# Interpretations of Modal Logic

## INTERPRETATIONS OF MODAL LOGIC

I

SOME of the notorious difficulties in connexion with modal concepts concern the relation of what we shall call higher order modalities to lower order modalities.

It is a ‘received truth’ of modal logic that if a proposition is true, then it is also possibly true. (*Ab esse ad posse* *valet* *consequentia*.) By an application of this principle to the case when the proposition concerned is to the effect that a further proposition is possibly true or not possibly true, we conclude:

If (it is true that) a proposition is possibly true, then the proposition in question is also possibly possibly true, and if (it is true that) a proposition is not possibly true, then the proposition in question is also possibly not possibly true.

It cannot be a universally valid truth of modal logic that if a proposition is possibly true, then it is (certainly) true. In the special case, however, when the possibly true proposition is to the effect that a further proposition is possibly true, it might be thought that possibility entails truth. In other words: it might be suggested that, if a proposition is possibly possibly true, then (it is true that) the proposition in question is possibly true.

In view of what was concluded above from the *ab esse ad posse* principle, this suggestion would entail an identity between what is possible and what is possibly possible. We shall refer to this as the First Identity.

It cannot be a universally valid truth of modal logic that if a proposition is possibly false, then it is (certainly) false. In the special case, however, when the possibly false proposition is to the effect that a further proposition is possibly true, it might be thought that possible falsehood entails falsehood. In other

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words, it might be suggested that, if a proposition is possibly not possibly true, then (it is true that) the proposition in question is not possibly true.

In view of what was concluded above from the *ab esse ad posse *principle, this suggestion would entail an identity between what is impossible (not possible) and what is possibly impossible. We shall refer to this as the Second Identity.

Possibility and necessity are interdefinable. What is necessarily true is what is not possibly false, and vice versa. The reader can easily satisfy himself that, if we prefer to speak in terms of necessity rather than in terms of possibility, then our First Identity becomes an identity between what is necessary and what is necessarily necessary, and our Second Identity becomes an identity between what is not necessary and what is necessarily not necessary.

It can be shown that the First Identity is weaker than the Second. This means: We can accept the First Identity without accepting the Second, but we cannot accept the Second Identity without also accepting the First. (Cf. below, Sections V and VII.)

The two identities are consistent with what we shall call the ‘received system’ of modal logic. The notion of a ‘received system’, however, is not clear in itself. We might identify it with Lewis's *S*2. I shall, however, here identify it with a more comprehensive system, which includes *S*2 and which differs substantially from *S*2 only in that it allows us to conclude from any given theorem of the system to the necessary truth of that theorem. I call this system ‘received’, because I do not think that anyone would seriously question the necessary and universal validity of its theorems for the notions of possibility and necessity.^{1}

The identity of what is possible with what is possibly possible is the constitutive feature of the system of modal logic known as S4, and the identity of what is impossible with what is possibly impossible is the constitutive feature of *S*5.

The test of truth for the two identities is thus not a test of consistency with other, accepted truths. The question then arises:

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On what grounds, if on any, can it be decided whether the two identities are valid, or not?

Our ‘logical intuitions’, apparently, give no strong indication in favour of any definite answer to the question of truth here. One of the main reasons for this, it seems to me, is the fact that higher order modal expressions like ‘possibly possible’ or ‘possibly impossible’ have hardly any *use *at all in ordinary or scientific discourse (outside modal logic).

A problem of primary importance, therefore, is to *invent a use *or some kind of ‘equivalent’ of a use for the expressions in question (outside modal logic). This done, the problem of truth mentioned above can be tackled on a firmer basis. We shall return to this problem in the concluding section of the present paper.

II

We shall first consider a geometrical interpretation of (‘classical’) propositional logic.

The variables of the calculus are *p*, *q*,* r*, . . .* *(an unlimited multitude). The constants which we are going to use are ~ for negation, & for conjunction, v for disjunction, → for material implication, and ↔ for material equivalence. The class of well-formed formulae (wff: s) is defined recursively in the usual way. Rules for brackets are given.

We consider an arbitrary line of unit length 1. All wff: s can be represented by segments of the unit line in accordance with the following rules of interpretation:

i. The variables are represented by arbitrary segments. (A segment may consist of parts which are disconnected.) For example:

ii. The negation of a wff. is represented by that part of the unit line which is not covered by the segment representing the wff. itself. We shall call the segment representing the negation of a wff. the *complement *of the segment representing the wff. itself.

