Logic and Games Stanford Encyclopedia of Philosophy

It makes no sense at all to ask for example whether 8 is dependent on 37. By the beginning of the twenty-first century it had become widely accepted that games and logic go together. The result was a huge proliferation of new combinations of logic and games, particularly in areas where Logic Games is applied. Many of these new developments sprang originally from work in pure logic, though today they follow their own agendas. One such area is argumentation theory, where games form a tool for analysing the structure of debates.

For example Catarina Dutilh Novaes makes a detailed defence of the view that obligationes present “a remarkable case of conceptual similarity between a medieval and a modern theoretical framework”. But whatever view we take on this question, these debates have inspired one important line of modern research in logical games. Various Löwenheim-Skolem Theorems of model theory can be proved using variants of the Forcing Game. In these variants we do not construct a model but a submodel of a given model.

Roland Fraïssé, a French-Algerian, was the first to find a usable necessary and sufficient condition. It was rediscovered a few years later by the Kazakh logician A. D. Taimanov, and it was reformulated in terms of games by the Polish logician Andrzej Ehrenfeucht. The games are now known as Ehrenfeucht-Fraïssé games, or sometimes as back-and-forth games. They have turned out to be one of the most versatile ideas in twentieth-century logic.

Almost as in the medieval obligationes, the Opponent wins by driving the Proponent to a point where the only moves available to her are blatant self-contradictions. Several medieval texts describe a form of debate calledobligationes. At the beginning of a session, the disputants would agree on a ‘positum’, typically a false statement. The job of Respondens was to give rational answers to questions from Opponens, assuming the truth of the positum; above all he had to avoid contradicting himself unnecessarily. The job of Opponens was to try to force Respondens into contradictions. So we broadly know the answer to the Dawkins question, but we don’t know the game rules!

But what if you said, “If I asked you if two plus two equals five, would you answer ja? ” If ja means yes, Truth would answer da, as would False; if ja means no, they’d also both answer da. So, you know that if the embedded question is correct, Truth and False always answer with the same word you use; if the embedded question is incorrect, they always answer with the opposite word. You also know they always answer with the same word as each other.

Logic games involve solving various puzzles using logic. Many logic puzzles are abstract and require a sharp mind to think outside the box. Other logic games require more simple problem-solving skills to overcome various obstacles in the game. Ideas of this kind lie behind the dialectical games of Paul Lorenzen. In a gesture towards medieval debates, he called \(\exists\) the Proponent and the other player the Opponent.

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