Title:

Modalities and formal systems

This work is in three parts. The first considers the most popular symbolic approach to the modalities and finds crucial shortcomings in it. The second explores the nature of symbolization and of logical form, and the third, drawing on the conclusions of the first two, sketches a symbolic treatment appropriate for certain types of talk about "necessity". The systems considered in Part 1 are those of C. I. Lewis and his followers. It is argued that they are unsuitable for use in symbolizing everyday modal talk, (i) because they do not have constants such as we should need for representing particular modalized utterances, and (ii) because they represent the valuation applied to those utterances which can be symbolized as being dependent upon features of symbolic systems rather than upon features of the symbolizanda. In Part 2 attention is focussed on various levels of symbolization, ranging from the symbolic use of a knot in one's handkerchief to the construction of a fullyfledged logical calculus complete with a truthtable formulation. Further the nature of logical form is explored, and it is concluded from these two investigations that the first step in symbolizing any new subject matter should be to seek out some range of valuations applicable to, and interesting for, that subject matter. The final part considers what is involved in claiming that something is necessary  whether in an everyday or a scientific context  and what it is for such a claim to "be "acceptable". It is argued that many  and probably the most important  uses of the term "necessary" are in utterances which should be symbolized as ad hoc postulates prefixed to normal truthfunctional calculus symbolizations This application of the approach argued for in Part 2 is complemented by some notes on other approaches to necessity and possibility, and the discussion is rounded off with the suggestion that further application of the "valuational" approach offers the best prospect for substantial advances in modal logic. Part 1 In Symbolic Logic Lewis sets out to show that his connective '→' is superior to the '⊃' of the truthfunctional prepositional calculus ('KC'). The superiority lies in the fact that for a "particular p and q" 'p → q' holds if and only if q is deducible from p; that is, Lewis claims, if 'p⊃q'is a tautology. It is unclear whether 'p' and 'q' are constants or variables. They are spoken of as "particular" and equated with individual utterances, yet they are, as arguments of a tautologous formula, presumably variables of KC. Further, the aspects of arguments of '→' which determine whether the relationship holds are the particular structures of these arguments and not the particular valuations assignable to simple variables of Lewis's system. The attempt to specify the conditions under which a formula consisting of two constant expressions bound by '→' should be assigned a true utterance as interpretation leads to a bizarre conclusion even when allowance is made for Lewis's personal view of tautologousness and for various readings of his argument. The matter is to be decided, it seems, only by taking utterances such as would be symbolized by the two arguments of '→', symbolizing these utterances in some other calculus (presumably KC) and deciding whether the generalization of the resulting formula is a tautology. Lewis's aims and methods are the root of the troubles. Although he condemns KC as unable to represent deducibility properly, he wishes his system to contain KC. This he hopes to achieve by retaining the existing apparatus of KC, adding a new operator and, inevitably, accepting the lack of any finite general method for determining the value of a formula. Were this new operator to have had any influence upon what can be proved, then the individual symbols of the resulting system would have ceased to be comparable to KC symbols; for to choose a KC symbol as a means of representing an utterance is to accept that all and only its truthfunctional relationships are worth symbolizing, but to choose the other sort of symbol would be to take more or different relationships as importantwith the result that a quite different structure might turn out to give the most suitable representation of the utterance concerned. In KC there is a simple general rule laying down what value is assigned to the formula composed of a material implication sign and two constants of given value. It is our knowledge of such rules which allows us to decide that KC is a suitable calculus for a given application. We are able to see that when we use designated constants to symbolize (all and only) true utterances, the fact that a given material implication is a designated formula has a certain significance. The lack of such a general rule makes it a step in the dark if we decide to apply the Lewis.systems and take designated constants to symbolize the true utterances we are considering. Furthermore it is clear that in order to assure ourselves of a strict implication holding for a given pair of expressions, we must complicate the structure of our symbolizations of them until the corresponding material implication, regarded as a KC formula, instantiates some theorem; or else we must introduce the desired strict implication directly as a designated (constant) formula. Since one is forced to introduce those strict implications which one wishes to employ later, as a deliberate part of the choice of symbolization for the individual utterances to which the system is to be applied, the symbol '→' may be seen as little more than a deviantly manipulated predicate, or 'P → Q' may be seen as a single constant whose use is restricted to cases where 'P' and 'Q' occur separately in such a way that 'P⊃Q' is designated. One may thus agree with Lewis's comment that "the only case in which any truthimplication is likely to have any value in application, as the basis of inference, is the case in which it coincides with strict implication." The most satisfactory view of his system is as a metalogic designed to deal with deducibilityrelations within KC, and itself behaving as a truthfunctional predicate calculus but lacking any distinction between variables of the meta logic and variables of the objectlogic, (that is, the 'p's and 'q's making up "particular" KC formulae dealt with). A consideration of several writers who have attempted to extend and interpret Lewis systems reveals neither a solution to the problem of the nature of their constants, nor recognition of the philosophical significance of the absence of any finite general truthtable treatment for '→'. This in no way depreciates the mathematical and logical ingenuity with which models for Lewis systems have been devised, but does leave the question of how the modalities should be symbolized a much more open one than might have been supposed. The question is best approached by considering what is involved in symbolizing a particular subject matter and what one should expect from a successful symbolization. Part 2 In examining various levels of symbolic apparatus and the criteria for preferring one system to another at each level, we are led to the conclusion that logical symbolization should always be seen as relative to some "valuation" of the symbolizanda, and that when given the task of constructing a logic for something one should begin by seeking the most interesting and important valuations applicable to it. It is further argued that logical form can only be determined in the light of knowing in which calculus and to which depth it is intended to conduct the symbolization.
