Lewis's triviality result

In the mathematical theory of probability, David Lewis's triviality result is a theorem about the impossibility of systematically equating the conditional probability with the probability of a so-called conditional event, .

Conditional probability and conditional events

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The statement "The probability that if  , then  , is 20%" means (put intuitively) that event   may be expected to occur in 20% of the outcomes where event   occurs. The standard formal expression of this is  , where the conditional probability   equals, by definition,  .

Beginning in the 1960s, several philosophical logicians—most notably Ernest Adams and Robert Stalnaker—floated the idea that one might also write  , where   is the conditional event "If  , then  ".[1] That is, given events   and  , one might suppose there is an event,  , such that   could be counted on to equal  , so long as  .

Part of the appeal of this move would be the possibility of embedding conditional expressions within more complex constructions. One could write, say,  , to express someone's high subjective degree of confidence ("75% sure") that either  , or else if  , then  . Compound constructions containing conditional expressions might also be useful in the programming of automated decision-making systems.[2]

 
Fig. 1 – A diagram of  ,  , and  . The   symbol is not assumed to represent any particular operation. Specifically, it is not assumed that   can be identified with  .

How might such a convention be combined with standard probability theory? The most direct extension of the standard theory would be to treat   as an event like any other, i.e., as a set of outcomes. Adding   to the familiar Venn- or Euler diagram of   and   would then result in something like Fig. 1, where   are probabilities allocated to the eight respective regions, such that  .

For   to equal   requires that  , i.e., that the probability inside the   region equal the   region's proportional share of the probability inside the   region. In general the equality will of course not be true, so that making it reliably true requires a new constraint on probability functions: in addition to satisfying Kolmogorov's probability axioms, they must also satisfy a new constraint, namely that   for any events   and   such that  .

Lewis's result

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Lewis (1976) pointed out a seemingly fatal problem with the above proposal: assuming a nontrivial set of events, the new, restricted class of  -functions will not be closed under conditioning, the operation that turns probability function   into new function  , predicated on event  's occurrence. That is, if  , it will not in general be true that   as long as  . This implies that if rationality requires having a well-behaved probability function, then a fully rational person (or computing system) would become irrational simply in virtue of learning that arbitrary event   had occurred. Bas van Fraassen called this result "a veritable bombshell" (1976, p. 273).

Lewis's proof is as follows. Let a set of events be non-trivial if it contains two possible events,   and  , that are mutually exclusive but do not together exhaust all possibilities, so that  ,  ,  , and  . The existence of two such events implies the existence of the event  , as well, and, if conditional events are admitted, the event  . The proof derives a contradiction from the assumption that such a minimally non-trivial set of events exists.

  1. Consider the probability of   after conditioning, first on   and then instead on its complement  .
    • Conditioning on   gives  . But also, by the new constraint on  -functions,    . Therefore,  .
    • Conditioning on   gives  . But also,    . (The mutual exclusivity of   and   ensures that  .) Therefore,  .
  2. Instantiate the identity   as    . By the results from Step 1, the left side reduces to  , while the right side, by the new constraint on  -functions, equals  . Therefore,  , which means that  , which contradicts the stipulation that  . This completes the proof.

Graphical version

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Fig. 2 – A diagram of disjoint   and  , and  .

A graphical version of the proof starts with Fig. 2, where the   and   from Fig. 1 are now disjoint and   has been replaced by  .[3] By the assumption that   and   are possible,   and  . By the assumption that together   and   do not together exhaust all possibilities,  . And by the new constraint on probability functions,    , which means that

(1)  

Conditioning on an event involves zeroing out the probabilities outside the event's region and increasing the probabilities inside the region by a common scale factor. Here, conditioning on   will zero out   and   and scale up   and  , to   and  , respectively, and so

(2)   which simplifies to  

Conditioning instead on   will zero out   and   and scale up   and  , and so

(3)   which simplifies to  

From (2), it follows that  , and since   is the scaled-up value of  , it must also be that  . Similarly, from (3),  . But then (1) reduces to  , which implies that  , which contradicts the stipulation that  .