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iii. The conjunction of two wff: s is represented by the common part, if any, of the segments representing the wff: s themselves.

It follows from these rules that the disjunction of two wff: s is represented by the part of the unit line covered jointly by the segments representing the wff: s themselves; that the implication of an antecedent and a consequent wff. is represented by the part of the unit line covered jointly by the segments representing the consequent wff. and the negation of the antecedent wff.; and that the equivalence of two wff: s is represented by the part of the unit line covered jointly by the segments representing the conjunction of the two wff: s and the conjunction of the negations of the two wff: s.

A wff. is a tautology of propositional logic, if and only if its representation in accordance with Rules i.–iii. is the unit line, independently of the choice of segments to represent the variables. For example: the segment representing *p* v ~*p* is the part covered jointly by an arbitrary segment of the unit line (representing *p*) and its complement. Hence the representation of *p* v ~*p* must be the unit line itself.

If and only if the representation of the implication of two wff: s is the unit line, the representation of the antecedent is a segment of the representation of the consequent. For suppose, e.g., that the representation of *p* → *q* is* *the unit line. This means that the representation of ~*p *covers at least those portions of the unit line which are not covered by the representation of *q*.* *Hence the representation of *p* ‘falls inside’, i.e., is a segment of, the representation of *q*.

If and only if the representation of the equivalence of two wff: s is the unit line, the wff: s are represented by the same segment.

As a name of the unit line itself we shall use the symbol I. As a name of the segment representing the negation of a wff., the representation of which is the unit line, we shall use the symbol 0.

III

To the calculus of propositional logic we add one new constant *M*. It is prefixed to wff: s in the same way as ~ is prefixed. We modify the definition of the class of wff: s so as to include mention of *M*. No new rules for brackets are needed.

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Just as we by the negation of a given wff. understand the wff. which we get, when ~ is prefixed to the given wff., we shall by the *modal *of a given wff. understand the wff. which we get, when *M* is prefixed to the given wff.

All wff: s can be represented as segments of a unit line, if―in addition to the three rules of the previous section―we adopt the following new rules of interpretation:

iv. The modal of a wff. is represented by an arbitrary segment of the unit line which covers the segment representing the wff. itself. For example:

v. The modal of the disjunction of two wff: s is represented by that part of the unit line which is covered jointly by the segments representing the modals of the two wff: s themselves.

vi. If two wff: s are represented by the same segment, then the modals of the two wff: s will have to be represented by the same segment as well.

vii. The representation of the modal of a wff., the representation of which is 0, is 0.

A wff. will be called a *M*-tautology or modal tautology, if and only if its representation in accordance with Rules i.–vii. is the unit line, independently of the choice of segments to represent the variables and the modals.

(Since the wff: s now under consideration include among themselves the wff: s of propositional logic, it follows that the modal tautologies include among themselves the tautologies of propositional logic.)

Let us convince ourselves, relying upon the geometrical interpretation of wff: s, that *M*(*p *&* q*)* → Mp *&* Mq *is a *M*-tautology. (‘If propositions are mutually consistent, they are self-consistent.’)

Remembering what was said above about the representation of implications, the task can be reduced to one of showing that the representation of *M*(*p *&* q*)* *must be a segment of the representation of *Mp *&* Mq*.

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We start from the picture

and ask ourselves the following question: How is the representation of *M*(*p & q*) to be fitted into this picture?

The segment representing *p* is the same as the segment representing *p *&* q *v* p *&* ~q*.* *According to Rule vi., the modal of *p* must then be represented by the same segment as the modal of *p* &* q *v* p *&* ~q*.* *But in virtue of Rule v., the modal of *p* & *q *v* p *&* ~q *is* *represented by the same segment as the representation of the disjunction of the modal of *p *&* q *and the modal of *p* &* ~q*.* *It follows that the representation of *M*(*p *&* q*)* *must be a segment of the representation of *Mp*, which is in the picture.

By an exactly similar argument, it is shown that the representation of *M*(*p *&* q*)* *must also be a segment of the representation of *Mq*,* *which is in the picture.

The two conditions, thus deduced, which the representation of *M*(*p *&* q*)* *has to satisfy, can be simultaneously fulfilled if and only if the representation of *M*(*p *&* q*)* *is* *a segment of the representation of *Mp *&* Mq*,* *which is in the picture.

A second example will be given. Let us convince ourselves that *M*(*p → ~M~*^{,}*p*) is* *a *M*-tautology.

