Action and ability (Q749516)

The author starts from a logic of ability that incorporates a requirement of reliability. He has developed a non-normal system of modal logic, using minimal models, and with a modal oprator \(\xi\). A minimal model is a triple \(M=<W,R,P>\), where W is a nonempty set of possible worlds, R is a relevance relation which relates worlds not to single worlds but to subsets (clusters) of W; and P is a valuation function. \(\xi\) A is true at a possible world \(\alpha\) in a model M iff there is some relevant cluster K in W, such that \(\alpha\) RK, and for any world \(\beta\) in K, A is true at \(\beta\) in M. Because two quantifiers, \(\exists \forall\), are involved in the specification of truth conditions for \(\xi\), this operator behaves a little like a package of two modal operators in one. Correlative operators are \(\eta\), \(\iota\), and \(\kappa\), corresponding to \(\exists \exists\), \(\forall \exists\), and \(\forall \forall\), respectively. \(\xi\) and \(\iota\) are duals, as are \(\eta\) and \(\kappa\), but \(\eta\) and \(\kappa\) are not definable in terms of the other two, and vice versa. The features common to the worlds in a cluster relevant to a world \(\alpha\) correspond to actions the agent is able to perform in \(\alpha\). ``\(\xi\) A'' means that the agent is able to bring it about that A. ``\(\eta\) A'' means that the agent is able to perform an action consistent with the (subsequent) truth of A. - Assuming a language \({\mathcal L}\) for sentential logic with \(\xi\) and \(\eta\) added, and with \(\iota\) and \(\kappa\) defined as duals of those, the author specifies a system \({\mathcal V}\) in \({\mathcal L}\) which is a minimal logic of ability, designed to capture just the formulas of \({\mathcal L}\) valid in the class of all minimal models. The author uses this logic of ability to construct a logic of action, \({\mathcal V}A\), based on the idea that actions are exercised abilities. The critical element of reliability is thereby transmitted to the derived logic of action. - Next, the author works in the opposite direction. Starting from the logic of action, he tries to build a logic of ability by construing ability as sheer alethic possibility of action. He treats \({\mathcal V}A\) simply as the beginning of a theory of action, adds to it an alethic possibility operator \(\diamondsuit\), and considers whether the combination \(\diamondsuit \xi\) (translated as ``can do'') behaves as we should expect an ability operator to do. A system \({\mathcal V}A/KT\) is considered, consisting of \({\mathcal V}A\) with additional axiom schemata and rules as in the normal modal system KT for the alethic operator \(\diamondsuit\). The original logic of ability is now recovered in a new notation (replacing \(\xi\) by \(\diamondsuit \xi\) etc.) as a subsystem of the new hybrid logic of possibility and action. This combined logic provides the opportunity within a single system to speak systematically of actions, abilities, and sheer alethic possibilities. The author concludes by listing a number of merits for such a combined system.