Hans Reichenbach's Probability Logic

Publisher Summary

Reichenbach states that an inductive logic cannot be built up entirely from logical principles independent of experience, but must develop out of the reasoning practiced and useful to the natural sciences. Inductive inference system needs to be built on some solid to guide scientific methodology. This chapter describes Reichenbach's reasons for stating the inverse approach for inductive logic. Instead of “a priori” foundation of inductive logic, Reichenbach's approach to induction is largely axiomatic. Reichenbach distinguishes deductive and mathematical logic from inductive logic. The former deals with the relations among tautologies, whereas the latter deals with truth in the sense of truth in reality. Deductive and mathematical logic are built on an axiomatic system. In contrast to the formal relations that are of interest in deductive logic, inductive logic is concerned with the determination of whether various relations among quantities are true in the world.

Introduction

Any attempt to characterize Reichenbach's approach to inductive reasoning must take into account some of the core influences that set his work apart from more traditional or standard approaches to inductive reasoning. In the case of Reichenbach, these influences are particularly important as otherwise Reichenbach's views may be confused with others that are closely related but different in important ways. The particular influences on Reichenbach also shift the strengths and weaknesses of his views to areas different from the strengths and weaknesses of other approaches, and from the point of some other approaches Reichenbach's views would seem quite unintelligible if not for the particular perspective he has.

Reichenbach's account of inductive reasoning is fiercely empirical. More than perhaps any other account it takes its lessons from the empirical sciences. In Reichenbach's view, an inductive logic cannot be built up entirely from logical principles independent of experience, but must develop out of the reasoning practiced and useful to the natural sciences. This might already seem like turning the whole project of an inductive logic on its head: We want an inductive inference system built on some solid principles (whatever they may be) to guide our scientific methodology. How could an inference procedure that draws on the methodologies of science supply in any way a normative foundation for an epistemology in the sciences?

For Reichenbach there are two reasons for this “inverse” approach. We will briefly sketch them here, but return with more detail later in the text: First, Reichenbach was deeply influenced by Werner Heisenberg's results, including the uncertainty principle, that called into question whether there is a fact to the matter – and consequently whether there can be certain knowledge about – the truth of propositions specifying a particular location and velocity for an object in space and time. If there necessarily always remains residual uncertainty for such propositions (which prior to Heisenberg seemed completely innocuous or at worst subject to epistemic limitations), then – according to Reichenbach – this is reason for more general caution about the goals of induction. Maybe the conclusions any inductive logic can aim for when applied to the sciences are significantly limited. Requiring soundness of an inference – preservation of truth with certainty – may not only be unattainable, but impossible, if truth is not a viable concept for empirical propositions. Once uncertainty is built into the inference, deductive standards are inappropriate for inductive inference not only because the inference is ampliative (which is the standard view), but also because binary truth values no longer apply.

Second, the evidence supporting Albert Einstein's theory of relativity, and its impact on the understanding of the nature of space and time revealed to Reichenbach the power of empirical evidence to overthrow truths that were taken to be (even by Reichenbach himself in his early years) necessarily true. The fact that Euclidean space had been discovered not only to be not necessary, but quite possibly not true — despite Immanuel Kant's transcendental proofs for its synthetic a priori status — called the state of a priori truths into question more generally. The foundations of any inference system could no longer be taken to be a priori, but had to be established independently as true of the world. Reichenbach refers to this confirmation of the correspondence between formal structures and the real world as “coordination” (although “alignment” might have been the more intuitive description of what he meant).

Einstein's and Heisenberg's findings had their greatest impact on Reichenbach's views on causality. Influenced by the Kantian tradition, Reichenbach took causal knowledge to be so profound that in his doctoral thesis in 1915 he regarded it as synthetic a priori knowledge [Reichenbach, 1915]. But with the collapse (in Reichenbach's view) of a synthetic a priori view of space, due to Einstein, Reichenbach also abandoned the synthetic a priori foundation of causality. Consequently, Reichenbach believed that causal knowledge had to be established empirically, and so an inductive procedure was needed to give an account of how causal knowledge is acquired and taken for granted to such an extent that it is mistaken for a priori knowledge. But empirical knowledge, in Reichenbach's view, is fraught with uncertainty (due to e.g. measurement error, illusions etc.), and so this uncertainty had to be taken into account in an inductive logic that formalizes inferences from singular (uncertain) empirical propositions to general (and reasonably certain) empirical claims. Heisenberg's results implied further problems for any general account of causal knowledge: While the results indicated that the uncertainty found in the micro-processes of quantum physics is there to stay, macro-physics clearly uses stable causal relations. The question was how this gap could be bridged. It is therefore unsurprising that throughout Reichenbach's life, causal knowledge formed the paradigm example for considerations with regard to inductive reasoning, and that probability was placed at its foundation.

