Single-case Probability and Logical Form Colin McGinn Mind, New Series, Vol. 88, No. 350. (Apr., 1979), pp. 276-279. Stable URL: http://links.jstor.org/sici?sici=0026-4423%28197904%292%3A88%3A350%3C276%3ASPALF%3E2.0.CO%3B2-T Mind is currently published by Oxford University Press.
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Single-case Probability and Logical Form COLIN MCGINN
My starting-point is a puzzle about singular probability statements concerning particular events. Suppose a fair coin is tossed and falls heads uppermost. We would judge that the probability of this outcome for that trial event is 0 . 5 . But it may be objected that this estimate cannot really be objectively true, since-granted determinism-if we had known in detail all the causally relevant facts about the particular trial and the laws of nature, we would have been able to predict with certainty that . the outcome be heads, so that the objective probability must have been unity. If we wish probability values intermediate between zero and unity, the argument is apt to continue, we had better construe such estimates either as in some way subjective, reflecting perhaps the estimator's propensity to gamble in a certain way, or as somehow relational, perhaps to our current state of knowledge. In this note I shall try to dispel the puzzle by proposing an account of the logical form (semantics) of singular probability statements. Take as paradigm sentence 'It is 0.5 probable that the next toss of this coin will fall heads'. Ignoring tense, indexicality and irrelevant complexity the natural way to render the form of this sentence is: where 'H' describes the outcome, '(?e)(Fe)' denotes the particular trial event and 'Pn' represents the assigned probability value. We said that when 'F' is replaced by 'toss of a fair coin' (and other devices to secure unique reference) the correct value of 'n', where 'H' is 'heads', is 0 . 5 . The determinist argued, in effect, that supplied with further descriptions of the event we could be in a position to assign a different value to 'n', viz. I. On pain of inconsistency we must reject one of these estimates, and it seems more plausible to reject the first, since it is based on less information about the event. The underlying difficulty is that the single case can be variously described-i.e. satisfies a large number of descriptions, more or less full-and that these dictate different assignments of probability in respect of the instantiation of some specified outcome. But from the fact that the value of 'n' varies according to which description is selected (in practice or principle) it follows neither that we have an inconsistency, forcing us to choose some one among the true descriptions as the 'designated description', nor that singular probability statements are elliptical, being relational in logical form. For it may simply be that the probability context is referentially opaque. The supposition that i t ~ i sopaque is well-motivated, inasmuch as it offers an answer to some of the puzzles that afflict single-case probabilities; but can it be independently substantiated? Well, consider 'It is likely (but not certain) that the next world war will be the last', 'It
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was improbable that the evening star should be identical with the morning star', 'The chances of Mister Gay winning the Grand National are slim'. If we suppose that in fact the next war is the last, that the evening star is the morning star, and that Mister Gay was, against all odds, the winner of the Grand National, then substitution of descriptions on the basis of those facts will not leave truth-value unchanged. Similarly, there will exist descriptions of our original coin-tossing substitution of which for 'the tossing of a fair coin by p at t' will not yield truth where 'n' is assigned 0.5. I t seems, then, that singular probability statements deal in particular events and objects only under this or that description; probability estimates for the single case are geared to sentences or propositions, and not extensionally to the entities sentences are about. We are prone to forget that different descriptions yield different probabilities because we typically have a special interest in some one among all the descriptions satisfied by the event, perhaps through paucity of knowledge. Discerning opacity gives a way of allowing a certain description to be 'preferred' in deciding upon a probability estimate, without denying that other descriptions exist with respect to which a different estimate is indicated.l I t is important to this account that the definite description in (I) be accorded narrow scope; for, as with other opaque contexts, transparency results upon broadening the scope of the description to include the context with respect to which it occupied, on its narrow scope reading, opaque position. But it is a question what should be said of intermediate probability values when the description has wider scope than the probability operator. So consider, expanding the description Russell-style: ~robabilitvstatement'. What are its Let us call this a 'de re sineular " truth-conditions? Intuition does not, I think, suggest a clear answer, perhaps because such narrow scope readings of the probability context are seldom intended. But the most plausible view, it seems to me-and there may be an element of stipulation in this-is that, according as whether e does or does not instantiate H, (2) is true only where 'n' is assigned unity or zero, respectively. That is, for the de re reading truth and maximum probability coincide. This view is best appreciated by comparing the case with what one wants to say about nomological contexts. So now consider 'It is determined by natural law that'. Since individual events instantiate laws onlv as described in vocabularv itself fit for incorporation in to a general law, the context just mentioned is o p a q ~ e But . ~ if this operator is given narrow scope with respect to the I
2
An independent advantage of this proposal is that unless we find the probability context opaque we shall have to contend with the well-known argument that, if a context is referentially transparent and invariant in truth-value under substitution of logically equivalent sentences, then it must be truth-functional; which probability contexts patently are not. (For a statement of the argument see my 'A Note on the Frege Argument', Mind, July 1976.) Cf. D. Davidson, 'Mental Events', in Experience and Theory, L. Foster and J. W. Swanson (eds.) (London: Duckworth, 1970)~p. 89.
