COMPLETENESS O F SOME QUANTIFIED MO DA L LOGICS. James W. GARSON

Transcription

1 COMPLETENESS O F SOME QUANTIFIED MO DA L LOGICS James W. GARSON O. Introduction We give here simple completeness proofs fo r a wide range of quantified modal logics. The systems in question have the axioms a n d rules o f some standard system o f propositional modal logic, axioms f o r intensional identity, and quantificational principles drawn from free logic. The semantics we give is more general than usual, and perhaps that explains the simplicity of the completeness results. Usually a semantics f o r quantified mo d a l lo g ic in vo lve s the introduction of a function D which assigns to each possible world d the set D(d) of individuals which «exist» in that world. We may think of D as the intension of a corresponding existence predicate 'E' in the object language. In our semantics, we introduce a function Ill, wh ich assigns to each possible wo rld d a set W(d) o f individual concepts, o r world lines (i.e., functions fro m the set o f possible worlds to the set of individuals). In this wa y we agree wit h Montague's [5] and Bressans [1] more general treatment of the intension of a predicate in our description o f the intension o f 'E'. ( result, the te rm position in 'E' becomes opaque, and not subject to substitution of identites. ( ( and by Bressan [1] on p. 24 and following. ( semantical treatment of 'E' t o the systems of this paper. The only changes required inv olv e res tric tion of the substitution of identities t o t erm -places of predicates whic h are n o t giv en t he more general interpretation.

2 154 J A M E S W. GARSON 1. Fo rma l Motivation Wh y should we want to generalize the treatment of the in - tension of 'E' in this wa y? The first set of reasons are formal. When D is used, there are several ways of introducing the truth definition f o r th e quantifiers, a n d i t i s n o t e a sy t o decide which is best. () A t least one o f these results in a definition of validity which is not axiomitizable (system Q2). On another definition (system Q3 ) t h e completeness results a re h ig h ly sensitive t o t h e strength o f th e mo d a l operator. Strategies which wo rk (say) t o show th e completeness o f S4 strength quantified modal logics (fo r example those used to show completeness o f Q3-S4) are not applicable to the weaker systems such as Q3-M o r Q3-K. That difficulty does not arise with the more general interpretation o f the quantifiers w e g ive here. One simple strategy serves to provide results for any quantified modal logic whose modal fragment is determined by some nonempty class of Kripke frames, and fo r which a standard Henkin style completeness proof is available. In short, the methods of this paper yie ld simple completeness proofs f o r quantified versions o f th e va st ma jo rit y o f studied systems o f mo d a l logic. 2. I n f o rma l Motivation Are there any reasons other than formal fo r accepting the approach o f this paper? I believe there are philosophical in - tuitions which recommend it, particularly when the quantified modal logics at issue are tense logics. I t is generally felt that it is not part of the province of logic to determine which of the competing ontological theories is correct. I t follows that semantics f o r quantified modal logics should n o t ru le o u t a n y reasonable ontological theory. ( discusses s imilar matters f o r another intens ional logic.

