No Future Author(s): Leon Horsten and Hannes Leitgeb Source: Journal of Philosophical , Vol. 30, No. 3 (Jun., 2001), pp. 259-265 Published by: Springer Stable URL: http://www.jstor.org/stable/30226728 . Accessed: 12/04/2011 18:24

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http://www.jstor.org LEONHORSTEN and HANNES LEITGEB

NO FUTURE

(Received16 October2000)

ABSTRACT.The difficulties with formalizing the intensional notions necessity, knowabil- ity andomniscience, and rational arewell-known. If thesenotions are formalized as predicatesapplying to (codesof) sentences,then from apparently weak and uncontroversial logicalprinciples governing these notions, outright contradictions can be derived.Tense logic is oneof thebest understood and most extensively developed branches of intensional logic. In tense logic, the temporalnotions future and past are formalizedas sentential operatorsrather than as predicates.The question therefore arises whether the notions that areinvestigated in tenselogic canbe consistentlyformalized as predicates.In this paperit is shownthat the answerto this questionis negative.The logicaltreatment of the notions offuture and past as predicatesgives rise to paradoxesdue the specificinterplay between bothnotions. For this reason, the tense that will be presentedare not identical to theparadoxes referred to above.

KEYWORDS: tense logic, tense predicates, diagonalization,

1. THE PARADOXES

Thederivations of theknown paradoxes concerning necessity, knowability andrational belief arediagonal arguments that resemble the derivationof the Liarparadox. These derivations all makeessential use of the T axiom -- scheme [p +o of . Montague's paradoxuses the axiom N; --+ a, where N is a necessity predicate and Wis the numeraldenoting the G6delnumber of the formulaap. Similarly, the so-calledparadox of theknower due to Kaplanand Montague uses the correspondingprinciple K -*- o, whereK is a knowabilitypredicate. Thomason's paradox about rationalbelief makes use of the principleRR(T --+ o), where R is a ratio- nal belief predicate.This latterprinciple is a weakeningof the T axiom. Thusone obtainsonly an "internal"inconsistency, i.e. one cannotderive I butonly RI. In the classicalsystems of (propositional)tense logic the notionsof futureand past areinvestigated. The language of propositionaltense logic contains,aside from the usual Booleanpropositional connectives and a stockof propositionalvariables, two sententialoperators G, H, of which

Journalof Philosophical Logic 30: 259-265, 2001. a 2001 KluwerAcademic Publishers. Printed in the Netherlands. 260 L. HORSTENAND H. LEITGEB

theintended interpretation is 'at every time in thefuture' and 'at every time in thepast', and their duals F, P whichmay be readas 'atsome time in the future'and 'at sometime in the past', respectively.The familiarsystems of tense logic do not containa versionof T as an axiom.It is therefore not initiallyobvious whether a diagonalargument can be set up which generatesa paradoxfor tenselogical systems in whichtemporal operators arereplaced by tensepredicates.' Nevertheless a paradoxcan be produced for reasonabletense whichtreat tense notionsas predicates.This is unfortunate,because the replacementof operatorsby predicateswould enableus to expressquantifications over an infiniteset of sentencesin a straightforward way, which would prima facie be an attractivefeature. Assumea first-orderlanguage X whichcontains unary predicates G, H, F, P such that G(H) is true of apif qpwill be (was) true at every momentin the future(past), F(P) is trueof apif -pwill be (was) trueat some momentin the future(past).2 Furthermore, ,d is assumedto contain the standardlanguage of arithmetic,so thatit cantalk and reason about its own syntax(via coding). Wewill considerthe following systems Kt andS, whichare formulated in X. K7 andS bothcontain the axiomsof predicatelogic with identity, and the axiomsof Robinson'ssystem of arithmeticQ.3 In Q, all recur- sive functionsare representable. Therefore Q can proveGbdel's diagonal lemma.4 Furthermore,K7 contains all instancesof the axiomschemes G1-3 and H1-3: (G G#), G1 Gp------H1 -+ -+ (Hi~ - GV), G2 a -+ HF-a, H2 p GP;5, -- G3 Gi "F-V , H3 H; -,P-,,p (forall y, / E ). S containsall instancesof G2, G3, H3 andthe schemeD*: D* H --+ P6 (for all pEE). L Apart from the axioms given, Kt and S have no other axioms. Both systemsare closedunder Modus Ponens and the followingtwo rules of : F-G R1 -ap=- 6, R2 Hp =- F-H (for all L). NO FUTURE 261

Thepropositional counterpart to K* is the minimaltense logic K,,5 whichseems unobjectionable. The propositionalcounterpart to S is a properfragment of K, plusa well-knownaxiom which is not implied by K,.6It says, loosely, that if somethinghas always been the case, then it hasbeen the case at some time in thepast. This statement does embody a substantialassumption concerning the structure of time,namely that time doesnot have a firstmoment.7 Insteadof employingaxiom schemes for K* andfor S, we mightas wellhave used single axioms quantifying over (codes of) sentences;but, as willbe seen,specific instances of theschemes above suffice to derive theparadoxes.

