Two comments on this  Berry does not specify which particular encoding of the Booleans you should choose only that one must be assumed beforehand which is completely fair as otherwise you would never know what true would be and what false would be So highertype whatever that would be in the untyped lambdacalculus is covered by the theorem  There is adifference between a parallel execution which exists for the lambda calculus in the form of parallel reductions sharing graphs etc and b a true parallel program which of which POR is the prime example just start evaluating both arguments and when one eventually evaluates reduces to true we dont need to evaluate the other argument any further before we move on to the next part of our program Berrys theorem shows that this we dont need to evaluate the other argument any further is impossible Unfortunately the most common reading of parallel is b and to make things worse at lot of people assume concurrency is synonymous with parallelism