Later developments

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In a follow-up article, Lewis (1986) noted that the triviality proof can proceed by conditioning not on   and   but instead, by turns, on each of a finite set of mutually exclusive and jointly exhaustive events   He also gave a variant of the proof that involved not total conditioning, in which the probability of either   or   is set to 1, but partial conditioning (i.e., Jeffrey conditioning), by which probability is incrementally shifted from   to  .

Separately, Hájek (1989) pointed out that even without conditioning, if the number of outcomes is large but finite, then in general  , being a ratio of two outputs of the  -function, will take on more values than any single output of the function can. So, for instance, if in Fig. 1   are all multiples of 0.01 (as would be the case if there were exactly 100 equiprobable outcomes), then   must be a multiple of 0.01, as well, but   need not be. That being the case,   cannot reliably be made to equal  .

Hájek (1994) also argued that the condition   caused acceptable  -functions to be implausibly sparse and isolated from one another. One way to put the point: standardly, any weighted average of two probability function is itself a probability function, so that between any two  -functions there will be a continuum of weighted-average  -functions along which one of the original  -functions gradually transforms into the other. But these continua disappear if the added   condition is imposed. Now an average of two acceptable  -functions will in general not be an acceptable  -function.

Possible rejoinders

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Assuming that   holds for a minimally nontrivial set of events and for any  -function leads to a contradiction. Thus   can hold for any  -function only for trivial sets of events—that is the triviality result. However, the proof relies on background assumptions that may be challenged. It may be proposed, for instance, that the referent event of an expression like “ ” is not fixed for a given   and  , but instead changes as the probability function changes. Or it may be proposed that conditioning on   should follow a rule other than  .

But the most common response, among proponents of the   condition, has been to explore ways to model conditional events as something other than subsets of a universe set of outcomes. Even before Lewis published his result, Schay (1968) had modeled conditional events as ordered pairs of sets of outcomes. With that approach and others in the same spirit, conditional events and their associated combination and complementation operations do not constitute the usual algebra of sets of standard probability theory, but rather a more exotic type of structure, known as a conditional event algebra.

Notes

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  1. ^ Hájek and Hall (1994) give a historical summary. The debate was actually framed as being about the probabilities of conditional sentences, rather than conditional events. However, this is merely a difference of idiom, so long as sentences are taken to express propositions and propositions are thought of as sets of possible worlds.
  2. ^ Reading "If  , then  " as "Not  , unless also  " makes compounding straightforward, since   becomes equivalent to the Boolean expression  . However, this has the unsatisfactory consequence that  ; then "If  , then  " is assigned high probability whenever   is highly unlikely, even if  's occurrence would make   highly unlikely. This is a version of what in logic is called a paradox of material implication.
  3. ^ A proof starting with overlapping   and  , as in Fig. 1, would use mutually exclusive events   and   in place of   and  .

References

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  • Hájek, Alan (1989). "Probabilities of conditionals – Revisited". Journal of Philosophical Logic. 18 (4): 423–428. doi:10.1007/BF00262944. JSTOR 30226421. S2CID 31355969.
  • Hájek, Alan (1994). "Triviality on the cheap?". In Eells, Ellery; Skyrms, Brian (eds.). Probability and Conditionals. Cambridge UP. pp. 113–140. ISBN 978-0521039338.
  • Hájek, Alan; Hall, Ned (1994). "The hypothesis of the conditional construal of conditional probability". In Eells, Ellery; Skyrms, Brian (eds.). Probability and Conditionals. Cambridge UP. pp. 75–111. ISBN 978-0521039338.
  • Lewis, David (1976). "Probabilities of conditionals and conditional probabilities". Philosophical Review. 85 (3): 297–315. doi:10.2307/2184045. JSTOR 2184045.
  • Lewis, David (1986). "Probabilities of conditionals and conditional probabilities II". Philosophical Review. 95 (4): 581–589. doi:10.2307/2185051. JSTOR 2185051.
  • Schay, Geza (1968). "An algebra of conditional events". Journal of Mathematical Analysis and Applications. 24 (2): 334–344. doi:10.1016/0022-247X(68)90035-8.
  • van Fraassen, Bas C. (1976). "Probabilities of conditionals". In Harper, W.; Hooker, C. (eds.). Foundations and Philosophy of Epistemic Applications of Probability Theory. Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science, Volume I. D. Reidel. pp. 261–308. ISBN 978-9027706171.