Aqvist Lógica Con Predicados Modales

LENNART AQVIST

MODAL LOGIC WITH SUBJUNCTIVE

CONDITIONALS AND DISPOSITIONAL

PREDICATES

1. INTR~D~JOTION

Modern Kripke-style semantics for modal logic in terms of possible worlds, accessibility etc. has given us a framework within which an ade- quate analysis and logic of subjunctive and counterfactual conditionals may hopefully be developed. The first published attempt to handle conditionals on the basis of such a semantics is , to the best of my knowledge, that of Stalnaker (1968); other important contributions are Stalnaker and Thomason (1970), and David Lewis (1970) (as yet unpublished). Among earlier treatments of subjunctive conditionals based on modal logic, we like particularly to emphasize the significance of Burks (1955), which fo- cusses on causal dispositional statements, and Burks (1951).

To facilitate comparison between the approach to conditional logic that we intend to work out in this paper and those of Stalnaker, Thomason, and Lewis, we quote the following passage from Goodmans celebrated (1947) :

A counterfactual is true if a certain COMectiOn obtains between the antecedent and the consequent. But as is obvious from examples already given, the consequent seldom follows from the antecedent by logic alone. (1) In the 6rst place, the assertion that a connection holds is made on the presumption that certain circumstances not stated in the antecedent obtain. When we say If that match had been scratched, it would have lighted, we mean that conditions are such - i.e. the match is well made, is dry enough, oxygen enough is present, etc. - that That match lights can be inferred from That match is scratched. Thus the connection we athrm may be regarded as joining the consequent with the conjunction of the antecedent and other statements that truly describe relevant conditions. Notice especially that our assertion of the counterfactual is nof conditioned upon these circumstances obtaining. We do not assert that the counterfactual is true ifthe circumstances obtain; rather, in asserting the counterfactual we commit ourselves to the actual truth of the statements describing the requisite relevant conditions. The first major problem is to define relevant conditions: to specify what sentences are meant to be taken in conjunction with an antecedent as a basis for inferring the consequent. (2) But even after the particular relevant conditions are specitied, the connection obtaining will not ordinarily be a logical one. The principle that permits inference of That match lights from That match is scratched. That match is dry enough. Enough oxygen is present. Etc. is not a law of logic but what we call a natural or physical or causal law. The second major problem concerns the deilnition of such laws (p. 16f).

Journal of Philosophical Logic 2 (1973) l-76. All Rights Reserved Copyrkht Q 1973 by D. Reiakl Publishing Company, Dorrlecht-Holkznd

2 LENNART AQVIST

A common idea on which our approach as well as those of Stalnaker and Thomason, and Lewis are based seems to be the following (disregarding divergencies as to details): let K be some set of possible worlds (situations), let k,, be a member of K representing the actual world, let P (k,) be the set of those worlds in K that are possible (in some requisite sense) with respect to k,, and let A ( CP (k,) E K) be the set of those worlds in P (k,,) where the antecedent (say, That match is scratched) is true. Defined on P (k,,) is a preference ordering 2 which is intended to reflect some relation is at least as similar to k, as. Let f(A) (CA) be the set of those worlds in A that are R-maximal in the sense of being most, or perhaps just su$iciently, similar to k,. In the Goodman example under discussion, one may reasonably assume that circumstances like the match being well made, dry enough, etc. obtain in every world belonging to f(A). Now, if B is the set of worlds in P (&) where the consequent (say, That match lights) is true, then Goodmans counterfactual is said to be true in k,, iff f(A) E B; and false in k, otherwise.

If we accept this account as at least heuristically plausible, how are we best to generalize it into a wieldy logical theory of conditionals? Both Stalnaker and Lewis suggest enriching some language of classical propositional or predicate logic with a primitive subjunctive-conditional-forming connective (say,+) and laying down a condition for the truth in a model of=> -sentences in the spirit off (A) E B above. Different assumptions may then be made concerning the choice function f, giving rise to different systems of conditional logic. Also, in such a language, certain one-place modalities of necessity and possibility turn out to be definable in terms of =2-.

In spite of being basically in sympathy with this approach, of course, we feel that there are certain definite advantages to be gained from somehow reversing the procedure just outlined, for reasons bound up both with general methodology and with applications. Thus, we suggest taking as our starting point a language with certain primitive one-place modalities (of necessity), the interpretation of which can be left open for the moment. Let us then introduce into that language a primitive logical operator (say, *) that matches the choice-function f in our semantical models; a formula *A may be read, tentatively, as A under the circumstances, A other things being equal, A ideally, A relevantly, or something of the sort. Finally, we then define the conditional connective *in terms

MODAL LOGIC WITH SUBJUNCTIVE CONDITIONALS 3

of *, some requisite available modality of necessity, and material implication. Returning to our match-example, we could formalize the Goodman conditional by 0 (+A-+ B) (abbreviated A*B), where A is that match is scratched, B is that match lights, where + expresses material implication, where *A is true in exuctZy those worlds that belong to f(A), and where(is some modality, say, of causal necessity such that any formulaUC is true in the actual world k, iff C is true in every world belonging to the set P (k,). In this formalization, then, the job done by * is to capture the relevant circumstances supposed to obtain in k, and somehow to remain constant when A is considered; and the job done jointly by q and+ is of course simply that of expressing the subset-relation as restricted to the power-set of P (b).

A nice thing about the suggested approach is that it enables us to make a fairly direct assessment of Goodmans own way of formulating the problem of counterfactual conditionals in the truly brilliant passage quoted above. He takes the problem of relevant circumstances to be that of speci- fying what sentences are meant to be taken in conjunction with an antecedent as a basis for inferring the consequent. This attitude may very well involve an overestimation of the available linguistic resources, though: applied to the match-example, it presupposes that the set f(A) must be capable of being characterized as the set of those worlds where a certain conjunction is true, viz. one having the sentence A as a conjunct and whose other conjuncts are all extensional (?), Perhaps f(A) is such that it resists any such characterization; if we use a language of the kind just outlined, it may very well turn out that the onZy way of characterizing f(A) is by applying to the sentence A some special logical operator like our *. We must emphasize here that our use of * makes the reference to obtaining circumstances not stated in the antecedent indirect or indexical: we do not explicitly state what the relevant circumstances are on the basis of which we athrm the counterfactual, nor do we even, in our opinion, state explicitly that there are relevant circumstances serving as such a basis. What the indexical reference to circumstances amounts to, more positively, can be nicely explained along lines indicated by Stalnaker (1968): just as in the semantics for classical predicate logic we give the quantifiers a single meaning and make the varying domain of discourse a parameter of the interpretation, we give the *-operator a single meaning and make the varying choicefunction a parameter of the interpretation. (If this sounds

4 MODAL LOGIC WITH SUBJUNCTIVE CONDITIONALS

mysterious to any reader, he should just consult our subsequent develop- ments to grasp what is meant.) Also, note Goodmans careful wording in this connection: he says that the assertion of a counterfactual is made on the presumption that certain circumstances etc. We take his use of this quite appropriate locution to be indicative of his awareness that the reference to circumstances may somehow be indexical in the sense just explained.

We heartily agree with Goodmans important contention that we do not assert that the counterfactual is true if the circumstances obtain; rather, in asserting the counterfactual we commit ourselves to the actual truth of the statements describing the requisite relevant conditions. In Section 6 below, we shall try to pinpoint and do justice to this observation in dealing with certain extensions of our rather weak basic logic of * to be developed in Sections 2-5.

As for the famous problem of natural or causal laws (as opposed to merely accidental generalizations) announced at the end of the quoted Goodman passage, we fully agree with the assessment made by Stalnaker and Thomason (1970) :

We have talked about laws of nature, but we might instead have talked directly about the relationships expressed by laws. One of the advantages of the semantic approach is that it allows us to focus directly on the world rather than on linguistic expressions purporting to describe it. One might, for example, give an analysis of explanation in terms of possible worlds and s-functions (i.e. selection or choice functions) rather than indirectly in terms of laws. This way, one could avoid the embarrassment of unstated or unknown laws, and one might tind an analysis which handles more plausible the every- day cases of explanation.

It is now high time for us to tell what is the program for the present paper. In its first major part, we present a propositional modal logic A with circumstantials (i.e. sentences of the form *A) within which connectives for subjunctive conditionality are definable in the suggested way. The axiomatic version of this system will be proved sound and complete with respect to the semantics or model-theory laid down for it. Two things are of great importance as far as the technical development of the theory is concerned :

(i) In the semantics for A, the basic set-theoretical entities will be certain structures calledframes; like Kripke (1963), we take a frame to con- sist not only of a set K of possible worlds together with an accessibility relation R on it, but also of a token k, (Kripke: G) for the actual world.


Now, it is well known that as long as we are dealing with so-called normal modal logics (like T, S4, S5, etc.) as opposed to nonnormal ones (like S2, S3, etc.), we do not really need this guy. However, in spite of the fact that all our A-modalities apart from * are normal, it is precisely the *-operator that requires us to include k, in our frames. Furthermore, k,, plays a crucial role in the elaboration of the quantificational extension QA of A to be studied in the second major part of the paper : the announced notion of transworld identity in QA is based on the presence of k, in frames.

(ii) The big problem to be faced when one attempts to construct a viable logic of * is of course the following: what constraints are we to impose on choice-functions f ? Now, it is quite clear to me that the correct answer to this basic question is implicit in Bengt Hansson (1968) (of which a mimeographed version appeared already in (1967)). Assume that the preference relation 2 is a weak ordering of P (k,,), i.e. that 2 is transitive and strongly connected in P (k,,), and let Expr be the set of all those subsets of P(b) that can be characterized as truth-sets or extensions of sentences in our language. Then our Hansson-inspired answer amounts to the require- ment that the pair (Expr, f) is to be a choice-structure on P (k,) in a sense closely related to the one discussed in his paper (in particular, see Section 3 below). It is almost obvious that if k, as interpreted above, is taken to be a weak ordering of P (kJ and if the choice-function f is interpreted accordingly, then f will satisfy the axioms laid down for it in Section 3. Some doubt may be expressed, though, as to the validity of our axiom (3) to the effect that for each non-empty member A of Expr, f(A) is non- empty as well (we register that Lewis is worried about this in his (1970), as I was in my (1969)). Apparently, the misgivings are due to ones taking 2 to be of high discriminating power and considering intiuite As having no element that can be top-ranked according to 2. As Hansson (1970) points out, however, we may assume only that every non-empty A has a member on some minimum level and think of f(A) as the set of those elements in A that are on or above that minimum level (that are suffi- ciently similar to k,, as we said above). This means that another preference-ordering replaces the original one and differs from it only in that levels on or above the minimum one are identified. Thus, we wish to protect axiom (3) by not assuming in general that the k-ordering is one of terribly high discriminating power.

