An algorithm for the nonlinear analysis of compound bifurcation

This paper attempts to provide both an elaboration and a strengthening of the thesis of Popper & Miller (Nature, Lond. 302, 687f. (1983)) that probabilistic support is not inductive support. Although evidence may raise the probability of a hypothesis above the value it achieves on background knowledge alone, every such increase in probability has to be attributed entirely to the deductive connections that exist between the hypothesis and the evidence. We shall also do our best to answer all the criticisms of our thesis that are known to us. In 1878 Peirce drew a sharp distinction between `explicative, analytic, or deductive' and `amplifiative, synthetic, or (loosely speaking) inductive' reasoning. He characterized the latter as reasoning in which `the facts summed up in the conclusion are not among those stated in the premisses'. The Oxford English Dictionary records that the word `ampliative' was used in the same sense as early as 1842, and that in 1852 Hamilton wrote: `Philosophy is a transition from absolute ignorance to science, and its procedure is therefore ampliative.' This was the background to our letter to Nature on 21 April 1983. It was there shown that, relative to evidence e, the content of any hypothesis h may be split into two parts, the disjunction h $\vee $ e (read h or e) and the material conditional h$\leftarrow $e (read h if e); and the `ampliative' part of h relative to e was identified with this conditional h$\leftarrow $e; that is, with the deductively weakest proposition that is sufficient, in the presence of e, to yield h. We then proved quite generally that if p(h, eb) $\neq $ 1 $\neq $ p(e, b) then s(h$\leftarrow $e, e, b) = -ct(h, eb) ct(e, b) < 0. Here s(x,y,z) = p(x,yz) - p(x,z) is a measure of the support that y gives to x in the presence of z, and ct(x,z) = 1 - p(x,z) is a measure of the content of x relative to z. Relative to b, both h's content and its support by e may be added over the two factors h $\vee $ e and h$\leftarrow $e: ct(h, b) = ct(h $\vee $ e, b) + ct(h$\leftarrow $e, b), (0.2) s(h, e, b) = s(h $\vee $ e, e, b) + s(h$\leftarrow $e, e, b). (0.3) What (0.1) establishes is that the (`ampliative') part of the hypothesis h that goes beyond the evidence e is invariably counter supported by the evidence. In other words, probabilistic support is not inductive support.