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event description(s), then the natural view of the sentence's truthconditions is that it is true if there exists some description-it need not alreadv occur in the sentence-under which the denoted events instantiate a law. That is, a sentence affirming nomic determination of certain described events admits of two readings: when the descriptions have narrow scope the sentence is true only if the predicates comprised in the descriptions themselves feature in the relevant covering law; but when the scopes are reversed the sentence is true irrespective of whether the predicates in the descriptions are such as to feature in the covering law whose existence is entailed-all that is required is that there exist some description substitution of which into the original sentence yields a truth when accorded narrow scope. Now I wish to maintain that something similar holds for (objective) probability contexts: if there is some description under which the event has a unity probability of instantiating the predicated outcome, and there will exist such a description if determinism is true, then the de re reading of the singular probability sentence is true if 'n' is assigned I, and not otherwise. In short, granted determinism, there are no 'objectively true' intermediate value de re singular probability statements.l Nor is it very surprising that intermediate values should attach only to the de dicto probability statement, given the close connection between (at least one notion of) objective probability and natural laws. I have claimed that the probability value to be attached to an event in respect of a given outcome is sensitive to how the event is described. But how exactly do the various event descriptions determine a unique probability value? Here I wish my account of logical form to be neutral, but it is worth noting how readily it consorts with accounts of probability for event types in terms of relative frequency or statistical laws. However we represent the logical form of a statement of relative frequency-as a probabilistically modified universal conditional or as a two-argument functor-we can say the following about the relation between the singular and the general probability statement. If the relative frequency of outcome H in population F is n, then a singular sentence formed by describing a particular event with the predicate 'F' (and associated uniqueness devices) and predicating 'H' of that event has probability n too. A particular event may be described in ways which relate it to many such relative frequency statements, and it will inherit the probabilities of the general statements to which it is thus related. But no inconsistency threatens because the context is opaque. Nor are we obliged to resort to claiming ellipsis in the singular statement, as if it were short for some sort of conditional statement whose antecedent consisted of the relative frequency conditional or again some evidential statement. We can cleave to sentence forms we already understand. Note that this claim is conditional. If determinism is not true-i.e. there are events, e.g. quantum phenomena, which satisfy no description under which they instantiate a (deterministic) law-then we can allow intermediate value de re probability statements, their value being fixed by the probability value of the corresponding irreducibly statistical law. This suggests a distinction of strong and weak indeterminism based on the availability of non-zero non-unity probability values for de re and de dicto readings of probability sentences.
,
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But now it may be protested that we don't understand the logical form I have so far been suggesting, because we don't understand opacity. More exactly, it might be asked how the logical form exhibited in (I), with its non-extensional sentence operator, is to be accommodated in an adequate semantics. My reply is that the account suggested is tailor-made for truth-theoretic treatment. We can treat probability sentences as Davidson treated sentences of ovatio obliqual: construe what is in surface structure one sentence as two, the particle 'that' serving, on an occasion of utterance, to refer demonstratively to the succeeding sentence-token, thus I t is 0.5 probable that. The next toss will be heads. Non-extensiona!ity dissolves, though its effect remains, since substitution of co-referential terms in the second sentence, while leaving its truth, value unchanged, may be counted upon to change the truth-value of the first sentence. This Davidsonian analysis fits probability sentences out for incorporation into a Tarskian truth-definition, and it gives a clear sense to our earlier contention that probability estimates attach primarily to sentences and not to the entities of which sentences speak (which is not to say that they are merely 'linguistic'). There is much about single-case probabilities on which I have not touched, but I would urge, echoing Davidson, that unless and until we get clear about the logical form of a certain kind of sentence we shall not be in a position to be clear about that kind of sentence. UNIVERSITY COLLEGE LONDON I
In 'On Saying That', Synthese, xix (1968-9).