3 COMPLETENESS OF SOME QUANTIFIED MODAL LOGICS 155 There is a venerable and pervasive ontology, which we might call ordinary onotology, o r Aristotelian ontology, which seems to require the approach we have taken to the intension of the existence predicate. I t is basic to this position that there are objects, and that objects change. So the formal counterpart of an object in a tense lo g ic semantics must be a time -wo rm o r individual concept, and not a slice o f a time wo rm, since i t makes no sense to say of the latter that it changes. In a tense logic then, a n object should be treated as a function f ro m times in t o temporal slices o f objects (o r point-events), a n d hence objects a re th e intensions, n o t th e extensions o f the terms. Now what does this have to do with whether the intension of 'E' should yield at time t a sot of individuals o r a set o f individual concepts? We kn o w that there are ma n y predicats P which are extensional, which means that the calculation of the truth-value o f 'Pn' (sa y a t t) depends so le y o n whether the time slice (individual) refered to b y 'n' at t fails in the extension of 'P' (at t), and does not otherwise involve the intension of 'n'. Couldn't it turn out that 'E' is one o f these extensional predicates? I think there are stong intuitions which rule against this and wh ich show that at least one rather basic treatment of existence i n temporal situations requires th e intensional approach. To see this, imagine what we would say (at t) i f a temporal slice (o f finite, b u t small duration, i f yo u like ) o f an object (say a slice of Gerald Ford) were to be present at t, in a context where the preceding and subsequent temporal slices o f Gerald Ford were absent. No w it is true that we might be in - clined to say that something existed a t t, namely a temporal slice of Gerald Ford, o r a fleeting «apparition» o f Gerald Ford. It would be strange, however, to claim that Gerald Ford existed at t under these circumstances, fo r part o f what makes us want to say that Gerald Ford exists a t t, is the existence o f previous and subsequent stages of Gerald Ford arranged in a certain coherent fashion. This result should not be surprizing. The ordinary ontology takes objects as basic, so questions about existence and non-

4 156 J A M E S W. GARSON existence of objects should be questions about objects and not about certain o f th e ir temporal parts. So i t fo llo ws th a t i f objects are to be treated as functions from times to times slices, then the referent o f 'Er a t a time t should be a set o f these sorts of functions, and not a set of time slices. 3. T h e predicate 'E' The predicate 'E' is included as a p rimitive of our systems. Its p rimit ive inclusion is important t o the generality o f the results. I n free logc, and in modal logics where the domain of quantification shifts from possible wo rld to possible world, the usual a xio m o f universal instantiation is invalid, since i t is possible f o r V xa t o b e t ru e a t wo rld d, a n d f o r A n i x t o be false when 'n ' denotes something outside th e domain o f quantification for d. If we conditionalize the consequent of this axiom, a va lid formula results, wh e n the rig h t condition is chosen. The condition we want is a formula wh ich expresses that 'n' refers to some «object» in the domain of quantification for d. I n free lo g ic ' 3 xx n' serves that purpose. However, this formula will not wo rk in an intensional logic which gives '--' th e intensional o r we a k interpretation o f identitiy, a n d which includes our approach to existence. That is because we now need a condition which says that the function refered to by 'n' falls into the domain 1 But ' 3 xx n' says something weaker, namely that there is a function g such that g(d) is the extension of n at d. It does not improve matters to replace ' 3 xx n' wit h ' x x = -n ', unless the u n d e rlyin g mo d a l syste m i s a s stro n g a s Szl. ( ' 3 xelx n' says o n ly that there is a function g T ( d ) wh ich agrees with the intension of n at those arguments in the range of the Kripke relation R at d. ( n n F r i ) bec omes v a l i d o n o u r semantics, i n s p it e o f t h e f a c t t h a t x 0 x n does not express that the func tion refered to b y ' n' at d falls in.411(d).

5 COMPLETENESS OF SOME QUANTIFIED MODAL LOGICS Rig id it y of variables One o f th e ma jo r simp lifyin g assumptions ma d e i n Q3, Thomason's [6] approach to quantified modal logic, is to assume that the intensions o f the variables a re constant functions, that is to say, that their extensions do not change f ro m possible wo rld to possible world. On the other hand, he allows the terms t o be non-rigid. Th is leads t o fo rma l and philosophical akwardness. The formal akwardness is that the domain of quantification becomes the set of constant functions, or as Thomason believes he should ca ll them, the substances. This invalidates the usual axiom fo r universal generalization. B y adding an instantiation condition H x x n', a valid axiom A4' V xad(3xdx ndanx) results, b u t o n ly when the underlying modal lo g ic is S4 o r stronger. I n logics o f weaker modality, A4 i s invalid, and it is difficult to specify the instantiation condition that is needed for a sound and complete system. Hin tikka [4] describes this matter in detail and gives the complicated axioms wh ich must be used. Thomason's refusal t o consider we a ke r modalities saves h im fro m considerable complications, and even so, the rules fo r his system Q3 are elaborate. Insisting on the rig id ity of variables leads to formal complications, b u t I doubt th a t i t i s acceptable o n philosophical grounds either. Thomason motivates his decision to «rigidify» the variables b y claiming that this allows us to incorporate a concept of substance into modal logic. In choosing our variables to be rig id designators, we are supposed to get the effect of quantifying ju st o ve r the substances. But wh y should we conclude th a t th e substances a re e xa ctly th e intensions o f rigid designators. It is at the ve ry least inconvenient to do so. First, the insistence that a term for a substance refers to the same individual in each possible wo rld immediately poses the problem of providing id e n tity conditions fo r individuals across possible worlds. Is i t sensible to assume that Plato (wh ich I

