Abstract
The hypothesis with which this chapter is concerned is that, in a sense to be explained, counterfactual conditionals like
If that bird were a canary then it would be yellow.
function as a kind of ‘epistemic past tense’, and in particular their probabilities at the time of utterance equal the probabilities which were or might have been attached to corresponding indicative conditionals like
If that bird is a canary then it will be yellow.
on real or hypothetical prior occasions. This hypothesis, which was earlier advanced in Adams [5] and independently by Skyrms in [50], will prove in the end to be untenable or at best dubious in complete generality. Nevertheless, it offers simple and plausible explanations for such a wide variety of logical phenomena involving the counterfactual that it merits detailed consideration in that one may reasonably expect a hypothesis like the present one to be central to any satisfactory general theory of the counterfactual.
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Notes
The implicit claim here, that premises of inferences should be reasons for arriving at conclusions, obviously takes premise’ in a narrower sense than is common in logical theory, where assumptions (such as occur in Reductio ad Absurdum arguments) are normally called premises.
The Bayesian theory of probability change, an illustration of which is given at the end of this section, is a standard part of elementary probability theory, and is usually given in the first two or three chapters of texts such as Feller [19] or Uspensky [58].
This version of Bayes’ Theorem, which generalizes to arbitrary hypotheses which may be neither mutually exclusive nor exhaustive in place of —C and C, is particularly useful in its neat separation of hypothesis likelihood ratios and inverse conditional probability ratios. It is appropriate to call this case of the inverse inference formula the probabilistic Modus Tollens formula because it appears to give a probabilistic generalization of Modus Tollens. We will encounter an analogous probabilistic Modus Ponens formula in Section 9.
In fact, the present urn and ball example can be looked upon as a stochastic model of the inference in the canary example. Analogous stochastic models can be given for many kinds of reasoning where there is a reason to pay attention to implicit uncertainties, and they give considerable insight into the question of how these uncertainties may affect the uncertainty of the conclusion.
It must be conceded that the subjunctive is rarely, if ever, used in textbook descriptions of inverse conditional probabilities (which are commonly described as ‘conditional probabilities’ simpliciter,or in some such form as “the probability of A,assuming that B”,as in Uspensky ([58], p. 61). I would hazard the following contentious conjecture concerning this. This is that probability theorists either do not conceive of, or ignore the possibility that conditional probabilities should be just as susceptible to change ‘in the light of new evidence’ as unconditional probabilities. At any rate, this would explain why ‘proofs’ such as those of I. J. Good [22], and Raiffa and Schlaiffer [44] that making new observations (as against refraining from observing) is always desirable (because it decreases expected entropic uncertainty and increases expected utility of the results of acting) normally go unquestioned in spite of obvious counterexamples which arise when the observer’s subjective conditional probabilities are mistaken in the sense of Section 11I.7 (cf. an example appearing in my review [6]).
This may be an example of ‘inference to the best explanation’ as considered in the writings of Harmon (e.g., [281). My attention was first drawn to such uses of the counterfactual by a nice instance in The Hound of the Baskervilles ([15], p. 684), where Sherlock Holmes begins an explanation with the words “if that were so, and it seems most probable,…”.
This ‘epistemic past tense’ reading of “it should have been (should be) the case that…” looks at least prima facie to be very close to Ryle’s analysis of the actual past tense in Concept of Mind (chapter on Memory, see also Shwayder [48]). If, as I think there is, there is something in Ryle’s view, this may show that the actual past is not to be analyzed independently of the counterfactual.
The identification of the deontic with the practical ought’ is borrowed from Vermazen [59], to which I am indebted for most of the little I know about Deontic Logic.
See, e.g., Vermazen (59), p. 17 or Nozick and Routley [43].
There are possible exceptions to the rule that future counterfactuals are equivalent to their corresponding indicatives, at least if the going to’ construction is regarded as future. Knowing for certain that B isn’t going to play in such and such a game, one might well affirm if A were the manager then B would be going to play“ but deny ”if A is the manager then B is going to play “On the other hand, this may be taken as one more bit of evidence that” is going to is logically present and not future. In any case, I have been unable to construct any other pure future’ counterexamples to the thesis that future counterfactuals are equivalent to indicatives.
