a note on comparing probabilistic and modal logics of conditionals

A note on comparing probabilistic and modal logics of conditionals

by

E R N E S T W. A D A M S

(University of California, Berkeley)

THOUGH Stalnaker’s modal senzmtics for conditional propositions (see ( 1 1) and (13), as well as David Lewis’ theories in (6) and (7) , with which it will be most convenient for us to work here)’ appears to differ radically from my probabilistic semantics (see ( l ) , (2), and (4)),2 nevertheless the theories of deductive soundnes.r. based upon the two kinds of semantics exhibit striking similarities. The purpose of this note is to state the exact relation between the two deductive soundness theories, and then to speculate briefly on a possible explanation for their similarity.

The formal expressions of concern to us are built from some unspecified list of atomic formulas, compounded by the standard sentential connectives c-’, ‘&’, ‘ v’, and ‘3’ for negation conjunction. disjunction, and the material conditional, respectively), plus

’ We shall ignore here the question of whether these modal theories are properly applicable to indicative, conditionals, which Stalnaker takes them to be. or to counter- fuctuals (subjunctives), which Lewis takes them to be. Since I hold my theory to apply to indicatives (with a variant prior prohuhilitj, version applying at least to some counterfactuals, as explained in Chapter IV of (4)), the most appropriate comparison is with the indicative conditional application of the modal theory. Note that the objection raised by Nute (9) to the Stdlnaker-Lewis theory is specifically to its counterfactual interpretation.

There are minor variations among the semantics described in these works, chiefly as they relate to the treatment of conditionals whose antecedents have probability 0. 1 will here adopt the convention of the earlier articles. and stipulate that such conditionals have probability 1. The alternative theory described in section 9 of (2). and the still different one described in (4) less strongly resemble the Stalnaker-Lewis theory. In spite of this. though, I will utilize a modified version of the completeness proof given for the latter system, and whose application to the ‘antecedent probability 0 implies conditional probability 1‘ semantics was originally given in (3) .

LOGICS OF CONDITIONALS 187

(to use Stalnaker’s symbolism) the special symbol ‘>’ for the ‘ordinary language conditional’. My probabilistically interpreted formal language comprises those of the above formulas without any occurr- ences a t all of ‘>’ (fuctual,formulus), or with just one such occurrence, which must be as the main connective (simple conditionalformulus). For the purpose of comparing the two theories we will consider their application to simple condition~tl inferences, which are inferences all of whose premises and whose conclusions are either factual or simple conditional formulas.

Using David Lewis’ ‘spheres of possible worlds’ semantics to determine the soundness of inferences, it will be sufficient for our purposes to consider ,finite, cenrered, sphere-system models, ( I , S>, where I is a finite set ofpossible ~~)rlds~ and S is a function mapping possible worlds i in I into centered, nested increasing sequences of subsets of 1: i.e., into sequences

s‘= s;, s;, . . . where Sl={if (centering) and S ~ C S ~ + ~ for k = 1 , 2. . . . (nested increasing property). Given a sphere-system we also have associated interpretation functions, [ J, mapping formulas A into subsets [ A ] of I , subject to the usual requirements for sentential compounds (e.g., [ - A] = I - [ A], and [ A & T?] = [ A]n[ B ] ) , plus the special interpretation rule for ordinary language conditionals:

for all i in I , i ~ [ A > ’ i ] if and only if either [A]n($US;U ...) is empty, or [A] f lSb.~[ t3 ] for the smallest n such that [A]nS;t is nonempty.

The latter is a formal way of saying that A> i3 is true in possible world i if and only if either A is not true in any possible world, or else B is true in all worlds ‘sufficiently close to i‘ in which A is true. Given this semantics, then premises A , , ..., A, m-mtuil (modal- model entail) a conclusion T? if there is no sphere-system model

’ Restriction to finite sets of possible worlds is justified by Lewis‘ results, pp. 134-136 of (7). showing that if a counterexample exists to an inference with a finite number of premises. then a counterexample with a finite number of possible worlds exists. The finiteness restriction also justifies the simplified representation of systems of spheres given here.

188 ERNEST W. ADAMS

( I , S), interpretation function [ ] for this model, and possible world i in I such that i€ [Aj ] f o r j = 1, . .., n, but i$[B].

