FOM: Re: FoM: Boole, Probability, and Material Implication
a.hazen at philosophy.unimelb.edu.au
Sat Jun 23 02:25:55 EDT 2001
Steve Stevenson asked a question which broades out to: What is the
relation between conditional probability and the probability of a
conditional statement? There has been a lot said about this in the
philosophical literature over the past few decades, some of it quite good,
and some of it containing (small, easy, but surprising) technical results:
bits of it would be appropriate "enrichment reading" or side topic in a
mathematical logic or probability course. Here's a (lightly annotated)
(1) In ordinary conversation, our willingness to say "If A is true, then
B is" seems to go along with our regarding the conditional probability of
B on A as high. This was noted by Ernest W. Adams. Subsequent literature
has referred to "Adams's Thesis," that the "assertibility" (technical
term: roughly the legitimacy of asserting. It's dependent on your
information, so it is NOT the same thing as truth) of a
(non-counterfactual) conditional statement is to be measured by the
conditional brobability of its CONSEQUENT on its ANTECEDENT. Not sure
Adams's original publication; there is discussion in his "A Primer of
Probability Logic" (Stanford: CSLI Publications, 1998).
(1a) It is intuitively plausible that something is assertible by an
omniscient speaker just in case it is true. Leading to the thought that
the truth value of a compound, and in particular a conditional, statement
might be found by by watching what happens to its assertibility as the
probabilities of its atomic constituents approach 1 and 0. Since the
conditional probability P(C/A) is undefined when P(A)=0, this thought seems
to motivate a THREE-VALUED interpretation of conditionals: they have the
truth value of the consequent when the antecedent is true, and are
undefined otherwise. This interpretation is mentioned (under the name
CONDITIONAL ASSERTION) in Quine's textbook "Methods of Logic." It is
discussed by Nuel Belnap in "Nous" (philosophical journal) vol. 4 (1970),
pp. 1-12. It is defended as an interpretation of (non-counterfactual)
"if-then" statements of English by W.S. Cooper in an article, "The
propositional logic of ordinary discourse," in "Inquiry" vol. 11 (1968),
pp. 295-320. (I believe that Cooper was a student of Adams's; he later
published a book on topics related to this.)
(2) Robert Stalnaker and David Lewis (independently) proposed a
"possible worlds" semantics for conditionals: "If A then B" is true just in
case B is true in the possible world most like the real world among those
in which A is true (ROUGHLY and APPROXIMATELY!). Best source for this is
Lewis's 1973 monograph (corrected edition: Blackwell, 1986)
"Counterfactuals." (150 pages, an expository gem: LOTS of ideas and a few
theorems) Lewis proposed this semantics only for COUNTERFACTUAL
conditionals; Stalnaker though it applied to all conditionals.
(Historical footnote: C.S. Peirce, of course. cf. his "Reasoning and the
Logic of Things," Harvard U. Press 1992, pp. 125-126.) Stalnaker initially
conjectured that the PROBABILITY of a conditional would be the same as the
conditional probability of its consequent on its antecedent....
(3) David Lewis refuted this initially plausible-seeming conjecture with
his "Trivialization" result: under quite weak assumptions about the logic
of the conditional, probabilities of conditionals can only be equated with
conditional probabilities on trivial probability distributions:
distributions on which every proposition has one of a small finite number
(4, as it it happens) of probabilities. D.K. Lewis, "Probabilities of
Conditionals and Conditional Probabilities," in "Philosophical Review"
vol. 85 (1976), pp. 297-315. Reprintedin W.L. Harper, R. Stalnaker, and G.
Pearce, eds., "Ifs" (Dordrecht: Reidel, 1981). Reprinted, with a four-page
"Postscript," in Lewis's "Philosophical Papers, v. II" (Oxford U.P., 1986).
COMMENT: Lewis's proof is not difficult, but I always find it surprising.
Stalnaker and his friends were surprised and somewhat dismayed by Lewis's
(3a) The Harper et al. "Ifs" collection is a worth looking at if you
are interested in the semantics of conditionals or in their relevance to
"causal decision theory" it collects a number of the best and most
important papers on these topics.
(3b) Lewis's trivialization result has been improved (weakening
assumptions, simplifying argument) by a number of writers:
--David Lewis, "Probabilities of conditionals and conditional probabilities
"Philosophical Review" vol. 95 (1986), pp. 581-589; repr. in Lewis's
"Papers in Philosophical Logic" (Camb. U.P., 1998).
--Alan Hajek, in "Journal of Philosophical Logic," vol. 18 (1989), pp. 423-428.
--Alan Hajek, "Triviality on the cheap," in Ellery Eells, ed., "Probability
and Conditionals" (New York: Cambridge U.P., 1994).
--Ned Hall and Alan Hajek, "The hypothesis of the conditional construal of
conditional probabilities," ibid.
(At a guess, the first Hajek paper in the Eells volume MIGHT be the
clearest short account, but I'm not very confidehad nt in saying this.
Hajek and Hall were both graduate students at Princeton, so Lewis as a
(3b) Lewis's view is that the (non-counterfactual) conditional in
ordinary speech (according to Cooper, non-counterfactuals outnumber
counterfactuals something like 4 to 1 in scientific writing) has the TRUTH
CONDITIONS of the material conditional (false if antecedent true and
consequent false; true on the other three combinations), but has the
ASSERTABILITY CONDITIONS claimed in Adams's thesis. Perhaps the best
exposition of this position is Frank Jackson, "On Assertion and Indicative
Conditionals," in the "Philosophical Review," vol. 88 (1979), pp. 565-589;
repr. (along with the two Lewis "CP&PC" papers in Frank Jackson, ed.,
"Conditionals" (Oxford U.P., 1991). NOTE: this is a conceptual paper with
very little technical content.
(4) The Lewis "Trivialization" arguments, or close cousins of them, have
played a role in the "Desire as Belief" debate. Basic idea: can we
identify "Wanting X to be the case" with "Believing that [something like]
it would be good if X were the case." Background assumption: rational
belief and desire obey the principles of Bayesian decision theory. Result:
(on at least one interpretation) Desire as Belief is in logical trouble.
--David Lewis, "Mind" vol. 97 (1988), pp. 323-332
--John Collins, "Mind" vol. 97 (1988), pp. 332-342
--Costa, Collins, and Levi, "Analysis"* vol. 55 (1995), pp. 2-5.
--David Lewis, "Mind" vol. 105 (1996) pp. 303-313
--Alex Byrne & Alan Hajek, "Mind" vol. 106 (1997), pp. 411-428
* There are at least three journals, in different disciplines, called
"Analysis." This one is the British philosophy journal that specializes in
(PERSONAL COMMENT: The logic in the literature listed in section 4 is
related to that of the conditional probability / probability of
conditionals discussion, which is why I list it here. The attempt to
analyze desire as a special case of belief seems to me to be implausible
as psychology, and the results in these papers are probably not conclusive
against it anyway: there are probably other interpretations of "Desire as
Belief" still open to the philosophical/psychological theorist attracted to
Apologies for going on so long. Ihope this will be of use/interest to someone!
(interests: philosophy of mathematics, general logic)
University of Melbourne
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