1.1. TheInternal Inconsistency of K7 SinceQ provesGddel's diagonal lemma, and Q is containedin K*,one canfind sentences a, P of X s.t. * K - aotHF-oa, * KI- Nowa andP areused to provethat no instantof timehas a futureor a past,8i.e.:

THEOREM1.1. K* F HI A GI.

Thereforewe saythat K* is internallyinconsistent. We devide the proof of Theorem1.1. intotwo subproofs:

LEMMA 1.2. K* F- HI. Proof In K*, supposefor reductiothat --a. Then--H F-"a. Therefore, by G2, a follows. Contradiction.So K* F a. Therebyalso (i) K* F- HF--a. But since Kt I- a, R1 yields GU and thus --F--,a by G3. But now R2 yields H--F--a. I.e., we have both HF-na and H--F--a, and thereforeK, I- HI. O

* LEMMA1.3. K - GI. Proof.For symmetry reasons (f is the temporalmirror image of a). O

The proofof the internalinconsistency of K* cannotbe strengthenedto a proofshowing the inconsistency of K7, sinceif theextensions of G andH 262 L. HORSTENAND H. LEITGEB areset identicalto a (andthe extensions of F andP areset identicalto 0), a modelof K* is triviallyobtained.

1.2. The Inconsistencyof S Usinga one obtains:

THEOREM1.4. S - 1. Proof Justas before,derive (i) S HF--,a, andalso H--F--,Fa.Finally, D*is usedto obtainP-,F--a, andwe have--H F--a by H3;therefore, with (i), S I. 0o

2. DISCUSSION OF THE PARADOXES

2.1. Strengtheningsof the Liar Paradox It is well-known9that adding the Tarskibiconditional scheme T;5 - Po to weak systemsof arithmeticyields an inconsistency.This is of course the formalizedversion of the Liarparadox. Essentially, the paradoxesof Kaplanand Montaguelo result from assuming only the left-toright direc- tion of the Tarskibiconditional scheme (the so-calledReflexivity Axiom T --+- a) outright.The conversedirection of the Tarskibiconditional scheme is weakenedby Kaplanand Montagueto a rule (the so-called NecessitationRule op/Tv). Inthis sense, their paradoxes can be considered as strengtheningsof theLiar paradox. Thomason" has in a sensestrength- enedthe Liarparadox even further.He has essentiallyshown that even if one also weakens the Reflexivity Axiom to T T -- p (and in addition assumes the TransitivityAxiom T- -+ T T-), an internalinconsistency follows. It is clearthat the "purepast" fragment Sp of Kt*U S, consistingof Q plus HI, H3, D* andR2 is consistent,and indeed internally consistent. It is easilyseen that Sp is a subtheoryof Friedmanand Sheard's system F, as it is describedin Sheard[8, pp. 1048-1049];F is knownto be consistent andinternally consistent.12 Analogously, the "purefuture" fragment of K* is consistentin any respect,even if an axiomanalogous to D* is added. But if Sp were strengthenedby the TransitivityAxiom HA - H H A, we would have an inconsistencyagain.13 McGee [5] has shown thatextending Sp by the Barcan formula, which can be seen as a formalized omega- rule, leads to omega-inconsistency.The temporalparadoxes discussed in this paperconcern systems that contain neither the transitivityscheme nor NO FUTURE 263 the Barcanformula. All theseinconsistency results can be consideredas furtherstrengthenings of the Liarparadox.

