Semantic games are best explained through an analogy to board games with a white and a black player where ties are impossible. We choose a start position c which is any legal position that is reachable from the standard start position of the board game. We have a group of players whom we ask the question: is there a winning strategy for white starting from c?
Let's assume some players say yes and others say no. We choose a player who said "yes" as white and a player who said "no" as black. They play the game. If white (black) loses she has made a mistake and the black (white) player wins a point.
Let's assume all players make the same choice, e.g., they all say "yes". Then we choose a player to be white and we force another player to be black, taking on the role of devil's advocate. If black wins white must have made a mistake and black wins a point. If white wins, we cannot blame black because she was forced.
Here the correspondence to semantic games:
Board Game Semantic Game start position claim white verifier black falsifier winning strategy for white winning strategy for verifier game logical dialog mistake mistake, contradiction game rules semantic game rules game depends on start position game depends on claimWhite claims that problem p has a solution of quality q that cannot be improved. Black shows a solution of quality q'>q. White has made a mistake.