6 LENNART AQVIST

The following is a more large-scale objection which might be brought against the very basis of our approach; in our opinion, it illustrates rather persuasively a current misconception of the capacities of philosophical logic. We have suggested founding the logic of circumstances on some relation of similarity to the actual world k,; but this proposal, it is claimed, is entirely useless as long as we are not told in what respect the worlds in f (A) are more like k, than those in A - f(A). For instance, are we dealing with similarity to k, in as much as the intrinsic circumstances (enduring properties, internal states) of objects in k, are concerned (as has often been suggested in the case of dispositional statements, see e.g. Burks (1955))? or rather with similarity as to extrinsic ones, of a more tran- sientcharacter? or perhaps both, somehow weighted against eachother, or what? The answer to this supposedly devastating objection is simply that, although it be admittedly relevant to applications of the logic of circumstances here defended, we need not worry about it when construct- ing that basic logic itself: as far as this latter task is concerned, the only thing that matters is the fundamental abstract pattern of a weak order relation and a resultant choice-structure, which must be common to all models of our language. The distinctions and refinements to which the objection appeals then become relevant in special contexts of application, where they may give rise to different superstructures over this basic pattern.

We should add that our logic of * admits of various interpretations besides the one here proposed. A good case in point worth mentioning is the logic of dyadic conditional obligation given in Hansson (1969), from which my own (1970) profited considerably.

As we have already intimated, the second main part of this paper intro- duces a quantificational extension QA of the propositional modal logic A, and proves it to be sound and complete relative to the semantics set forth for it in Section 8 below. The major novel feature of QA (in addition to circumstantials and conditionals) consists in its capacity to formalize various notions of transworld identity by means of a single one-place intensional (modal) predicate Q. Moreover, the system seems to combine basic simplicity and naturalness with the expressive power needed to do justice to many valuable suggestions given in the literature as to how the well known difficulties in quantied modal logic are to be handled. As one cannot judge of this until the detailed technical development of the theory is completed, closer discussion of QA from a general viewpoint will be


postponed to Section 14, where comparisons will be made with the modal quantification theories of some outstanding logicians in this area of in- quiry.

Nonetheless, a few remarks on the general idea behind QA are perhaps not out of place here. It seems to me that the tist really crucial problem to be faced when one starts doing quantified modal logic is the celebrated Morning Star Paradox and its cognates, and that further elaborations depend in a decisive way on which solution is adopted to this puzzle; for one thing, this means that we tend to view the much emphasized free logic-aspect as somewhat less important than the Morning Star one, at least at the outset, without wanting, of course, to deny its obvious relevance. (Note incidentally that the Morning Star puzzle does not con- front us until we try to incorporate identity in our stystem; on our view, modal predicate logic without identity is a largely uninteresting business.) Now, the only acceptable solution to this puzzle that I can think of consists in adopting a semantics where assignments of individuals as denotations or values to singular terms (be they individual constants or variables) are made relative to the members of some system of possible worlds, like our K above; also, assignments to predicate letters must be similarly relativized. Thus, if K and k, are as in our previous example, if k is a possible world in K distinct from k,, if a, b are individual constants in our language (the Morning Star, the Evening Star, if you like), and if V(a, k,,), V(b, k,), V(a, k), and V(b, k) are individuals assigned to a, b relative to k, and k by some binary valuation-function V of the kind recommended, then the following might well be the case: V(a, k,,)= =V(b, k,) so that the formula (a = b) is true in k, relative to V; whereas V(a, k) #V(b, k) so that (a = b) is false in k relative to V, whence q (a = b) is false in k, relative to V for appropriate necessities 0. This explanation of Morning-Star-type situations appears indeed to work nicely enough.

Now, the price we have to pay for this solution is quite different in nature from what it is ordinarily taken to be. It is a sudden increase in and need for notions of identity, and it seems to me that all known languages of modal predicate logic, as they stand and as a plain matter of fact, simply lack the syntactical resources for expressing many of these identity relations. Their expressive power is in effect limited to what can be said in terms of = and available modalities, i.e., broadly speaking, we can express in them horizontal identities of the general format: V(a, k) =

8 LENNARTAQVIST

= V(b, k) either for some particular k in K, or for each k in some subset of K. But how about vertical identities of e.g. the type: V(a, k)= =V(b, k) for each k in K bearing relation R to k? or V(a, k)=V(b, k,,)? or simply V(a, k)=V(a, k,,)? In order for such relations of truly transworld identity to be expressible in modal languages, the latter have to be enriched with some new logical machinery. To that purpose we have introduced in the system QA the predicate Q obeying the truth-condition : a formula Qa is true in world krelative to valuation V iff V(a, k) =V(a, &), where, as we recall, the token k, is fixed in any frame (and model). Let us see what can be done in terms of this predicate, how it associates with ordinary identity, quantification and modality etc., and postpone further informal discussion to Section 14.

So much for the general motivation of QA. In a final section, we intend to introduce into it certain devices for forming dispositional predicates as well as some further items of philosophical concern.

THE PROPOSITIONAL LOGIC A

2. LANGUAGEOPA

Our alphabet consists of (i) denumerably many proposition letters,

(ii) brackets, (iii) two categories of primitive logical constants, viz.

(a) classical (or Boolean): I (for absurdity) and + (for material implication) ; (b) modal : q (for universal, or perhaps, analytic necessity), q (for what we call k-necessity, or distinguished necessity), 0 (for some third, as yet unspecified, brand of necessity), and * (for maximality, or ideality).

The set of wellformedformulas of A, (kwffs, or wffs) is defined as the smallest set S such that

(i) every proposition letter is in S, (ii) I is in S

(iii) if A, B are in S, then so are (A + B), HA, q A, @A, and *A. Symbols for negation (-), conjunction (&), disjunction (v), and

material equivalence (w) are defined as usual; three possibility-operators 0, @, and 0 are respectively defined as - q - , - a-, and - a-.

Also, we introduce into the language of A a subjunctive-conditional

MODALLOGICWITHSUBJUNCTIVECONDITIONALS 9

operator =S by the delmition:

where A =S B is read as if it were the case that A then it would be the case that B. Its dual, defined by -m r[ (*A+ -B), i.e. @ 0 (*A&B), could perhaps be read as it might be the case that B when A, where the when conceals some idea of conjunction rather than one of implication.

3. SEMANTICSFORA

A frame is a structure F = (K, k,,, R) where (i) K is a non-empty set (heuristically, of possible worlds).

(ii) k,EK (a designated member of K, heuristically, the actual world). (iii) RS K x K (a binary relation on K, heuristically, of alternative-

ness or accessibility) A vaZuution (of the A-language) on a frame F = (K, k,, R) is any binary

operation V such that for each proposition letter P and for each koK,

is defined and is either 1 (truth) or 0 (falsity). A choice-function on a frame F = (K, k,,, R) is a function f where Domain (f) cB{koK: k,Rk) (the power-set of the set of R-alternatives

to I Range(f) EB{koK: k,Rk}

(We shall return to and be more explicit about this notion in a moment). An A-quasi-model is an ordered triple M= (F, V, f) where F=

= (K, k,, R) is a frame, V is a valuation on F, and f is a choice-function on F. We now define the notion of truth of a wflA at a point (world) k in an

A-quasi-model M (in symbols: F k

A). Given any M= (F, V, f) with

F = (K, k,, R), then, for all k in K:

(1) !k! P iff V(P, k)= 1 (forall proposition letters P).

10 LENNART;~QvIST

(3) -i;(A+B) iff if r

FA then FB

(4) FmA iff PA foreachkinK

(5) /$& ifl EA

@I ED iff F A for each k in K such that kRk

We adopt the following notational convention: if A is any wff and M is any quasi-model, we set

llAl/ = {kczK: k,,Rk and k A}. /M

In clause (7) above, we assume, of coirse, that f is dejinedfor l/All. In order to guarantee that this is in general the case, we only have to parallel clauses (1) through (7) with inductive stipulations as follows: f is defined for 11 PII M, for each proposition letter P, as well as for II I II M; for all wffs A, B: if f is defined for /jAll and for ~~B~~~, then f is defined for I](A+B)\]~,

IIW4I~ IIE141~ lln4I~ and f-or II*41~ Next, for any quasi-model M, we define the set of expressible subsets of

(kEK:k,Rk) as the collection of all sets IjAIl where A is any wff. In symbols,

Expr={IIAIIM:Ais a wff).

Clearly, Exprr is always a subset of B{koK: k,,Rk), and f is defined for every member of Expp.

Again, an A-model is a triple M = (F, V, f) with F = (K, k,, R), where (i) M is an A-quasi-model (ii) f is a function from Expp into Exp?

(iii) The pair (Expp, f) is a choice-structure on {koK: k,Rk) in the Hansson-inspired sense that for all wffs A, B:

(0 If IIAll = ll~ll then f llAllM = II*BII


(2)

(3) (4.1) (4.2)

fllAll c PII f IlA]j = 0 only if lj~ll = 0

fn ~~B~~ = fllA&BnM If f(llAll) n IlBll#S, then fjA&BIIM c f(l(AI(% IlBll.

Evidently, conditions (4.1) and (4.2) in this definition can be lumped together into a single condition:

(4) If f(llAII)n ]IBll#S, then f(llAII)n ~~B~~ =fllA&BII.

A set S of wfG is said to be (simultaneowly) &satisfiable, iff, there is an

A-model M and a member k of K such that for all wffs A in S, F k A.

