
Logic & Philosophy of Science Colloquium
Helge Rückert

The Conception of Validity in Dialogical Logic 
Abstract:
Semantic approaches in logic can be divided into two broad categories: denotational and usebased approaches. Within the latter, a further division is important: according to prooftheoretic semantics meaning is captured via rules for how to use expressions in proofs, whereas according to gametheoretic semantics meaning is captured via rules for how to use expressions in language games. The two most important gametheoretic semantic frameworks are Hintikka's GTS and Dialogical Logic.
Both GTS and Dialogical Logic are based on twoperson zerosum games: two players argue about an initial formula by alternately making moves according to certain rules. For each logical particle there is a rule that determines how a corresponding formula can be attacked and defended. At a certain point every play of a game reaches the level of atomic formulas, and it is here that GTS and Dialogical Logic part ways. GTS assumes a model according to which the atomic formulas are evaluated and the result of the play is accordingly determined. Truth in a model can then be defined as the existence of a winning strategy for the first player given this model, and validity, as usual, as truth in every model, i.e. as the existence of winning strategies for the first player given any model. I will call this conception "validity as general truth".
So far, all these games are games with perfect information. But GTS becomes especially interesting when games with imperfect information are introduced, because this idea leads Hintikka to his IFlogic: in games about formulas with nested quantifiers a player might have to decide about how to attack or defend a certain quantifier without knowing how the other player attacked or defended another quantifier earlier on. In this sense the second quantifier is independent of the first and only uniform strategies with respect to the second quantifier work as winning strategies.
The distinguishing feature of Dialogical Logic, the socalled formal rule, can best be explained by another transition to games with imperfect information. Suppose that the first player lacks information about the atomic level: he does not know utterance of which atomic formulas leads to a win or a loss. Is it still possible for the first player to have a winning strategy? Yes, if he is able to employ the socalled copycat strategy at the atomic level. Broadly speaking, this strategy consists in copying the moves of the other player in order to make him indirectly play against himself. This is the motivation behind the formal rule in Dialogical Logic, according to which the first player is only allowed to assert an atomic formula if the other player has already asserted this atomic formula at an earlier point in the play.
Thus there is at the heart of Dialogical Logic a conception of validity that differs from validity as general truth: validity as the existence of a winning strategy for the first player in a game with imperfect information in which he lacks information about the atomic level and is only able to use the copycat strategy. This might be called validity as formal truth, since it amounts to validity as the existence of a winning strategy when the formal rule is in effect. To conclude my talk, I will point out some consequences of this conception of validity or the formal rule in Dialogical Logic, especially concerning the possibility of formulating different nonclassical logics within the dialogical framework.