PETER QUALITATIVE GARDENFORS PROBABILITY AS AN INTENSIONAL LOGIC* I The main aim of this paper is to study the logic of a binary sentential operator ‘z=‘, with the intended meaning ‘is at least as probable as’. The object language will be simple; to an ordinary language for truth-functional connectives we add ‘&’ as the only intensional operator. Our choice of axioms is heavily dependent of a theorem due to Kraft et al. [8], which states necessary and sufficient conditions for an ordering of the elements of a finite subset algebra to be compatible with some probability measure. Following a construction due to Segerberg [ 121, we show that these conditions can be translated into our language. When giving semantical models of this formal language we will start from a universe U, interpreted as the set of possible worlds. We then assume that the probabilities of other worlds, from the standpoint of a given world, can be evaluated. We do this by associating with each world a probability measure on the universe. When a standard truth-valuation of propositional letters is added, we can then define the set of worlds where a given formula is true. The central clause here is that the truth-value at a world x of a formula of the form ‘A + B’ is determined from the probabilities at x of the sets of worlds where ‘A’ and ‘B’ are true. This mechanism also provides a simple way of handling higher order probability statements, i.e. statements about probabilities of probabilities. As we will see, some tentative restrictions on the probability measures in the models will reflect different approaches to the logic of higher order probability sentences. For the semantics of classical one-place modal operators, the alternativeness relation on the universe has proven to be useful and clarifying. However, as soon as one turns to more complicated intensional operators as e.g. counterfactuals, preferences, conditional obligations and probabilities, such a relation between points in‘the universe seems as mysterious as the operators themselves In our opinion, a more fruitful way to deal with these operators semantically Journal of Philosophical Logic Copyright 0 1975 by D. Reidel 4 (1975) Publishing 171-185. All Rights Reserved Company, Dordrecht-Holland 172 PETER GARDENFORS is rather to use different kinds of measures (as e.g. degrees of similarity, utility and probability measures). Following this program, we also outline how other kinds of qualitative orderings, such as additive semiorderings and conditional probability orderings, can be treated in much the same way. Finally, we define the ordinary necessity operator from ‘3’ and show which modal logics can be derived within our different probability logics. The logic derivable from our basic system will be D, which has been introduced as the logic for one-place deontic operators. II The alphabet of our object language contains countably many propositional letters, the binary sentential connectives ‘+’ (implication) and ‘+’ (the probability-operator), the nullary connective ‘0’ (falsity) and parentheses. The set of formulas are defined as follows: (9 (ii) (iii) (iv) ‘0’ is a formula, every propositional letter is a formula, if ‘A’ and ‘B’ are formulas, then ‘(A + B)’ and ‘(A 3 B)’ are formulas, nothing else is a formula Note that we allow formulas to contain nestings of the + -operator’. One of the advantages of using an intensional language is that we can deal with higher order probability statements without much ado. Our semantics will also treat the different types of formulas in a uniform manner. For the sake of greater perspicuity we introduce the following abbreviations: 1 =dfo’o, -A =dfA’O. The usual definitions of &, A-B =df(ABB)&(B?=A), A+B =~f(.42=B)&-(BbA). V and *. As a convention for the use of parentheses we will regard the propositional connectives as binding more strongly than the probability-operators. QUALITATIVE PROBABILITY AS AN INTENSIONAL LOGIC 173 Our language differs from that of Segerberg [ 121 in that it does not include the necessity-operator 0. Whether such an operator ought to be included in a language for qualitative probability is a non-logical problem. However, the necessity-operator might be regarded as superfluous, since one may introduce it by defining ‘it is necessary that p’ as ‘p is equally probahle as 1, i.e. truth’. We will return to this topic in the sequel. For future needs we also want to introduce a powerful abbreviatory device (originating in Segerberg [ 121) as follows. For m 2 0, let A0 , . . ., A,, BO, . . ., B, be a fixed set of formulas in ourlanguage. Form all conjunctions doAl-J& . . . Ld,A, &eoBo &. . .