A wff A is said to be A-valid (in symbols: t A), iff, { -A} is not A-satis-

fiable, i.e. iff F

A for all A-models M and for all k in K. Note that this k

criterion of validity (satisfiability) is stronger (weaker) than the old

Kripke one which required instead: A for all M (there is an M such

that for all A in S, E

E 0 A).

Finally, let us pay some attention to the following question: can the modalities q and q be characterized in terms of suitable alternativeness relations on K, and their truth-conditions be accordingly formulated in the style of (6)? The answer is of course a&rmative: in the case of a, the corresponding alternativeness relation is simply K x K, in that of q , the relation is {(k, k) : koK and k= k,}, i.e. that constantfunction whose value is uniformly k, for all arguments koK. For the purpose of our subsequent completeness proofs, it is now advisable to bear the following fact in mind :

Let R,,sK x K be any relation on K. Then R, satisfies conditions (rl)-(r4)belowifF%=(:keKandk=kJ.

0-1) W&o WI For all k in K there is a k in K such that kR,k (R, is serial in

K)

12 LENNART iiQVIST

63) For all k, k, k in K, if kR& and kR,k then k= k (RO is functional in K)

b-4) For all k, k, k in K, if kR,k then kR,k

(in the presence of (r2)-(r3), this condition makes for the constancy of R, in K).

The proof of this fact is obvious. As a corrollary we then obtain the following alternative truth-condition for wffs of the form (OlA:

(5) E aA iff EA for each k in K such that kR,k,

where R, is any relation on K satisfying (rl)-(r4). Thus, (5) and (5) pro- vide equivalent characterizations of q .

4. THE AXIOMATIC SYSTEM A

The system A is determined by the following rules of inference and axiom schemata :

Rl. A, A+B

B

R2.

AO. Al. A2. A3. A4. AS. A6. A7. A8. A9. AlO. All.

All tautologies


A12. *A+(mB-+B) ~.13 ~Y~((A-B)-, I,~(*Aw*B) A14. *A+A

A16.1 (*A&B)-+*(A&B) A16.2 q O) in S such that B1 &...&B,+A is A-provable. Again, S is &consistent iff Sfl; otherwise, S is A-inconsistent, i.e. iff Sl-l.

Next, we say that a set S of wffs is k-saturated (or maximal A-consistent), iff, S is A-consistent, and for each wff A, either AES or - AES (negation-completeness of S). In the sequel, we appeal inter alia to the foIlowing obvious properties of any A-saturated set S:

(i) S is closed under detachment (if AE S, and A + BE S, then BES) (ii) Every A-provable wff is in S (iii) -AES iff A#S (iv) A+BES,iff,if AES then B&3.

5. SEMANTIC SOUNDNESS AND COMPLETENESS OF A

The main result of this section is the following THEOREM 1. Let S be any set of A-wffs. Then the following condi-

tions are equivalent :

(1) (2)

S is A-consistent There is an A-model M = (F, V, f) with F = (K, k,, R) such

M that for some kin K,

t ; A for every wff A in S; in other words,

S is simultaneously A-satisfiable.

To see that condition (2) of the theorem implies condition (I), we ob-

14 LENNART /JQVIST

serve that schemata AO-A16.2 are all valid and that Rl and R2 preserve validity. Hence, it is readily verified by induction on the length of proof that for all A, if t A then != A. Suppose then that some S meets (2) but fails to meet (l), so that there are B,, . . . . B, in S such that F B, &...&B, -1. Hence, ,= B, &. . . &B, 41. But this means that for some point k in the model M, whose existence is guaranteed by S meeting (2), we have

r -k-B,&...&B, as well as .-k

F - (B, &. . . & B,,) - a flat contradiction. Hence

the desired result. As for the proof that condition (1) implies condition (2) - strong seman-

tic completeness of A - we begin by considering the set 1 of all A-saturated sets, and let R, be the relation on Z defined as follows : for all S and S in Z, SR,S iff for each wff A, if q AES then AES (or, if you prefer, (A:(uAoS}G). N ow, it is a familiar fact that the presence of all instances of A2 in every member of .?Y (c$ saturation property (ii) above) makes R, reflexive in X, that the presence of A3 makes it transitive in X, and that the presence of A4 makes R, symmetric in Z. Hence, R, is an equivalence relation on Z which partitions I; into disjoint equivalence classes (as usual, the R,-equivalence-class of S in Z will be denoted by C%>.

Again, let S be any A-consistent set of wffs. By an obvious adaptation of Lindenbaums Lemma to modal logic, there is an A-saturated set S+ EX with SE S+ (see e.g. Makinson (1966, p.381f)).

We then define the canonical A-model generated by S as the structure

where M= ( (K, km Rh V,f > ,

(i) K=[S+],U={S~C:{A:~A~S+)cS) (ii) k,, = the unique k in K such that {A:mAoS}sk (see below) (iii) R = {(k, k):k, koKand {A:Dok}ck} (iv) V = the valuation on (K, k,, R) defined as follows: for each prop-

osition letter P and for each ko K, V(P, k) = 1 iff Pok (v) f = the function defined as follows: for each wff A,

f((keK:k,,Rk and Aok)) = {koK:k,,Kk and *Aok}. Notational convention. In connection with M, we set

lAlM = {ksK:km and AE k} (for each wff A)


and SpeP = {/AI:A is a wff) .

Thus, according to (v),fis a function from SpecM into itself. Now, in order to complete the proof of Theorem 1, we must establish

three lemmata: LEMMA 1. As defined, (K, k,, R) is a frame.

LEMMA 2. For each wff A and for each k in K, F

AiffAEk. k

LEMMA 3. The pair (Expp,f) ( = (Spe@J) by Lemma 2) is a choice structure on {kEK: k,,Rk).

Let us first consider how these lemmata together yield the result that condition (1) of Theorem 1 implies condition (2). Lemma 1, Lemma 3, and clause (iv) in the definition of M clearly imply that M is an A-model. Then,

A4 by Lemma 2, we obtain in particular that for each wff A,

M

1

I= S+ A itIA&.

Hence, since S E S , we have S+

A for every A in S. In other words, assum-

ing S to be any A-consistent set of wffs, we have exhibited an A-model M,

viz. M, such that for some k in M, viz. S+, F k

A for each A in S, as desired.

To establish our lemmata, we need the following distinctively modal Saturation Lemma for A. Let S be any A-consistent set of wffs and

let M be the structure ( (K, k,, R), V, f) as defined above. Let R, E K x K be the relation {(k, k): k, k~Kand {A:(ok}sk}. Then K is such that for all wffs A and kf K:

(4 w Ek iff A& foreachkinK (b-l FJ Ek iff A& for all kin Ksuch that kR,k

(4 p Ek iff AE~ forallkinKsuchthatkRk

Proof. Case (a). The only if part is obvious by the definition of K. As for the

if part, suppose that w#k. Hence, by saturation property (iii), 43 Ek. Then, the set k* = {B:mEk} u { -A) is A-consistent by an application to A of Lemma 3 in Makinson (1966, p. 382) ; Al and R2 are utilized in the application. Now, kA has an A-saturated extension (k,,)+

16 LENNART AQVIST

with B~(kp3+ for each wff B such that q Bek, and we know k to be in K. Thus, since K= [SlR, = [k-JR, and kR,(kJ+, we have (kJ+EK and -WkJ+, since (kJ+ is saturated and contains -A as a member. Hence, there is a k in K, viz. (kJ+, such that A$k, our desired result.

Case (b). The argument parallels the preceding one with the following alterations. In applying Makinsons lemma to show the A-consistency of

kA = {B:lBEk} u (-A}, A

we appeal to A6 and the rule __ lolA

which is

readily seen to be derivable from R2 and A5, using Rl. zo, we must show that (k,J is in K, which is obviously the case iff kR,(kJ+. Well, since by A5, UA-) @Aok for each A, we get (B:mBEk)G(B: :aB o k} = k, s (kd)+, whence kR,(k,J+, as desired.

Case (c). In applying Makinsons lemma to kA, we appeal to Al 1 and

to the easily derived rule -

AlO. ;A*

In showing (kJ+ to be a member of K, use

The proof of the Saturation Lemma is complete. Proof of Lemma 1. The non-trivial task is to justify the definition of k,

(clause (ii) in the definition of M>. In view of our remarks on the truth- conditions (5) and (5) in Section 3, we gather that our definition of k, will be justified iff the relation R, = {(k, k):k, koKand {A:mAEk}=k) satisfies conditions (rl)-(r4) with respect to K.

To begin with (r2), we see that the presence of every instance of A7 in every member of 1y makes R, serial in K (notice that clause (b) in the Sa- turation Lemma must be relied upon in this case).

As for (r3), suppose that RO is not functional in K, i.e. that for some k, k, k in K we have kR,k, kR,k, but k #k. Now, the last condition means, of course, that for some wff A, A& while A $ k, whence -A E k. Appealing to an equivalent definition of R,, as {(k, k):k, ~EK and {@A:Aok) E k}, we conclude from the first two conditions, respectively, that @Ao k and that a- Ao k. By A8, @A + w is in k, whence DA and -@-A are in k. So @-A$k, which gives us a contradiction.

Next, suppose that R, fails to satisfy (r4), i.e. that for some k, k, k in K we have kR,k but not k%k. Thus, for some wff A, @Aok but A#k.ByA9,(A+-,b/Aok, h w encem(lAok.Sincek,kEKsothat kR,k, we obtain (olAok, whence, because k&k, Aok: contradiction.

As for (rl), finally, since S+R&,, by the definition of k,, we conclude


from the fact that R, satisfies (r4) that k&k,. We may also verify this result in the following way: we are to show that for all wffs A, if )olAEkk,, then Aok,. Now, we are readily seen to have m(bb + A) as an A-thesis and as a member of S+, for each A. Then, since S&k,, aA + AEk, for all A, whence the desired result is immediate by saturation property (iv).

Knowing that RO satisfies (rl)-(r4), we have the following R,,-k, result: & = {(k, k):k, koKand(A:@AEk)~k) = {(k, k):koKandk = k,}.

Lemma 1 is obvious in view of this result. Proof of Lemma 2. By induction on the length of A. Basic step: A is either (a)l, or (b) some proposition letter P.

(a) k F

I and 1$k (for each koK; since every k&is A+

consistent)

(b> By the definition of V.we have F k

Piff V(P, k)=l iffPEk.

