Sunday, April 03, 2005
Game-theoretical Semantics as a Challenge to Proof Theory
Game-theoretical Semantics as a Challenge to Proof Theory: "Abstract:
Game-theoretical semantics (GTS) is but a systematization of the familiar ``epsilon-delta'' technique of mathematicians. It relies essentially on the game-theoretical notion of strategy for its truth definition. Since we want to express all possible patterns of dependence among variables by means of our logic, we must replace the received first-order logic by an independence-friendly (IF) logic. This logic is semantically incomplete. Hence we cannot characterize the meanings of logical notions by their inference rules. This changes the role of logic in mathematics from fallacy policing to a search for stronger logical principles. Since a typical mathematical problem is equivalent to the validity of some IF first-order formula, this makes a (reoriented) proof theory the engine of discovery in mathematics."
Game-theoretical semantics (GTS) is but a systematization of the familiar ``epsilon-delta'' technique of mathematicians. It relies essentially on the game-theoretical notion of strategy for its truth definition. Since we want to express all possible patterns of dependence among variables by means of our logic, we must replace the received first-order logic by an independence-friendly (IF) logic. This logic is semantically incomplete. Hence we cannot characterize the meanings of logical notions by their inference rules. This changes the role of logic in mathematics from fallacy policing to a search for stronger logical principles. Since a typical mathematical problem is equivalent to the validity of some IF first-order formula, this makes a (reoriented) proof theory the engine of discovery in mathematics."
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