&e,B, such that exactly p of the di’s and p of the ei’s are the negation sign (where 0 <p < m + l), the rest of them being the empty string of symbols. Define C’ as the disjunction of all such conjunctions. We now introduce the abbreviation symbol E (which may be read as ‘are generalized equivalent to’) by the scheme: A,,A1.. .A, E BoBI. . B, =df(COV . ..C.+,)-1. To give an idea of how this construction works, we note that A E B iff (A & B v -A & -B) - 1, i.e. iff (A ++B) - 1. Or, to state the meaning of E in so far vague semantical terms: AoAl * * *A, E BoBI * * *B, is true iff exactly as many of the Ai’S as of the Bi’S come out true, no matter how we assign truth values to the compounds of the Ai’s and Bi’S. III In order to make our axiomatic system intelligible, we next present a theorem originally due to Kraft et al. [8] . Their paper is rather difficult to read (partly due to their cumbersome notation), but a much more lucid presentation of their conditions and a simpler proof can be found in Scott [ 1 l] . We state Scott’s version of the Kraft-Pratt-Seidenberg theorem without proof. It is formulated for an ordering on the subsets of a set, but we will see how the theorem can be applied to the formulas in our language. By x we denote the characteristic function of the set X, i.e. the function that assigns 1 to the elements which belong to the set and 0 to the elements outside the set. THEOREM. Let S be a finite set and > an ordering of the subsets of S. 174 PETER GiRDENFORS Necessary and sufficient conditions that there,exists a probability measure P such that X a Y iff flX) > P( Y) are that the following hold for all subsets Xand YofS: (1) xz 8, (2) XaY (3) S > 8 (i.e. not @> S), (4) for all natural numbers m and all subsets Xe ,Xx,. . .X,,, ,Ye, or Y r...Y,,, YhX, ofS,ifXi> YiforO<i<mandXe(~)+Xr(~) + . . . +Xm(.z) = Ye(z)+ . ..+Y~(z)forallzeS. then Y, > X,. Condition 4 may be hard to understand, but the algebraic equation means that any element of S belongs to exactly as many Xi’s as Yi’s. To see what this condition excludes let S be {a,b,c,d,e} and consider the following relations: (Q,b} > {C}, {Q,C} > {hd} {kc} > {Q,e}a {de) > {Q,hC)- It was shown by Kraft, Pratt and Seidenberg that this set of relations can be extended to an ordering of all subsets of S which satisfies the conditions proposed by de Finetti [2], but there does not exist any probability measure on S which agrees with these relations. They also violate Condition 4 since every element of S belongs to as many of (a,b), (Q c}, @,c>, @,e) as of (cl, {hd}, {QS}, {Q,hC) so, according to Condition 4 if the first three relations hold, the fourth should not. The axiomatic system we now introduce is in essencea translation of these conditions into our language. Our basic logic, called QP,has as its axioms all truth-functional tautologies and all instances of the following schemes: : (AO) (A+tB- (Al) A+O, 642) (A z=B)v(B (-43) 1 b 0, for all m 2 1 l)&(C*D-l)+((A >A), + c)++(B -‘D)), QUALITATIVE (A4(m)) PROBABILITY AeAr.. AS AN .A,,, E&-Jr.. .B, (Am.el + B,-,)+(B, INTENSIONAL LOGIC 175 & (A,-, * BO) & . . . & ?=A,). The rules of QP are modus ponens and the following: WI if t A, then t A * 1. To illustrate the power of the A4(m)axioms we outline a derivation of the scheme(A z=B) & (B?=C’)+(A +c): (1) ABC E BCA (from tautologies, via RN and the definition of El, (2) A z=B (assumption), (3) B ?