Induction step: A is either (a) q B, or (b) HB, or (c) q B, or (d) *B, or (e) B + C (for some wffs B, C). Lemma 2 is assumed to hold for B, C and all k in K.

(a) A is q B. The case will be clinched if we establish the following chain of equivalences : ~~J%iK F B for each k in K, iff, B& for

each k in K, iff, q BE k. But the first iff is true by definition, the second is given by the inductive hypothesis, and the third iff is simply Case (a) of the Saturation Lemma. Thus, we are done.

(b) A is o B. The following chain of equivalences suffices for this case:

r s i; q B, iff, k B, iff, Bok,, iff, BEk for each k such that kR,k, iff, q Bo k. Of theie iff s, the fist one is again true by definition, the second is given by the hypothesis of induction, the third by our R,,-k,, result, and the fourth iff is just &se (b) of the Saturation Lemma; so we are done.

(c) A is q B. The argument proceeds just as under (a) above. The third X is Case (c) of the Saturation Lemma.

(d) A is *B, and the inductive hypothesis guarantees that llB]= lBI.

In this case we establish the following chain of 8s:

18 LENNART AQVIST

kof(llBll), iff, kEf(lBI), iff, ko{koK:k,,Rk and *Bek), iff, k,Rk and *BE k, iff, *BE k. Of these equivalences, the first, the third and the fourth one hold by defhrition, the second by the inductive hypothesis, whereas the fifth one requires some argument. The non-trivial part is of course this : for each kin K, if *Bok then k,,Rk. Suppose then that *Bok. By Al2 and saturation properties (i) and (ii) we then have FnC + C in k, for all wffs C; hence, by saturation property (iv), we have for all C, if p]lCEk, then Cok. By Case (b) of our Saturation Lemma and the 4-k, result we have mnC!ok iff q CEk,, whence we conclude that for each wff C, if 0 CEk, then Cok. Since k, k,EK, this amounts to k,Rk, as desired.

(e) A is B + C. This case is left to the reader. The induction is complete. As an obvious corollary of Lemma 2 we

record that for all wffs A, IIAIl = IAIM, so that ExpP = SpeP. Proof of Lemma 3. We are to verify that for all wffs A, B: (Expp, f)

(= (SpeW9>) meets conditions (1)-(4.2) in the definition of a choice- structure on {koK:k,Rk}.

Condition (1): Assume that llAllM = IIBII. By Lemma 2, then, lAlM = = IJAM, which result is readily seen to amount to q (Aw B&k, and hence (by Case (b) of the Saturation Lemma and the h-k,, result) to mn(A++B)ok for all k in K. By Al3 and saturation property (Q-o-(i), we then obtain mm< A * t, *B) E k for all k in K, which result amounts to /*AIM = I*BIM (by the same token as above). Now, since /*AIM= =flAIM =fllAll (by the definition off and Lemma 2) and I*BI= = II*B/I~ (by Lemma 2), the desired conclusionf jA/j = j*BIjM is immediate.

Condition (2): We observe that fjlAlj"s []A/]" holds iff I*AI~IAIM. Now, the latter is an immediate consequence of the fact that every instance of Al4 is in every ke K, so we are done.

Condition (3): Assume that ll~ll( = /AIM) # 0, which amounts to lo( n ~~B~~ = I*AW, and, by the same token, f/A&BIIM = = I*(A&B)I. By A16.1, the first set is clearly a subset of the second, which proves the case of (4.1). As for (4.2), the first set being non-empty


amounts to the antecedent of A16.2 being a member of every k in K, so that (by saturation property)(i)) the consequent of A16.2 is a member of every k in K. But this amounts to the second set being a subset of the first one - our desired result.

The proof of Lemma 3 is complete, and so is that of Theorem 1. As an immediate corollary of the latter, we have the following weak soundness- and-completeness result :

THEOREM 1.1. For all wffs A, A is provable in A (I- A) iff A is A is A-valid (!= A).

6. MODAL EXTENSIONS OF A; A+-SYSTEMS

The system A is capable of being extended in at least four main different directions, all of which may be relevant in connection with philosophical applications. An extension can result from (i) adding quantifiers to A, (ii) adding to A additional axiom schemata governing 0 (obviously, we cannot do anything with q and q , because their logic is fixed, so to speak), (iii) adding to A additional axiom schemata governing *; also, we may consider (iv) the possibility of generating the system A and suitable extensions of it within systems that lack * as a primitive logical constant but are such that * is definable in them.

The result of adding quantifiers to A will occupy us at considerable length in subsequent sections. In the present one, we make some comments on the three remaining ways of extending A that were just indicated.

Ad (ii). Since we want to leave open the question as to the exact interpretation of q and its matching alternativenessrelation R (different applications and purpose may require different interpretations and logics of q ), we have deliberately adopted a kind of minimal modal logic of 0, viz. the system nowadays generally known as K (for Kripke)

A and axiomatized by AO, All, Rl and -

D( necessitation with respect to

I. Now, it is well known that there is a great proliferation of type (ii) extensions of the basic system K (see e.g. Lemmon and Scott (1966) and Makinson (1966) ; particularly important are the so-called normuZ extensions of K, i.e. those admitting the necessitation rule in addition to substitution and Rl. As far as the logic of 0 is concerned in the context of A, let us indicate it by subscripts as follows: Ax is the system

20 LENNART d;QVIST

A as we know it, A, is A with the schema IA-+ A added as a new axiom, As4 is A, plus m + Ii-IA; and so on. On the semantical level, the now familiar completeness results directly tell us what is the right notion of model that matches the system in question: e.g., an &model is an A-model with reflexive R, an &,-model is an A-model with reflexive and transitive R, and so forth. Thus, when dealing with this kind of extensions of A, we really have access to an impressive store of standard knowledge in modal logic.

Ad (iii). Consider the effect of adding any of the following schemata to A:

A151 Irr] (OB+B)-+(aA-+-,*A)

Al52 HA + !-+A .

It is fairly obvious that these schemata correspond respectively to these semantical restrictions on models and their choice-structures:

(3-l) If k,EllAllM then kOEfllAIIM

A then &Ef[/AII.

We note that (3.2) is stronger than (3.1): (3.2) implies that k,,Rk,,, whereas (3.1) does not. On the other hand, in any model where R is such that k,Rk, (including those with reflexive R), the distinction between (3.1) and (3.2) obviously vanishes. Note also here that some condition of the present sort must certainly be imposed on any choice-structure (in any model) where the underlying preference-ordering is intended to reflect the relation is at least as similar to the actual world (i.e. k,,) as: what could be more similar to k, than k,, itself?

The task of working out the appropriate semantic completeness proofs of the systems A plus A15.1, and A plus A15.2, may be left to the reader.

Ad (iv). We are now going to consider some propositional Iogics the language of which is like that of A except that a unary operator + replaces * as a primitive logical constant (for that reason they will be referred to as A+-systems, and the weakest logic in this family is called A+). The purpose of the A+-systems is to isolate logically the idea, implicit in the *-notions, of circumstances, whether merely possible,


favorable, intrinsic, extrinsic, realized in the actual world, or whatnot. The maximality operator * is defined in these systems by

so that the subjunctive-conditional operator-is def?nable in the A- systems as well.

As for the semantics of A, an A-quasi-model is a triple M=(F,V,g) where F = (K,k,,R) is a frame in our previous sense, V is a valuation on F in our previous sense, and where g is a function with domain and

range included in B(keK: k,Rk}. The notion of F

is defined for all k

wffs A as in the case of A-quasi-models except that clause (7) is replaced by

(7+) iff kogllA1l.

We guarantee that g is defined for IIAll, for all A and M, in the same way as in the case of A; thus, the notion of Exp? becomes available in A+ as well.

Again, an A-model is a triple

22 LENNART AQVIST

Turning now to the axiomatic version of A+,it has the same rules ofproof as A, and schemata AO-All are common to both systems. In A+, A12-A16. l-2 of A are replaced by the following:

Al2+ +A*(l-qr(B+B) A13+ mr/(AHB)-,mr[(+Ao +B) Al5+ q OA--$JO(A&+A) A16.1+ A&+A&B~A&B&+(A&B) ~16.2+ q O(A&+A&B)~~~I~(A&B&+(A&B)-,

-+A&+A&B).

Using our definition of * in terms of + and &, we easily deduce in A+ the schemata A12-A16.1-2 of A. Thus, as one would expect, A contains A.

Strong and weak semantic completeness of the system A is then readily established. The proof parallels that of Theorem 1 with obvious alterations like the following:

In the definition of the canonical A+-model M (generated by S), one takes g to be the function defined as follows: for each wff A, glAI= = 1 +AI= {keK:kJ?k and +AEk}. In case (d) of the induction step in the proof of Lemma 2, where A is +B (for some B), the job previously done by Al2 is now done by A12+. And so on.

We may also consider adding e.g. any of these schemata to A:

A14+ q O+A A14.1+ q +A .

Clearly, these schemata correspond respectively to the following restrictions on A+-models:

@+I g IIAll~0 (2-l) kow 11 AlI

for all wffs A. (2+) guarantees that the circumstances are always possible relative to the actual world, even when associated with impossi- bilities (note here that (3+) and A15+ only guarantees the possibility of circumstances associated with possible events). The stronger condition (2.1) guarantees that the circumstances are always realized in the actual world; such a notion of circumstances plays an important role


in Goodmans discussion of counterfactuals, for instance. We also observe that, given the A+-definition of *, the system A+ plus A14.1+ contains the system A plus A15.2.

Type (ii) extensions of our A+-systems arise just as they do in the case of A and require no further comments. Finally, it should be mentioned in this connection that a fruitful treatment of certain counterfactuals and dispositional statements may well have to be based on sophisticated combinations of the A- and A+-systems so far considered. As this topic concerns the application of our logics to the philosophy of science, we reserve it for another paper.

6a. Some Formal Properties of *

We take the opportunity here to point out a few peculiarities of the conditional connective => as defined in Section 2 above. Some comparisons with the conditional logics of Stalnaker and Thomason, and Lewis will also be made in this connection.

