A New system of modal logic
I. INTRODUCTION
As far as I know, modal logic from Aristotle to modern times has only dealt with modality (necessity, possibility, impossibility) as a monadic predicate or ‘property’ of propositions.
The system presented in this paper differs from the traditional systems of modal logic in that it studies the various modalities as dyadic predicates or ‘relations’ between propositions.^{1} Such modalities may also be called relative modalities.
We introduce a symbol ‘M(p/q)’. It can be read: p is possible given q. Instead of ‘given q’, we may also read ‘on conditions q’ or ‘relatively to q’. Accordingly, ‘~M(p/q)’ can be read: it is not the case that p is possible, given q, or: p is impossible, given q, ‘~M(p/q)’ can be read : it is not the case that not-p is possible, given q, or: not-p is impossible, given q, or (by verbal convention): p is necessary, given q. (This, by the way, is not the only reasonable interpretation of necessity in terms of negation and possibility.)
The letters ‘p’ and ‘q’ stand for arbitrary propositions. The letter ‘t’ will be used to represent an arbitrary tautology of propositional logic (PL).
Our notation must not mislead a reader into thinking that the letter ^{‘}M’ and the stroke ‘/’ represent two logical constants. Relative possibility is the one indefinable modal concept of our system. Instead of the stroke ‘/’, we might have used the comma ‘,’. This would have been in conformity with a usual notation in the logic of relations. There are, however, some practical reasons for preferring the stroke.
By the absolute possibility of a proposition we shall here understand its possibility on tautologous conditions. ^{‘}M(p/t)’ thus says that p is absolutely possible. In a similar manner,
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we define absolute impossibility [~ M(p/t)] and absolute necessity [~ M(~p/t)].
It will be shown later (p. 97) that absolute possibility means possibility on some (absolutely) possible conditions, that absolute impossibility means possibility on no (absolutely) possible conditions, and that absolute necessity means necessity on all (absolutely) possible conditions.
Absolute possibility (impossibility, necessity), as here defined, is thus a species (limiting case) of the generic notion of relative possibility (impossibility, necessity).
It would be rash to maintain that our notions of the absolute modalities were the same as the modal notions of the traditional systems. For, it is not clear what views the authors of those systems would have taken of the ‘meaning’ of their monadic concepts in relation to our dyadic modalities. (Some might have wished to regard the two kinds of modal concepts as being entirely different.)
By the ‘classical’ systems of modal logic I shall here understand the systems S1-S5 of C. I. Lewis, the system T of R. Feys, the systems M, M’, and M’’ of von Wright, and the system developed by O. Becker. The systems T and M may easily be shown to be equivalent. The systems M’ and S4 are equivalent, and so also M’’ and S5. The system of Becker is somewhat weaker than T and M.
It is an interesting fact that the laws (axioms and theorems) of the ‘classical’ systems of modal logic can be interpreted as laws of absolute modalities within our new system. In a sense, therefore, the ‘classical’ systems may be regarded as special or limiting cases of the logic of dyadic modalities. The precise meaning of this contention will be given later. (Section 8.)
Among the traditional systems of modal logic, a peculiar position is held by the system (or systems) developed by J. Łukasiewicz. Some comments on it will be found in an appendix to the present paper.
2. COMPOSSIBILITY, MODAL COMPATIBILITY, AND SELF-CONSISTENCY
It is important that our notion of relative possibility should not be confused with the notion of compossibility of the ‘classical’
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systems. In other words: the meaning of ‘M(p/q)’ ought to be distinguished from the meaning—using C. I. Lewis’s symbolism for possibility—of ‘◊(p & q)’. There is a significant difference between the two notions. ◊(p & q) is false, i.e. p and q are not compossible, as soon as one of the two propositions, say q, is impossible.
~ ◊ q→ ~ ◊(p & q) is a law of ‘classical’ modal logic. But M(p/q) may be true, even though q is (absolutely) impossible. ~ M(q/t) → ~ M(p/q) is not a law of our system.
We shall distinguish between the compossibility of two (or more) propositions, and the compatibility or consistency of one proposition with another proposition. We shall say that ‘M(p & q/r)’ expresses the compossibility of p and q on conditions r, and ‘M(p & q/t)’ the absolute compossibility of p and q. And we shall say that ‘M(p/q)’ expresses the compatibility or consistency of p with q. As will be seen presently (p. 100f.), the relation of compatibility is not unrestrictedly symmetrical.
Consider the three propositions ‘there are exactly 23 primes between 0 and 100’ and ‘there are at least 21 primes between 0 and 100’ and ‘there are 25 primes between 0 and 100’. Shall we not say that the second is compatible with the first, and that the third is incompatible with the first? Namely, considering that the first entails the second and contradicts the third. But the first proposition is false. If we regard truths of arithmetic as necessary truths, we shall have to regard the first proposition as not merely false, but impossible. We shall, moreover, have to regard the second and third propositions as being necessary.
It seems to me that this is an example of an impossible proposition, with which one necessary proposition is compatible and another necessary proposition is incompatible—in a perfectly good sense of ‘compatible’.^{2} Yet neither one of the two necessary propositions is compossible with the impossible
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proposition. This would show that compatibility and compossibility ought to be distinguished. The fact, therefore, that the traditional modal systems are unable to make this distinction may be regarded as an insufficiency of those systems.
Compatibility, in our sense, is a ‘typically relational’ notion. Whether one proposition is compatible with another depends upon their mutual modal relation and cannot be established from the modal character (say impossibility) of one of them alone.
In our system, we can further distinguish between absolute possibility and self-compatibility or self-consistency of a proposition. The absolute possibility of p is expressed by ‘M(p/t)’ and the self-compatibility (self-consistency) of p by ‘M(p/p)’. As will be seen presently (p. 97), the two notions are not formally equivalent. The former is stronger than the latter. All absolutely possible propositions are self-consistent, but not all self-consistent propositions are absolutely possible.
It may again be regarded as an insufficiency of the traditional systems that they are unable to distinguish between propositions which are ‘simply’ impossible and propositions which are self-inconsistent. The latter are a sub-class of the former.
3. M-EXPRESSIONS
M-expressions are defined recursively as follows :
By an M-expression of order n(1 ≤ n) we understand an atomic M-expression of order n or a molecular complex of atomic M-expressions, at least one of which is of order n and none of which is of higher order than n.
By an atomic M-expression of order n(1 ≤ n) we understand an expression of the form ‘M( / )’, in which one of the blanks is filled by an M-expression of order n-1 and the other blank by an M-expression of order n-1 or of lower order.
By an M-expression of order zero we understand a propositional variable p, q, r, . . . or the tautology variable t or a molecular complex of propositional variables (or t).
It follows from the above definitions that atomic M-expressions of the first order are expressions of the form ‘M( / )’, in which the blanks are filled by M-expressions of order zero.
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It follows further that M-expressions of the first order are atomic M-expressions of the first order or molecular complexes consisting of atomic M-expressions of the first order and/or M-expressions of order zero.
An M-expression of the first order will be called homogeneous, if it is either an atomic M-expression of the first order or a molecular complex of atomic M-expressions of the first order.
M-expressions are the well-formed formulae of the various calculi of the new modal logic which we are going to present.
4. THE CALCULUS M_{d}1
In this section we shall be studying, in some detail, a particularly simple modal calculus. We shall label it the system M_{d}1. It constitutes the ‘core’ which is common to all the other modal calculi which we are going to mention. The system M_{d}1 may be called the core of dyadic modal logic.
It is the distinguishing feature of M_{d}1 that its well-formed formulae are the homogeneous M-expressions of the first order.
The axioms of the calculus are the following three formulae:
A1. M(p/p) → ~ M(~p/p)
A2. M(p/q) v M(~p/q)
A3. M(p & q/r) ↔ M(p/r) & M(q/r & p)
The theorems of the calculus are all formulae which (recursively) satisfy one of the following four conditions :
i. The formula is obtainable from a tautology of propositional logic by substituting for all its propositional variables homogeneous M-expressions of the first order.
ii. The formula is obtainable from an axiom or theorem of our calculus by substituting for some propositional variables in it some M-expressions of order zero.
iii. The formula is the consequent in a material implication formula, which is itself an axiom or a theorem and whose antecedent is also an axiom or a theorem of our calculus.
iv. The formula is obtainable from an axiom or theorem of our calculus by substituting for some M-expression of order zero in it another M-expression of order zero, which is tautologously equivalent to the first.
