Effectivity Functions and Systems of Logic
The notion of a model structure plays a crucial role in the semantics of both DLA and DLA*. As we have seen, a model structure can be understood as a description of histories of the world. A formula is said to be satisfiable if it fits within the description: there is a history of the world in which the formula is true. Obviously, some types of formulas do not fit into some types of model structures. Consider, for instance, the set C of all non-cooperative CGTs in which the game forms which are used to determine what an individual can and cannot do — the feasible game forms — all assign exactly one strategy to each individual. Or, stated differently, consider those noncooperative CGTs in which each of the feasible game forms can be played in only one way. It is easy to see that every formula of the form ‘∀ s [CanDo s (t i ,φ) → MustDo s (t i ,(φ)]’ is a C-valid formula of DLA: it is true in every model structure in which the CGT is an element of C. If every individual has been assigned only one strategy, then every individual has only one option. He or she must choose that strategy. If an individual does not have the option of smoking a cigarette, then he must see to it that he is not smoking, or more concisely, he must refrain from smoking. On the other hand, given the existence of model structures in which some of the game forms assign more than one strategy to individuals, seeing to it that something is the case does not always imply that one must see to it that it is the case; consider the individual who has both the option of smoking and the option of not smoking. Similarly, the characteristics of the admissible game forms of a model structure determine which types of deontic formulas are satisfiable. If, for instance, all individuals have also only one admissible strategy, then the formula ‘∀ s [MayDo s (t i ,φ) → ShallDo s (t i ,φ)]’ is also a C-valid formula of DLA.
KeywordsEffectivity Function Game Form Game Tree Admissible Strategy Basic Proposition
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