Modalities and formal systems

General
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 fully-fledged logical calculus complete with a truth-table 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 truth-functional 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 truth-functional 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 truth-functional relationships are worth symbolizing, but to choose the other sort of symbol would be to take more or different relationships as important-with 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 truth-implication 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 meta-logic designed to deal with deducibility-relations within KC, and itself behaving as a truth-functional predicate calculus but lacking any distinction between variables of the meta- logic and variables of the object-logic, (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 truth-table 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. Given that information the problem becomes "What aspects of this symbolizandum could, by being otherwise than they are, force me to alter my valuation of it?" - where the valuation referred to is that upon which the calculus concerned is based.
The lowest level of symbolization considered is that of signs; these are no more than asystematically selected objects used as ad hoc reminders or representatives of other things. When there is some system in the way the representing objects are chosen, we may speak of a sign-system; the system need be no more than there being some aspect of the represented things by reference to which we can generate (the idea of) the representing things.
Where in a sign-system the representing things are in fact symbols for which an ordering is laid down in other contexts, it is a symbolism. The ordering concerned may or may not be parallel to a discerned ordering in that aspect of the symbolizanda by reference to which we have chosen the signs (symbols); such a parallellism is, however, preferable.
To the signs of a symbolism may "be added other, complex, signs. Where this occurs and these complex signs are always equivalent to some simple signs, we talk of a sub-logic. The reducibility of complex signs to simple ones is pointful only when the symbolizanda can also be observed in groups or structures to which the same descriptions can be applied as apply to individual symbolizanda. A sub-logic is most useful where the descriptions satisfied by complexes pf symbolizanda are functions of the descriptions satisfied by their members, where satisfaction of such descriptions is the aspect of the symbolizanda on the basis of which the signs can be generated, and where the ordering of the signs parallels that ordering which we find it natural to impose upon satisfaction of the various descriptions.
In a calculus, the highest level of symbolic system considered, the signs are supplemented by variables, and formulae involving variables and, possibly, signs occur as a part of the system. The series of complex signs resulting from the application to a formula of all possible sets of substitutions of signs for the distinct variables it contains, is called the set of "realisations" of that formula.
A calculus is endowed with an assertion sign which is applicable to any formula all of whose realisations are equivalent to some given ('designated') sign or signs. The selection of a designated sign or signs, however, is only pointful when we have symbolized the subject matter with respect to a range of descriptions certain of which enjoy a special status. Such-a range of descriptions results from what we describe as a "valuation" of that subject matter. It is such a valuation which should be sought when setting out to construct a logic for modal utterances.
Part 3
In the final part "necessity" is taken as an example of the type of subject matter we 'would like to symbolize. Leaving aside other attempts to devise logics of this notion, consideration is given to various cases of "saying that something is necessary" and to the extent and manner in which the utterances involved can be represented in a logical system.
The most common type of utterance of this sort is that by means of which someone, in order to spare himself the need to provide factual evidence for an assertion, or in order to enforce some convention or definition, invokes a necessity. The valuations 'true' and 'false' are inappropriate in this case, but we might well use 'acceptable' and 'unacceptable'.
To decide upon the applicability of such a valuation, however, tore must consider the utterance in relation to its particular context; it is by variation in the context not in the utterance itself or in any factual matters that we might be forced to alter our valuation, for the acceptability depends upon whether any such convention, definition or assumption as is invoked does apply in that particular context. If we were to proceed with symboliz.ation taking this source of variability in the valuation used, as our starting point, then we should have to take contexts as individuals and produce a representation of the internal relations within such a context. If however we take a more traditional approach to logic, we shall prefer a system in which the individual utterance is the basic unit, and one in which the rules are context-independent. One way of achieving this and catering for the symbolization of invocations of necessity, is to add to the axioms present in the symbolization of some context, a set of ad hoc postulates - an 'assumption-set' ; an invocation of necessity is then an utterance which purports to be a reiteration of some member of the assumption-set or of some consequence of it.
Contrasted with invocations of necessity are assertions of necessity. These are straightforward - 'true' or 'false' - claims that certain things are accepted as being necessary. They are to be seen as referring to particular groups of persons and particular contexts, subject to the usual practice of dropping such references where it is obvious that everyone or every context is intended.
Confusion between the truth of an assertion of necessity and the acceptability of an invocation of necessity, can account for the feeling that it makes sense to ask someone for his evidence that something is necessary, while it is quite clear that if a given thing is necessary no evidence should have any influence upon our acceptance of it. Again the attraction of the rule that what is necessary is true, can "be traced to the obvious oddity in the "behaviour of anyone who, after assenting to a universally-framed assertion of necessity (which implicitly claims that he too accepts the necessity in question in all contexts) displays the falsity of what he has assented to by himself refusing to accept the corresponding invocation of necessity or -factual claim.
An analogous approach to natural necessity is possible. The assumptions made in a given context - the theories and laws, in this case - determine which natural necessities are acceptable; these can then be invoked and made the basis for predictions. Assertions of natural necessity deal with the tenets of various theories, and the laws which are assumed to hold in various contexts - or disciplines.
It is not paradoxical that the most empirical type of knowledge is hereby mixed up with 'linguistic' matters and ad hoc conventions. To establish a law we must consider it first in a context in which it is not assumed; thereafter, however, it need not constantly be re-established every time we wish to apply it - it is in practice taken as a premiss which in conjunction with observations of a particular situation allows us to make deductions about that situation. It seems farther that 'lawlike' statements - those from which subjunctive conditionals may be derived - can be explained as being those statements which make up the assumption-set for a given argument.
The present treatment-of necessity is not intended to "be definitive; it is rather to "be seen as a demonstration of the fruitfulness of the approach to symbolization which has "been argued for, and as an indication of the direction in which logical study of the modalities might go. Two related topics relevant to such future work are considered in "brief and without the presupposition of the results of the rest of Part 3.
The evidential interpretation of the modalities can be seen as an attempt to range the modalities along a scale of "strength" which would parallel that of the strength of evidence. Although it is difficult to carry through this programme for all. modalities, it does seem to contain the seeds of some valuation which might yield a system for symbolizing the interrelations amongst the modalities.
Consideration of the various paraphrases of "necessary" in terms of possibility, and in particular the notion of "only possibility" suggests that a symbolization for necessities might emerge as a byproduct of a system for enumerating possibilities. "Possibility", however, like "alternative", seems to be a word which is used for categorizing rather than for describing or characterising things; recognition of this removes much of the temptation to think of "possibilities" as concrete things which can be counted, or as attributes which can be spoken of along with the other things discussed in modal utterances.
In conclusion it is suggested that much more work on the modalities is c-alled for, and that the intuitive semantic approach - especially when it employs the 'valuational' method - offers the most promise for worthwhile advances in modal logic.