The crumbling support for such central notions as space, time and causality, also led Reichenbach to change his view on the foundations of deductive inference. Though he does not discuss the foundations of logic and mathematics in any detail, there are several points in Reichenbach's work in which he indicates a switch away from an a prioristic view. The a prioristic view takes logic to represent necessary truths of the world, truths that are in some sense ontologic. Reichenbach rejects this view by saying that there is no truth “inherent in things”, that necessity is a result of syntactic rules in a language and that reality need not conform to the syntactic rules of a language [Reichenbach, 1948]. Instead, Reichenbach endorsed a formalist view of logic in the tradition of David Hilbert. Inference systems should be represented axiomatically. Theorems of the inference system are the conclusions of valid deductions from the axioms. Whether the theorems are true of the world, depends on how well the axioms can be “coordinated” with the real world. This coordination is an empirical process. Thus, the underlying view holds that the axioms of deductive logic can only be regarded as true (of the world) and the inference principles truth preserving, if the coordination is successful – and that means in Reichenbach's case, empirically successful, or useful. In the light of quantum theory, Reichenbach rejected classical logic altogether [Reichenbach, 1944].

Instead of an a priori foundation of inductive logic, Reichenbach's approach to induction is axiomatic. His approach, exemplified schematically for the case of causal induction works something like this: We have causal knowledge. In many cases we do not doubt the existence of a causal relation. In order to give an account of such knowledge we must look at how this knowledge is acquired, and so we have to look closely at the methodologies used in the natural sciences. According to Reichenbach, unless we deny the significance of the inductive gap David Hume dug (in the hole created by Plato and Sextus Empiricus), the only way we will be able to make any progress towards an inductive logic is to look at those areas of empirical knowledge where we feel reasonably confident that we have made some progress in bridging that gap, and then try to make explicit (in form of axioms) the underlying assumptions and their justification (or stories) that we tell ourselves, why such assumptions are reliable.

There are, of course, several other influences that left their marks on Reichenbach's views. Perhaps, most importantly (in this second tier), are the positivists. Their influence is particularly tricky, since Reichenbach was closely associated with many members of the Vienna Circle, but his views are in many important ways distinctly “negativist”: Reichenbach denies that there can be any certainty even about primitive perception, but he does believe — contrary to Karl Popper — that once uncertainty is taken into account, we can make progress towards a positive probability for a scientific hypothesis. We return to the debate with Popper below.

Second, it is probably fair to say that Richard von Mises, Reichenbach's colleague during his time in Berlin and Istanbul, was the largest influence with regard to the concept of probability. Since probabilistic inferences play such a crucial role in scientific induction, Reichenbach attempted to develop a non-circular foundation and a precise account of the meaning and assertability conditions of probability claims. Reichenbach's account of probability in terms of the limits of relative frequency, and his inductive rule, the so-called “straight rule”, for the assertability of probability claims — both to be discussed in detail below — are perhaps his best known and most controversial legacy with regard to inductive inferences.

As with any attempt to describe a framework developed over a lifetime, we would inevitably run into some difficulty of piecing together what exactly Reichenbach meant even if he had at all times written with crystal clarity and piercing precision — which he did not. On certain aspects, Reichenbach changed or revised his view, and it did not always become more intelligible. However, areas of change in Reichenbach's account are also of particular interest, since they give us a glimpse into those aspects that Reichenbach presumably deemed the most difficult to pin down. They will give us an idea of which features he considered particularly important, and which ones still needed work. As someone, who in many senses sat between the thrones of high church philosophy of his time and (therefore?) anticipated many later and current ideas, Reichenbach's views are of particular interest.