The segment representing *p* → ~ *M~*^{ }*p *is the same as the segment representing ~*p *v *~M*~*p. *According to Rule vi., the modal of *p* → ~*M*~*p *must then be represented by the same segment as the modal of ~*p *v* ~M*~*p. *But in virtue of Rule v., the modal of ~*p *v* *~*M*~*p *is* *represented by the same segment as the representation of the disjunction of the modal of ~*p *and the modal of ~*M*~*p*.

The representation of *M~p *v* *~*M*~*p *is the unit line. Since, according to Rule iv., the representation of ~*M*~*p *is a

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segment of the representation of *M*~*M*~*p*,* *it follows that the representation of *M*~*p *v* M~M~p* must be the unit line too. *Q.E.D.*

For any given wff. of the class under consideration can be effectively decided, whether or not it is a *M*-tautology. The proof will not be given here.

The class of *M*-tautologies includes within itself the class of theorems of S2. The proof will not be given here.

The class of *M*-tautologies is identical with the class of what is called in this paper the 'received truths' of modal logic.

IV

We consider the same class of wff: s as in the previous section, but add a new rule for the interpretation of modals:

viii. The representation of the modal of the modal of a wff. is the same as the representation of the modal of the wff.

The rule answers to what we called above the First Identity (what is possible = what is possibly possible).

A wff. will be called a *M'*-tautology, if and only if its representation in accordance with Rules i.–viii. is the unit line, independently of the choice of segments to represent the variables and the modals. (It follows trivially that *M*-tautologies are also *M'*-tautologies.)

Let us convince ourselves, relying upon the geometrical interpretation of wff: s, that *M*(*Mp *& *M~p*)* ↔ Mp *&* M*~*p *is a *M'*-tautology. ('The contingent is the same as the possibly contingent.')

Remembering what was said above about the representation of equivalences, the task can be reduced to one of showing that the representation of *M*(*Mp *& *M*~*p*)* *must be the same segment as the representation of *Mp *& *M*~*p*.

We start from the picture

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and ask ourselves the following question: How is the representation of *M*(*Mp* & *M*~*p*) to be fitted into this picture?

We know from what was shown in the preceding section that the representation of *M*(*Mp* & *M*~*p*) must be a segment of the representation of *MMp* & *MM*~*p*. In virtue of Rule viii., however, the representation of *MMp* is the same segment as the representation of *Mp*, and the representation of *MM~p*, the same segment as the representation of *M*~*p*. Consequently, the representation of *MMp* & *MM*~*p* is the same segment as the representation of *Mp* & *M*~*p*, which is already in the picture.

We have now shown that the representation of *M*(*Mp* & *M*~*p*) must be a segment of the representation of *Mp* & *M*~*p*. In virtue of Rule iv., on the other hand, the representation of *Mp* & *M*~*p *must be a segment of the representation of *M*(*Mp* & *M*~*p*.) These two features of the representation of *M*(*Mp* & *M*~^{,}*p*) can be reconciled only if the representation of *M*(*Mp* & *M*~*p*) in the picture is the same segment as the representation of *Mp* & *M*~*p*.

For any given wff. of the class under consideration can be effectively decided, whether or not it is a *M'*-tautology. The proof will not be given here.

The class of *M'*-tautologies is identical with the class of theorems in S4.

V

We consider the same class of wff: s as in Sections III and IV, but add a further rule:

ix. The representation of the modal of the negation of the modal of a wff. is the same as the representation of the negation of a modal of the wff.

This rule answers to what we called above the Second Identity (what is impossible = what is possibly impossible).

A wff. will be called a *M''*-tautology, if and only if its representation in accordance with Rules i–vii. and ix. is the unit line, independently of the choice of segments to represent the variables and the modals.

It can be shown that *M'*-tautologies are also *M''*-tautologies.

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Consider the picture

We ask the question: How is the representation of the modal of the modal of *p*, i.e. *MMp*, to be fitted into the picture?

In virtue of Rule ix., the representation of the modal of the negation of the modal of *p*, i.e. *M*~*Mp*, is the same as the representation of the negation of the modal of *p*, i.e. ~*Mp*, which is in the picture. In virtue of Rule vi., if two wff: s have the same representation, then their modals have the same representation too. Thus the representation of *M~M~Mp* is the same as the representation of the modal of the modal of *p* i.e. *MMp*. But in virtue of Rule ix., the representation of *M~M*~*Mp* is the same as the representation of ~*M~Mp* which was already shown to be the same as the representation of *Mp*. Thus we conclude that the representation of the modal of the modal of *p*, i.e. *MMp*, is the same segment as the representation of the modal of *p*, i.e. *Mp*, which is already in the picture.