=C (assumption), (4) A B C (from 1,2 and 3 by A4(3)). We also note that from A0 and RN we can derive the following replacement rule : (W if t A ++B and D is like C except for containing B in some place where ChasA, then1 C-D. Instead of the conditions in the Kraft-Pratt-Seidenberg theorem, many other sets of conditions on the ordering > have been shown to guarantee tha the ordering agrees with a probability measure (see e.g. Savage [IO] or Fine [5]). These have all had two disadvantages. Firstly, they are sufficient but not necessary for an ordering to agree with a probability measure. Secondly, if one wants to translate them into some kind of intensional language, propositional quantification seems to be a necessary ingredient, while the set of conditions we use only demands a very meagre language. IV We next turn to the appropriate formal semantics for QP. A probability frame, or frame for short, is a structure tU,P) where (9 Uis a set, (ii) P is a function on U, 176 (iii) PETER GiRDENFORS for any x in I/, the value P, of P is a probability measure on the set of all subsets of U. In this general definition, P, is allowed to be any probability measure. In particular, we do not demand that P, is countably additive. In the sequel we will consider two tentative restrictions on the probability measures and give the corresponding axioms. A probability model, or model, is a structure W,P, V) where W,P) is a frame and I’ a function which assignsto every propositional letter A a subset V(A) of U. For a given model M = W,P, V), we now recursively define a function II IIM from formulas to subsets of U: (0 if A is a propositional letter, then IhI IP = V(A), (ii) II0IP = Ql) (iii) llA-,BlI’ = (U- (iv> IIAz BllM = (~‘3 IIAII”) U llBll”, Px(llAIIM)~P,(IIBIIM)}. We will drop the superscript referring to the model, when it is clear which model we refer to. We say that a formula A is true (in M) at x, iff x E IIA IP. A Formula A is uaZid in Miff IIA IIM = U. Finally, a formula is valid in a frame F iff it is valid in all models on the frame. Our next aim is to show that the set of formulas valid in all models are exactly the set of formulas derivable in QP. The completeness theorem per se will not add much to our understanding of a logic for qualitative probability, but as a by-product we can prove that QP has the finite model property and is decidable. First of all one needs to show that QP is sound with respect to the models. CONSISTENCY THEOREM. The theorems of QP are valid in all probability models. Proofi All rules preserve validity and for all axioms except those generated from A4(m) it is completely trivial to show that they are valid in all models. For the A4(m)-axioms it suffices to show that if AOA 1. . .A,,, E BeBr . . .B, is true at x in a model M = W,P, I?, then P,.(IIAe II) + . * . P,( IIA, II) = P,. llBo 11)t . . . + Px( IIB, II). The argument for this parallels Segerberg’s in [ 123 pp. 344-345 and will not be repeated here. QUALITATIVE PROBABILITY AS AN INTENSIONAL LOGIC 177 We now turn to the proof of the converse of the Consistency Theorem. Let A be any formula which is not provable in QP. We want to construct a finite frame FA = (U, ,PA >such that A is not valid in FA . Define a relation =Bp on the set of formulas by A B iff A of B is derivable in QP. Clearly p is an equivalence relation. Denote the equivalence classof A by 1A 1.If we define operations ‘-’ and ‘v’ on the set of equivalence classesby 1A1v IB 1= 1A v B 1and -1A 1= l-A I we obtain a Boolean algebra B with I1I and IO 1as unit and zero elements. To show that this construction is well-defined one makes use of A0 and RN’. It follows that A is derivable in QP iff IAI = Ill. The rank of a formula C, denoted rank (C), is the maximal number of nestings of the k -operator in the formula. If a formula does not contain + , we say that it has rank 0. Let 3, be the set of formulas C which are built up from some subset of the propositional letters in A and for which it holds that rank(C) < rank(A). Let SA be the set of formulas built up from the same propositional letters for which rank(C) d rank(A) + 1. Let B, be the subalgebra of B which has as its elements exactly the equivalence classesof the formulas in S, . Similarly, define BA as the subalgebra of B which consists of the equivalence classesof the formulas in Sd. Clearly B, is a subalgebra of Bi. Since A has a finite rank and contains only finitely many propositional letters, it can be shown by e.g. a normal form argument that BA contains only finitely many elements. In fact, an upper bound of the size of BA can easily