My probabilistic semantics are defined precisely in the articles cited, and will only be informally described here. We may begin as before with a nonempty finite set I of possible worlds and interpretation functions [I, which, however, only assign ‘possible world extensions’ to factual formulas. Sphere-systems are not needed because we do not attach truth-conditional interpretations to conditional formulas. Instead we work with possible prohuhility functions, p , associated with I and [ I which are generated by arbitrary probability distributions over I , and which assign probabilities to factual and simple conditional formulas according to the rules: if cp and $ are factual formulas then

and

Now factual and simple conditional premises A, , . . ., A, are defined to p-entail (probabilistically entail) a conclusion i3 of the same kind in case the following ‘certifiability of the conclusion’ requirement is satisfied: either there exists 6 > 0 such that there is no nonempty set I , interpretation function [ 1, and probability function p associated with I a n d [ I such tha tp (Aj )z l -6 f o r j = l , ..., n; or for all E > O there exists 6 > 0 such that for any probability function p, if p( A j ) z 1-6 for, j= I , . . ., n, then p( i3)z 1 -E. The intuitive meaning of this condition is that if premises p-entail a conclusion then it is possible to assure arbitrarily high probability (short of certainty) in the conclusion by requiring sufficiently high probabilities in the premises (short of certainty).

THEOREM. Let A,, . . ., A, and i3 be,factual and/or simple conditional formulas. Then A,, . . ., A, p-entail i3 i f und only if they iwentuil a.

Proof: Suppose first that A , , ..., A, p-entail a. I t is shown in (3)


that this can only be the case if there is some subset of A,, . . ., A, whose quasi-conjunction p-entails El. The quasi-conjunction of simple conditional formulas cpl > t,hl, . . ., cp, > $,,, is defined to be the simple conditional

(40, v”’vcpm)>((cp1=$1) “ ‘ k ( c p m 2 $ r n ) ) ,

and this operation is extended to factual formulas cp by the device of identifying cp with T>cp, where T is an arbitrary tautology, and where the quasi-conjunction of the empty set of formulas is stipulated to be T. Furthermore this quasi-conjunction can only p-entail B if either: (1) B is p-entailed by T, or (2.1) the material counterpart of E (obtained by replacing ‘ >’ by 2’ in 8) is a tautological consequence of the material counterpart of the quasi-conjunction and (2.2) the conjunction of the antecedent and consequent of the quasi-conjunction tautologically implies the conjunction of the antecedent and consequent of B.

It is elementary both that any finite set of formulas (simple conditional or not) m-entails its quasi-conjunction, and that if the latter satisfies either of conditions (1) or (2.1) and (2.2) above, then it must m-entail B. To show the former, suppose possible world i belongs to [ ~ p ~ > $ ~ ] f o r j = l , ..., m. Either [cpj]fl(SiUS&J ...) is empty for all , j= 1, . . ., in, in which case clearly i will belong to the extension of the quasi-conjunction, or there will be some minimal n such that for some,;, [cpj]nSi is nonempty. Then, in virtue of the nested increasing property of Sl, Si, . . . and of the fact that i E [ c p j > $ j ] f o r j = l , ..., itfollowstriviallythat[cp, v . . . vcp,]nSi~[(cp,=$,)& . . . &(40,=$,)] -i.e., i is in the extension of the quasi-conjunction.

Similarly, suppose that the quasi-conjunction is abbreviated cp > $ and B is the conditional q>p. If B is p-entailed by T, it is shown in ( 3 ) that this can only happen if q ~ p is tautological, in which case clearly B is m-entailed by the empty set of premises. If cp>$ and q > p satisfy (2.1) and (2.2) then cp 2 $ tautologically implies q 2 p and cp & $ tautologically implies q & p. Suppose i ~ [ c p > $1, and Si is minimal such that [q]nS, is nonempty (if this minimum doesn’t exist clearly i will belong to [ i I > p ] ) . Either [cp] f lSj is empty or not. In the former case Si G [ - cp] E [cp 3 $1, and since cp 3 $ tautologically implies q>p , S ; C [ V ~ ~ ] , hence [q ]nSb~[pL] , so i E [ q > p ] . If [cp]nSl

190 ERNEST W. ADAMS

is nonempty, let k s n be minimal such that [cp]nSk is nonempty. Since iE[cp>$], [cp]ns:~[t)]. Since cp & I) tautologically implies q & p , hence [cp]n[t)]c[q]fl[p], i t follows that [q]nSi is nonempty. Hence k = n since n was minimal such that [ q ] n Si was non-empty. Furthermore, given that [cp]n[t)]~[q]n[p] and Sh= Si, [q]nSh~[p]- i.e., i ~ [ q > p ] . We have shown that if A,, . . ., A, p-entail B then they m-entail 8.

