“They say that Understanding ought to work by the rules of right reason. These rules are, or ought to be, contained in Logic; but the actual science of logic is conversant at present only with things either certain, impossible, or entirely doubtful, none of which (fortunately) we have to reason on. Therefore the true logic for this world is the calculus of Probabilities, which takes account of the magnitude of the probability which is, or ought to be, in a reasonable man’s mind.”
James Clerk Maxwell, quoted in Jeffreys (1967) p 1.
Scientific methods have always made abundant use of evidence, from which they infer the truth or untruth [or, if you will, utility or dis-utility] of hypotheses and forecast patterns of future experience (see, for an illustration, Shimony 1970 & Jaynes 2003). This fact is most obvious in the post-Kuhnian times in which we find ourselves. Therefore, it is natural that a perennial concern within the philosophy of science is how and in what manner are statements of evidence related to hypotheses. [Note: I shall use ‘hypothesis’ and ‘beliefs’, and their cognates, synonymously and therefore interchangeably.] In other words, what constitutes the conditions, necessary and sufficient, for the following:
(1) A statement, e, is evidence for a hypothesis, h.
Attempts at an answer often come in the appearance of formal schemas, wherein the evidence statements serve as premises in an argument that has for its conclusion a logical consequence that follows from the evidence (let us call these formal structures evidential schemas). In that evidential schemas are arguments intended to explicate the desired logical relationship between evidence and theory, we may speak of them as being valid or invalid. A valid argument confers certain, demonstrable truth upon the conclusion via only its form and the meanings of the premise(s) and conclusion, which is to say that if the premises are true, then the conclusion must by necessity be true. The domain of study of valid arguments is of course deductive logic. However, no matter how close the approximation to deductive logic, evidential schemas are not, nor can they be, deductively valid. Rather, insofar as evidential schemas seek to explicate the dynamics of belief change, they are inductive enterprises. This, of course, is to understand induction in the most general sense of the word. As used here, it means simply any schema that employs inferences of a non-deductive nature, which, rather than guaranteeing truth, lend varying degrees of support to beliefs.
Thus, per above, science is in essence an inductive endeavor. If the the nature of scientific inferences- that is, the relationship between evidence and hypotheses- is to be understood, it must be done so via a logic of induction, and the Bayesian account, which has met with much popularity in the past twenty years, appears to be the most promising candidate. It operates as a measure of a state of knowledge and an algorithm by which epistemic agents update beliefs given the addition of evidence (E.T. Jaynes 2003). The algorithm (various forms of Bayes’ Theorem) itself is premised upon an axiomatized system, Kolmogorov’s axioms of probabilities, mapped onto a Boolean algebra. Hence, Bayesian probability is quite literally a logic of uncertain inference.
The standard Bayesian account of evidence is the increase-in-probability (positive relevance) view (to be defined below). Positive relevance is taken by many (see Howson and Urbach 2006, Jaynes 2003, Roush 2005, to name only a few) to be at least a necessary condition to an account of evidence, and thus an essential part in scientific inference. However, Peter Achinstein contends positive relevance is not a necessary condition for evidence, let alone an essential part of science. He makes his argument through the use of a putative counter-example, wherein he attempts to show that some statement can be evidence for a hypothesis without increasing the likelihood of the hypothesis; indeed, his example purports to show that a statement can be evidence for a hypothesis even if it decreases the probability of the hypothesis. Achinstein remains perhaps the most vocal critic of the positive relevance view, and it is necessary that his criticism be answered. Below, I will attempt to do just that and show why his counter-example does not give us good reason to reject the positive relevance view. [Note: Dr Achinstein has offered other counter-examples which will not be addressed here.]
The positive relevance view of evidence is an account of confirmation. Confirmation- the organized testing of hypotheses via observation and/or experimentation- has played a primary role in the history and methodology of science, and will continue to do so for the foreseeable future. Indeed, in many important respects, the systematic practice of confirmation and dis-confirmation are the defining scientific practices.