The conditions which formulae have to satisfy in order to qualify as theorems could alternatively be described as rules
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of inference or transformation. Condition i. simply means that in our calculus, as indeed in most deductive theories, the laws of (‘classical’) propositional logic are taken for granted. Conditions ii. and iii., the Rules of Substitution and of Detachment (modus ponens) may be called basic rules of all formalized reasoning whatever. Condition iv. has a somewhat more special status. It may be called a rule of the Substitution of Identities or of Extensionality. ^{‘}Identity’ is here defined as meaning tautologous equivalence in propositional logic.
There is a (well-known) difference between ‘substitution’ in the sense of i. and ii., and ‘substitution’ in the sense of iv., of which it may be well to remind the reader. If an expression is being substituted for a variable, it is necessary that the substitution should be carried out in all places where the variable occurs in the formula in question. But if an expression is being substituted for another expression, the two expressions being tautologously equivalent, then it is not necessary that the substitution be carried out in all places where the expression occurs in the formula in question.
For the sake of notational convenience we shall introduce a special symbol ‘N’ for (relative) necessity. ‘N(p/q)’ can be read: p is necessary, given q. In view of the elucidation, given above (p. 89), of relative necessity in terms of relative possibility and negation, we shall regard ‘N(p/q)’ as an abbreviation of ‘~M(~p/q)’.
The axioms call for some comments:
A1 says that if a proposition is self-consistent, then its negation is inconsistent with it. Using the symbol ‘N’, we can write A1 in the form M(p/p) → N(p/p). The axiom thus also says that a self-consistent proposition is self-necessary.
The notion of self-necessity is not without interest. Some comments on it are given in Appendix I of the present essay.
A2 says that either the ^{‘}positive’ or the ‘negative’ of any other arbitrary proposition p is compatible with an arbitrary proposition q. The formula can also be written ~M(~p/q) → M(p/q), or N(p/q) → M(p/q). The axiom in question thus also says that, if a proposition is necessary relative to another, then it is possible relative to it. (‘Necessity entails possibility’.)
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A3 strikes us as an analogue in the logic of the relative modalities to the so-called multiplication-principle in the theory of probability. This analogy will be explored in further detail later. (Section 10.)
We now proceed to the proof of some theorems.
The proofs will be presented in the form of a successive enumeration of formulae which are themselves theorems of M_{d}1. It will be left to the reader to verify that the enumerated formulae satisfy the conditions i.-iv. above of theoremhood. The verification is usually quite simple; in order to facilitate it references to formulae are sometimes added within brackets.
T1. M(p/q) ↔ M(p & r/q) v M(p & ~r/q)
Proof:
(1) M(p/q) ↔ M(p/q)
(2) M(p/q) ↔ M(p/q) & [M(r/q & p) v M(~r/q & p)]
(3) M(p/q) ↔ M(p/q) & M(r/q & p) v M(p/q) & M(~r/q & p)
(4) M(p/q) ↔ M(p & r/q) v M(p & ~r/q) Q.E.D.
T2. M(p v q/r) ↔ M(p/r) v M(q/r)
‘The possibility of a disjunction is a disjunction of possibilities.’
Proof:
(1) M(p v q/r) ↔ M[(p v q) & p/r] v M[(p v q) & ~p/r] (T1)
(2) M(p v q/r) ↔ M(p/r) v M(~p & q/r) [Since (p v q) & p is equivalent to p alone, and (p v q) & ~p to ~p & q.]
(3) M(p v q/r) ↔ M(p & q/r) v M(p & ~q/r) v M(p & q/r) v M(~p & q/r)
(4) M(p v q/r) ↔ M(p/r) v M(q/r) Q.E.D.
T3. N(p & q/r) ↔ N(p/r) & N(q/r)
‘The necessity of a conjunction is a conjunction of necessities.’
This theorem is the ‘dual’ of T2. To prove it, replace N-expressions with that for which they are an abbreviation.
T4. M(q/q) → [M(p/q) ↔ M(p & q/q)]
If q is self-consistent, then the possibility of p, given q, is equivalent to the possibility of p and q, given q.
Proof:
(1) M(p & q/q) ↔ M(q/q) & M(p/q) (A3)
(2) M(q/q) → [M(p/q) ↔ M(p & q/q)] Q.E.D.
T5. M(q/q v r) v M(r/q v r) →
[M(p/q v r) ↔ M(q/q v r) & M(p/q) v M(r/q v r) & M(p/r)]
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Proof:
(1) M(q v r/q v r) → [M(p/q v r) ↔ M(p & (q v r)/q v r] (T4)
(2) M(q/q v r) v M(r/q v r) → [M(p/q v r) ↔ M(p & q/q v r) v M(p & r/q v r)]
(3) M(q/q v r) v M(r/q v r) →
[M(p/q v r) ↔ M(q/q v r) & M(p/q) v M(r/q v r) & M(p/r)] [Since (q v r) & q is equivalent to q and (q v r) & r to r.] Q.E.D.
T6. M(p/t) → M(p/p v q)
Proof:
(1) M[p & (p v q)/t] → M(p v q/t) & M(p/p v q)
(2) M(p/t) → M(p/p v q) Q.E.D.
T7. M(q/t) & M(r/t) → [M(p/q v r) ↔ M(p/q) v M(p/r)]
This theorem says that a proposition is possible relative to a disjunction of (absolutely) possible alternatives, if and only if it is possible relatively to some (at least one) of those alternatives individually.
Proof:
(1) M(q/q v r) & M(r/q v r) → M(q/q v r) v M(r/q v r)
(2) M(q/q v r) & M(r/q v r) → [M(p/q v r) ↔ M(q/q v r) & M(p/q) v M(r/q v r) & M(p/r)] [(1) and T5.]
(3) M(q/q v r) & M(r/q v r) → [M(p/q v r) ↔ M(p/q) v M(p/r)]
(4) M(q/t) & M(r/t) → M(q/q v r) & M(r/q v r) (T6)
(5) M(q/t) & M(r/t) → [M(p/q v r) ↔ M(p/q) v M(p/r)] Q.E.D.
T8. M(q/t) & M(r/t) → [N(p/q v r) ↔ (N(p/q) & N(p/r)]
This theorem says that a proposition is necessary relative to a disjunction of (absolutely) possible alternatives, if and only if it is necessary relatively to all those alternatives individually.
T8 is the ‘dual’ of T7. To prove that T8 is ‘but another form of’ T7, replace N-expressions with that for which they are an abbreviation.
This is a corollary of T7:
T9. M(q/t) & M( ~ q/t) → [M(p/t) ↔ M(p/q) v M(p/~q)]
And these are corollaries of T8:
T10. M(q/t) & M(~q/t) → [N(p/t) ↔ N(p/q) & N(p/~q)]
T11. M(q/t) & M(~q/t) → [ ~ M(p/t) ↔ ~ M(p/q) & ~ M(p/~q)]
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T9-T11 may easily be generalized so as to become valid for the case in which, instead of the two alternatives q and ~q, we have an arbitrary number n of mutually exclusive and jointly exhaustive alternatives q1, . . ., qn. The theorems may then be rendered in words as follows:
The absolute possibility of a proposition is its possibility under some possible circumstances, i.e. on some (at least one) alternative in any arbitrary set of mutually exclusive and jointly exhaustive (absolutely) possible conditions. The absolute necessity of a proposition is its necessity under all possible circumstances, i.e. on all alternatives in any arbitrary set of mutually exclusive and jointly exhaustive (absolutely) possible conditions. The absolute impossibility, finally, of a proposition means that the proposition is possible under no possible circumstances, i.e. the proposition is not possible on any alternative in any set of mutually exclusive and jointly exhaustive (absolutely) possible conditions. (See above p. 90.)