For any given wff. of the class under consideration can be effectively decided, whether or not it is a *M''*-tautology. The proof will not be given here.

The class of *M''*-tautologies is identical with the class of theorems in *S*5.

VI

We shall now consider physical models which are ‘isomorphic’ with the above geometrical models of propositional logic and various systems of modal logic.

We conceive of the variables *p*, *q *, *r*, … as the ‘physical’ parts of some ‘physical’ whole. (The parts need not necessarily be spatio-temporally continuous.) For example: *p*, *q*, *r*, . . . can be represented by segments of a metal rod, geographical areas on the earth's surface, populations of human beings, etc. All the wff: s of propositional logic can be interpreted as parts of such wholes, if we adopt the rule that the negation of a wff.

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is represented by that part of the whole which remains, when the part representing the wff. itself has been removed, *and *the rule that the conjunction of two wff: s is represented by the common part, if any, of the parts, representing the wff: s themselves.

Consider next some ‘activity’ or ‘process’ which may occur in the parts of a whole under consideration. For example: an electric current of a certain strength running through a segment of a metal rod, rainfall in a district, a certain disease occurring among (not necessarily all) members of a population, etc.

Relative to such an assumed activity, we decide to call the parts *fields of activity*.

The activity under consideration may have some 'influence' or 'effect', extending from a part in which the activity is supposed to occur to other parts of the same whole. For example: an electric current running through a segment of a metal rod may have a thermic effect on the surrounding portions of the rod; a change of climate in one country may produce a change of climate in neighbouring countries; a disease among members of one population may spread by contagion to the members of another population, etc.

Relative to an assumed activity and an assumed influence, we decide to call the part, to which the influence may extend, when the activity occurs in a given part, the *field of influence *of (or associated with) the given field of activity.

It will be convenient to understand the concept of a field of influence in such a way that any given field of influence *includes *the associated field of activity.

Relative to an assumed activity and an assumed influence, we further decide to call the part, to which the influence will *not *extend, when the activity occurs in a given part, the *field of resistance *of the surrounding (of the given field of activity).

The activities and influences to be considered are assumed to obey the following three laws:

* L*1. If a field of activity is divided into parts (not necessarily mutually exclusive), then the fields of influence of the parts conceived of as fields of the same activity, are jointly identical with the field of influence of the original field of activity.

* L*2. With the same field of activity is always associated the same field of influence.

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*L*3. If the field of activity is non-existent (‘the o-field’), then the associated field of influence is also non-existent.

Activities which obey *L*1 can conveniently be called *additive *with regard to their influences. Far from all processes in nature possess this feature of additiveness. But some obviously possess it. For example: if an epidemic occurs among the members of two populations, then the members of the surrounding population which may become infected are exactly those which may get the disease from members of the one *or *the other of the two populations, whenever the disease occurs in them.

We can now give a new interpretation to the class of wff: s considered in Sections III–V (the wff: s formed of the symbols of propositional logic and the symbol *M*).

We call the parts of a given whole, fields of activity. The variables, negation, and conjunction are interpreted as before. The modal of a wff. is interpreted as the field of influence of the field of activity representing the wff. itself.

(Since, under the terminology adopted, any part of the given whole is called a field of activity, it follows that the fields of influence (and of resistance) are a sub-class of the fields of activity.)

A wff. is a *M*-tautology, if and only if its representation is the whole, of which the fields of activity are parts, independently of the choice of fields of activity to represent the variables and fields of influence to represent the modals.

VII

Processes in nature which are additive with regard to their effects in the sense of *L*1 above, can be divided into the following three categories:

* A*. Processes which are such that, if a field of their influence becomes itself a field of activity for the same process, then the field of influence of this new field of activity extends beyond the boundaries of the original field of influence.

* B*. Processes which are such that, if a field of their influence becomes itself a field of activity for the same process, then the field of influence of this new field of activity does *not *extend beyond the boundaries of the original field of influence.

* C*.* *Processes which are such that, if a field of resistance of the

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neighbourhood of a field of their activity becomes itself a field of activity for the same process, then the field of influence of this new field of activity does not extend beyond the boundaries of the original field of resistance.

Processes of the third kind are also processes of the second kind, but not necessarily vice versa.

The fields of activity, influence, and resistance of processes of the kind *B* offer a model of a system of modal logic in which the First Identity holds.

The fields of activity, influence, and resistance of processes of the kind *C* offer a model of a system of modal logic in which the Second Identity holds.