Perhaps a consideration of the use of counterfactuals in descriptions of fair damages (e.g. damages equal to what an injured person would have been able to earn, if the injury had not occurred) would be useful in this connection. It is interesting that this type of counterfactual use also occurs in the rules of certain games, as in the rule prescribing the penalty for obstructing a runner or batter-runner in baseball, where it is stipulated that the runners shall advance to the bases they would have reached, in the umpire’s judgment, if there had been no obstruction ([9], p. 23).
For this reason it is appropriate to regard Equation (7) as the probabilistic generalization of the Modus Ponens inference rule.
This QR approximation theory is plausibly adaptable to the situation in which new information is acquired by observation,where the act of asking question Q is replaced by that of making an observation (e.g., looking to see what color some object is), and the response’ is replaced by the direct result of the observation (e.g., to learn that the object at least seems to have such and such a color). The model becomes, perhaps, philosophically suspect in this interpretation, because of its close connections with dubious sense data epistemologies.
The present QR approximation model should be contrasted with an approximate conditionalization model proposed by Jeffrey [34], and developed in detail by Harper [29], according to which the posterior probability, pi(’), arising from the acquisition of a somewhat uncertain new premise.1 is given by where a is an uncertainty parameter’ close to but generally somewhat less than 1. The Jeffrey model is inconsistent with the QR model in the sense that if a differs from 0 and 1 and Q and R are in the domain of definition of po, then the Jeffrey posterior probability given above cannot always equal the QR posterior probability, Po(Q & R.) (in particular, these probabilities cannot be equal for t’=Q & R). The two models are compatible in the weaker sense that, given any Jeffrey posterior probability as defined by the above weighted average over a fixed class of propositions C, it is possible to extend the domain of definition of po by adjoining new propositions Q and R to it in such a way that the QR posterior probability will equal the Jeffrey posterior probability for all propositions in the original domain of po. If we think of Q and R as being, in a sense, subjective hidden variables’, which should not be part of the public domain’ of po, perhaps the foregoing is enough to reconcile the two models.
Equation (8), together with the interpretations just given of its four terms, usefully pinpoints what goes wrong in certain alleged counterexamples to the rule of probability change by conditionalization, such as the following striking paradox of three job-seekers’ described in Gardner [20]. Three men A B and C, have applied for one job with a priori equal chances of getting it. The management has decided who is to get the job, but hasn’t yet announced its decision. Candidate A approaches the manager and asks whether he got the job, but is told that the decision can’t be made public yet. A then says “if you won’t tell me whether I got the job, at least tell me the name of one person besides myself who didn’t get it.” The manager relents to the extent of telling A that B didn’t get the job, whereupon A feels somewhat better because he now thinks his chances have improved (from 1/3 to ½) for getting the job. These improved chances are those which would follow if posterior probabilities were computed by conditionalizing on the new premise.B= −B (B didn’t get the job), yet it takes little thought to see that in fact being told that B didn’t get the job should not lead A to alter his prior estimate of his chances — because he could have foretold a priori that the manager would be able to tell him the name of someone besides himself who didn’t get the job. The trouble here lies in the relation between the question (or demand) that the manager should tell A the name of someone besides A who didn’t get the job, and the manager’s response that B didn’t get the job. The analogue of the informedness and helpfulness premise does not hold in this case, because it is not highly probable a priori that if this question is asked and B didn’t get the job, then the manager will respond that B didn’t get the job (if neither B nor C got the job, the manager might just as easily have replied that C didn’t get the job). In fact, here the informedness and helpfulness uncertainty u o(Q & B → R)can be as high as ¼, hence the proper QR posterior probability that A got the job, which in this case equals the prior probability of 1/3, can differ from the B-approximation (which is ½) by as much as ¼, even if all other uncertainties are 0.
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Adams, E.W. (1975). A Hypothesis Concerning Counterfactuals; Probability Change Aspects of Inference. In: The Logic of Conditionals. Synthese Library, vol 86. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7622-2_4
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