The purpose of the present note is to advertise an interesting conjecture concerning a well-known translation in modal logic, by confirming a (highly restricted) special case of the conjecture

French, Rohan

Humberstone, Lloyd

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Partial Confirmation of a Conjecture on theBoxdot Translation in Modal LogicR FS  P  B,M U,V, 3800Arohan.french@gmail.comL HS  P  B,M U,V, 3800ALloyd.Humberstone@arts.monash.edu.auReceived by Greg RestallPublished July 29, 2009http://www.philosophy.unimelb.edu.au/ajl/2009© 2009 Rohan French and Lloyd HumberstoneAbstract: The purpose of the present note is to advertise an interesting conjec-ture concerning a well-known translation in modal logic, by confirming a (highlyrestricted) special case of the conjecture.1  We work with the conventional language of (mono)modal logic, taking 2 andsome functionally complete set of boolean connectives as primitive, to gener-ate the set of formulas from the set of propositional variables, p1, . . . , pn, . . .For p1 and p2 we generally write p and q. Given a formula A,  A abbreviatesthe formula 2A∧A, and a formula in which all occurrences of 2 are in subfor-mulas of the latter form will be called a  -formula. The boxdot translation, τ ,from modal formulas to modal formulas replaces all occurrences of 2 with  .According to a well-known result, this translation embeds KT faithfully in (or‘into’) K in the sense that for all formulas A:`KT A if and only if `K τ (A).Rohan French and Lloyd Humberstone, “Partial Confirmation of the Boxdot Conjecture”, Australasian Journal of Logic (7) 2009, 56–61http://www.philosophy.unimelb.edu.au/ajl/2009 57Because of its simplicity, the boxdot translation has been a traditional favouritein the philosophical literature on the significance of such embeddings, for ex-ample, [7], [4] (and more briefly, by the same authors, in Example 2.9 of [8]and §2 of [5]. Recent technical studies in which it figures prominently includeGoris [3] and Litak [6]; see also Zolin [10], Lemma 5.9. ([3] and [10] includereferences to the literature on modal provability logic, in which again   andτ  have enjoyed considerable currency.)Now, experience with a considerable number of cases prompts the con-jecture that the range of (normal: see below) modal logics in which τ  faith-fully embeds KT—in the sense that reference to them can replace that to Kin the biconditional inset above—comprises precisely the logics S for whichK  S  KT.1 While it is not hard to see that τ  embeds KT faithfully intoall the logics between K and KT, and also (since `KT A $ τ (A) for all A) thatno proper extension of KT enjoys this status as a candidate ‘target’ for the em-bedding (with KT as ‘source’), it is not so easy to see that only logics betweenK and KT are suitable candidates. (We have informally circulated this problemamongst several logicians with a special interest in modal translations, but nonehas so far been able to provide a solution.) What, in particular, is there to ruleout the existence of a logic -incomparable with KT, in which KT is nonethe-less embedded by τ ? According to the conjecture just outlined, no such logicexists: τ  embeds KT only into sublogics of KT. “Logic”, here, means normalmodal logic;2 for these we use the labels of Chellas [1] where they exist, and ingeneral for a set of formulas Γ , S Γ is the smallest normal extension of S con-taining all formulas in Γ . When Γ = {A} for some formula A, we write “S  A”rather then “S  {A}”. In particular, then, K  A is the smallest normal modallogic containing the formula A. In view of the facts about τ  cited above, theconjecture amounts to the conjecture that for any non-theorem A, of KT ( =K  T, in the notation just introduced), there is a K-unprovable  -formula Bwith `KA B. As we may put it, the normal extension of K by any non-theorem1Considerations from the Kripke semantics point in a similar direction, but since these haveproved suggestive rather than conclusive for us to date, they will not be in play explicitly inwhat follows. The idea is that whenever A is not KT-provable, the reflexive closures of frameson which A is valid will validate some formulas—the A* of the proof of Corollary 4 below—notvalid on every reflexive frame. Note that we do not say that A is valid on a frame if and only ifA* is