All four of us do agree that the following principles should not in general be valid for * : so-called augmentation of premisses, transitivity, and contraposition. Thus, these schemata are easily seen to fail of A- validity (for obvious reasons) :

(1) (A=z.B)+(A&C*B)

(2) (A* B) + ((B* C) + (A* C))

(3) (A=>B)+(-B=> -A),

Note that (1) should be contrasted with

(14 (A* B) + (A==- (C-, B))

which is clearly A-valid, as well as

(lb) - (A* -C)+(l)

which tells us that, expressed somewhat differently, a necessary condition for the failure of (1) is that A=+ -C holds. Think of A and B as in the Goodman match example and of C as this match has been soaked in water overnight, and we see that this is a highly plausible result.

Again, in connection with (2), one may ask under which conditions it

24 LENNART biQVIST

can be restored; we note that

(24 (A=> *B) + (2) (2b) q O(*A &*B) + (2)

are both A-valid. The case of (2a) is of course plain; as for (2b), which looks prima facie much more interesting, we may resort to the semantical argument below (where M is any A-model and A,B,C any wffs.

Conditions (2), (4.1) and (4.2) in our definition of a choice-structure clearly imply the conditions

(44 GW

f I/AII=f(f llAfl> If IIAII~ IIBII and llAllnf IIB/#O, then f nAll=

= l141nf llBll~ where (4b) may be referred to as Arrows axiom, following Hansson 1968. Now, the antecedent in (2b) amounts to f IIAII nf /BII#O, and the first hypothesis in (2) to f IIA[IE /Bll. Keeping in mind that f llAll= II*AII, A rrows axiom then gives us f (f I/All)= =f llA/lnf ~~B~~. This set must be a subset of Ilcll since, by the second hypothesis in (2), f ll~ll~cI/~II; but by (4a) this set=f l]All, so we are done.

Let us also ask whether the validity of (3) may be restored under some non-trivial condition. This can be achieved in the system A+ in the following way, which may be of some interest in connection with applications. In A+, the antecedent of (3) abbreviates m(+A&A+ B) from which we may infer Im(+ -B & -B+ -A), i.e. the consequent of (3), provided that gllA[(=gll-BIIM; in other words

(3a) m(+A- + -B)+(3)

is valid in A+. We now turn to the schema

(3 (A* B) + @(A+ B)

which asserts that any subjunctive conditional entails that the corresponding material conditional holds at k,. It appears that (5) is valid in any A-extension that contains the system A plus A15.2, whereas it fails in weaker systems, like A itself. We note that analogues of (5) are valid in all the conditional logics of Stalnaker and Thomason and Lewis,


except the weakest Lewis system CO (we say analogues since their languages lack the m-operator). In logics where (5) is forthcoming, the following principle of * -detachment at k,, is clearly valid: from aA and A*B to infer HB.

We come next to some problem cases where my own intuitions (or prejudices) start to conflict somewhat radically with those of Stalnaker and Thomason and Lewis. They concern (i) the relation of conditionals to conjunctions true in the actual world, and (ii) the question as to what is the logical import of the direct negation (contradictory) of a conditional. (Throughout the subsequent discussion we allow ourselves a certain- amount of paraphrase of our colleagues so as to make their ideas expressible in our terminology.)

(i) David Lewis suggests that the correct logic of counterfactual conditionals should validate (5) above and consequently that models for this logic should satisfy an analogue of our condition (3.2) in Section 6. This is perfectly all right with me. He also suggests that the following schema ought to be valid in the correct logic of conditionals:

(6) q (A&B)+(A=-B). He proposes that models for the correct logic should meet the condition that no other world is as similar to a world as that world itself is, which amounts to the following if we let the world in question be k,,:

(3.3) If :A then f ~~A~~c (k,,]. t 0

We note that, given k,Rk,, (3.3) implies our (3.2) so that the E can be strengthened to e as soon as we have k,Rk,,. Clearly, assuming all models and their choice-structures to satisfy (3.3) obviously validates (6). An axiomatic counterpart to (3.3) in the A-language would be:

(7> @-+A+(B+~B))

from which (6) is easily derived. Now, we can find no reason to adopt (3.3) in general - but I understand that Lewis being in favor of it is due to his taking the underlying preference-ordering k to be always of the highest possible discriminating power (cf. Section 1). It seems to me that it is important that (6) fail to be valid on the following ground: suppose that the antecedent of (6) is true for some choice of A and B, where B is

26 LENNART AQVIST

a statement in need of explanation, so that B is an explanandum, in other words. Now, in order for A to count as an explanans of B it may well have to be such that A* B is true; the mere fact that A (as well as B) is true at k, does not entitle us to claim that (the truth of) A explains (that of) B. But, if (6) is accepted as valid, we automatically deprive ourselves of the possibility to treat the connection expressed by=> as an essential ingredient in explanation (as I indeed think it is )over and above mere true conjunction that may well be accidental or coincidental. This consequence of accepting (6) as valid can hardly be a desirable one from an applicational point of view.

(ii) Turning now to the Stalnaker & Thomason systems, we find that not only (6) is valid in them but the following as well:

(8) (A-B) v (A* -B).

Since also an analogue of

(8a) q OA+((A+B)-+ -(A* -B)) is valid in their logics, we obviously obtain as valid

(Xb) q oA+( -(A=+B)c+(A* -B)) which amounts to the dictum of Stalnaker (1968), who credits it to Good- man and Chisholm in their early papers on counterfactuals: the normal way to contradict a counterfactual is to contradict the consequent, keeping the same antecedent.

In our opinion, (8) and (8b) are definitely too strong to be acceptable. However, let it be observed that we have no misgivings about (8a): it is easily proved as an A-thesis, using Al5 and elementary principles in the logic of q and 0. As for (8), it clearly says that f 11Al], unless empty, is always a singleton or unit set. More precisely, it is readily verified that (8) is the correct axiomatic counterpart to the following condition on all models and their choice-structures:

(3.4) If kofl/AIIM and kof llAl[, then k = k (for all A andallk,kinK).

Incidentally, we may note here, as Lewis does, that schema (6) is derivable from (8) and (5) by elementary m-principles, but that it is possible to accept the validity of (6) without being committed to that of (8).

On our view, then, the direct negation of a conditional A * B is equi-


valent to the possibility-statement m(*A & -B). For instance, to deny If the match had been scratched, it would have lighted is to aflirm what can perhaps be expressed by It is possible that the match (still) had failed to light (even) when it was scratched, where we take the when to indicate conjunction rather than implication (as was suggested in Section 2 above). Furthermore, we understand the locution Even if the match had been scratched, it (still) would not have lighted ordinarily to be stronger than the direct negation of A* B and to be adequately formalized by A =- - B rather than by -(A =+ B) ; by (8a), the former implies the latter, provided only that A is possible. Thus, we think that Goodman (1947) is mistaken in claiming that the logical import of a semifactual (i.e. conditional with false antecedent and true consequent, like the even-if statement just instanced) is different from its literal import. He asserts, quite correctly in our opinion, that literally a semifactual and the corresponding counterfactual are not contradictories but contraries, and both may be false (p.15 n.2, our italics). Yes, this will precisely be the case if we adopt our suggested formalization of semifactuals and use reasonable extensions of the system A where (8) is not countenanced as valid. But we dont agree with him in supposing that the role of auxiliary terms in semifactuals like even and still is to indicate idiomatically that a not quite literal meaning is intended. Rather, the situation seems simply to be this: the consequent -B is by assumption true in the actual world; now, changing the actual world in such a way that A be true of it instead of -A is not going to affect the truth of -B, i.e., -B would remain a truth about the actual world regardless of the truth-status of A. This is the type of claim, we suggest, that is idiomatically indicated by particles like even and Stir.

Our mode of formalizing semifactuals necessitates a few comments on the Stalnaker (1968) denial that the conditional can be said, in general, to assert a connection of any particular kind between antecedent and consequent; and on the related view of Chisholm expressed as follows in his celebrated (1946) :

What is the nature of the connection on the basis of which we derive q (= the consequent of some conditional with antecedent p)? We shall go astray if we confme ourselves to a search for thekonnection which must hold befween p andq. This is confIrmed by the fact that we afErm many subjunctive conditionals in order to show that there is 110 relevant connection between antecedent and consequent; e.g., Even if you were to sleep all morning, you would still be tired.

28 LENNART d;QVIST

As far as we can judge, there is a true and important observation involved in these claims of Chisholm and Stalnaker; from which, however, they seem inclined to draw a wrong conclusion. The important observation is this. Let M = ((K, 16, R), V, f) be any A-model where the relation R is interpreted as one of causalpossibility with respect to (so that q expresses some notion of causal necessity). Call a wff A causally irrelevant to a wff B relative to M just in case fllAllc IIBII andfll--AIIC ~~B/~. Now, we take the important suggestion of Chisholm and Stalnaker to be one to the effect that the truth of A * B in a model M of the sort must not exclude A being causally irrelevant to B relative to M, or simply that there are models M and wffs A and B such that A is causally irrelevant to B relative to M. As an illustration, think of A as You are not going to sleep all morning, B as You will be tired, and take M to be such that m, q B are true as well as A + B and -A + B. The present notion of causal irrelevance is interestingly related to the familiar one of probabilistic independence (or irrelevance) in ways worthy of exploration (see e.g. Car- nap (1950, ch. vi)) ; we also believe the notion to be a useful starting point for finer analyses of causation (e.g. in the law; c$ Suppes (1967, ch.5)).

If this is correct, we see that the phrase there is no relevant connection etc. in the last sentence of the Chisholm passage quoted above had better read there is a relation of (causal) irrelevance etc. in order to be unam- biguous and true. Now, it is of course highly misleading to convey the fact that some As bear the relation of causal irrelevance to some Bs relative to some Ms by denying that conditionals assert a connection of any particular kind between antecedent and consequent, as at least Stal- naker seems to be doing. For, who would deny that if A+ B affirms, relative to M, that f llAllM E IIBII, a perfectly good connection is asserted between antecedent and consequent, whether or not f II -AIIM G ~~B~~ holds true as well?