Examples of processes of each of the three kinds can be given. Here we shall consider only one example.

Let us conceive of *p* as a certain population of human beings. Let the activity, of which *p* is a field, be an epidemic disease. (We need not suppose, however, that all members of the population actually have or even can get the disease.) ~*p* is represented by the population, of which a man is a member if and only if he is not a member of the population representing *p*. For the sake of convenience, we shall speak of the 'population' and its ‘neighbourhood’ (population).

Of *Mp *we conceive as the population itself *and *those members of the neighbourhood, who for some reason or other may catch the disease from members of the population. The representation of *Mp *can thus conveniently be called the field of influence of the epidemic in the population.

It follows that ~*Mp *is represented by those members of the neighbourhood, who for some reason or other cannot catch the disease from members of the population. The representation of ~*Mp *can thus conveniently be called the field of resistance (‘immunity’) of the neighbourhood.

If to the field of influence are added those members, if any, of the neighbourhood who, although they cannot catch the epidemic from members of the population, yet may for some reason or other catch it from such of their fellow members of the neighbourhood as may themselves catch it from members of the population, we get a representation of *MMp*.

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of the field of influence who, for some reason or other, may catch the disease from a member of the field of resistance, we get a representation of *M* ~*Mp*.

As to the relative magnitude of the population representing *Mp *and *MMp**, *two different situations might occur:

1. The members of the field of resistance are totally immune from the disease and cannot, consequently, catch it from anybody. In this case, the populations representing *Mp *and *MMp *coincide.

2. The members of the field of resistance are secure from catching the disease from members of the population and yet not secure from catching it from infected fellow members of theirs of the neighbouring population. (This could be the case for biological reasons, but also because, say, the members of the field of resistance are segregated from contact with members of the population.) In this case, the population representing *MMp *would be larger than the population representing *Mp*.

If the fields of resistance of all populations concerned consist entirely of members, if any, which cannot catch the disease in question from non-members of those fields, we obtain a model of a modal logic, in which the First Identity holds.

As to the relative magnitude of the populations representing ~*Mp *and *M***~***Mp*,* *two different situations might also occur:

1. The members of the field of influence are secure from catching the disease from members of the field of resistance. (This would trivially be the case, if the members of the field of resistance are totally immune and, consequently, unable to acquire the disease. It would non-trivially be the case, e.g., if members of the field of resistance could acquire the disease, though only in a non-catching form.) Under this alternative the populations representing ~*Mp *and *M*~*Mp *coincide.

2. The members of the field of influence are not secure from catching the disease from members of the field of resistance. (It may, for example, happen that a member of the field of resistance catches the disease from a fellow member of the neighbouring population, who has himself caught it from a member of the population, and that subsequently another member of the field of influence catches the disease from this infected member of the field of resistance.) Under this alternative, the

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population representing *M~Mp* would be larger than the population representing ~*Mp*.

If the fields of influence of all populations concerned consist entirely of members, if any, who cannot catch the disease in question from non-members of those fields, we obtain a model of a modal logic, in which the Second Identity holds.

It is easy to see that a model in which the Second Identity holds (universally) is also a model in which the First Identity holds (universally). If the members of the population representing ~*Mp* cannot infect the members of the population representing *Mp*, then the second population equals its own field of resistance and the first its own field of influence. But if it is universally true of fields of influence that they consist entirely of members who cannot catch the disease from non-members, it follows that the population representing *MMp* must coincide with the population representing *Mp*.

VIII

We have seen that the 'logical behaviour' of natural processes which are, in a defined sense, *additive* with regard to their effects is governed by the same formal rules as the concept of possibility. These processes—or their fields of activity, influence,and resistance—can be said to afford models or illustrations of the possible. In the language which speaks about these processes there is a use for formal equivalents of higher order modalities and thus also for formal equivalents of the two identities with which we have been concerned in this paper. The criteria of truth at hand are such as to show that for some additive natural processes neither of the identities holds, for some both identities hold, and for some only the first identity holds.

It seems to me that the existence of such illustrations as these among natural processes shows that neither of the identities in question can claim universal validity for the concept of the possible (and the necessary) in modal logic, nor can either of them be rejected as contrary to the 'true' nature of modality.

It would be of some interest to investigate the 'logical behaviour' of non-additive natural processes with a view to the question whether this behaviour also illustrates some general notions of logic.

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^{1}A ^{ }more detailed study of this system will be found in my publication *An Essay in Modal Logic, *Amsterdam, 1951.