valid on its reflexive closure—by analogy with the well-known fact that τ (A) is valid on aframe iff A is valid on its reflexive closure. In fact, there can be no function g() from formulasto formulas with the property that for all formulas A, A is valid on a frame iff g(A) is valid onits reflexive closure, since different frames with the same reflexive closure need not validate thesame formulas.2We are taking a normal modal logic to be a set of formulas in the language described above,containing all the theorems of K and closed under Necessitation, Modus Ponens, and UniformSubstitution. For such a logic S we write “`S A” for “A 2 S” and read this as: “A is provablein (or: is a theorem of ) S”. Because of the Uniform Substitution condition, it doesn’t matterwhether we think of KT as the smallest normal modal logic containing the formula 2p ! p oras the smallest such logic containing all instances of the schema 2A! A. Whereas Chellas [1]uses T as the name of the schema, we use it as the name for the formula just mentioned.Rohan French and Lloyd Humberstone, “Partial Confirmation of the Boxdot Conjecture”, Australasian Journal of Logic (7) 2009, 56–61http://www.philosophy.unimelb.edu.au/ajl/2009 58of KT is  -nonconservative. (Note that the  -formulas comprise precisely theimage of the translation τ .)Massaged into the form just given, the conjecture clearly admits of no coun-terexample in which A is of modal degree 0, since for such an A /2 KT, KA isthe inconsistent logic (as A is then a 2-free formula which is not a two-valuedtautology). In the following section, we shall extend this to cover the case offormulas of modal degree 1—formulas, that is, in which no occurrence 2 lieswithin the scope of another occurrence of 2. (On this convention a formula ofmodal degree n is also of modal degree n + 1.) Evidently this is a considerablelimitation. The normal proper extensions of K axiomatizable by one-variablefirst degree formulas are precisely KD, KDc, KD!, KT, KTc, KT!, K2? and the in-consistent logic. Amongst first-degree formulas considered as axioms by whichto extend K without restriction as to the number of variables, the most famousare perhaps those from the infinite sequence of formulas Altn (n 2 N) fromSegerberg [9], p. 52. This section concludes with some remarks on the generalcase, without any restriction as to modal degree.Sometimes when A /2 KT, we can see that K  A is not  -conservativebecause its theorems include τ (A), and 0K τ (A). Examples include the casesin which A is 4 ( = 2p ! 22p) or B ( = p ! 23p, where 3p is ¬2¬p), or Tc( = p ! 2p). In other cases, 0KA τ (A), such as when A is 2?, 2? ∨32?,or Dc ( = 3p! 2p, a minor variant of Segerberg’s Alt1); but here it is not hardto find  -formulas other than τ (A) which are not K-provable but which areof the form τ (B) for KT-unprovable B. (In the cases just mentioned, take B asTc, 3(2p∨2¬p), and p∨2(p! q) ∨2(p! ¬q), respectively.) The functionf described in the following section applies to any first degree A to provide a Bwith the desired behaviour.2     In any normal modal logic, a formula A is provably equivalent to a conjunc-tion of disjunctions of formulas of the form ¬B ∨ ¬2C ∨ 2D1 ∨    ∨ 2Dnin which B is 2-free, and for the case of A of modal degree 1, the formulasC,D1, . . . , Dn are also 2-free;3 we will write such ‘basic disjunctions’ in theimplicational form:(B∧2C) ! (2D1 ∨   ∨2Dn),in which the consequent is understood as ? when n = 0 and either or bothof the conjuncts in the antecedent may be absent.4 (If both are, we may think3Fine [2] is the locus classicus for normal forms in (normal) modal logic. Fine takes 3 asprimitive rather than2, and (essentially) works with disjunctive rather than (as here) conjunctivenormal forms.4We don’t need to write “2C1 ∧    ∧2Cm” in the antecedent, since by normality this sim-plifies to 2(C1 ∧    ∧ Cm). In terms of the disjunctive formulation, this is why we have just¬2C rather than ¬2C1 ∨   ∨ ¬2Cm.Rohan French and Lloyd Humberstone, “Partial Confirmation of the Boxdot Conjecture”, Australasian Journal of Logic (7) 2009, 56–61http://www.philosophy.unimelb.edu.au/ajl/2009 59of the antecedent as > or, equivalently, identify the conditional with its conse-quent.)Whether or not a given A is a first degree modal formula, if A is writtenas a conjunction A1 ∧    ∧ Ak with each