From A3 we easily obtain:
T12. M(q/t) → [N(p/t) → N(p/q)]
T13. M(q/t) → [~M(p/t) → ~M(p/q)]
In words: a (absolutely) necessary proposition is necessary relatively to any (absolutely) possible proposition. An (absolutely) impossible proposition is impossible relatively to any (absolutely) possible proposition.
T14. M(p/t) → M(p/p)
Proof:
(1) M(p & p/t) ↔ M(p/t) & M(p/p) (Since t & p is equivalent to p.)
(2) M(p/t) → M(p/p) Q.E.D.
The theorem says that, if a proposition is absolutely possible, then it is self-consistent. The theorem cannot be converted. Thus, according to the laws of our modal logic, a proposition may be self-consistent without being absolutely possible. The class of self-inconsistent propositions, in other words, is a sub-class of the class of absolutely impossible propositions. (See above p. 92.)
To what extent the distinction between self-contradictory and impossible propositions is an important and useful one will in the last resort depend upon considerations which fall outside
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the scope of the present inquiry. What is noteworthy so far is only that the system provides for this distinction. Examples of self-inconsistent propositions would be ‘this square is round’ or ‘he is a female brother’. On the face of things, it looks as though ‘there are exactly 23 primes between 0 and 100’ might be an example of an impossible but not self-contradictory proposition. Whether it really is an example cannot be decided before one has given criteria, other than those provided by the calculus itself, of absolute impossibility and self-inconsistency.
T15. M(t/t)
The tautology is self-consistent (and absolutely possible).
Proof:
(1) M(p v ~p/t) ↔ M(p/t) v M(~p/t) (T2)
(2) M(p/t) v M(~p/t) (A2)
(3) M(p v ~p/t)
(4) M(t/t) Q.E.D.
T16. M(t/~t)
Proof:
(1) M(p v ~p/~t) ↔ M(p/~t) v M(~p/~t)
(2) M(p/~t) v M(~p/~t)
(3) M(p v ~p/~t)
(4) M(t/~t) Q.E.D.
T17. M(p/p) ↔ N(p/p)
‘Self-consistency and self-necessity are one and the same.’
Proof:
(1) M(p/p) → N(p/p) (Al)
(2) N(p/p) → M(p/p) (A2)
(3) M(p/p) ↔ N(p/p) Q.E.D.
T18. N(t/t)
Proof:
(1) M(t/t) ↔ N(t/t)
(2) M(t/t)
(3) N(t/t) Q.E.D.
T18 can also be written ~ M(~t/t). The tautology is thus absolutely necessary and the contradiction absolutely impossible.
T19. ~ M(~t/~t)
Proof:
(1) ~ M(~t/~t) ↔ ~ N(~t/~t) (From T17)
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(2) ~ N(~t/~t) ↔ M(t/~t)
(3) M(t/~t) (T16)
(4) ~ M(~t/~t) Q.E.D.
T19 can also be written N(t/~t).The tautology is thus necessary (even) relative to the contradiction.
T20. ~ M(p/p) ↔ N(~p/p)
‘Self-inconsistency is the same as necessity of the negation relative to the proposition.’
Proof:
(1) ~ M(p/p) ↔ ~ M(~~p/p)
(2) ~ M(p/p) ↔ N(~p/p) Q.E.D.
T21. N(p/~p) → N(p/t)
Proof:
(1) ~ M(~p/~p) → ~ M(~p/t) (From T14 by contraposition.)
(2) N(p/~p) → N(p/t) Q.E.D.
This theorem says that if a proposition is necessary relatively to its own negation, then it is absolutely necessary. (Related to the so-called consequentia mirabilis.)
T22. M(p/p) v M(~p/~p)
Proof:
(1) M(p/t) → M(p/p) (T14)
(2) M(~p/t) → M(~p/~p)
(3) M(p/t) v M(~p/t) → M(p/p) v M(~p/~p)
(4) M(p/t) v M(~p/t) (A2)
(5) M(p/p) v M(~p/~p) Q.E.D.
Thus any given proposition either is itself self-consistent or has a negation that is self-consistent. Negatively speaking: it must not be the case that both a proposition and its negation are self-inconsistent.
By antinomic propositions one may understand propositions such that the truth of the propositions themselves and of their negations could be established by means of logic. Of such antinomic propositions it is usually characteristic that one could also establish the self-inconsistency both of the propositions themselves and of their negations. That is: one may show that the propositions and their negations entail their own contradictories. Thus propositions which give rise to antinomies break the laws of our modal logic in much the same way as they break
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the laws of ‘classical’ propositional logic and ‘classical’ modal logic.
It is important to observe that this is not a theorem of our modal logic: M(p/q) ↔ M(q/p). Compatibility is not (unrestrictedly) symmetrical. q may be compatible with p, and yet p is not compatible with q.
But is not this in flagrant conflict with our ‘logical intuitions’? That the conflict is apparent only, will, I think, be seen from the following considerations:
Let us consider under what conditions the notion of compatibility is, in our modal logic, symmetrical. We easily prove the following theorem:
T23. M(p/t) & M(q/t) → [M(p/q) ↔ M(q/p)]
Proof:
(1) M(p/t) & M(q/p) ↔ M(q/t) & M(p/q) (From A3.)
(2) M(p/t) & M(q/t) → [M(p/q) ↔ M(q/p)] Q.E.D.
The relation of compatibility is thus symmetrical for (absolutely) possible propositions. In other words: if and only if (at least) one of two propositions is (absolutely) impossible, may it happen that the first is compatible with the second and yet not the second with the first. We can also prove the following theorem:
T24. M(p/t) & M(q/p) ↔ M(q/t)
Proof:
(1) M(p/t) & M(q/p) ↔ M(p & q/t)
(2) M(p & q/t) → M(q/t)
(3) M(p/t) & M(q/p) → M(q/t) Q.E.D.
With an (absolutely) possible proposition there can thus be compatible only propositions which are themselves (absolutely) possible. But not any proposition which is compatible with an (absolutely) impossible proposition need itself be (absolutely) impossible. ~ M(p/t) & M(q/p) → ~ M(q/t) is not a theorem of our modal logic.
Consider again (cf. supra, p. 91) the propositions ‘there are exactly 23 prime numbers between 0 and 100’ and ‘there are at least 21 prime numbers between 0 and 100’. The first is an impossibility and the second a necessity. Thus the second proposition is also possible. The first proposition, moreover, entails the second. If we take the view, which is very natural indeed,
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that the logical consequences of a proposition are compatible with it,^{3} then we shall have to say that the second proposition is compatible with the first. But is there any reason—apart from the presumed asymmetry of the relation of compatibility—for saying that the first proposition is compatible with the second? It may be doubted, whether there is one. But there certainly is a reason against saying thus, viz. the provability of a modal law to the effect that with possible propositions only other possible propositions can be compatible.
The theorems T16 and T18 offer another example of the occasional asymmetry of the relation of compatibility.
One may say that, in our system of modal logic, the relation of compatibility is ‘normally’ symmetrical, and only in ‘extreme cases’ not symmetrical. It may be suggested that the fact that the relation is not unrestrictedly symmetrical strikes us as being contrary to our ‘logical intuitions’ only because we forget about these extreme cases. Or could anyone maintain with perfect confidence that even in those cases he had an ‘intuition’ of the symmetry of the relation? And what would an appeal to ‘intuition’ be worth here?
A ‘dual’ of M(p/q) ↔ M(q/p) is N(p/q) ↔ N(~q/~p): p is necessary, given q, if and only if ~ q is necessary given ~ p. This, of course, is not a theorem. From the fact that the relation of compatibility is not unrestrictedly symmetrical it follows that the operation of ‘contraposition’ (in the sense of the above formula) is not unrestrictedly valid in our modal logic. It is, however, ‘normally’ valid, i.e. valid under the conditions indicated by the following theorem which is a corollary of T23:
T25. M(~p/t) & M(q/t) → [N(p/q) ↔ N(~q/~p)]
Thus if ~ p and q are (absolutely) possible, the relation of necessity of p, given q, can be ‘contraposed’.
Considering what was said above on the symmetry of the relation of compatibility, this restriction on contraposition can hardly be said to conflict with our ‘logical intiutions’.