Now suppose that A,, ..., A, do not p-entail a. It is shown in (3) that in this case there must exist a nested decreasing sequence of subsets of the premise set a = { A , , . . ., A,}, say P , , . . ., P, , and a corresponding sequence of truth-assignments to factual formulas, t , , . . ., t,, with the following properties: (1) each f j , j = 1, . . ., k - 1 , verifies (in the sense of making both antecedent and consequent of a conditional formula true) at least one formula in P P and does not Julsifv (in the sense of making antecedent true and consequent false) any premise in this set; (2) P j + ,, j = 1, . . ., k - 1, is the set of all premises in P j not verified by t j ; (3) t j , , j= 1, . . ., k - 1, does not verify the conclusion 8; (4) t , does not falsify any formula in Qk, but it does falsify the conclusion, B. Intuitively, if we rank the truth assignments t , , ..., t , ‘from the earlier to later’, then any premise which is ever falsified has been previously verified, while the conclusion is never verified and it is ultimately falsified.

Now we construct our modal-model counterexample by taking I to be the set { t l , . . ., t,} above, and by taking our sphere-sequences to be the ‘linear sequences’

Si= { l j ) , { t j , t j + ,), . . ., ( t j , . . ., r,,. As our interpretation function [ I we may take the natural one generated by the truth-assignments, where for each atomic cp, [cp] is the set of t j such that cp is true in t j , and the extensions of compound formulas (including the ordinary language conditionals) are defined by the rules already stipulated. Now taking t , as our possible world, it is straightforward to verify that r l ~ [ A j ] for j = 1, ..., n, while I , #[B]-hence B is not m-entailed by A , , ..., A,. This concludes the proof.

The problem of explaining the convergence between the deductive soundness theories considered is one deserving detailed consideration,


but I will only make some brief speculations on it here. I would sug- gest that, at least in its application to simple conditional propositions and inferences, the Stalnaker-Lewis theory of ‘nearest possible worlds’ can be interpreted as a theory of worlds ‘nearest in probability’, and that under this interpretation this logic is the logic, not of truth, but of high probability, or ‘acceptability’. Recall the sequence t , , . . ., 1, of truth-assignments, used in proving the Theorem to give a modal counterexample to the inference of B from A,, ..., A,. This same sequence can be used to construct a ‘probabilistic counterexample’ to the inference by assigning probabilities

p(r l )= 1 --E,

p(tj)=&j-’(I--E)forj=l, ..., k-1, P ( t k ) = E k .

Under this probability-assignment, and granted that t , , . . ., t , satisfy conditions (1)-(4) stipulated in the proof for the construction of t , , . .., tk , p( A j ) z 1 --E f o r j = 1, ..., 12, but p( B)=O-i.e., with these probability-assignments the premises can be arbitrarily highly probable while the conclusion has probability 0. But observe that here the world closest in probability to any t j is t j + , , and that this is the ‘nearness ordering’ reflected in the sphere-system in the modal co~nterexample.~ Furthermore, conditonal propositions are Stal- naker-Lewis ‘true’ in t , just in case the nearest world t j in which they are verified precedes the nearest one in which they are falsified-i.e., they are ‘true’ if they are more likely to be verified than falsified, which is precisely what is required for their conditional probabilities to be high. In fact, this is just what seems to me to provide the intuitive rationale for the Stalnaker semantics in its application to indicative conditionals.

Of course, the foregoing suggests that a searching evaluation of the Stalnaker-Lewis theory as disfincf from the probabilistic one, should focus on propositions more complicated than simple conditionals, and on more complex inferences than simple conditional

It is worth noting here the formal identity between the ’possibility orderings’ described by Lewis in (7). pp. 52-56, and the P-ordering (probability orderings) which I introduce in (2). p. 284, in an early version of the completeness proof in the probabilistic logic.

192 ERNEST W. ADAMS

inferences, the study of which seems to me to be a neglected task. I will only observe here that the question of which propositions of these sorts are true or probable, and which inferences are sound, seems to me to be somewhat more controversial than is the analogous question concerning simple conditionals and their inferences5

In concluding I would like to observe that the apparent convergence between the modal and the probabilistic soundness theories in application to simple conditional inferences has its limits. To the extent that probabilities as against truth-values matter at all in application to the propositions entering into these inferences, it is probably far more important practically in assessing these patterns to know h o ~ . improbable conclusions can be, given bounds on the probabilities of the premises, than it is to know that the premises do or do not p-entail (or m-entail) the conclusion. In investigating this matter in application to inferences involving factual premises alone, Howard Levine and I (5) introduce what I will here call uncertainty transmission ,functions, which, given an inference with premises A,, ..., A, and conclusion a, specify the maximum conclusion uncertainty,

max{u(B): u(A,)SE,, ..., u ( A , ) ~ E , )

(where uncertainty is 1 minus probability-not to be confused with entropic or information-theoretic uncertainty), compatible with each premise, Aj’s having an uncertainty no more than E ~ . What is striking is that in application to simple conditional inferences, the uncertainty bounds given by the probabilistic theory yield, or appear to yield, radically different results from those ‘computed’ on the basis of the modal semantics. The Modus Tollens pattern, with premises

Lewis’ striking comment: “The principal virtue and the principal vice of Stalnaker’s theory is that it makes valid the law of the Conditional Evcluded Middle: (cp > $) v (cp > - $).” ((7), p. 79).

seems symptomatic of the present situation with regard to compounds with conditional constituents (cf. also Nute’s objection to this law in (9)). Brian Skyrms (personal communication) has told me that in developing a second-order probability theory applicable to compounds with conditional constituents, he can show that alniost no inference patterns involving such constructions are universally valid (according to his probabilistic semantics).


