Abstract
In a recent survey of the literature on the relation between information and confirmation, Crupi and Tentori (Stud Hist Philos Sci 47:81–90, 2014) claim that the former is a fruitful source of insight into the latter, with two well-known measures of confirmation being definable purely information-theoretically. I argue that of the two explicata of semantic information (due originally to Bar Hillel and Carnap) which are considered by the authors, the one generating a popular Bayesian confirmation measure is a defective measure of information, while the other, although an admissible measure of information, generates a defective measure of confirmation. Some results are proved about the representation of measures on consequence-classes.
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Notes
Broad (1926), p. 67.
These are all variations on the following: that any attempt to infer anything about the future from the past will necessarily beg the question. Howson (2013) argues that this argument translates straightforwardly into a theorem of the probability calculus, and hence that attempts to challenge it while remaining within a probabilistic framework are doomed to failure.
An earlier technical report by the two authors (Carnap and Bar Hillel 1952), of which their 1953 paper is a more concise published version, is useful for its additional explanatory remarks and will be cited often in the present paper.
In Bar Hillel and Carnap’s paper \(\hbox {M}(a)\) is finite for each a: it is the set of state-descriptions admitted by a when a is formulated in a finite monadic predicate language without equality (informally speaking, the state-descriptions are the finest partition of possibility definable in the language).
Actually they called it the Content of a (upper case ‘c’).
This is admittedly a very brisk and nuance-free summary. The reader is recommended to consult Bar Hillel and Carnap’s own exposition.
Crupi and Tentori (2014), p. 81. It will sometimes be useful in what follows to regard the algebra of propositions as the so-called Lindenbaum sentence algebra of a language L, where logically equivalent sentences are in effect identified (thus there will be just one tautology, T, and one contradiction \(\bot \)). But there are many contexts where standard finitary languages are incapable of expressing the propositions of interest: for example, the infinite disjunction ‘a ticket numbered 1 will win or a ticket numbered 2 will win or a ticket numbered 3 will win or ....’, in the context of a countably infinite lottery whose tickets are labelled by the (numerals for) the natural numbers.
Crupi and Tentori (2014), p. 84. The ‘R’ is for ‘ratio’, the ‘D’ for ‘difference’; we shall see why in the following section.
Bar Hillel and Carnap allow b to be a class of statements, like a body of background assumptions, but if, as is usually assumed, the class is finite it can of course be represented by a single conjunction.
Thus conditionalisation is implicitly assumed.
Milne shows that a set of postulates based on this heuristic principle implies that the information added to b by a, in his sense, is a decreasing function f of \(\hbox {p}(a {\vert } b)\) with \(\hbox {f}(1) = 0\), and he points out that both \(-\hbox {logp}(a {\vert } b)\) and \(1- \hbox {p} (a {\vert } b)\) qualify in this role. His defence against the charge that the notion of a proportion in infinite possibility-spaces, where no limiting procedure exists, is meaningless is that
a probability distribution P is a (weighted) measure of possibilities and so, with a measure of possibilities in place, we restore propriety by saying that a proposition adds the more information to b (according to the distribution P) the smaller \(P(a \wedge b)/P(b)\), i.e., the smaller \(P(a {\vert } b)\). (ibid.)
This defence puts more weight on ‘weight’ than it should probably be asked to bear: a Bayesian probability, for example, is just a coherent degree of belief having nothing to do with ‘proportions’ of possibilities.
See also Crupi and Tentori (2010).
Note that this feature is inherited also by instantiating \(\hbox {inf}(h)\) – \(\hbox {inf}(h {\vert } e)\) with Bar Hillel and Carnap’s measures \(\hbox {cont}(h)\) and \(\hbox {cont}(h {\vert } e)\) respectively where \(\hbox {cont}(h {\vert } e)\) is defined by (1): the result is easily seen to be equal to \(\hbox {cont}(e \hbox {v} h)\) and thence, when h entails e, to \(\hbox {inf}(e)\). It might seem surprising therefore, given their endorsement of (1), that a principal objective also of Bar Hillel and Carnap was to exhibit a close connection between confirmation and semantic information. The mystery is dispelled once we recall that by ‘degree of confirmation’ Bar Hillel and Carnap meant nothing more than conditional probability where the probability was what Carnap called ‘logical probability’. Carnap called \(\hbox {bc}_{\mathrm{D}}\) ‘incremental confirmation’, i.e. the change in Carnapian degree of confirmation.