6 158 J A M E S W. GARSON take to be a substance for the present discussion) in this wo rld is exactly identical to the Plato of some other possible wo rld? Well, not unless Plato's identity does not depen don his accidental properties. But if trans-world identification is to depend only on essential properties, h o w can we be sure that there aren't possible wo rld s wh e re t wo substances (sa y Aristo tle and Plato) have exactly the some essence, and so are identical in that world, with the result that they must be ruled identical in all worlds? The inconvenience becomes mo re apparent when we g ive modal logics a tense interpretation. We would like to quantify over Gerald Ford as a member of our ontology, yet the extension o f 'Gerald Ford' is not constant across times, since each time slice of his has its own peculiar properties. Insistence on the myth that 'Gerald Ford' is after all a rig id designator provokes a fruitless search fo r «something» wh ich remains constant in all these time slices. The difference between substances a n d o th e r «things» i s undoubtedly too subtle to be captured via the simple notion of the rig id it y o f designators. I n fact, i n order to be fa ir to any system o f ontology, th e best th in g t o d o wo u ld b e t o make no assumptions at all about the semantical properties o f the terms that happen to refer to substances. We now find another reason for choosing the general approach of this paper. O u r existence predicate E has a set o f in - dividual concepts as its extension, and this set can serve admirably as the boundary between substances and non-substances for those ontologies which bother to make the distinction. Since there are no conditions on this set in our definition o f a model, any sort of ontology of substances may be accomodated. Since the substances so identified are not necessarily constant functions, the problem o f trans-world identification does n o t need t o bother us unless th e ontology in question requires such a concept. In this wa y we leave the problem to the ontologist, not the logician. Special logics designed t o capture assumptions about substance o r any other interesting sorts fo r quantification can be constructed b y putting appropriate conditions o n the seman-

7 COMPLETENESS OF SOME QUANTIFIED MODAL LOGICS 159 tical behavior of E in our definition of a model. So Thomason's approach can be reconstructed with in ours. If we want a logic which does n o t close ontological issues, however, we should choose the generality of the approach of this paper. 4. A xio ma t ic Features of MQ We w i l l demonstrate th e completeness o f a system MQ, written i n a morphology L f o r quantified mo d a l lo g ic wit h identity, wh ich includes the primative predicate constant 'E' for 'exists'. M Q contains the principles o f some propositonal modal logic M for which a standard Henkin-style completeness proof is available. (By standard, we mean to include that the definition o f the Krip ke relation R in the canonical model is given in the standard way: Rdc if f {A : 0 A E d }c c. ) I t includes as we ll axioms for intensional identity: (A=) t = t (AS) H (s = t D (F(s) F ( t ) ), where F(s) is atomic and F(t) is the result of replacing t fo r some occurrence o f s in F(s). A formula is atomic just in case it contains no logical constants. We consider 'E' to be a logical constant, so substitution guaranteed b y (AS) does not apply to 'E'. M Q also contains the following quantificational principles d ra wn f ro m free logic: (VG) H ( A D (Ex DB)) H ( A D V xb) (VI) ( V xa D (EtDAt/x)). At/x i s t h e re su lt o f replacing a n y t e rm t p ro p e rly f o r x wherever x appears in A. 'When l'' is a set of formulas, 'F H A' means that either A is provable in MQ, o r there is some conjunction G o f members of I ' such that H 31Q