It is a further consequence of the restriction on the symmetry of the relation of compatibility that, although M(p/q) v M(~p/q) (A2) is a truth of our modal logic, M(q/p) v
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M(q/~p) is not unrestrictedly true. This, however, is a theorem:
T26. M(p/t) & M(~p/t) & M(q/t) → M(q/p) v M(q/~p)
T26 is easily proved from A2 in combination with T23.
If a proposition and its negation are both (absolutely) possible, we call the proposition (absolutely) contingent. T26 thus says that any arbitrary (absolutely) possible proposition is compatible with either the ‘positive’ or the ‘negative’ of any (absolutely) contingent proposition.
T27. M(p/t) → [M(q/p) & N(r/q) → M(r/p)]
Proof:
(1) M(p/t) & M(q/p) ↔ M(p & q/t)
(2) M(p & q/t) ↔ M(p & q & r/t) v M(p & q & ~r/t)
(3) M(p/t) & M(q/p) ↔ M(p & q & r/t) v M(p & q & ~r/t)
(4) M(q & ~r/t) ↔ M(q/t) & M(~r/q)
(5) ~M(~r/q) → ~M(q & ~r/t)
(6) N(r/q) → ~ [M(p & q & ~r/t) v M(~p & q & ~r/t)]
(7) N(r/q) → ~M(p & q & ~r/t) & ~M(~p & q & ~r/t)
(8) [M(p & q & r/t) v M(p & q & ~r/t)] & ~M(p & q & ~r/t) & ~M(~p & q & ~r/t) ↔ M(p & q & r/t) & ~M(p & q & ~r/t) & ~M(~p & q & ~r/t)
(9) M(p/t) & M(q/p) & N(r/q) → M(p & q & r/t) & ~M(p & q & r/t) & ~M(~p & q & ~r/t)
(10) M(p & q & r/t) & ~M(p & q & ~r/t) & ~M(~p & q & ~r/t) → M(p & q & r/t)
(11) M(p & q & r/t) ↔ M(p/t) & M(q & r/p)
(12) M(q & r/p) ↔ M(r/p) & M(q/p & r)
(13) M(p & q & r/t) → M(r/p)
(14) M(p/t) & M(q/p) & N(r/q) → M(r/p)
(15) M(p/t) → [M(q/p) & N(r/q) → M(r/p)] Q.E.D.
T27 says that if p is (absolutely) possible, q possible relatively to p, and r necessary relatively to q, then r is possible relatively to p, and, by virtue of T24, also absolutely possible.
A corollary of T27 is:
T28. M(p/t) & N(q/p) → M(q/t)
What is necessary relatively to an (absolutely) possible proposition is itself (absolutely) possible.
T29. M(p/t) → [N(q/p) & N(r/q) → N(r/p)]
Proof:
(1) M(p/t) & M(~r/p) ↔ M(p & ~r/t)
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(2) M(p & ~r/t) ↔ M(p & q & ~r/t) v M(p & ~q & ~r/t)
(3) M(p & q & ~r/t) v M(p & ~q & ~r/t) → M(p & ~q & r/t) v M(p & ~q & ~r/t) v
M(p & q & ~r/t) v M(~p & q & ~r/t)
(4) M(p & ~q & r/t) v M(p & ~q & r/t) v M(p & q & ~r/t) v M(~p & q & ~r/t) ↔
M(p & ~q/t) v M(q & ~r/t)
(5) M(p & ~q/t) → M(~q/p)
(6) M(q & ~r/t) → M(~r/q)
(7) M(p & ~q/t) v M(q & ~r/t) → M(~q/p) v M(~r/q)
(8) M(p/t) & M(~r/p) → M(~q/p) v M(~r/q)
(9) M(p/t) → [M(~r/p) → M(~q/p) v M(~r/q)]
(10) M(~r/p) → M(~q/p) v M(~r/q) ↔ N(q/p) & N(r/q) → N(r/p)
(11) M(p/t) → [N(q/p) & N(r/q) → N(r/p)] Q.E.D.
T29 says that the relation of relative necessity among (absolutely) possible propositions is transitive.
A corollary of T29 is:
T30. N(p/t) & N(q/p) → N(q/t)
What is necessary relatively to an (absolutely) necessary proposition is itself (absolutely) necessary.
T31. M(p & q/p & q) → N(p/p & q)
Proof:
(1) M(p & q/p & q) ↔ N(p & q/p & q) (T17)
(2) N(p & q/p & q) ↔ ~M(~p v ~q/p & q)
(3) ~M(~p v ~q/p & q) ↔ ~[M(~p/p & q) v M(~q/p & q)]
(4) ~[M(~p/p & q) v M(~q/p & q)] ↔ ~M(~p/p & q) & ~M(~ q/p & q)
(5) M(p & q/p & q) → N(p/p & q) Q.E.D.
T31 says that any member of a self-consistent conjunction is necessary relatively to the conjunction.
T32. M(p/p) → N(p v q/p)
Proof:
(1) M(p/p) → N(p/p)
(2) N(p/p) → N(p/p) v N(q/p & ~p)
(3) N(p/p) v N(q/p & ~p) ↔ ~[M(~p/p & M(~q/p & ~p)]
(4) ~[M(~p/p) & M(~q/p & ~p)] ↔ ~M(~p & ~q/p)
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(5) ~M(~p & ~q/p) ↔ N(p v q/p)
(6) M(p/p) → N(p v q/p) Q.E.D.
T32 says that the disjunction of a self-consistent proposition with any arbitrary proposition is necessary relatively to this self-consistent proposition.
In the monadic systems of modal logic the notion of a strict implication means a necessary material implication. If we accept the notion of an absolutely necessary material implication as that which ‘corresponds’ in dyadic modal logic to the notion of a strict implication in the monadic systems, we can prove that relative necessity is stronger than strict implication. For we have the theorem:
T33. N(q/p) → N(p → q/t)
Proof:
(1) M(p & ~q/t) ↔ M(p/t) & M(~q/p)
(2) M(p & ~q/t) → M(~q/p)
(3) ~M(~q/p) → ~M(p & ~q/t)
(4) N(q/p) → N(p → q/t) Q.E.D.
T33 cannot be ‘converted’, i.e. N(p → q/t) → N(q/p) is not a theorem of our modal logic.
As is well known, it has been claimed that the notion of a strict implication would be an adequate ‘formalization’ of the notion of entailment. In view of the so-called Paradoxes of Strict Implication, however, these claims must be regarded as futile. The paradoxes consist in the (dually related) facts that an impossible proposition strictly implies any proposition and that any proposition strictly implies a necessary proposition.
To the Paradoxes of Strict Implication correspond, in dyadic modal logic, the following two theorems :
T34. ~M(p/t) → N(p → q/t)
T35. N(q/t) → N(p → q/t)
The theorems follow immediately from the equivalences M(p & ~q/t) ↔ M(p/t) & M(~ q/p) and ~(p & ~q) ↔ (p → q).
T34 says that if a proposition is absolutely impossible, then any material implication of which it is the antecedent, is absolutely necessary. And T35 says that if a proposition is absolutely necessary, then any material implication, of which it is the consequent, is absolutely necessary.
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These theorems are, of course, ‘paradoxical’ only if it is claimed that absolutely necessary material implication and entailment are one and the same thing.
It is noteworthy, that there are no corresponding ‘paradoxes’ for the notion of relative necessity. For, neither is it the case that any proposition is necessary relatively to an absolutely impossible proposition, nor is it the case that any absolutely necessary proposition is necessary relatively to any proposition. Neither ~M(p/t) → N(q/p) nor N(q/t) → N(q/p) are theorems of our modal logic.
5. THE CALCULUS M_{d}0 + M_{d}1
We shall next briefly consider a modal calculus, the distinguishing feature of which is that its well-formed formulae are all M-expressions of the first order. We may call it the calculus M_{d}0 + M_{d}1.
The axioms of this extended modal calculus are the axioms A1-A3 of M_{d}1 and one new axiom:
A4. p → M(p/t)
A4 says that if a proposition is true, then it is also absolutely possible. This is what corresponds in the logic of relative or dyadic modalities to the well known ‘ab esse ad posse’-principle of traditional modal logic.