Though it is endorsed by Milne (1996): one of Milne’s key assumptions in his derivation of this confirmation measure is that that if e has the same probability given h and \(h'\) then h and \(h'\) are equally confirmed by it (desideratum 5, p. 21). He remarks ‘Viewed correctly, I submit, [the assumption] is ... utterly compelling’ (1996, p. 22). Utterly compelling it may seem, until viewed in both green and grue light.
This of course raises the question of how \(\hbox {p}(e {\vert } h)\) is to be understood if \(\hbox {p}(h)\) is zero, given that the usual rule \( \hbox {p}(e {\vert } h) = \hbox {p}(e \& h) /\hbox {p}(h)\) breaks down. Fortunately there are satisfactory axiomatisations of conditional probability in which \(\hbox {p}(e {\vert } h)\) is meaningful when \(\hbox {p}(h) = 0\), so long as h is not a contradiction (among others De Finetti 1974, vol. 2, p. 339; Coletti and Scozzafava 2002, p. 76; Popper 1959, Appendix *iv; the measures satisfying Popper’s axioms are known as Popper functions).
It is also not convex in (1/2, 1): I will argue in Sect. 6 that this is an important defect.
Good notes that Turing himself thought of the log-odds measure, scaled in what he called ban units, as analogous to that of sound, whose bel unit is the logarithm to base 10 of a ratio of sound-intensities (Good 1950, p. 63).
Bar Hillel and Carnap themselves noted the curious weakness of \(e\rightarrow h\) (1953, p. 151; they use i and j for h and e).
Miller and Popper themselves argued for \(e\rightarrow h\) as carrying the excess information of h over e on the ground that \(e\rightarrow h\) is the weakest statement which, conjoined with e, delivers h as a consequence. Cf Hintikka (1968, p. 313).
‘cont(i) is offered as one ... explicatum of the ordinary concept “amount of information conveyed by i” ’ (Carnap and Bar Hillel 1952, p. 149).
The following report in Aubrey’s Brief Lives about the philosopher Thomas Hobbes is an amusing illustration:
He was 40 years old before he looked in on Geometry; which happened accidentally. Being in a Gentleman’s Library, Euclid’s Elements lay open, and ‘twas the 47 El. libri I. He read the Proposition. By God, sayd he (he would now and then swear an emphaticall Oath by way of emphasis) this is impossible! So he reads the Demonstration of it, which referred him back to such a Proposition; which proposition he read. That referred him back to another, which he also read. Et sic deinceps that at last he was demonstratively convinced of that trueth. This made him in love with Geometry. (Aubrey 2015)
This is in fact one of Carnap and Bar Hillel’s basic rules for measures of semantic information (1952, p. 12).
Carnap considered two possibilities: the class of nontautologous consequences versus the class of all consequences, but regarded a choice between them as unnecessary once the decision was taken in favour of the sets \({}^*\hbox {M}(a)\) of possibilities excluded by a (see immediately below).
Since the set of sentences of L is assumed to be countable (hardly a restrictive assumption) there are also countably additive measures on the full power set (since the existence of measures depends only on cardinality, this is equivalent to saying that there are countably additive measures on the power set of \({\mathbb {N}}\), the set of natural numbers). But countably additive measures even on measure-spaces of this cardinality are very restrictive: there can be no uniform distributions over countably infinite partitions, unlike the case of finitely additive measures (where every singleton, and every finite subset, receives the value 0 if the measure is finite). Kadane and O’Hagan (1995) investigate three familiar types of finitely additive uniform probability measure on \({\mathbb {N}}\).
Carnap and Bar Hillel also added a third, that the information measure of a contingent sentence be positive, but this is now usually regarded as too restrictive.
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Howson, C. Does information inform confirmation?. Synthese 193, 2307–2321 (2016). https://doi.org/10.1007/s11229-015-0918-7
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DOI: https://doi.org/10.1007/s11229-015-0918-7