Searching for the General Science of Evidence: Venn on Probability and Induction

Abstract

In this paper Venn's account of probability inference and induction is examined, tracing their differences as well as how they ‘co-operate’ in inferences from particulars to particulars. We discuss the role of mathematical idealizations in making probability inferences, the celebrated rule of succession and we delve into the nature of the reference class problem arguing that for Venn it is a common problem for both induction and inference in probability. Our approach is both historical and philosophical attempting to sketch Venn's position both in the philosophy of probability and induction of his time and in relation to the twentieth-century frequentism.

KEYWORDS:

• Induction
• probability
• John Venn
• idealization

Notes

1 See also, Psillos and Stergiou (2022).

2 This reconstruction is due to Verburgt 2014.

3 A place-selection is the construction of a subsequence of a sequence where the decision to keep the nth element in or reject it depends either on the ordinal number n of this element or on the attributes manifested in the (n-1) preceding elements. The decision does not depend on the attribute exhibited by the nth or by any subsequent element (von Mises 1964, 9).

4 A formulation of the theorem in modern probability theory is the following: Let $\mu$ be the number of occurrences of an event $A$ in $n$ independent trials and $p$ the probability of occurrence of event $A$ in each of the trials. Then, for any $ϵ>0$, $\underset{n\to \mathrm{\infty }}{lim}P\left\{\left|\frac{\mu }{n}-p\right|<ϵ\right\}=1$ (Gnedenko 1969, 199) It states that it is almost certain (probability equals 1) that the difference between the relative frequency and the probability $p$ of occurrence of an event $A$ in a sequence of independent trials becomes arbitrarily small as the number of trials tends to infinity.

5 Venn 1888, 92. Venn's criticism of Bernoulli's theorem in terms of its realistic presuppositions was not commonplace for the 19^th century adherents of frequentism. Despite being highly critical if not dismissive of Bernoulli's views, Robert Leslie Ellis, an English mathematician, editor of the works of Francis Bacon and proponent of a frequentist account of probabilities, was a realist who believed that probability to be made science requires us to accept the existence of universals. His conception of probability rested on the thesis that the genus – subordinate species relation between universals and particulars implies relations of numerical nature among the particulars (or, more generally, the subordinate species) which are captured, by the mind, when certain idealizations are made, exempting the contingency and limitation from the actual sequences of events. These ideal numerical relations are expressed by probabilities. (For insightful comparisons of Venn's and Ellis's views, consult Verburgt 2013, Zabell 1991, Kilinç 2000).

6 Hájek's criticism is not addressed exclusively to Venn but also to von Mises and Reichenbach.

7 As Verburgt, correctly, notices: ‘[Venn] at least said that probability should in principle be grounded on the “potential experience” of the real world’ (2022, 111).

8 We use symbols $\mathcal{A}$, $\mathcal{B}$ to denote averages and attributes of averages, respectively, and common letters A, B to denote attributes of individual things.

9 Augustus de Morgan was a British mathematician and logician. His work Essay on Probability (1838) has been one of the books that Venn read on recommendation of his mathematical coach Isaac Todhunter, a former student of de Morgan, to be introduced to the problematic of probability theory (Verburgt 2022, 104). De Morgan was a proponent of the conceptualist view of logic which, contrary to Venn's material view, claimed that ‘logic, including probability, deals with the ‘laws of our mind in thinking about things’’ (Verburgt 2022, 106) and he adhered to the subjectivist view of probabilities as degrees of belief, contrary to Venn's frequentism.

10 For a discussion of the distinction between inductive and physical probability, the interested reader may consult (Maher 2006).

11 The standard formulation of the problem is due to Hans Reichenbach: ‘If we are asked to find the probability holding for an individual future event, we must first incorporate the case in a suitable reference class. An individual thing or event may be incorporated in many reference classes, from which different probabilities will result. This ambiguity has been called the problem of the reference class’ (1935, 374). More recently, Hajek (2007) argued that the reference class problem is not a problem for the frequency interpretation of probability only. Some versions of other interpretations are also vulnerable to variants of the reference class problem.

12 Actually, Salmon's definition of homogeneity is different from Venn's definition. Salmon stipulated that in a homogeneous class, no partition would provide subclasses in which the probabilities are different among each other; leaving open whether this probability would be equal or not to the probability in the original class.

13 Salmon 1984, 61. Salmon suggests to read von Mises's condition of invariance of the limiting relative frequency under place-selections as a homogeneity condition.