be computed. Since BA is a finite Boolean algebra it is isomorphic to the subset algebra of the set of its atoms. We define U, , the first component in our desired frame, as the set of atoms in BA . We note that BA is also a finite Boolean algebra. Each atom in B, may in general consist of several atoms in Bd , since B, may be (and in general is) a proper subalgebra of BL . In the sequel we will make tacit use of the fact that a finite Boolean algebra is isomorphic to a subset algebra on a finite set, and not always distinguish between Boolean and set-theoretical uses of the elements. For every element x in U, , we now introduce an ordering >, on the elements of BA by stipulating that I C I>, ID 1iff x E 1C 3 D I. This is well-defined since I C k D I is in Bi. LEMMA 1. The ordering 2, of elements in B, satisfies conditions 1 - 4 of the Kraft-Pratt-Seidenberg theorem, for any x E U, . =Qp =Q 178 PETER G;i RDENFORS Proof It is handwork to verify that Conditions 1 - 3 are satisfied given the presence of axioms (Al) - (A3), and the proof is omitted. For the verification of Condition 4, which is the central part in the completeness proof, let L-4, I, IAi I, . . . IA, I, 1~01, IB1 I, . . . IB, l&. elements in BA (these elements may be considered as sets of atoms in BA). Furthermore, assume that the algebraic equation in Condition 4 holds for these elements, which is the same as assuming that every atom of BA belongs to exactly as many of the I~il’s as the 1~~1’s.Finally, assume that IA,-,I >, I&, I, . . . IA,-~ I >, l&-r I. Any atom w in Bd is included in exactly one of the atoms of BA since these are disjoint. It follows that w belongs to the same number, say p, of the 1~~1’sas the IBiI’s. Therefore w E Iq I, where C’ is defined as in connection with the definition of E. By the definition of the Booleanoperations, w E IC, v C, v . . . v Cm+, I. This holds for any atoinwinBA,hence Ic~v...vc’,,,+~ I = II I. From the rule RN it follows that I&AI...& EBoB1.. .B, I = I1 I. This holds independently of which probability measure function and valuation we add’to U, in order to get a model. From the presence of the axiom scheme A4(m) we derive that I@, +B,) & . . . & (Am-r & Bmml)+(B, SA,)I= lll.Fromour earlier assumptions and the definition of the equivalence classeswe conclude that x f IB, + A, I, i.e. IB, 12, IA, I. This ends the proof of the lemma. According to the Kraft-Pratt-Seidenberg theorem, we now know that, for every x in U,, there exists a probability measure P, defined on all elements of BA , such that Px( ICI) > Px( IO I) iff ICI 2, IO I. This probability measure is completely determined by the magnitudes of the measures of the atoms in BA . We next show how to extend P, to a probability measure PL defined on all elements in BA . Let a be any atom in BA and suppose P,(a) =P.Ifa;,ai, . . . aA are the atoms in BA which are included in a, let Pi be any probability measure such that P!Jai) + P:(ai) + . . - P:(aA) = p. Since every atom in Bd is included in exactly one atom in Bi, such a measure always exists. Defining P: in this manner for every atom in Bi determines a unique measure of every element in BA . Clearly, P:(a) = P,.(a) for any element a in BA . We can now define the second component of our frame. Let PA be the function which assignsPi to every x in lJ, . Let FA be the frame (U,, PA ). The crucial feature of this construction is revealed in the next lemma. QUALITATIVE PROBABILITY AS AN INTENSIONAL LOGIC 179 LEMMA 2. If B is a formula in S, , then B is derivable in QP iff B is valid in FA. Proof: If B is derivable in QP, then B is valid in FA by the consistency theorem. For the converse, let V, be any valuation such that, for any propositional letter C in S, , V,(C) = ICI. This equivalence class is clearly in BA and hence denotes a subset of U, . We show by induction on the length of B that IBI = IIBII, where II II is the function generated from VA. If B is a propositional letter it holds by definition. If B is 0, then trivially lOi= 11011. IfBisC+D,then IBI= IC+Di=-IClv lDl,which,ifwechange the notation to set-theoretical for 17, , according to the induction hypotheses is the same as (iI, - IICII) U IID II, which by definition is IlC+ DII. Finally, ifBisCBD,thenwehave IBI= ICkDI={w E VA: w E ICBDI}= {w E UA:Pw(~C~)S-Pw(tD~)}={w E U,:P,(~IC~~)>P,(~~D~~)}= IIC + D Il. It follows that if B is not derivable in QP, then IB i # 11I and hence IIB II # VA in the model defined above. Therefore, B is not valid in FA . This completes the proof of the lemma. COMPLETENESS THEOREM. Every formula which is valid in all probability models is derivable in QP. Proof: Let A be a formula which is not derivable in QP. Since A belongs to S, , it follows from Lemma 2 that there is a frame where A is not valid and therefore there is a model where A is not valid. DECIDABILITY THEOREM : QP is decidable. Proof: The upper bound of the number of elements in VA can be computed from the number of propositional letters in A and rank (A). It is true that for any finite set U there are infinitely many probability measures on U, but these generate only finitely many orderings on U. As a corollary to the Kraft-Pratt-Seidenberg theorem we know how to decide whether an ordering agrees with a probability measure. So in order to find out whether a given formula is a theorem of QP, we only have to check finitely many valuations for finitely many orderings on finitely many finite sets. A highly impractical procedure, but entirely mechanical. V In the deftnition of a probability model we allowed the values P, of P to be 180 PETER GiRDENFORS any probability measures on the subsets of U. We next turn to a discussion of some constraints on P, where we also make use of our abilities to deal with higher order probability statements. A very rough motivation for the first constraint is that if x is the actual world, then certainly x is possible. It might therefore be argued that there is some chance that x occurs, given that x is the actual world, so a natural condition on P would be that 6) U~xl> > 0% for all x E U. Of course, we do not want an interpretation of P, such that P,( (x)) = 1. This would be tantamount to the fatalistic mistake that whatever happens is necessary. The axiom which corresponds to the requirement that (i) holds is: (A9 (A - 1)+/l. A straightforward reading of this axiom is that if something is as probable as the truth (or happens as often as the truth), then it is actually true. If our goal were to find axioms which are true for a subjective reading of the @ operator, (AS) seems inevitable. This is, however, in sharp contrast with quantitative probability theory, where it e.g. holds that the probability measure of the set of irrational points on the real line is the same as the measure of the entire line. So it is perfectly possible to have some world x where some stochastic variable has a rational number as its value. In this situation we would have Px({x}) = 0, since for the set of worlds S where the variable has an irrational value P,(S) = 1 holds according to standard measurement theory. It is obvious that the axiom (A5) is valid in any frame which satisfies (i). In fact, our proof of the completeness theorem can be modified to show that QP + (AS) is complete with respect to the set of such frames. In the same way as before it can be shown that the logic is decidable. We also note that (A3) can be derived from (A5) and the rest of the axioms. Some notions of probability are stronger than those we have discussed so far. According to what might be termed the ‘analytical’ notion of probability, a sentence saying that A is more probable than B is true only if it is necessarily true. Or in more precise semantical terms: If A + B is true at X, then A & B is true everywhere in CJ.This leads us to the next constraint of P: (ii) P,(S) = P,,(S) for any x and y in U and for any S E U . QUALITATIVE PROBABILITY AS AN INTENSIONAL LOGIC 181 Or more informally; the probability of a world is not dependent on which world it is evaluated from. The axiom scheme corresponding to (ii) is: W) ((‘4 *B)++((A* B)-1)) & (-(A +@*((A -‘R)-0)). As is easily checked, (A6) is valid in all frames satisfying (ii), and conversely, our completeness proof can be modified to show that QP + (A6) is decidable and determined by the class of all such frames. As a byproduct we get that (AS) and (A6) are independent of each other. We also note that every instance of the following schemes is derivable inQP+(A6): (1) ((A ?