MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

32 Robert Goldblattif α is atomic then not both α ∈ µ and ¬α ∈ µ,if α ∧ β ∈ µ, then α ∈ µ and β ∈ µ,if α ∨ β ∈ µ, then α ∈ µ or β ∈ µ,if ∃xα ∈ µ, then α(y/x) ∈ µ for some variable y,that are sufficient to guarantee that µ can be extended to a maximal model setwhich has all such closure properties corresponding to the conditions for satisfactionfor the truth-functional connectives and the quantifiers. 31Hintikka’s article [1957] gives a definition of satisfaction for formulas of quantifieddeontic logic using model sets whose conditionsmay be thought of as expressing properties of the set of all statements thatare true under some particular state of affairs.He notes [1957, p. 10] that his treatment derives from anew general theory of modal logics I have developed.This general modelling of modalities was published in [1961], where he views amaximal model set as the set of all formulas that hold in some state-descriptionin the sense of Carnap, and says thata model set is the formal counterpart to a partial description of a possiblestate of affairs (of a ‘possible world’). (It is, however, large enough a descriptionto make sure that the state of affairs in question is really possible.)The point of the last sentence is that for non-modal quantificational logic, everymodel set is included in µ M for some actual model M. Hence a set of non-modalformulas is satisfiable in the Tarskian sense if it is included in some model set.The 1957 article deals with a system that has quantifiable variables ranging overindividual acts, and dual modalities for obligation and permission, with formulasOα and P α being read “α is obligatory” and “α is permissible”, respectively.The paper makes very interesting historical reading, especially on pages 11 and12 where one can almost see the notion of a binary relation between model setsquickening in the author’s mind as he grapples with the question of what we meanby saying that α is permitted. His answer is thatwe are saying that one could have done α without violating one’s obligations.In other words, we are saying that a state of affairs different from the actualone is consistently thinkable, viz. a state of affairs in which α is done but inwhich all the obligations are nevertheless fulfilled.Thus if the actual state is (partially) represented by a model set µ, then to representthis different and consistently thinkable state we need31 In fact it is assumed that formulas are in a certain normal form, but we can overlook thetechnicalities here.