8 160 J A M E S W. GARSON contradiction o f propositional logic. Let T be any set o f terms o f L, and le t F(T) be the set o f formulas o f the morphology like L save that i t has its o n ly terms the members of T. Let T b e the set of terms that appear in the set of formulas A set o f formulas A i s a MQ-se t if f f o r e ve ry formula A of F(TA) j. A 4 V x A E A. A MQ-set is ju st a saturated set with respect to the set F(TA) of formulas of the morphology which contains only terms found in A, where saturation is defined as is appropriate fo r free logic. MQ-set Lemma. I f F is a set o f formulas such that r and there is an infinite set of terms T of L, none o f which is in T F (T Proof: Simila r t o th e p ro o f o f the Lindenbaum Lemma. MQ-sets obey the usual properties o f saturated sets wh e n attention is restricted to the formulas o f th e ir corresponding morphologies. In particular P. Wh e n A is a MQ-set, the fo llo win g properties hold fo r all A, B F ( T 1. i f A 2. A E A iff A E A. 3. ( A D B ) E A iff A(4 A o r BE A. 4. V x A L if f for all terms tett,, ( E t D A t x ) L. 5. Semantics for MQ An M-model U fo r L is a quintuple < D, R, I, u >, where <D, R > is a member of the set of Kripke frames which deter- (5) Following Hans s on a n d Gardenfors [31 a K rip k e f ra me i s a p a i r < D, R > c ons is ting of a non-empty domain D (of possible worlds ) and a binary relat ion R on D. A propos itional modal logic M is determined b y

9 COMPLETENESS OF SOME QUANTIFIED MODAL LOGICS 161 mine M, ( which assigns to each d D a subset o f the set o f functions from D into I, and u is an interpretation function which assigns to each term t a function u(t) fro m D into I, and which assigns to each j-a ry predicate letter Pi (other than E) a function fro m D into the power set o f V. (We le t P null set, so that in the case o f a propositional variable 1: u (1 We define ' U d 0) U d Et iff u(t) (1)(d) 1) U d Pit 2) U s = t iff u(s) (d) is u(t)(d) 3) U d A iff not U d A 4) U ( A B ) iff not U A or U B 5) U 1 d V x A iffuf/x or fix A for all f U ( d ) (Ufix the model identical to U save that its interpretation function assigns f to x.) 6) U D A iff for all e I D, if Rde, then U e A set o f formulas F o f morphology L is MQ-satisfiable i n L just in case there is an M-model fo r L en wh ich each o f the members o f F is true. Valid ity is defined fro m satisfaction in the usual way. It is a simple matter to show by induction that S. U " (t) / 6. T h e completeness of MQ Let us define a canonical model U quintuple < D, R, I, 1 there is an infinite set of terms o f L not in T {A : D A c d} c, and iii. u(t)(d) is { s : s = t c.1}, and iv. u(pl)(d) a class K of Kripk e frames jus t in case f o r ev ery member < D, R > o f K, every model f o r < D, R > satisfies a ll and only the theorems of M.