To make one further point about the negation of conditionals in relation to the just explained notion ofirrelevance, let A, B be as in the above example and assume aA, q B and A => B to be true in requisite M. If we now assert It is false that even if you were to sleep all morning you would still be tired, formally -(-A G-B), we are obviously denying that A is causally irrelevant to B relative to M in the sense defined, or, in other words, claiming that your not sleeping all morning is causally relevant to your being tired. But this assertion of relevance is perfectly consistent, we

MODALLOGICWITHSUBJUNCTIVE CONDITIONALS 29

hold, with its being possible that you still be tired in spite of sleeping all morning, i.e. with the negation of -A + -B; in making the relevance claim we are only affirming that circumstances are such that it is (also) possible that you sleep all morning and are not tired. In short, we want to defend the distinction between -(-A => B), which is the direct denial of an even-if...would statement in this example, and the stronger -A* =S -B, which does not directly deny any even-if.. .would statement at all, but at most an even-if...might one. Thus, there seem to be good reasons, philosophical as well as idiomatic, for rejecting (8) and (8b).

Finally, a few words about iterations of conditionals on our analysis as well as about their contingency status. By the logic of @ and q we verify that, for instance,

(9) (A+(B*C)) v ((B=+C)*D)

(94 (B*C)+(A*(B+C))

W) (B*C)ti(B=>(B*C)) (SC) q (A=+B) v q -(A-B)

are A-valid (while, for sure, the converse of (9a) fails). Among the results here that concern iterations (of which the material counterparts are also valid) (9b) looks plausible whereas (9) and (9a) are liable to reasonable doubt. As for (SC), it asserts that all conditionals are non-contingent with respect to universal necessity, i.e., either true at every world in K or none. (SC) is likely to be the worst item on the above list. Although it may surely be said that these results are difficult to assess in view of the vagueness and ambiguity of subjunctive conditionals given in ordinary language, we take them to indicate that the machinery of A should be enriched in one specific respect for the purpose of our obtaining a better and more general treatment of conditionals. This task will be undertaken in the Appendix to the present paper. Meanwhile, let us deal with the system QA of modal quantification theory with transworld identity and subjunctive conditionals.

7. LANGUAGESOFQA

A language L of the quant#cational extension QA of A (or a Qklanguage) is a structure made up of the following disjoint components:

(i) A denumerably Smite set Var, of indiviual variables. Syntactic notation : x, y, z.

30 LENNART ilQVIST

(ii) A denumerable set Cons, ofindividualconrtants. Syntacticnotation: a, b, c.

(iii) For each nonnegative integer n, a denumerably infinite set Pred, of n-adic predicate letters. Syntactic notation: P.

(iv) Two categories of primitive logical constants, viz. (a) classical: I, -+, V (for universal quantification), = (for ordinary identity), and E (for actual existence; this is a monadic predicate); (b) modal: q , q , 0, *, and Q (for a sort of transworld identity; this is a monadic, highly modal predicate).

We say that A is an atomic formula of a Q.&language L iff either (1) A is I, or (2) A is a proposition letter (i.e., zero-place predicate letter) PO, or (3) A is Pal, . . ., a, (n>O) for some a,, . .., a, in Cons, and for some P in Pred:, or (4) A is Ea, for some a in Cons,, or (5) A is Qa, for some a in Cons,, or (6) A is (a = b), for some a, b in Cons,.

The set WL of w@s of L is then defined as the smallest set S such that (i) every atomic formula of L is in S;

(ii) if A, B are in S, so are (A+B), UA, q A, DA, and *A; (iii) if x is in Var,, a is in Cons,, and if A is in S and contains a but

not x, then VxAx/a is in S. (Here AX/a is the result of replacing every occurrence of a in A by one of x.)

The detiitions of negation, possibility, the subjunctive-conditional operator, etc. are taken over from A; as usual, 3 (for existential quantification) is defined as -V- . Furthermore, various modes of restricted quantification of obvious importance in modal logic are definable in QA-languages; to examplify a bit, let x be in Var,, a in Cons,, and let AEW~ contain a but not x:

VxA/a = ,,Vx(Ex+A/a) V06exAX/a = ,,Jx (@Ex+A/a)

L,xAla =dfvx(pq rlQx-+A/a) V onq, .,=W = dx

MODALLOGICWITHSUBJUNCTIVE CONDITIONALS 31

The first two of these universal quantifiers are closely related to what may be called the free logicians favourite; the remaining express different modes of what has pertinently been labelled substunce-quantification (see e.g. Thomason 1969, and my own 1970).

8. SEMANTICSFORQA

A (QA-)frame is an ordered quadruple F = (K, k,, R, D) where the first three terms are as usual, and D is a non-empty domain of individuals. A valuation of a Q&language L on a frame (K, k,,, R, D) is any binary operation V which, for each kEK, assigns:

(1) To each a&ons, a member V(a, k) of D; (2) to each P EPredE a truth-value V(P, k) in { 1, O> ; and to each

PEPred: (n>O) a subset V(P, k) of the Cartesian product D; (3) to the predicate E a subset V(E, k) of D (intuitively, the set of

those individuals in D that actually exist with respect to the world k). Let V be a valuation (of L) on a frame F= (K, &, R, D). We say that

V is canonical on F iff (i) every kEK is such that for each deD there is an a&ong with d=V(a, k), (ii) every koK is such that for each a&ons, there is b&onsL with V(a, k)=V(b, k)=V(b, k,,), and (iii) every kEK is such that for each a&ons, there is bECons, with V(a, kJ=V(b, k,)=V(b, k). (Condition (i) in this definition obviously requires V to be onto D at each kEK.)

Next, if F= (K, k,, R, D) is a frame, we let the set of all functions from K into D be denoted by DK, as usual; a world-line wl on F is then any function in DK. Now, it is clear that any valuation V on F determines a set of world-lines in DK as follows: for each a&ons, and for each koK we set

V,(k) = Vh k)

so that V,ED~ for each a&ons,. Thus, we may think of any V, when its first domain of arguments is restricted to Cons,, as being split up into a family of world-lines indexed by Cons,. More precisely, then, let us define the set Act(V) of world-lines accepted by V, for any V on F, as the set of all V,, where a&ons,. In symbols, Act(V) = {wl~ DK: wl =V, for some aocons,}.

Again, a Q&quasi-model (for L) will be an ordered quadruple M=

32 LENNART iiQVIST

= (F, V, a, f), where F is a frame, f a choice-function on F with domain and range included in the power-set of {kcK: k,Rk}, V a valuation (of L) on F, and where a is a set of world-lines with Acc(V)sa~D~ satisfying the conditions :

(i) If V is canonical on F, then a = Act(V) ; (ii) if V is not canonical on F, then either (iii) a= DK, or (iiii) a=

=Acc(V*), for some valuation VS on F such that (1) V* is canonical on F, (2) Act (V) E Act (V), (3) V*(P, k) = V (P, k) for each PE Pred: (n > 0) and for each ke K, and (4) V(E, k) = V (E, k) for each ke K.

We may note here that if V fails to be canonical on F for the reason that D is nondenumerably i&rite or that V does not meet (ii) or (iii) in our definition of canonicity, then a must be chosen as DK, since in such cases there cannot be any V on F satisfying (l)-(4).

The role of a in a quasi-model is to determine a set of admissible world-lines for V and its variants, or, abusing Wittgensteinian language a little, to determine the Spielraum of the latter in the quasi-model. We next introduce the notion of variant in the following rigorous way:

Let M = (F, V, a, f) with F= (K, k,,, R, D) be a quasi-model for L, let V be any valuation of L on F with Acc(V)za, and let a&ons,. We say that V differs from V at most with respect to a, in symbols: V = gV, iff,

(i) V(P, k)=V(P, k) for each keK and each proposition letter P E Pred:,

(ii) V(P, k)=V(P, k) for each keK and each PEPredL (for each n>O>,

(iii) V(E, k)=V(E, k) for each keK, and (iv) V (b, k) = V(b, k) for each keK and for each bECons, that is

distinct from a. Again, given any quasi-model M = (F, V, a, f) for L with F= (K, k,,

R, D), and any k E K, we define for each AE W, inductively as follows :

P iff V(P,k)=l


(3) iff (V(a,, k), . . . . V (a,,, k)) E V (P, k)

(4) Ea iff V(a, k)EV(E,k)

(5) iff V(a,k)=V(a,k,)

(6) iff V(a, k) = V(b, k)

(of L) on F such that Act (V) E 6! and V = ,V (we assume here that aoCons, is foreign to A, and that An/x is the result of substituting a for x everywhere in A).

The clauses for +, q , q , 0, and * go over unchanged from our A-definition of

M

I- k except that, of course, M and L are understood to

be more complicated in the present case. Also, the shift from QA-quasi-models to Qkmodels is effected in

perfect analogy with the A case and need not detain us. Again, the same thing goes for the notions of simultaneous QA-satisfiability and QA- validity.

9. THESUBSTITUTIONLEMMAFORQA

Before we proceed to consider the axiomatic version of QA, it is con- venient to establish here the following

Substitution Lemma for QA. Let AoWL, let a, b&ons,, let M=(F, V, u, f> with F = (K, &, R, D) be any QA-model (for L), let V be any valuation of L on F such that Acc(V) E tl, and let ko K. Then, if

(i) V = .V, and (ii) V (a, k) = V (b, k) for each ko K, or there are no occurrences of

a in A,

34 LENNART AQVIST

then

(F, V, a, 0

I

A iff (F, V, a,f>

k I__ k AbIas

where Ah/a is the result of replacing all occurrences of a in A by b (if a does not occur in A, then, of course, Ah/a is A).

This lemma is of course to be proved by induction on the length of A. Most of the details can be left to the reader; however, for the sake of illustration, we ought to consider a few important novel cases, viz. those relating to Q, *, and V:

Case A = Qc (for some c&ons,). Assume (i) and (ii) in the hypothesis of the lemma. Suppose that c is a, so that the fist disjunct in (ii) applies. Now, we have in particular that V (a, k) = V (b, k) and V (a, kJ = V(b, k,), whence V (a, k) = V (a, k,) iff V(b, k) = V(b, k,). But since c is a, this result is our desired one, by virtue of truth-condition (5). Again, suppose that c is distinct from a. Then, we have by (i) that V(c, k)=V(c, k) and V (c, k,,) =V(c, k,), so that we obtain V (c, k)=V (c, k,,) iff V(c, k) = = V (c, kJ, as desired.

