Paul Grice - Grice's Paradox

Grice's Paradox

In his book Studies in the Way of Words, he presents what he calls "Grice's Paradox". In it, he supposes that two chess players, Yog and Zog, play 100 games under the following conditions:

(1) Yog is white nine of ten times.
(2) There are no draws.

And the results are:

(1) Yog, when white, won 80 of 90 games.
(2) Yog, when black, won zero of ten games.

This implies that:

(i) 8/9 times, if Yog was white, Yog won.
(ii) 1/2 of the time, if Yog lost, Yog was black.
(iii) 9/10 times, either Yog wasn't white or he won.

From these statements, it might appear one could make these deductions by contraposition and conditional disjunction:

( from ) If Yog was white, then 1/2 of the time Yog won.
( from ) 9/10 times, if Yog was white, then he won.

But both (a) and (b) are untrue—they contradict (i). In fact, (ii) and (iii) don't provide enough information to use Bayesian reasoning to reach those conclusions. That might be clearer if (i)-(iii) had instead been stated like so:

(i) When Yog was white, Yog won 8/9 times. (No information is given about when Yog was black.)
(ii) When Yog lost, Yog was black 1/2 the time. (No information is given about when Yog won.)
(iii) 9/10 times, either Yog was black and won, Yog was black and lost, or Yog was white and won. (No information is provided on how the 9/10 is divided among those three situations.)

Grice's paradox shows that the exact meaning of statements involving conditionals and probabilities is more complicated than may be obvious on casual examination.