2.2. PhilosophicalSignificance of the TemporalParadoxes The perceivedphilosophical significance of Montague'sparadox about necessityderives from the fact that it concernsthe associatedpredicate system of a plausibleand time-honoredsystem of propositionalmodal logic. Montague'sparadox pertains to the predicatelogical analogueof Feys' systemT of modallogic. T was one of the firstsystems of modal logic to be axiomatizedand has remained one of themost basic systems of modallogic eversince. It is scarcelyan exaggeration to say thatT givesan exhaustivelist of theuncontroversial principles of modallogic, although it is of coursean extensionof the minimalmodal logic K. Similarremarks canbe madefor Kaplan and Montague's epistemic paradox. This epistemic paradoxconcerns the associatedpredicate logic of a plausibleand basic propositionalsystem for formalizingthe "absolute"notion of knowability. Withrespect to Thomason'sparadox concerning rational belief we want to expresssome reservations as to its philosophicalsignificance. For it is not immediatelyclear how the axiom R R -> qp("For any sentence apof the language,it is rationalfor X to believethat: if it is rationalfor X to believe po,then ap") can be convincinglyargued for as being intuitively valid. Now we see that in tense logic the situationis even worse than in modallogic. Ourresults show that the associatedpredicate system K, of Lemmon'sminimal tense logic K, is internallyinconsistent, whereas in the case of modallogic, at least K*, the associatedpredicate system of K, is consistent.The reasonfor this differenceis the higherdegree of expressivenesswhich the languageof tense logic has, comparedto the languageof usual modallogic. In tense logic, one cannotonly express in the objectlanguage some propertiesof the accessibilityrelation, as in the modallogic of necessity,but also some propertiesof the converseof the accessibilityrelation and of the relationbetween these two relations. Thisexplains why the logicalinteraction between the temporalpredicates is essentialto ourderivations of the paradoxes. Summingup, the distinctive features of the temporalparadoxes are: 1. K* is internallyinconsistent, whereas the predicateversion of T is inconsistent, 2. the temporalparadoxes do not completely parallel Montague's para- dox, 3. the temporalparadoxes do not affect the pure futurepart, or the pure past partof tense logic alone. 264 L. HORSTENAND H. LEITGEB

NOTES

1 In recentyears tense logical languages have been proposedwhich contain operators for which it is obviousthat if they are treatedas predicates,paradox ensues. For in- stance,one can investigatethe notion'until (and including) now'. Forthis notion,the T axiom schemeand the axiomsof the minimalmodal logic K seem clearlyvalid. Then the argumentof the paradoxesof Montagueand Kaplan can be invokedto yield a con- tradiction.An analogouspoint can be madefor the so-calledDiodorean tense modali- ties. 2 F and P mayas well be treatedas "defined"predicates, i.e. as metalinguisticabbre- viations. 3 Fora descriptionof the systemQ, see Boolosand Jeffrey [1], p. 107. 4 See Boolosand Jeffrey [1], Chapters17 and18. 5 See Prior[7], p. 176. 6 See Prior[7], p. 176,Axiom 5.2. 7 See Burgess[2], pp. 104-105. 8 Actually,we couldhave also used differentpairs of formulas,like ac' *+ GH--c', HG -"', for the same purpose. 9 See Tarski[9]. 10See [4] and[6]. 11See [10]. 12This is provedin Friedmanand Sheard [3], pp. 11-13. 13This is provedin Friedmanand Sheard [3], p. 14.

REFERENCES

1. Boolos,G. andJeffrey, R.: Computabilityand Logic, 3rd edn, Cambridge University Press,Cambridge, 1989. 2. Burgess,J.: Basictense logic, in D. Gabbayand F. Guenthner(eds.), Handbook of PhilosophicalLogic, Vol. II, Reidel,Dordrecht, 1984, pp. 89-133. 3. Friedman,H. andSheard, M.: An axiomaticapproach to self-referentialtruth, Ann. Pure Appl. Logic 33 (1987), 1-21. 4. Kaplan,D. andMontague, R.: A paradoxregained, Notre Dame J. FormalLogic 1 (1960),79-90. 5. McGee,V.: How truthlike can a predicatebe? A negativeresult, J. Philos.Logic 14 (1985), 399-410. 6. Montague,R.: Syntactic treatmentsof modality with corollaries on reflexion princi- ples and finite axiomatizability,Acta Philos. Fennica 16 (1963), 153-167. 7. Prior,A.: Past, Present and Future, Oxford University Press, London, 1967. 8. Sheard,M.: A guide to truthpredicates in the modernera, J. SymbolicLogic 59 (1994), 1032-1054. NOFUTURE 265

9. Tarski,A.: Der Wahrheitsbegriffin den formalisiertenSprachen, Studia Philosophica 1 (1935), 261-405. 10. Thomason, R.: A note on syntactical treatmentsof modality, Synthese 44 (1980), 391-395.

LEON HORSTEN CentrumLogica, filosofie wet. K.U. Leuven Kard.Mercierplein 2 B-3000 Leuven Belgium (E-mail:Leon.Horsten @ hiw.kuleuven.ac.be)

HANNES LEITGEB Dept. of Universityof Salzburg Franziskanergasse1 A-5020Salzburg Austria (E-mail: [email protected])