Ai (1 6 i 6 k) a basic formula (asinset above), we define f(A) to be the conjunction of the formulas f(Ai) wheref maps Ai, to the formula:(s∧ B∧ C) ! ( (D1 ∨ s) ∨   ∨ (Dn ∨ s)),in which s is a propositional variable not occurring in Ai (i.e. not occurring inthe given B,C,D1, . . . , Dn), and the  -notation is as explained in Section 1.So defined, f(A) is not unique, since various different conjunctive normalformsVki=1Ai of A can make a difference, as well as the choice of s fromthe countable list of propositional variables (see the opening sentence of Sec-tion 1), though the first difference is inconsequential and the second could beironed out by choosing s for Ai as the first pj in that enumeration which doesnot occur in Ai. Rather than making any such moves, however, we can sim-ply take f(A) to denote an arbitrarily selected formula satisfying the definingconditions. 1 For any formulaA: `K A! f(A).Proof: It suffices to observe that whereVki=1Ai is a normal form for A, wehave `K Ai ! f(Ai), for 1 6 i 6 k. 2 For any formulaA of modal degree 1, if `K f(A) then `KT A.Proof: As in the proof of Lemma 1, it suffices to check this for eachAi. TakeAias above, i.e., as (B∧2C) ! (2D1 ∨   ∨2Dn), so that f(A) is (s∧B∧ C) !( (D1 ∨ s) ∨   ∨ (Dn ∨ s)). Our hypothesis is that this is provable in K, soweakening the “ ”s in the consequent to simple “2”s, and unpacking the “ ”in the antecedent (together with some re-arranging), we have:`K ((s∧ B∧ C) ∧2C) ! (2(D1 ∨ s) ∨   ∨2(Dn ∨ s)).Since A, and therefore Ai, is of modal degree (at most) 1, s∧ B∧ C is a 2-freeformula, so, using the fact that K is a (fully) modalized logic in the sense of [10],we can infer that either (a) `K ((s∧ B∧ C) ! ? or (b) `K 2C! (2(D1 ∨ s) ∨  ∨2(Dn ∨ s)). In case (a) we have `K B! ¬C (after substituting B or > fors), and therefore (since `KT 2C ! C) `KT B ! ¬2C, from which we concludethat `KT Ai. In case (b) we have, substitutingVni=1Di for s and simplifying:`K 2C! (2D1 ∨   ∨2Dn), so `K Ai and therefore again `KT Ai.Putting these ingredients together settles the conjecture of Section 1 inso-far as it bears on first degree formulas: 3 SupposeA is a first degree modal formula not provable in KT. Then KAproves some -formula not provable in K.Rohan French and Lloyd Humberstone, “Partial Confirmation of the Boxdot Conjecture”, Australasian Journal of Logic (7) 2009, 56–61http://www.philosophy.unimelb.edu.au/ajl/2009 60Proof: Let A be as described. We may choose f(A) as the desired  -formula,since it is provable in KA, by Lemma 1 (and Modus Ponens), but not provablein K, by Lemma 2 (contraposed).For the first degree KT-unprovable formulas A listed at the end of Section1, the formulas B given there as having B provable in K  A without τ (B)provable in K are minor variations on what the definition of f(A) would deliver.For example, for the case of A as 2? we gave as a candidate B the formulap ! 2p. Since this A contains no propositional variables, taking s as the firstvariable not occurring in A means that s is p (alias p1), and f(A) is thereforep !  (? ∨ p), which is provably equivalent in K to p ! 2p. More generally,since we may have to make substitutions, we can say for the proffered examplesof B that B and f(A) are K-interducible in the sense that KB = Kf(A).5 Thishappens in the case of A = p! 2p in which again we gave B as p! 2p, whilef(A) is (p∧q) !  (p∨q) and we need to substitute p for q before proceedingfurther.Returning to the conjecture as first formulated in Section 1, we have thefollowing: 4 If S = K  Γ for a set Γ of first degree formulas and S 6 KT, thenτ  does not embed KT faithfully in S.Proof: Given that S 6 KT, pick A 2 Γ r KT, and the proof of Theorem 3 showsthat f(A) is a  -formula in K  A r K. Thus we have (1) 0K f(A) while (2)`S f(A). Since f(A) is a  -formula, replacing every   in f(A) with 2 gives aformula A* for which f(A) = τ (A*), and (1) and (2) can be reformulated as(1 0): 0K τ (A*), (2 0): `S τ (A*). From the fact (recalled in Section 1) that τ embeds KT in K, we infer from (1 0) that 0KT A*. If it were also the case that τ faithfully embedded KT in S, (2 0) would imply that `KT A*: a contradiction.What is wanted, then, is a way of lifting the restriction to first degree for-mulas from Theorem 3 (and Corollary 4)—or else of showing, with a coun-terexample, that it cannot be lifted. Should