The theorems of M_{d}0 + M_{d}1 are all formulae which (recursively) satisfy one of the following four conditions:
i. The formula is obtainable from a tautology of propositional logic by substituting for its propositional variables M-expressions of the first order.
ii. As in M_{d}1 (vide supra p. 93, ‘Our calculus’ now means M_{d}0 + M_{d}1.)
iii. As in M_{d}1.
iv. As in M_{d}1.
All theorems of M_{d}1 are also theorems of M_{d}0 + M_{d}1. We shall prove a few theorems which are peculiar to the extended system.
T36. p → M(p/p)
‘A true proposition is self-consistent’.
Proof:
(1) p → M(p/t) (A4)
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(2) M(p/t) → M(p/p) (T14)
(3) p → M(p/p) (T36) Q.E.D.
T37. p → N(p/p)
‘A true proposition is self-necessary.’
Proof:
(1) p → M(p/p) (T36)
(2) M(p/p) ↔ N(p/p) (T17)
(3) p → N(p/p) Q.E.D.
This theorem may also be written p → ~M(~p/p). ‘ A true proposition excludes its own negation.’
T38. N(p/t) → p
‘An absolutely necessary proposition is true.’
Proof:
(1) ~p → M(~p/t) (From A4).
(2) ~M(~p/t) → p
(3) N(p/t) → p Q.E.D.
T39. p & q → M(p/q)
‘All true propositions are compatible with each other.’
Proof:
(1) p & q → M(p & q/t)
(2) M(p & q/t) ↔ M(q/t) & M(p/q)
(3) p & q → M(p/q) Q.E.D.
It would have been a more elegant solution to the problem of extending the calculus dealing with homogenous M-expressions of the first order to a calculus dealing with all M-expressions of the first order, if we had been able simply to replace one of the original axioms A1-A3 by a new axiom ‘linking truth with modality’. I have not, however, been able to find such a solution.
6. THE CALCULUS M_{d}
The well-formed formulae of M_{d} are all M-expressions of whatever order. M_{d} may be called a ‘full’ system of dyadic modal logic.
The axioms of M_{d} are the axioms A1-A4 of M_{d}0 + M_{d}1.
The theorems of M_{d} are all formulae which (recursively) satisfy one of the following five conditions:
i. The formula is obtainable from a tautology of propositional logic by substituting for its propositional variables M-expressions.
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ii. The formula is obtainable from an axiom or theorem of M_{d} by substituting for some propositional variables in it some M-expressions.
iii. As in M_{d}1. (Vide supra, p. 93.)
iv. The formula is obtainable from an axiom or theorem of M_{d} by substituting for some M-expression in it another M-expression which is provably (in M_{d}) equivalent to the first.
v. The formula is of the form ‘N(F/t)’ (or ‘~M(~F/t)’) where F is an axiom or theorem of M_{d}.
All theorems of M_{d}1 amd of M_{d}0 + M_{d}1 are also theorems of M_{d}.
7. THE CALCULI M´_{d} AND M´´_{d}
The well-formed formulae of M´_{d} and M´´_{d} are the same as those of M_{d}.
The axioms of M´_{d} are the axioms A1-A4 of M_{d} and a fifth axiom:
R1. M[M(p/t)/t] → M(p/t)
‘If it is absolutely possible that a proposition is absolutely possible, then the proposition in question is absolutely possible.’ The axiom may also be written in the form N(p/t) → N[N(p/t)/t]. ‘If a proposition is absolutely necessary, then it is absolutely necessary that the proposition is absolutely necessary.’
The axioms of M´´_{d} are the axioms A1-A4 of M_{d} and a fifth axiom:
R2. M[~M(p/t)/t] → ~M(p/t)
‘If it is absolutely possible that a proposition is absolutely impossible, then the proposition in question is absolutely impossible.’
If in our formulation of the five conditions (i.-v., p. 106f) which formulae have to satisfy in order to qualify as theorems of M_{d} we replace ‘M_{d}’ by ‘M´_{d}’ and by ‘M´´_{d}’ respectively, we obtain the five conditions which formulae have to satify in order to qualify as theorems of M´_{d} and M´´_{d} respectively.
In M´´_{d} we can prove R1 or the distinguishing axiom of M´_{d}.
Proof:
(1) ~M(p/t) → M[~M(p/t)/t] [From A4 by substituting ~M(p/t) for p.]
(2) M(~M(p/t)/t) → ~M(p/t) (R2)
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(3) M(~M(p/t)/t) ↔ M(p/t)
(4) M(~M[~M(p/t)/t]/t ↔ ~M[~M(p/t)/t] [From (3) by substituting ~M(p/t) for p.]
(5) M[~~M(p/t)/t] ↔ ~~M(p/t) [From (4) by substituting for M(~M(p/t)/t) the provably equivalent formula ~M(p/t).]
(6) M[M(p/t)/t] ↔ M(p/t)
(7) M[M(p/t)/t] → M(p/t) Q.E.D.
Thus all theorems of M_{d} are also theorems of M´_{d} and all theorems of M´_{d} are theorems of M´´_{d}.
The systems M_{d}0 + M_{d}1, M_{d}, M´_{d}, and M´´_{d} are successive extensions of the system M_{d}1. They all contain M_{d}1 as their ‘core’. One may, however, also evolve systems of dyadic modal logic which are not extensions of M_{d}1, but ‘modifications’ of M_{d}1. One such modification, which may be of some interest, is obtained by weakening the axioms A2 and A3 so as to take the form M(q/t) → M(p/q) v M(~p/q) and M(p & r/t) → [M(p & q/r) ↔ M(p/r) & M(q/r & p)] respectively. The modified system thus obtained is contained in M_{d}1, i.e. all theorems of the system in question are theorems of M_{d}1, but not conversely.
8. RELATIONS BETWEEN THE DYADIC AND THE (‘CLASSICAL’) MONADIC SYSTEMS OF MODAL LOGIC
◊-expressions are defined recursively as follows:
By a ◊-expression of order n (1 ≤ n) we understand an atomic ◊-expression of order n or a molecular complex of atomic ◊-expressions, at least one of which is of order n and none of which is of higher order than n.
By an atomic ◊-expression of order n (1 ≤ n) we understand an expression of the form ‘◊ e’, where e is a ◊-expression of order n-1.
By a ◊-expression of order zero we understand the same as by an M-expression of order zero. (Vide supra, p. 92.)
A ◊-expression of the first order will be called homogeneous, if it is either an atomic ◊-expression of the first order or a molecular complex of atomic ◊-expressions of the first order.
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◊-expressions are the well-formed formulae of the ‘classical’ systems of monadic modal logic.
By the ‘translation’ of an arbitrary ◊-expression into the symbolism of M-expressions we shall understand the M-expression which we get when we successively replace all parts of the form ‘◊e’ in it by parts of the form ‘M(e/t)’.
For example: the ‘translation’ of ◊[p v ~◊(q → ◊ r)] is M(p v ~M[q → M(r/t)/t]/t).
We shall first consider the system of monadic modal logic which in my An Essay in Modal Logic is called M1. Its well-formed formulae are homogeneous ◊-expressions of the first order.^{4}
The axioms of M1 are the following two:
Al. ◊t
A2. ◊ (p v q) ↔ ◊ p v ◊q
The theorems of M1 are all formulae which (recursively) satisfy one of the following four conditions:
i. The formula is obtainable from a tautology of propositional logic by substituting for all its propositional variables homogeneous ◊-expressions of the first order.
ii. As in M_{d}1. (Vide supra, p. 93. ‘Our calculus’ now means M1. M-expressions of order zero, it should be remembered, are the same as ◊-expressions of order zero.)
iii. As in M_{d}1.
iv. As in M_{d}1.
As seen, the conditions i-iv. of theoremhood in M1 are essentially the same as the conditions i.-iv. of theoremhood in M_{d}1. (The only difference being that, in formulating the conditions for M1 we speak of ◊-expressions and of axioms and theorems of M1, and in formulating the conditions for M_{d}1 we speak of M-expressions and of axioms and theorems of M_{d}1.)