A, = ‘ A > B’ and A, =‘ - B’ and conclusion B=‘- A’, is a case in point. On the probabilistic semantics, the corresponding uncertainty transmission function is given by:6

In contrast, assuming the modal semantics and assuming thatprobabil- ities are defined as probablities of truth, and that all possible distributions of probabilities over possible worlds define possible probability functions, the uncertainty transmission function for this inference must be a linear function of the uncertainties of the premises (or a mi-

mum of such functions), and in fact, if both premises are essential t the inference, the function must simply be E~ +s2. Thus, the way rhich the conclusion depends on the conditional premise in Modus ’ollens comes out very differently when the inference is analyzed nodally from the way it does when analyzed probabilistically.’

References

[ I ] ADAMS, E. W. “The logic of conditionals”. Inquirj, vol. 8 (1965), pp. 166-197. [2] ADAMS, E. W. “Probability and the logic of conditionals.” In J . Hintikka and

P. Suppes (eds.), Aspects of inductive logic (North-Holland, 1966), pp. 265-316.

This is interesting in its own right. I t shows that for the purpose of drawing the conclusion of the Modus Tollens inference, it is sufficient that the first premise (the conditional premise) only be not improbable, rather than, as one might expect, that it should actually be probable. This changes radically, however, in the rwv-stugu Modus Tollens inference in which the conditional premise is learned first and then the factual premise is learned. This is discussed in section 4.1 of (4). ’ There are controversial questions implicit here, connected with the characterization of the ‘space’ of possible probability functions. In arriving at the second result 1 assumed all distributions over possible worlds were possible, and that the probability of a proposition equalled the measure of the class of possible worlds in which it was true. Stalnaker (12) implicitly rejects either or both of these assumptions, for, as David Lewis’ important triviuliry rrsulrs (8) show, one cannot simultaneously assume: ( 1 ) probabilities of conditionals are conditional probabilities (which Stalnaker does assume), (2) all probability distributions over possible worlds define probability functions. and (3) that probabilities of propositions are sums of the probabilities of the possible worlds in which they are true. Articles of Skyrms (10) and van Fraassen (14) throw interesting light on these issues.

13 ~ Theoria 3:1977

194 ERNEST W. ADAMS

[3] ADAMS, E. W. “The logic of ‘almost all’”, Journal of philosophical logic, vol. 3 (1974), pp. 3-17.

[4] ADAMS, E. W. The logic of conditionals: an application of probability to deductive logic. (D. Reidel, 1975).

[S] ADAMS, E. W. and LEVINE, H. P. “On the uncertainties transmitted from premises to conclusions in deductive inferences.“ Synthese, vol. 30 (1975), pp. 429-460.

[6] LEWIS, D. “Completeness and decidability in three logics of counterfactual conditionals.” Theoria, vol. 37 (1971), pp. 74-85.

[7] LEWIS, D. Counterfactuals. (Basil Blackwell, 1973). [8] LEWIS, D. “Probabilities of conditionals and conditional probabilities.” The

philosophical review, vol. 85 (1976), pp. 291-315. [9] NUTE, D. “Counterfactuals.” Notre Dame journal of,formal logic, vol. 16 (1975).

pp. 416-482. [lo] SKYRMS, B. “Can the inferential probability conditional survive David Lewis‘

Triviality Proof?” Manuscript. (Department of philosophy, University of Illinois, Chicago Circle, 1974).

[ I I] STALNAKER, R. C. “A theory of conditionals.’’ In N. Rescher (ed.), Studies in logicai theory (Basil Blackwell, 1968).

[I21 STALNAKER, R. C. “Probability and conditionals.’’ Phi1osoph.y qfscience, vol. 37 (1970), pp. 68-80.

[I31 STALNAKER, R. C. and THOMASON, R. H. “A semantic analysis of conditional logic.” Theoria, vol. 36 (1970), pp. 23-42.

[I41 VAN FRAASSEN, BAS C. “Probabilities of conditionals.” In W. Harper and C. A. Hooker (eds.), Foundations and philosophy of statistical theories (D. Reidel, 1976).

Received June 13, 1977.

a note on comparing probabilistic and modal logics of conditionals

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