=B) - 1) v ((A 3 B) - 0) ) (2) (A>B)++((ABB)?=(BzA)), (3) ((AzB)> (4) (A?=(B~A))-+(A~B), (5) ((A 2=B)?=B)-,(ABB). (C?=D))+,((C3=D)+(A z=B)), So, in a model which satisfies (ii), a formula containing ‘3’ as its main operator is either valid in the model or its negation is. Finally, note that since every non-theorem fails in a finite frame, the constraint that P, is countably additive adds nothing to our axiomatic system. Or in other words, there is no formula in our language which corresponds to a property that separates countably additive probability measures from finitely additive. VI In the Kraft-Pratt-Seidenberg theorem the ordering 2 was based on the condition X 2 Y iffP(X) > P( Y). We get another kind of ordering, a so called additive semiordering, if we introduce an ordering > by the condition X> YiffP(X)>P(Y)+eforsomefixedesuchthatO<e<l. The following representation theorem is proved in Domotor and Stelzer [41 : THEOREM : Let S be a finite set and > an ordering of the subset of S. Necessary and sufficient conditions that there exists a probability measure P 182 PETER GiRDENFORS on S such that X > Y iff P(X) > P( Y) + E, where 0 < E < 1, are that the following hold for all subsets X, Y and 2 of S: (1) S>Q, (2) not X> X, (3) ifXgY (4) for all natural numbers m and all subsets Xi, Yr, Zi, and Wi ofS,ifXi> andZ> Y, then.Z> X, YiandnotZr> WiforallisuchthatOGi<m3 and for all elementsx E S i:m(Xi(X) + @i(X)) = z (Yi(X) + ii(X)) holds, then if X, > Y,,, , then i<m z, > iv, 4. This theorem is thus a counterpart to the earlier theorem with an even more complicated Condition 4. These conditions can, however, also be translated into an intensional language with one binary operator. The only difficulty is to give a condition on formulas such that the algebraic equation of the characteristic functions is satisfied. And this can be done in a way similar to when we introduced E. Therefore we can develop an intensional logic for the additive semiordered qualitative probability in much the same way as we have done for the ordinary probability operator. The semantics will be the same except for using the threshold condition when defining [IA s B II. Also a completeness theorem for this semantics can be proved along the same lines as before. Moreover qualitative conditional probability can be handled in much the same manner. For a four-place relation X IY > ZIW defined on all subsets X, Y, Z and W of a set S (with the restriction that Y and W are such that YIS > 0 IS and WlS > 8 IS, Domotor [3] has proved that a certain set of conditions on the relation are necessary and sufficient to guarantee that there exists a probability measure P such that X I Y > Z I W iff P(X IY) > P(Z IW). These conditions can be formulated in an ordinary propositional language supplemented with one four-place operator, though the degree of complexity of the axioms increases considerably. Finally, we do not see any problems in proving a completeness theorem for the corresponding semantics. QUALITATIVE PROBABILITY AS AN INTENSIONAL LOGIC 183 VII The final topic in this paper is an investigation of the logics we can derive for the necessity operator when this is introduced with the help of our probability operator. What we have in mind is to define &4 as ‘it is maximally probable that A’ or ‘A is as probable as the truth’. Formally We can then introduce OA in the customary manner. Since -(--A - 1) f) (A Z=0) is a theorem of QP. we could as well have used the following: Od=df(A PO). From our semantical definitions it follows that DA is true at x iff A is true (in a finite model) at all points y such that P, (b}) > 0 and OA is true at x iff A is true at some such point. We say that a modal formula is a formula of our language which by use of the definitions can be written as a formula not containing z= , s or -. Our aim is now to characterize the set of modal formulas derivable in QP, QP + (AS) and QP + (AS) + (A6) respectively. Similar ideas can be found in Rescher [9] and Danielsson [ 1] . They prove that from their probability axioms the modal logics S5 and T respectively can be derived. They do not prove, however, that the set of