Case A= *B (for some BEWJ. Assume (i) and (ii) in the hypothesis of the lemma. Now, whether a occurs in *B or not, the hypothesis of induction gives us

(F, V, a, 0 B iff (F, V, a, f> k k

Bb/a

so that obviously

whence

IIB ,, = IIBb/all


have

(1)

(11)

(F, V, a, 0

k VxB, as well as

VxBb/a ,

By our truth-condition (7) for V, (I) and (II) respectively amount to

(F, V, a, 0 Q> k B/x for each V on F such that

kcc (V) E a and V = =V

(where we assume c to be foreign both to B and Bb/a and hence to be distinct from both a and b)

(Bb/a)/x for some V+ on F such that

kc(V+)Ca and V =,V.

Now, consider this V+, and define a valuation V* on F as follows: (1) V* (c, k) =V+ (c, k) for each k EK (2) V* (d, k) =V(d, k) for each k EK and for each d&ons, dis-

tinct from c (3) To the effect that, for each koK, V* makes the same assignment

as V, V and V+ to all PEPredL (n>O) and to E. We now readily verify the following results :

0) v* = J+ V* (a, k) = V+ (b, k)

(g v*=cv* for each koK

As for (ii), it clearly follows from the first disjunct in (ii) and the definition of V*. We also note that Acc(V*) E a follows from the fact that both Acc(V+)ca and Acc(V)Ea.

Given (i) and (ii), the hypothesis of induction yields:

(F, V*, a, f> k

BClx iff (F7 '+' ayf' (BClx)bla.

k

36 LENNART AQVIST

Since (Bc/x)b/a = (B/a)/x, we then conclude from (II) that

Be/x.

But from (iii), (I), and the fact that Acc(V*) E CI we obtain in particular (taking V* as V) that

(F, V*, Q, f> k

Be/x.

This contradiction proves the only if part of the conclusion of the lemma in this subcase. The proof of the if part is obtained by reversing the above argument in the obvious way.

Consider then the remaining possibility that a does not occur in VxB, so that the second disjunct in (ii) applies and B = Bb/a. Again, we assume (I) and (II), we define V* just above, and obtain (i) as well as (iii) ((ii) cannot be established this time, but is not needed). Armed with this information and the fact that a does not occur in B, we use the inductive hypothesis to derive a contradiction in much the same way as above. This establishes the only if part of the conclusion of the lemma in this subcase; the if part is done by an obvious reversal of the argument.

Finally, we may observe that, in proving the present case (where A=VxB), we dont have to worry about the exact nature of CI: the only thing that matters is the quite general feature of o! to the effect that both Act(V) and Acc(V) be subsets of it.

10. THEAXIOMATICSYSTEM QA

A deductive structure will now be imposed on Q&languages L by closing a set of axioms under certain rules. The axioms are determined by the stipulation that any instance of A0 through A16.2 is an axiom (with the understanding that A, B now range over W,, of course), as well as any instance of the following schemata:

A17. VxA + As/x where a is any member of Cons, A18. a=a A19. a= b-+(A+A//b)

provided that no occurrence of b in A that is replaced by


a falls within the scope of a modal constant (remember that * and Q are modal!); here, An//b is any result of replacing various (not necessarily all, or even any) occurrences of b in A by occurrences of a.

A20. a=b+(l(a=b)+(Qat,Qb))

A21. Qa + (Qb + (a = bo q (a = b))) A22. VX~Y (Y = x & QY) A23. VX~Y (R(Y = xl & QY) A24 Vx~A-+@xA A25 VxuA+ q VxA.

In addition to Rl and R2, QA has the rule of proof:

R3. A + B*/x A + VxB

where a E Cons, occurs neither in A nor in B .

The notions of Q&provability, QA-derivability, and Q&consistency are then deEned in just the same way as in the case of A.

Let us list without proof two important theorem-schemata:

Tl. VxmA++A

T2. @(a=b)+(A+A//b).

Since the logic of q is S5, a simple adaptation of the Prior 1956 proof to QA gives us Tl. Now, to accept the Barcan-formulae Tl, A24, and A25 for V is not to accept them for other quantifiers like, e.g., those examplified at the end of Section 7 above; our semantics enables us, though, to settle questions as to which versions of the Barcan formula are acceptable and which are not. As for T2, we note, in the spirit of Kanger (1957) and Scott (1970), that the fundamental condition for substitutivity in Q&languages is q (a = b), not (a = b).

Let us observe in this connection that AO-A25 are Q&valid and that Rl-R3 preserve QA-validity. We do not go into the tedious business of proving this in detail; note, however, the usefulness of the Substitution Lemma in validating Al7 and R3.

38 LENNARTAQVIST

11. THE SATURATIONLEMMAPORQA

Letting L be any Q&language, we say that a subset S of W, is QA-L- saturated (abbreviated L-saturated in the sequel) iff S is such that

(1) S is QA-consistent, (2) for each AoWL, AES or -A&5,

and (3) for each V~AEW,, if A/xES for each aoConsr, then VXAES. An equivalent rendering of (3) is this: (3) For each 3xA~Wr, if ZMES, then A/xG for some aoCons,. Clearly, any L-saturated set will satisfy the obvious QA-analogues of

conditions (i)-(iv) of Section 4. Following Thomason (1970), we now establish the following distinct-

ively modal : Saturation Lemma for QA. Let Z be the set of all L-saturated sets,

let R, be the relation on C such that for all Sr, SzoZ, S,R,S, iff {AEW,: q A&,) c S,, which, by the SS-axioms for q , is an equivalence-relation on X. Letting S be any L-saturated set, define

0) K= Nt (ii) R, = {(k, k):k, koK and {AoWL:mAok} E k) (iii) R= ((k, k):k, koK and {AoW,:aAok} 5 k}.

Then K is such that for all koK and all AEW,:

(a) BAok iff AEk for each kin K

(W 0 o Aok iff A& for each k in K such that k R,, k

(4 0 A ok iff A E k for each k in K such that k R k.

Proof. (The proof is not so obvious in the present quantificational case as it was in the propositional one, since the notion of L-saturation is more complicated.)

Case (a). The only if part is obvious by the definition of K. As for the if part, suppose that q A$k, and let S= {B: q Bok). We verify that S satisfies condition (3) in the definition of Q&L-saturation as follows: if A/xoS for each aoCons,, then q A/xok for each a&ons,, whence, as k is L-saturated, Vx mAok. Now, since every instance of

MODALLOGICWITHSUBJUNCTIVECONDITIONALS 39

Tl is in k, q VxAEk, so that VxAeS. Knowing that S satisfies Gsatura- tion condition (3), we easily see that S,, does, where S,, is the closure of Su{ - A} under Q&derivability (in symbols: S,= (Bow,: Su{ -A)

-B}). It is also readily verified that S, is Q&consistent (see Makinson, QA 1966; or Thomason, 1970).

Again, we record the following fact, analogously utilized by Thomason (1970) (p. 59): the proof given in Henkin (1957), pp. 3-4, of Theorem 3 of that paper can be directly applied to QA so as to yield an L-saturated extension (SJ of S, (and hence of Su{ -A}). Clearly, then, (SJ EK and A#(SJ+, so we are done.

Cases (b)-(c). The argument exactly parallels the preceding one, except that we appeal to the presence in k of all instances of A24, or of A25, as the case may be, in the verification that (B : q Brzk}u( -A} and (B: q Bok)u{ -A} satisfy L-saturation condition (3). Also, as in the case of A, we establish membership in K of the final L-saturated extensions by an appeal to A5 and AlO, respectively.

The proof of the Saturation Lemma is complete - note tha virtues of the Barcan formulae Tl, A24, and A25 in this connection!

12. SEMANTICSOUNDNESSAND COMPLETENESSOF QA

Main Lemma. Let L be a QA-language and let S c W,. Then S is L- saturated, if, and only if, there is a Q&model M= (F, V, ~1, f) with F = (K, k,, R, D) and a k E K such that V is canonical on F, a = Act(V),

and S={AoW,:

Proof. Let us begin with the only if part which is the hardest Suppose then that SC W, is L-saturated, define K, R,, and R as in the Saturation Lemma, and let

(iv) k, = the unique k in K such that S&k (v) f =the function defined as follows: for each AoWL, f ({koK:

k,,Rk and Aok})={koK: k,Rk and *Aok). We justify (iv) just as in the case of A: derive the R,,-k, result by

verifying that R, satisfies (rl)-(r4) of Section 3. In order to arrive at a

40 LENNART d;QVIST

model of the requisite kind, then, we must construct D, V, and u with the right properties.

To this crucial purpose, let us first consider any keK and the relation hk on Cons, such that a- kb iff (a= b)Ek; clearly, for each kM, -L is an equivalence relation which partitions Cons, into disjoint equivalence classes. Consider now the equivalence class [a]wk for arbitrary aECons, and kEK: since k is L-saturated, every instance of Al7 and of A22 is in k, whence, in particular, we have 3y(y= a & Qy)o k (k being closed under detachment), whence, by (3) in the definition of L-saturation, we have b=a & QbEk, for some beCons,. Call this b (or pick one of them, and call it) h([a]-J. What this argument shows is that for all a&ons, and all koK, the equivalence class [a] wk contains at least one member b (viz. h[a]-,) with Qbok; in other words, the function h is defined for each equivalence class [a] wk.

Again, consider the equivalence class [a] wk. for arbitrary a&ons,. Now, since every keK is L-saturated, every instance of Al7 and of A23 is in k, for each keK, whence in particular we get 3y(B(y=a) & Qy)Ek, for each keK; whence, by L-saturation property (3), we obtain the result that for each kEKthere is b&!onsL such that (m(b=a) & Qb)Ek. Again, by Case (b) of the Saturation Lemma for QA and the R,-k. result, this means that for each kEKthere is bECons, such that (b=a)ok, and Qbok. For any given kEK, call this b (or pick it and call it) j ([a] - kO, k). Clearly, we always have j([a]wkO, k)E[a]-,. And our argument shows that for each a&ons, and for each kEK, the class [aINk, contains at least one member b, viz. j([a]-,O, k), with Qbek, so that the function j is defined for each [a] wR, and each kEK.