The axioms and theorems of M1 are axioms or theorems of all the ‘classical’ systems of monadic modal logic, with the exception of the very weak system S1 of C. I. Lewis. M1 may also be called the ‘core’ of classical modal logic.
If we now ‘translate’ the axioms of M1 we get the formulae M(t/t) and M(p v q/t) ↔ M(p/t) v M(q/t) of M_{d}1. They are
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both theorems of M_{d}1. (The first is T15 above and the second an immediate corollary of T2.)
It follows from the sameness of the conditions of theoremhood in M1 and M_{d}1 that, since the ‘translations’ of the axioms of M1 are theorems of M_{d}1, the ‘translation’ of any theorem of M1 must be an axiom or theorem of M_{d}1.
Not any axiom or theorem of M_{d}1 is, of course, the ‘translation’ of some axiom or theorem of M1. M_{d}1 is an essentially richer system than M1.
In view of the above, M1 or the ‘core’ of classical modal logic may be regarded as a special or limiting case of M_{d}1 or the ‘core’ of the logic of relative modality. (Much in the same sense in which Euclidean geometry may be regarded as a limiting case of non-Euclidean geometry.)
We next consider the system of monadic modal logic which in my ‘Essay’ was called M0 + M1. Its well-formed formulae are ◊-expressions of the first order.
The axioms of M0 + M1 are the following two :
A1. p → ◊p
A2. As in M1.
The theorems of M0 + M1 are all formulae which (recursively) satisfy one of the following four conditions :
i. The formula is obtainable from a tautology of propositional logic by substituting for its propositional variables ◊-expressions of the first order.
ii. As in M1.
iii. As in M1.
iv. As in M1.
As seen, the conditions i.-iv. of theoremhood in M0 + M1 are the same as the conditions i.-iv. of theoremhood in M_{d}0 + M_{d}1. (Vide supra, p. 105.)
If we ‘translate’ A1 of M0 + M1 we get p → M(p/t). But this is A4 of M_{d}0 +M_{d}1.
It is now clear that the ‘translation’ of any axiom or theorem of M0 + M1 will be an axiom or theorem of M_{d}0 + M_{d}1 . But the latter system is essentially richer than the former.
The well-formed formulae of the system of monadic modal logic which in my ‘Essay’ was called M are all ◊-expressions of whatever order. M is thus a ‘full’ system of monadic modal logic.
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The axioms of M are the axioms of M0 + M1.
The theorems of M are all formulae which (recursively) satisfy one of the following five conditions:
i. The formula is obtainable from a tautology of propositional logic by substituting for its propositional variables ◊-expressions.
ii. The formula is obtainable from an axiom or theorem of M by substituting for some propositional variables in it some ◊-expressions.
iii. As in M1.
iv. The formula is obtainable from an axiom or theorem of M by substituting for some ◊-expression in it another ◊-expression which is provably (in M) equivalent with the first.
v. The formula is of the form ‘~◊~F’ where F is an axiom or theorem of M.
As seen, the conditions i.-v. of theoremhood in M are ‘essentially the same’ as the conditions i.-v. of theoremhood in M_{d}.
It follows that the ‘translation’ of any axiom or theorem of M is an axiom or theorem of M_{d}. But the latter system is richer than the former.
The well-formed formulae of M´ (=S4) and M´´ (=S5) are the well-formed formulae of M.
The axioms of M´ are the axioms of M and a third axiom:
R1. ◊◊p → ◊p.
‘Possible possibility entails possibility.’
The axioms of M´´ are the axioms of M and a third axiom:
R2. ◊ ~◊p → ~◊p
‘Possible impossibility entails impossibility.’
If in our formulation above of the five conditions which formulae have to satisfy in order to qualify as theorems of M we replace ‘M’ by ‘M´’ and ‘M´´’ respectively, we obtain the set of conditions which formulae have to satisfy in order to qualify as theorems of M´ and M´´ respectively.
As seen, the conditions of theoremhood in M´ and M´´ respectively are the same as the conditions of theoremhood in M´_{d} and M´´_{d} respectively. (Vide supra, p. 107.)
It follows that the ‘translation’ of any axiom or theorem of M´ is an axiom or theorem of M´_{d} and that the ‘translation’ of
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any axiom or theorem of M´´ is an axiom or theorem of M´´_{d}. But the dyadic systems are richer than the monadic ones.
Herewith has been shown, how the ‘classical’ monadic systems of modal logic are related to our dyadic systems, and in what sense the former may be regarded as limiting cases of the latter.
9. LOGICAL AND PHYSICAL MODALITIES
An interesting application of the logic of relative modalities is for the purpose of marking a formal distinction between logical and physical (‘natural’) modalities—between the logically necessary, possible, and impossible on the one hand, and the physically necessary, possible, and impossible on the other hand.
By the logical necessity, possibility, and impossibility of a proposition p we may understand that which we have called before its ‘absolute’ necessity, possibility, and impossibility. Thus we should have:
D1. ‘p is logically necessary’ = ‘N(p/t)’
D2. ‘p is logically impossible’ = ‘~M(p/t)’
D3. ‘p is logically possible’ = ’M(p/t)’
In view of T9-T11 above, we may also say that the logically necessary is that which is necessary in all possible worlds, the logically impossible that which is possible in no possible world, and the logically possible that which is possible in some possible world. These definitions are not circular inasmuch as the notion of necessity, impossibility, and possibility in possible ‘worlds’ refer to the generic dyadic modalities, which are here regarded as fundamental.
The next task will be to say what it is for a proposition p to be logically necessary (impossible or possible) relatively to another proposition q. This task is accomplished as follows:
We say that p is logically necessary relatively to q, if p is necessary relative to q and this necessity is a necessity in all possible worlds. In symbols: N(p/q) & N[N(p/q)t]. But, by virtue of T38, N[N(p/q)/t] entails N(p/q). Thus ‘N(p/q)’ becomes redundant, and we have the definition:
D4. ’p is logically necessary relatively to q’ = ‘N[N(p/q)/t]’. That p is logically impossible relatively to q should mean that
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the impossibility of p relatively to q is a necessity in all possible worlds, or—since N[~M(p/q)/t] entails ~M(p/q)—:
D5. ‘p is logically impossible relatively to q’ = ‘N[~M(p/q)/t]’
That p is logically possible relatively to q is simply the negation of the proposition that p is logically impossible relatively to q. But ~N[~M(p/q)/t] is the same as M[M(p/q)/t]. Thus we have:
D6. ‘p is logically possible relatively to q’ = ‘M[M(p/q)/t]’ That p is logically possible relatively to q thus means that in some possible world p is possible relatively to q.
One may distinguish between logical necessity (impossibility, possibility) in the absolute sense, meaning on no particular evidence or ‘as such’, and logical necessity (impossibility, possibility) in the relative sense, meaning on some evidence q.
It is noteworthy that logical necessity, impossibility and possibility in the relative sense are modal concepts of the second order.
We now proceed to the physical modalities.
The conditions, under which something can be physically necessary (impossible, possible), it would seem, must be logically contingent, i.e. they must be some conditions which (logically) can either be or not be. Otherwise we should get absurdities. (Vide infra.) Logical contingency is defined as follows:
D7. ‘p is logically contingent’ = ‘M(p/t) & M(~p/t)’
We introduce the abbreviation ‘Cp’ for ‘p is logically contingent’.
We now suggest that the physical necessity of p on (contingent) conditions q means (a) that p is not logically necessary, (b) that p is necessary relatively to q but (c) that this necessity is not a necessity in all possible worlds, i.e., is not logical necessity relatively to q. Considering that ‘~N(p/t)’ may also be written ‘M(~p/t)’ and that ‘~N[N(p/q)/t]’ may also be written ‘M[M(~p/q)/t]’, we can say that the physical necessity of p relatively to q means that, although p is necessary relatively to q, yet in some possible world the denial of p is possible relatively to q. Thus we have the definition:
D8. ‘p is physically necessary relatively to q’ = ‘Cq & M(~p/t) & N(p/q) & M[M(~p/q)/t]’
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The function of condition (a) is to rule out the absurdity of calling something, which is logically necessary ‘as such’, physically necessary relatively to q. If it should be thought that this is not an absurdity, one may drop condition (a).