derivable modal formulas is exactly the theorems of S5 and T. THEOREM : The modal formulas derivable in QP are exactly those derivable in the modal logic D. Proof: We first show that D is included in the set of modal theorems in QP. These certainly form a logic, closed under modus ponens and substitution, which contains all instances of truth-functional tautologies. All instances of the scheme 0 (A & B) * 0 A & OB is in the set since any instance of is a theorem in QP. Similarly any instance (A &B-l)++(A-:l)&(B-1) of q A + 0.4 is included since (A - 1) + (A h 0) is a derivable scheme. Finally, the rule of necessitation is an immediate consequence of the rule RN. We show the converse, that any modal theorem in QP is derivable in D, by semantical arguments. It is well-known that D is determined by the set of relational models ( U, R, V) where U is a set, R is a binary relation on U such that for every x in U there exists some y such that xRy, and I/ is a valuation 184 PETER GARDENFORS of the propositional letters. Given a finite relational model (U,R, I’) of this kind, it is always possible to construct a corresponding probability model (U,P, I’) such that P,(b)} > 0 iff xRy, for any x and y in I/. If A is not a theorem ofD, there is some finite model&f, = (UA,R,, VA) such that A is not valid in MA. If we construct a corresponding probability model (U,,P,, VA) as indicated above, it is easy to verify that A is not valid in this model and hence not a theorem in QP. Therefore any modal formula derivable in QP is derivable in D and our theorem is proved. That the modal formulas derivable in the logic QP + (A5) are exactly those in T, can be verified in a similar manner. That T is included is immediate given the previous theorem since oA + A is equivalent to (AS) by definition. To show the converse it is sufficient to note that in the above mentioned construction it is true that PX({x}) > 0 iff xRx. It is well-known that if A is not derivable in T, there is some finite reflexive relational model in which A fails. Therefore A also fails in a corresponding probability model where it is true that PX({x)) > 0 for all x. Hence A is not derivable in QP + (A5). Finally, the modal formulas derivable in QP + (A5) + (A6) are exactly those in S5. The modal scheme one has to add to Tin order to get S5 is OA + nOA. According to the definition this is the same as (A > 0) + ((A + 0) - 1) and this scheme is an immediate consequence of (A6). So S5 is included in the set of modal formulas derivable from QP + (A5) + (A6). The relation characteristic for an SS-model may be taken to be the universal relation. If (U,R, T/j is an SS-model, we define a corresponding probability model (U,P, V) by demanding that, for any x, y and z in U, PX({z}) = P,({z}) > 0. As long as U does not contain more than countably many elements it is always possible to find such a probability model. If A is a formula which is not a theorem of S5, there is some finite model in which A is not valid. So, as before, A is not valid in a corresponding probability model and therefore not a theorem of QP + (A5) + (A6). This proves that the modal theorems of QP + (A5) + (A6) are contained in S5. Princeton University and University of Lund NOTES * The research for this paper was supported by a grant from the American-Scandinavian Foundation. The author wishes to thank Zoltan Domotor, Bengt Hansson, Glenn Kessler, David Lewis and Krister Segerberg for valuable help and criticism. ’ In contrast to Segerberg [ 121. QUALITATIVE PROBABILITY AS AN INTENSIONAL LOGIC 185 2 The reader may consult Hansson and Gedenfors [ 61 and [ 71, if he is unfamiliar with the construction of Lindenbaum-algebras in intensional logic. ’ If m = 0, we regard this as satisfied. 4 As before, 1 denotes the characteristic function of the set X. BIBLIOGRAPHY [l] Danielsson, S., ‘Modal Logic Based on Probability Theory’, The&a 33 (1967), 189-197. [ 21 De Finetti, F., ‘La pr8vision: ses lois logique, ses sources subjectives’, Ann. Inst. Poincar2 7 (1937), l-68. Relational Structures and Their Applications, Techn. [ 31 Domotor, Z., Probabilistic Report 144 (1969), Inst. for Math. Studies in the Sot. 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