10 162 J A M E S W. GARSON is t < u (t 1 ) is i fo r some term t of L and d ED, and vi. f k lf (d ) if f u(t) is f and EtE d for some term t of L. If follows from this definition that E. E t E d iff u(t)eu(d) Proof: The proof from le ft to rig h t is trivia l. No w suppose that u(t)eu(d). Then fo r some t', u(t) is u(t') and Et' Ed. But u(t) is u (t) only when t is t', for when t is not t' there is always a MQ-se t ei D wh e re t E T, a n d t ' OT,, w i t h th e re su lt th a t u(t)(e) is not 0, while u(t')(e) is. It follows then that Et Ed. We are not ready to prove The Model Lemma F o r every formula A and each d ED, if A E F(T The proof is by induction on the form of A. Cases for formulas having fo rms other than O B a re easy t o p ro vid e g ive n P. S., and E.. So we tu rn to th e case f o r formulas having the form OB. Suppose d E D and E I B E F(T Then (I) f o r all ced, if Rdc, then U B. Now le t d b e { A : D A E d I, and suppose fo r reductio that d. 1 set T o f terms foreign to c, and since the members o f d U { B} are all in F(T Now f o rm t wo in fin ite d isjo in t sets T T that d U { B} c e and e c F (T 1=1 El (9 T o do this, order Ti then let T of the ordering.

11 COMPLETENESS OF SOME QUANTIFIED MODAL LOGICS 163 of terms (namely T Bee. Since E )B e F (T By the hypothesis of the induction and B e F (T clude that not U, 1 3. So there is a member e o f D such that Rde and not 1 1,B, which contradicts (1). Ou r reductio is complete, and so we know that d 1 B. I t follows that either or there are formulas D A & A )n B. I n the first case D B, hence D B ed b y P., and in the second case it follows b y the principles o f M (wh ich is as strong as K) that ( D A, & & DA ) D DB. So d 1 DB, and by P., OBed. Now let e be any member of D and suppose that O B e d and Rde. Then Bee, and so B e F (T the induction and this last result to obtain U h that U ( 1. Theorem. MO is strongly complete. Proof. We show that any consistent set of formulas is MCIsatisfiable. Let I' be any consistent set of formulas in morphology L. Let L b e the morphology that results fro m adding to the terms o f L an infinite se t T o f terms foreign t o L. N o w divide T into two disjoint infinite sets T Lemma, there is an extension A of F such that A F ( T U TO. No w consider U that its Krip ke frame < D, R > is a member o f the class o f Kripke frames wh ich determines M, b y reviewing the appropriate portion of the completeness proof for M, since the definition given for R in U c Now there is a n in fin ite se t (namely T which are not in T Lemma, A e A if f U B y forming the model like U that its interpretation function is restricted to the terms o f L, we produce an M-model fo r L wh ich satisfies the set o f fo r- mulas o f A wh ich are written in L. Since F is a subset o f this set, F ust be MO-satisfiable in L. We have been somewhat detailed in our exposition o f this proof. Th a t is because we have ru n a considerable ris k o f failure in introducing saturated sets wh ich do not contain a ll

12 164 J A M E S W. GARSON the terms of our language. It was therefore important to show that the proof works in spite of the use of such improvershed sets, and so we paid particular attention to h o w mention o f the language plays a role in this proof. University of Notre Dame J a m e s W. Garson BIBLIOGRAPHY [1] A. BRESSAN, A G eneral Interpreted M o d a l Calc ulus, Y a le Univ ers it y Press (1972). [2] J. GARSON, Indefinite t opologic al logic, J ournal o f Philos ophic al Logic, vol. 2. (1973), pp [3] B. HANSSON and P. airdenfors, A guide t o intens ional semantics, i n Modality, Mo ra lit y a n d O t h e r Problems o f Sense a n d Reference, Lund (1973). [4] J. FlorrnaA, Existential and uniqueness presuppositions, in Philos ophic al Problems in Logic (ed. K. Lambert), Reidel (1970). 151 R. MONTAGUE, T h e p ro p e r t reat ment o f quantific ation i n englis h, i n Formal Philosophy (ed. R.H. Thomason) Yale Univ ers ity Press (1974), pp [6] R. THOMASON, Mo d a l lo g ic a n d metaphysics, i n Th e Logic al W a y o f Doing Things (ed. K. Lambert), Y a le Univ ers it y Press (1969).