On the positive relevance view, then, a hypothesis, h, is confirmed if and only if
(a) p(h|e) > p(h), which reads, the posterior probability of h given the evidence, e, is greater than the prior probability of h when e was not considered.
Similarly, h is dis-confirmed by e if and only if
(b) p(h|e) < p(h)
And e is neutral in relation to h when
(c) p(h|e) = p(h); that is, when e neither increases nor decreases the prior probability of h
It will be helpful to note that within the Bayesian view probabilities may be objective when representing stochastic events. However, in respect to epistemic agents, probabilities are interpreted as degrees of belief, subjectively determined by an agent’s epistemic context. When the events are stochastic, such as coin flips, the degree of belief should map onto the objective probability. When the events are non-stochastic, like, for instance, the likelihood of your car being in the driveway as you read this paper, the degree of belief may differ from the objective probability (there in fact may be no objective probability, or at least none accessible to the agent). While the details of the positive relevance view have been refined with varying degrees of precision, it remains simple, extremely intuitive, and therefore explanatory. Of course, I have neglected treatment of the intricacies, but the above should set the groundwork for the remainder of the paper.
Positive Relevance Challenged
Peter Achinstein (2001 & 2005) rejects, as neither necessary nor sufficient, the condition of evidence stipulated by the positive relevance view, which may be defined as follows:
(a) A statement, e, serves as evidence for hypothesis, h, if and only if p(h|e) > p(h) [Note: p(x) should be interpreted as degrees of belief, to include beliefs in non- stochastic events.]
He argues for this claim via the following counter-example, which is as follows:
h = Bill Clinton will win the lottery
e1 = The New York Times (NYT) reports that Clinton owns all but one of the tickets.
e2 = The Washington Post (WP) reports that Clinton owns all but one of the tickets.
b = background information that the lottery is fair and the total amount of tickets is exactly 1000.
Achinstein purports e2 to be evidence for h given e1 & b, despite
(1) p(h|e2 & e1 & b) = p(h|e1 & b) = .999.
In other words, he contends that though e2 does not increase the probability of h, we are wont to consider it as evidence for h. Achinstein here quite obviously appeals to an unstated assumption, i.e., our degree of belief in the veracity of e1 and e2 equal 1. As Sherrilyn Roush (2005) notes, (1) can only be true if and only if e1 and b make our degree of belief in h2, viz., Bill Clinton in fact owns all but one of the lottery tickets, certain, which we may represent as:
(2) p(h2| e1 & b) = 1
However, considering the inductive risk involved in all empirical inferences, it is not the case that e1 is a perfect truth-bearer, thus (2) is false, and if (2) is false, then (1) is false. If (1) is false, then Achinstein’s counter-example fails to be a challenge to positive relevance. Nevertheless, even recognizing the falsity of (2), Achinstein reprises his counter-example, and provides two further versions.
In the first, the ideal case, we are to suppose the NYT and the WP to be infallible sources of evidence, (2) and (1) would both be true. If (1) and (2) are true, then Achinstein’s initial counter-example holds:
(3) Given the NYT report (e1), the WP report (e2) serves as evidence for h despite the fact that e2 fails to increase the probability that h.
In the second, the real case, let N = the NYT report is correct and W = the WP report is correct. Now, given N and W, Achinstein claims e2 is evidence for h,
(4) Given W & e1 & N & b, e2 is evidence for h
In short, given N and W, p(h|e2 & W & e1 & N & b) = p(h|W & e1 & N & b) = .999, which leads Achinstein to conclude that (4) violates the requirement for positive relevance. Moreover, in the ideal case, (3) likewise violates the requirement for positive relevance. Thus, per Achinstein’s counter-example, and despite the falsity of (2), positive relevance is neither necessary nor sufficient for a condition of evidence.
Positive Relevance Defended
Let us be clear about what Achinstein’s counter-example purports to do by first stating what is not disputed.
(i) In the real world, strictly speaking, (2) above is false because of the nature of inductive inferences.
(ii) Thus, because (2) is false, (1) is also false.
