On the logic of ability
It has been argued by philosophers that the `can' expressing a person's ability cannot be considered as a modal operator since it violates the characteristic laws of even the weakest \(\diamond\)-logics, in particular: \(\diamond (AvB)\to (\diamond Av\diamond B)\). (``I am able to draw a card which will have one of the colors red or black, but I cannot draw a red card, and I cannot draw a black card'', p. 2). The author suggests, however, to make the `can' of ability amenable to logical treatment by extending the standard possible worlds-semantics to so-called minimal models in which one has instead of the usual accessability relation, R, a ``relevance relation between a world and a set of possible worlds'' (p. 3). The truth-condition runs as follows: `X can bring it about that A' ``will be true at a given world iff there exists a relevant cluster of worlds, at every world of which A is true'' (p. 5). This semantic analysis reveals that the `can' of ability contains some elements of an ordinary \(\diamond -\) but also some of an ordinary \(\square\)-operator. In the formal part of the paper (pp. 8-22), axiom-systems, corresponding semantic requirements, and completeness-proofs are presented.
scientific article; zbMATH DE number 4064951