In the mathematical theory of probability, Lewis's triviality result (named after David Lewis) is a theorem about the impossibility of systematically equating the conditional probability P ( B ∣ A ) {\displaystyle P(B\mid A)} with the probability of a so-called conditional event, A → B {\displaystyle A\rightarrow B} .
Conditional probability and conditional events The statement "The probability that if A {\displaystyle A} , then B {\displaystyle B} , is 20%" means (put intuitively) that event B {\displaystyle B} may be expected to occur in 20% of the outcomes where event A {\displaystyle A} occurs. The standard formal expression of this is P ( B ∣ A ) = 0.20 {\displaystyle P(B\mid A)=0.20} , where the conditional probability P ( B ∣ A ) {\displaystyle P(B\mid A)} equals, by definition, P ( A ∩ B ) / P ( A ) {\displaystyle P(A\cap B)/P(A)} . Beginning in the 1960s, several philosophical logicians—most notably Ernest Adams and Robert Stalnaker—floated the idea that one might also write P ( A → B ) = 0.20 {\displaystyle P(A\rightarrow B)=0.20} , where A → B {\displaystyle A\rightarrow B} is the conditional event "If A {\displaystyle A} , then B {\displaystyle B} ". That is, given events A {\displaystyle A} and B {\displaystyle B} , one might suppose there is an event, A → B {\displaystyle A\rightarrow B} , such that P ( A → B ) {\displaystyle P(A\rightarrow B)} could be counted on to equal P ( B ∣ A ) {\displaystyle P(B\mid A)} , so long as P ( A ) > 0 {\displaystyle P(A)>0} . Part of the appeal of this move would be the possibility of embedding conditional expressions within more complex constructions. One could write, say, P ( A ∪ ( B → C ) ) = 0.75 {\displaystyle P(A\cup (B\rightarrow C))=0.75} , to express someone's high subjective degree of confidence ("75% sure") that either A {\displaystyle A} , or else if B {\displaystyle B} , then C {\displaystyle C} . Compound constructions containing conditional expressions might also be useful in the programming of automated decision-making systems.
How might such a convention be combined with standard probability theory? The most direct extension of the standard theory would be to treat A → B {\displaystyle A\rightarrow B} as an event like any other, i.e., as a set of outcomes. Adding A → B {\displaystyle A\rightarrow B} to the familiar Venn- or Euler diagram of A {\displaystyle A} and B {\displaystyle B} would then result in something like Fig. 1, where s , t , … , z {\displaystyle s,t,\ldots ,z} are probabilities allocated to the eight respective regions, such that s + t + ⋯ + z = 1 {\displaystyle s+t+\cdots +z=1} . For P ( A → B ) {\displaystyle P(A\rightarrow B)} to equal P ( B ∣ A ) {\displaystyle P(B\mid A)} requires that t + v + w + y = ( s + t ) / ( s + t + x + y ) {\displaystyle t+v+w+y=(s+t)/(s+t+x+y)} , i.e., that the probability inside the A → B {\displaystyle A\rightarrow B} region equal the A ∩ B {\displaystyle A\cap B} region's proportional share of the probability inside the A {\displaystyle A} 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 P ( A → B ) = P ( B ∣ A ) {\displaystyle P(A\rightarrow B)=P(B\mid A)} for any events A {\displaystyle A} and B {\displaystyle B} such that P ( A ) > 0 {\displaystyle P(A)>0} .
… excerpt ends here. Continue reading the full article.