the former turn out to be possible,perhaps the proof of Theorem 3 can be turned into the inductive step of aproof of the desired general result (by induction on the modal degree of for-mulas); however, we do not currently see how the details of such an argumentwould go.5Indeed, since the rule of necessitation is not required here, we could say more informativelythat K +B = K + f(A), where S + Γ is the smallest (not necessarily normal) modal logic extendingS and containing all formulas in Γ . In fact, everything said in Lemma 2 about KT applies to thequasi-normal modal logic K + T ( = K[T]0 from [9], p. 179), suggesting a variant on Corollary 4below in which the references to K Γ and KT are replaced by references to K + Γ and K + T, thequalification “first degree” is dropped, and reference to τ  is replaced by a one to the transla-tion τ0, where τ0(A) is the result of replacing only the outermost (i.e., not modally embedded)occurrences of2 inA by . We do not go into this in any detail because of the marginal interestof τ0 by comparison with τ .Rohan French and Lloyd Humberstone, “Partial Confirmation of the Boxdot Conjecture”, Australasian Journal of Logic (7) 2009, 56–61http://www.philosophy.unimelb.edu.au/ajl/2009 61R[1] B. F. Chellas, Modal Logic: An Introduction, Cambridge University Press,Cambridge 1980.[2] Kit Fine, ‘Normal Forms in Modal Logic’, Notre Dame Journal of Formal Logic16 (1975), 229–237.[3] E. Goris, ‘Interpreting Knowledge into Belief in the Presence of NegativeIntrospection’, Technical Report TR-2007005, CUNY Ph.D. Program inComputer Science, 2007.[4] Lloyd Humberstone,‘Béziau’s Translation Paradox’, Theoria 71 (2005), 138–181.[5] Lloyd Humberstone, ‘Weaker-to-Stronger Translational Embeddings inModal Logic’, pp. 279–297 in G. Governatori, I. Hodkinson and Y. Ven-ema, (eds.) Advances in Modal Logic, Vol. 6, College Publications, London2006.[6] Tadeusz Litak, ‘The Non-reflexive Counterpart of Grz’, Bulletin of the Sec-tion of Logic 36 (2007), 195–208.[7] F. J. Pelletier, ‘Six Problems in “Translational Equivalence”’, Logique et Anal-yse 27 (1984), 423–434.[8] F. J. Pelletier and A. Urquhart, ‘Synonymous Logics’, Journal of PhilosophicalLogic 32 (2003) 259–285; Correction: ibid. 37 (2008), 95–100.[9] Krister Segerberg, An Essay in Classical Modal Logic, Filosofiska Studier, Up-psala 1971.[10] E. E. Zolin, ‘Embeddings of Propositional Monomodal Logics’, Logic Jour-nal of the IGPL 8 (2000), 861–882.Rohan French and Lloyd Humberstone, “Partial Confirmation of the Boxdot Conjecture”, Australasian Journal of Logic (7) 2009, 56–61The Australasian Journal of Logic ( 1448-5052) disseminates articles that signifi-cantly advance the study of logic, in its mathematical, philosophical or computationalguises. The scope of the journal includes all areas of logic, both pure and applied totopics in philosophy, mathematics, computation, linguistics and the other sciences.Articles appearing in the journal have been carefully and critically refereed under theresponsibility of members of the Editorial Board. Only papers judged to be both sig-nificant and excellent are accepted for publication.The journal is freely available at the journal website athttp://www.philosophy.unimelb.edu.au/ajl/.All issues of the journal are archived electronically at the journal website.S Individuals may subscribe to the journal by sending an email, in-cluding a full name, an institutional affiliation and an email address to the managingeditor at ajl-editors@unimelb.edu.au. Subscribers will receive email abstracts of ac-cepted papers to an address of their choice. For institutional subscription, pleaseemail the managing editor at ajl-editors@unimelb.edu.au.Complete published papers may be downloaded at the journal’s website at http://www.philosophy.unimelb.edu.au/ajl/. The journal currently publishes in pdfformat.S The journal accepts submissions of papers electronically. To submitan article for publication, send the LATEX source of a submission to a member of theeditorial board. 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Partial Confirmation of a Conjecture on the Boxdot Translation in Modal Logic

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Partial Confirmation of a Conjecture on the Boxdot Translation in Modal Logic

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