the appropriate generalization of Necessitation, that κ itselfbelong to the logic.) The conclusion, ⊥, of an application of Porte’s rule tothis new axiom clearly does not belong to the logic, which contains exactlythe formulas (and ⊥ is not one of them) valid on all frames 〈W,R,W0〉with R ⊆W ×W and W0 ⊆W satisfies the condition (in which we denoteby R(x) the set of y ∈ W such that Rxy) that for all x ∈ W , R(x) ∩W0 6= ∅ and R(x) ∩ (W rW0) 6= ∅, truth in a model being defined withthe condition that κ is true precisely at elements of W0. (Validity on aframe is truth throughout W in every model on the frame.) The axiomtells us that it is contingent whether κ, and was used by Anderson in anearly version (see note 40 of [1]) of his reduction of deontic logic to alethicmodal logic – using κ to record the violation of an obligation so that “itought to be that A” could be thought of as (¬A → κ). More recently,Pizzi ([9], [10]) has used a similar idea to define necessity – understood assubsuming any interpretation of the  operator – in terms of contingency(or noncontingency), even in logics too weak to allow for a definition of in terms of ▽ (“it is contingent whether”). Of course, given the changeof primitives, he writes the above axiom as ▽κ rather than ♦κ ∧ ♦¬κ,but shows how to use κ alongside ▽ in an explicit definition of  even inthese weaker logics. For a review of this project, see [7], esp. Section 4,where further references and a contrast with a different kind of contingencyconstant (used by Meredith and Prior) may be found.2 The reason for2Pizzi [10] is there referred to as ‘Relative Contingency and Multimodal Logics,’ itsworking title as of the time of writing. The bottom paragraph of p. 1296 in [7] seemsto suggest that Prior [12] had observed that without the addition of a new constant30 Rohan French and Lloyd Humberstonegiving this example here is simply to show that there are naturally arisingextensions of the basic monomodal apparatus in which  remains normalbut Porte’s rule ceases to be admissible.References[1] A. R. Anderson, The Formal Analysis of Normative Systems, [in:] N. Rescher(ed.), The Logic of Decision and Action, University of Pittsburgh Press,Pittsburgh, PA 1967, pp. 147–213.[2] Johan van Benthem, Modal Logic and Classical Logic, Bibliopolis,Naples 1985.[3] W. J. Blok and P. Ko¨hler, Algebraic Semantics for Quasi-Classical ModalLogics, Journal of Symbolic Logic 48 (1983), pp. 941–964.[4] R. Fagin, J. Y. Halpern, and M. Y. Vardi, What is an Inference Rule?,Journal of Symbolic Logic 57 (1992), pp. 1018–1045.[5] Rohan French, Denumerably Many Post-complete Normal Modal Logics withPropositional Constants, Notre Dame Journal of Formal Logic 53(2012), pp. 549–556.[6] Robert Goldblatt and Tomasz Kowalski, The Power of a Propositional Con-stant, Journal of Philosophical Logic 43 (2014), pp. 133–152.[7] Lloyd Humberstone, Zolin and Pizzi: Defining Necessity from Noncontin-gency, Erkenntnis 78 (2013), pp. 1275–1302.[8] David Makinson, Some Embedding Theorems for Modal Logics, NotreDame Journal of Formal Logic 12 (1971), pp. 252–254.[9] Claudio Pizzi, Necessity and Relative Contingency, Studia Logica 85(2007), pp. 395–410.[10] Claudio Pizzi, Relative Contingency and Bimodality, Logica Universalis 7(2013), pp. 113–123.[11] Jean Porte, The Deducibilities of S5, Journal of Philosophical Logic 10(1981), pp. 409–422.[12] A. N. Prior, Time and Modality, Oxford University Press, Oxford 1957.[13] A. N. Prior, Review of various papers by A. R. Anderson, Journal of Sym-bolic Logic 24 (1959), pp. 177–178.nothing was provably contingent in the modal logics under consideration there, butthis observation was actually made by Prior in line 10 from the base of [13], p. 177.(The reference in that paragraph of [7] to note 4 of the [7] should instead be to note26 there, which mentions specifically Appendix D of Prior [12], rather than the morepertinent [13].)An Observation Concerning Porte’s Rule in Modal Logic 31[14] P. K. Schotch and R. E. Jennings, Modal Logic and the Theory of ModalAggregation, Philosophia 9 (1975), pp. 265–278.[15] Krister Segerberg, Post Completeness in Modal Logic, Journal of Sym-bolic Logic 37 (1972), pp. 711–715.Department of Philosophy,Monash University,Victoria 3800,Australia

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An Observation Concerning Porte’s Rule in Modal Logic

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An Observation Concerning Porte’s Rule in Modal Logic

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