That p is physically impossible relatively to q simply means that the denial of p is physically necessary relative to q.
D9. ‘p is physically impossible relatively to q’ = ‘Cq & M(p/t) & ~M(p/q) & M[M(p/q)/t]’
The physical impossibility of p relatively to q thus means that, although p (actually) is impossible relatively to q, yet in some possible world p is possible relatively to q. Or, in still other words: although p is impossible relatively to q, it is not logically impossible relatively to q.
There remains the physically possible. It is clear that it cannot be simply the negation of the physically impossible (as the logically possible is the negation of the logically impossible). For, the denial of physical impossibility covers four cases. If p is not physically impossible relatively to q, then it is the case either that q is not a contingent proposition (but logically necessary or logically impossible), or that p is possible (in the generic sense), or, finally, that p is logically impossible either ‘as such’ or relatively to q. Now, obviously, the physically possible cannot be logically impossible. Therefore the third and fourth alternatives ought to be excluded. It is equally plain that the physically possible ought to be possible in the generic sense (since it is a species of it). Thus the second alternative must be included. And since we have adopted the view that the physical modalities subsist relative to logically contingent conditions, the first alternative will have to be excluded. Thus physical possibility answers to the conjunction Cq & M/p/t) & M(p/q) & M[M(p/q)/t]. But Cq & M(p/q) entails M(p/t) (T24) and M(p/q) entails M[M(p/q)/t] (A4). Thus, omitting the redundant parts, we get the definition:
D10. ‘p is physically possible reltively to q’ = ‘Cq & M(p/q)’
We are now in a position to see why the physical modalities should be taken relative to contingent conditions. For, if we omitted the condition ‘Cq’ from the above definitions, physical possibility would become indistinguishable from possibility in the generic sense. Now it may happen that a proposition p
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is logically impossibly ‘as such’— ~M(p/t)—and yet possible relatively to some impossible proposition q. If p’s possibility relatively to q were enough to secure its physical possibility relatively to q, it would follow that a proposition could be at the same time physically possible relatively to another proposition and logically impossible. And this, I think, we should wish to reject as absurd.
The reader may easily satisfy himself that the following relations hold between the various logical and physical modalities under their above definitions D1-D10:
i. If p is logically or physically necessary relatively to q, then p is logically and physically possible relatively to q.
ii. Logical and physical necessity, relatively to one and the same proposition q, are exclusive.
iii. Logical and physical impossibility, relatively to one and the same proposition q, are exclusive.
iv. If p is physically possible relatively to q, then p is logically possible relatively to q and also logically possible ‘as such’.
10. MODALITY AND PROBABILITY
Suppose that it were, somehow, possible to measure the degree to which a given proposition p is possible relative to another proposition q. (That is: suppose we could attach some meaning to such ‘degrees of possibility’.)
In order to develop this idea, suppose further that, the degree to which a given proposition p is possible relative to another proposition is measured by a unique and not-negative real number. That the measure is ‘unique’ should mean that, for given p and q, there is one and one only numerical value of M(p/q). That the measure is a not-negative real number means that it is 0 or greater than 0. And suppose, finally, that if p is not possible given q, i.e. if ~M(p/q) is true, then the measure is 0.
By an atomic P-expression we understand an expression of the form’P( / )’, in which the blanks are filled by M-expressions of order zero. For example: ‘P(p/q)’ is an atomic P-expression.
By a P-expression we understand an atomic P-expression or an expression formed of atomic P-expressions and the arithmetical connectives ‘-‘, ‘+’, ‘*’, and ‘:’. (The full recursive
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definition may easily be supplied by the reader.) For example: ‘P(p/q) - P(p & q/r)’ is a P-expression.
By an atomic P-formula we understand a formula formed of P-expressions, numerical variables or constants, and the signs of equality or inequality (‘ =’, ‘>’, and ‘<’). For example: ‘P(p/q) > 0’ is an atomic P-formula and so is also ‘P(p/q) + P(p & q/~r) = P(q/s)’.
By a P-formula, finally, we understand an atomic P-formula or a molecular complex of atomic P-formulae, i.e. a formula formed of atomic P-formulae and the truth-connectives ‘&’, ‘v’, ‘→’ and ‘↔’. For example: ‘~[P(p/q)>0] → P(p/r) = P(q/r)’ is a P-formula.
A homogeneous M-expression of the first order may be regarded as a molecular complex of constituents of the form ‘~M( / )’. (By replacing unnegated atomic constituents in it by doubly negated ones.)
Let it now be that in a homogeneous M-expression of the first order we replace all constituents of the form ‘~M( / )’ by atomic P-formulae ‘P( / ) = 0’, the M-expressions of order zero filling the blanks being left unchanged. On this ‘principle of translation’ the homogeneous M-expression is being turned into a P-formula.
If we apply this translation-rule to the axioms A1-A3 of M_{d}1 we get:
A1. ~[P(p/p) = 0] → P(~p/p) = 0
A2. ~[P(p/q) = 0] v ~[P(~p/q) = 0]
A3. ~[P(p & q/r) = 0] ↔ ~[P(p/r) = 0] & ~[P(q/r & p) = 0]
A1 can also be written P(p/p) = 0 v P( ~p/p) = 0.
A2 can also be written ~[P(p/q) = 0 & P( ~p/q) = 0].
A3 can also be written P(p & q/r) = 0 ↔ P(p/r) = 0 v P(q/r & p) = 0.
On the basis of these three axioms and after a slight modification of the rules of inference i.-iv. of M_{d}1 we could develop a calculus of P-expressions. This calculus, however, would hardly be of much interest, and we shall not stop to study it here.
Consider A3 in its above ‘translation’. In it, three numerical quantities are involved. Let us call them, in order, x, y, and z.
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A3 says that x is equal to zero, if and only if, at least one of the two other quantities, y or z, is equal to zero.
Now we may raise the following problem: supposing that we wanted x to be a function of y and z, such that for given values of y and z the value of x would be uniquely determined, which could this function be considering the condition imposed in A3? One answer is: x could be the arithmetical product of y and z. For, if x = y * z, then x is zero, if and only if y or z is zero. This is, of course, not the only answer to the question. The function x = 2yz, for example, would also satisfy the requirement. But one is probably entitled to say that x = yz is the simplest function which answers to the condition imposed in A3.
Consider next A2 in its above ‘translation’. In it, two numerical quantities are involved. Let us call them x and y. A2 says that at least one of them is different from zero. In other words: if one of them is zero, the other is not zero.
We raise the following problem: Supposing that we wanted x to be a function of y (and y of x), which could this function be considering the condition imposed in A2? An answer is that x and y would be functionally related in a way answering to the requirement of A2, if the sum of x and y were a constant which is different from zero. One such possible constant is the value 1.
The constant, which is the sum of x and y, must also be the maximum value which the measure of a degree of possibility may attain. For, if x were greater than this constant, then y would be negative. And this, on our assumptions, cannot be the case.
Leaving A1 unchanged, we now replace A2 and A3 above by new axioms stating functional relationships between measures of degrees of possibility, conforming to the conditions which A2 and A3 impose on such relationships. We might then get the following set of axioms:
A1. P(p/p) = 0 v P(~p/p) = 0
A2. P(p/q) + P(~p/q) = 1
A3. P(p & q/r) = P(p/r) * P(q/r & p)
(We shall not here consider alternative axiom-systems which also satisfy the requirements.)
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This axiom-system has a very interesting property: It is an axiom-system of (elementary^{5}) probability-theory. (It is indeed the simplest axiom-system of this theory which I know of.) This means that the notion of probability is a notion with exactly those structural properties required by the axioms of modal logic for a numerical measure of degrees of relative possibility.
We shall call the calculus of elementary probability-theory the calculus P.
The theorems of P are all P-formulae which (recursively) satisfy one of the following six conditions:
i. The formula is obtainable from a tautology of propositional logic by substituting for all its propositional variables P-formulae.
ii. As in M_{d}1. (Vide supra, p. 93.) (‘Our calculus’ now means ‘P’.)
iii. As in M_{d}1.
iv. As in M_{d}1.
v. The formula is obtainable from a true formula of arithmetic by substituting for some number variables P-expressions.
vi. The formula is obtainable from an axiom or theorem of P by substituting for some P-expression in it another P-expression which is provably (in P) identical with it.