Next, consider the set U of all equivalence classes [a]-k determined by any kEK and a&onsL, in symbols: U= {[a]-,: a&ons, and kEK}. Define !Z as the relation on U such that for each a, bECons, and each k, kEK: [a]-k=[b]-L, iff there is coCons, such that j([c]NkO, k)e E [al -k andj (F4 -key k) E [b] -k. We now verify that x is an equivalence relation on U:

Symmetry: Trivial. Reflexivity: Consider any a&ons, and koK. Now, we have the wff

(h[alNk=j([h[a]-,]-,,, k))oko, so I(h[a]Nk=j([h[a]Nk]Nk., k)) as well as Qh [a] - L and Qj (fi [a] -,J - k,, k) are all in k. Hence, since every instance of A21 is in k, we obtain (h[a]-,=j([h[a]NL]-k,,, k))ek. But,


as (a=h[a]-,&k, we have (j([h[a]-,]-,O, k)=a)Ek as well. Thus, since a and k are arbitrary, we get the result that for each a&ons, and each kEK there is c (viz. h [a] -&Consr such that j @I - Co9 k)o [a] -k, which obviously amounts to z being reflexive on U.

Transitivity: Suppose that (i) [a]-,= [b]Nk,, and (ii) [b]-,.= [d]-,. From (i) we obtain that some c~Cons, is such that

(1) j ([Cl - k,, k) = j @I N k,, k) = c& (2) j([CINk.,, k) = ad (3) j @I - A,* k) = bok

and from (ii) we gather that some c ~Cons~ is such that

(4) (5) (6)

j ([c] N ko, k) = j ([cc] N k,, k) = c+ Ek,, j([c+]-b, k) = bEk j@+l- k,, k) = dEk.

From (3) and (5) we obtain that j ([cl - k,, k) =j ([c+] - kO, k)ok, Qj([+ko> k)Ek, and Qj([c+]wkO, k)Ek. By A21, then, we get q (j(klNkO9 k)=j([C+]rco, k)kk, SO j([CINk,+ k)=j([c+]ko, kW,. Hence, by (1) and (4), c= c+ ek,, so that [c] - k, = [c] N k& From this we obtain that j([c]-ko, k) and j([c+]-kOP k) are one and the same constant, picked from one and the same equivalence-class [c]~~,; the same thing is true of j([c]Nk,, k) and j([c+]Mk,, k). Hence, (2) and (6) gives us the result that for some coCons, (viz. our c of (2), or our c+ of (6)), j @I - k,, k) E [aI -k and j([c]- kOP k))o[d]-kn, which is to say that [al- k = [d] -k, as desired.

xbeing an equivalence relation on U, it partitions U into disjoint equivalence classes. Now, for each a&ons, and koK, let II be the function defined by

da, k)= [[alhlc, = ={[bl-k:[bl-k~bl-k9 where bECons, and koK} .

We now define D in our desired model by (vi) D={u(a, k,,):aKons,}

42 LENNART AQVIST

And we define the valuation V on the frame F= (K, k,, R, D) as follows: (1) For each aECons, and koK: V(a, k)=u(a, k)=[[a]-,I, (2) For eachPoPredE and keK: V(P, k)= 1 iff PEk; and for

each P E Pred;l (n > 0) and each k E K: V(P, k) = (vii) = ( and that V is canonical on F. To that purpose, define for keK

. D,={u(a, k):k=k and a&ons,},

.e., Dk is the domain as determined at k, whereas D, as defined in (vi), is the domain as determined at k,,. If we show that for all keK, Dk = D, we have clearly shown (i) that the values V(a, k) are all in D, which settles the first point, and (ii) that V is onto D at each koK, which settles a good deal of the second.

To see that DL c D for all kE K, make the counterassumption that for some a&ons, and some krzK:u(a, k)EDk but u(a, k)#D. Now, we readily verify that [a] wk x [h[a] -J - k z [h[a] wk] - k0 (as for the first x, appeal to the reflexivity of = and to the fact that a= h[a] -JOE k; as for the second %, take h [a] -k as the desired c and use Case (b) of the Satura- tion Lemma, the Ro-k,, result, and the fact that every instance of A21 is in k). Hence, by the deBnition of u, u(a, k) = u (h [a] -k, k) = u (h [a] -k, k,,). But, by (vi), u(h[a]- k, k,,) is clearly a member of D, whence u(a, k)cD, contradicting our hypothesis.

To see that D c Dk for all kEK, make the counterassumption that for some a&ons, and koK:u(a, k&D but u(a, kO)$Dk. Now, we have [al- ko= ti([+k,,, k)l-ko~Y[j(M-k,,~ k)] IVk (the argument parallels the one just given; as to the second =, take j ([a] - ,+,, k) as the desired c and mobilize the same machinery as above). Thus, we obtain u(a, k,)= = 0 ci ([aI - kop k), kJ=~(j(bl-ko~ k), k). But, by the definition of D,, u(jW-ko9 k), k)o D,, whence u(a, k,,)ED,, contrary to hypothesis.

Finally, we notice that our argument shows that V, as defined in (vii),


clause (l), meets the remaining conditions in our definition of canonicity on F: for each k o K and for each a o Cons, there is b E Cons, with V(a, k) = = V(b, k) = V(b, k) - satisfied because h [a] wk does the job and is defined for all a and k ; and, for each koK and for each ao Cons, there is bo Cons, with V(a, k,) = V(b, k,,) = V(b, k)-satisfied because j ([a] - kO, k) does the job and is defined for all a o Cons, and all k o K.

We have now completed the definition of a structure M= = ((K, k,, R, D) V, u, f) which we intend to serve as the desired QA- model of our Main Lemma (only if part). We know that the initial quadruple is a frame, that D is at most denumerable, and that V is canonical on this frame so that a has to be equated with Act ( V). In order to finish the proof of the Main Lemma (only if part), then, it is clearly sufhcient to establish two sublemmata:

SUBLEMMA I. For each AE W, and for each koK, F

A iff Aok. k

SUBLEMMA II. The pair (Expp,f) is a choice-structure on (kcK:k,,Rk).

Again, we first consider how these lemmata together yield the only if part of our Main Lemma: applying Sublemma I to the particular case where k is S, i.e. the L-saturated starting point for our construction, we

conclude that S = {AE WL : F S

A}, as desired. It only remains to verify

that M, as defined, is a Q&model : in view of our just recorded knowledge about M, this is guaranteed by Sublemma II. We add that, given Sub- lemma I, Sublemma II is established exactly as in the propositional case of A. We then turn to the

Proof of Sublemma I. By induction on the complexity of A. The more interesting cases are as follows.

Case A is (a= b), for some a, bczcons,. We are to show that V(a, k) = = V(b, k) iff (a=b)ok. Suppose first that (a=b)ok. Then, [a]-,= [b]NL. Hence, [a] wLz [b]-,, since R is reflexive in U. Hence, [[a] -,& = = [[bl-,I,, i.e. V(a, k)= V(b, k), as desired. Conversely, suppose that (a=b)$k so that (a#b)ok and [a] -L # [b] N k. If we now establish that bl-kW4-k, the desired conclusion V(a, k) # V(b, k) is immediate. Make the counterassumption that [a] wL % [b] N L. Then there is coCons, such that j([c]Nkg9 k)o[a]-, and j([c]k,,, k)o[b]-k, i.e., we have

44 LENNART AQVIST

j @I - ko9 k)=aek as well as j([c]wko, k)=bak. But in view of agbek and A19, we then obtain j([c] N k,, k) # j([c]w,, k)Ek, contrary to k being L-saturated.

Case A is Qa (for some aKons,). We are to show that v(a, k)= = V(a, k,) iff Qaek. Suppose first that Qaok, and consider [a]-kO. As alWayS, j([a]-kO, k)o[a]-kO, SO that j([a]mkO, k)=arzk, and q


F such that Accra and V = ,V, iff, (6 v, a,f>

k Bb/x for each

bECons,, iff, Bb/xok for each b&onsr, iff, VxBok. (As usual, aoCons, is foreign to B.) Of these iffs, the first is true by definition, the third by the inductive hypothesis, the fourth by the fact that k is L-saturated, whereas the second one requires some argument.

To begin with, let us point out that the Substitution Lemma, although it was stated for Q&models in Section 9, is in fact valid for the wider class of QA-quasi-models as well, as is readily verified. Now, we certainly know our structure M to be a quasi-model, although we cannot know it to be a model until Sublemma I is established (since the latter is needed for the proof of Sublemma II). This being so, we can apply the Substitu- tion Lemma to M in the following way so as to derive our second iff above :

Zfpart: An obvious application of the Substitution Lemma to our quasi-model M yields :

(1) CC K a,f>

k k Bb/x only if [V = gV and V(a, k)= V(b, k) for each

k& only if (F, V', a,f>

k B/x] ; where we have taken B/x (a&ons,

foreign to B) as A, so that Ah/a becomes (B/x)b/a, i.e. Bb/x, and where V is any valuation of L on F with Acc(V) c Acc( V) (= a), and b is any member of Cons,.

Suppose then that the hypothesis of (1) holds for all bECons,. Then, since (1) holds for all beCons, and all V on F with Acc(V)s Act(V), the consequent of (1) must hold for all such V and all bECons,; in other words,

(2) For each V on F such that Acc(V)E Act(V) and V = .V, if there is boConsr with V(a, k)= v(b, k) for each kEK, then

W V, a,f > k

B*/x.

Reflecting now for a moment on the import of the condition that Acc(V)c Acc( V), we find that it amounts to the following: each world- line V: is to be identical with some world-line V,, or, more precisely: for each aoConsr there is boConsL such that Vi is identical to V,. Now, this

46 LENNART /JQVIST

identity clearly amounts to Vi (k) = V (a, k) = Yb (k) = V (b, k) for each k&, whence the following result ensues:

(3) Acc(V) E Acc( V), iff, for each a&ons, there is beCons, such that for each k&, V(a, k)= v(b, k).

In view of (3), then, we see that the if-clause in (2) is

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Aqvist Lógica Con Predicados Modales