(iii) If taken separately, either the NYT report or the WP report ought to be considered evidence for our belief that Bill Clinton will win the lottery. This intuition is of course endorsed by the positive relevance view. The probability of Bill Clinton winning a 1000-ticket lottery is undoubtedly increased by the NYT report that he owns 999 tickets. Consider the following. Suppose we were confidant in the fact that Bill Clinton owned one ticket and were aware of the relevant background information (b), then our degree of belief in h would be .001. Now, given the NYT report, we would appropriately update our belief in h. We will, however, update our belief in h in proportion to our confidence in the veracity of the NYT report. This is so because the report would improve our epistemic context in relation to h (i.e., would give us more reason than we had before to believe h is true). All things being equal, we would in all likelihood measurably increase our degree of belief in h. Thus, the p(h|e1 & b) would be greater than the p(h|b) = .001; likewise for the WP report.
Now, having stated (i) – (iii), it seems odd that Achinstein asserts the falsity of (2) while at the same time asserting the truth of (4). Indeed, both are real-world cases involving the evidential merit of e2 given h, b, and e1. Had Achinstein not introduced in the ideal case the further supposition that both the NYT and the WP always correct in their reports, he would have contradicted himself in asserting (4) and not (2). Moreover, without (3), which is derived from the ideal case, then (4) loses its relevance. Thus, it is to the ideal case that scrutiny must be given if Achinstein’s counter-example is to be defeated.
In the ideal case, he has us assume that the NYT and the WP “always [get] it right”. Although being important to the case at hand, however, Achinstein does not inform us if we know that the NYT and the WP are ultra-reliable truth tellers. It appears that we have only two options: either we know that the NYT and the WP are ultra-reliable (the known-case), or we do not (the unknown-case).
If we do, then if we read, say, the NYT report first, we would have no need to read the WP report because our degree of belief would have been raised to the highest degree permitted under the possible available evidence. In other words, the epistemic value of either e1 or e2 in the known-case would at the very least equal or exceed the epistemic value of the conjunction of e1 and e2 in the unknown-case; that is, insofar the epistemic value is determined by approximating a degree of belief of .999 in h. [Note: This is analogous to the fact that scientific experiments may lend support to a hypothesis to a limit, whereupon, when reached, other experiments or observations must be devised.] If we read the NYT report first, the WP report would become superfluous and thus add nothing to the epistemic context. Hence, we also find in the known-case that even though (2) obtains,
(2) p(h2| e1 & b) = 1, where, recall, h2 = “Bill Clinton in fact owns all but one of the lottery tickets”
(1) p(h|e2 & e1 & b) = p(h|e1 & b) = .999 does not, since e2 would both on intuitive and formal grounds not be evidence for h
If we do not know that the NYT and the WP always get it right, Achinstein may be right that the objective p(h|e1 & b) = .999, but the subjective p(h|e1 & b) would not. Rather, depending upon an agent’s epistemic context, the degree of belief in h conditional on e1 and b may be between .001 and .999, in which case e2 would further confirm h and thus the p(h|e1 & e2 & b) > p(h|e1 & b), resulting in the fulfillment of positive relevance.
Achinstein, P. (2001) The Book of Evidence. New York: Oxford University Press.
Achinstein, P. (2005) “A Challenge to Positive Relevance Theorists: Reply to Roush,” in P. Achinstein (ed.), Scientific Evidence: Philosophical Theories & Applications. Baltimore: Johns Hopkins University Press. 91-94
Howson, C. and Peter Urbach (2006) Scientific Reasoning: The Bayesian Approach. Chicago: Open Court Press.
Jaynes, E.T. (2003) Larry Bretthorst (ed.) Probability Theory: The Logic of Science. Cambridge: Cambridge University Press.
Jeffreys, H. (1967) Theory of Probability. Oxford: Oxford University Press.
Shimony, A. (1970) “Scientific Inference,” in R.G. Colony (ed.), The Nature and Function of Scientific Theories. Pittsburgh: The University of Pittsburgh Press: 79-172
Roush, S (2005) Tracking Truth: Knowledge, Evidence, and Science. Oxford: Oxford University Press.