Condition v. may be regarded as an extension of condition i. This means that in probability-theory, beside propositional logic, arithmetic also is taken for granted.
Condition vi. may be regarded as an analogous extension of condition iv. This means that in probability-theory not only tautologically equivalent but also arithmetically identical expressions are interchangeable.
We shall only prove some few theorems as illustrations :
T1. P(p/q) = P(p & r/q) + P(p & ~r/q)
Proof:
(1) P(p/q) = P(p/q)
(2) P(p/q) = P(p/q) * [P(r/q & p) + P(~r/q & p)]
(3) P(p/q) = P(p/q) * P(r/q & p) + P(p/q) * P(~r/q & p)
(4) P(p/q) = P(p & r/q) + P(p & ~r/q) QE.D.
T2. P(p v q/r) = P(p/r) + P(q/r) - P(p & q/r)
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Proof:
(1) P(p v q/r) = P[(p v q) & p/r] + P[(p v q) & ~p/r]
(2) P(p v q/r) = P(p/r) + P(~p & q/r)
(3) P(~p & q/r) = P(q/r) - P(p & q/r)
(4) P(p v q/r) = P(p/r) + P(q/r) - P(p & q/r) Q.E.D.
T3. P(~p/p) = 0 → P(p/p) = 1
Proof:
(1) P(p/p) + P(~p/p) = 1
(2) P(~p/p) = 0 & P(p/p) + P(~p/p) = 1 → P(p/p) = 1
(3) P (~p/p) = 0 → P(p/p) = 1 Q.E.D.
T4. P(p/p) = 1 v P(~p/p) = 1
Proof:
(1) P(~p/p) = 0 → P(p/p) = 1
(2) P(p/p) = 0 = 1
(3) P(~p/p) = 0 v P(p/p) = 0 → P(p/p) = 1 v P(~p/p) = 1
(4) P(p/p) = 0 v P(~p/p) = 0
(5) P(p/p) = 1 v P(~p/p) = 1 Q.E.D.
It should be observed that P(p/p) = 1, which states that p is probable to degree 1 relatively to itself; is not a theorem of P. (Just as M(p/p) or that p is possible relatively to itself is not a theorem of M_{d}.)
If we accepted P(p/p) = 1 as an axiom of P we should get a contradiction in the system. This is easily shown. If any proposition is probable to degree 1 relatively to itself, then also the contradictory proposition p & ~p. Thus we should have P(p & ~p/p & ~p) = 1. But by virtue of A3, P(p & ~p/p & ~p) = P(p/p & ~p) * P(~p/p & ~p) . Thus P(p/p & ~p) * P(~p/p & ~p) = 1, from which follow P(p/p & p) = 1 and P(~ p/p & p) = 1. But from A2 and P(p/p & ~p) = 1 follows P(~ p/p & p) = 0. Thus we have both P(~p/p & p) = 1 and P(~p/p & ~p) = 0 and a contradiction is produced.
It may be suggested that one could accept P(p/p) = 1 as an axiom and avoid the contradiction by restricting substitutability so as to make p & ~p an impermissible substitution for p. This way of proceeding would have certain technical advantages. Al would become provable and could be dropped from its position as an axiom. It would now, moreover, be possible to prove ‘unconditionally’ a number of theorems which in the
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calculus, as it stands, can be proved only as consequents of implications, the antecedents of which contain conditions of the form P(p/p) = 1. For example: in the present calculus we can prove P(q/q) = 1 → P(q/q) = P(p & q/q), whereas in the modified calculus we could prove P(p/q) = P(p & q/q).
Against the suggested modification one may, however, raise the objection that it is only a disguise for making certain assumptions, which the calculus in its present shape clearly displays whenever these assumptions are involved.
That P(p/p) = 1 is not an axiom or theorem of probability-theory is no ‘intuitive absurdity’. When we think of P(p/p) = 1 [or of M(p/p)] as being, somehow, ‘self-evident’, it is only because we forget about such odd cases as those prevented by self-contradictory propositions. Can anyone claim self-evidence for the proposition that the probability of the proposition ‘this square is round’ on itself as evidence is 1? The truth seems to be that an appeal to ‘intuition’ in such cases is useless.
It may finally be observed that one can embellish the calculus P with a new axiom.
A4. p → ~[P(p/t) = 0]
corresponding to that A4 of Md. From this axiom and A1-A3 we can easily prove the theorem p → P(p/p) = 1. This theorem says that, if a proposition is true, then it is also probable to degree 1 relative to itself. In other words: on its own truth as evidence any proposition is probable to degree 1. This would also hold good for the proposition p & ~p. Yet no contradiction would arise. [Since the deduction of P(~p/p & ~p) = 0 and P(~p/p & ~p) = 1 would now be conditional on the assumption p & ~p.]
It goes without saying that principles such as A1 or A4 play a very insignificant role for the further developments of probability-theory. For most purposes it would be wuite sufficient to have a calculus with A2 and A3 as the only axioms.
The logic of probability could be called a numerical modal logic. Modal logic, one may say, is turned into probability-theory by the introduction of a metric for degrees of possibility. The relation between probability and modality which we have here been investigating is, it should be observed, purely ‘formal’. It exists quite independently of any specific way of defining
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probability and possibility. In particular, it is independent of the well-known ‘modal’ definition of probability as a ratio among a number of (‘favourable’ and ‘unfavourable’) possibilities.
APPENDIX 1. THE NOTION OF SELF-NECESSITY IN ARISTOTLE´S LOGIC
It has occurred to me that the logic of relative modality may prove useful for the purpose of reinterpreting in modern terms some arguments, doctrines, and systems of ancient and medieval modal logic.
In De interpretatione 19^{a}23-24 Aristotle says: Το μεν ουν ειναι το ον οταν η, και το μη ον μη ειναι οταν μη η, αναλκη. As the schoolmen put it: omne quod est, quando est, necesse est esse, or: unumquodque, quando est, oportet esse. This has sometimes been taken to mean that, if a proposition is true, then it is necessary.^{6} In symbols: p → ~◊ ~p. This interpretation, however, is implicitly rejected by Aristotle 19^{a}25-26, where he observes that: ο γαρ ταυτον εστι το ον απαν ειναι εξ αναγκης οτε εστι και το απλως ειναι εξ αναγκες.^{7} Thus it is different for something to be necessary, when it is, (οτε εστι), and to be necessary simply (απλος). Aristotle´s observation has, in turn, been interpreted as a distinction between ‘it is necessary that, if p, then p’ and ‘if p, then it is necessary that p’.^{8} In symbols: ~◊ ~(p → p) and p → ~◊ ~p. The first formula is indeed a truth of ‘classical’ modal logic, whereas the second is not. But it may be doubted, whether the symbolic interpretation is altogether adequate to the distinction as intended by Aristotle.
It seems to me natural to relate the distinction which
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Aristotle makes in the quoted passage to the distinction here made between absolute and relative modality. Aristotle in several places^{9} distinguishes between that which is necessary absolutely or simply (απλως) and hypothetically or relative to suppositions (εξ υποθεσεως).^{10} Of the necessity which belongs to that which is, when it is, he says in the quoted passage from De interpretatione that it is not a necessity απλως. One would therefore think that it must be a case of necessity εξ υποθεσεως. I would understand Aristotle´s thought thus: relative to the hypothesis (supposition) that it is true, a proposition cannot be but true (is necessarily true). Thus not: if a proposition is true, then it is (absolutely) necessary. But: if a proposition is true, it is self-necessary.^{11} In symbols: p → N(p/p). And this, as we have seen (p. 106), is a truth of the logic of relative modalities.
APPENDIX II. SOME REMARKS ON THE MODAL LOGIC OF ŁUKASIEWICZ
A singular position in the family of modal logics is held by the systems developed by J. Łukasiewicz. A fair appreciation of their interest and value is made difficult by the fact that Łukasiewicz thinks of modal logic as a many-valued logic. But I think it can be shown and should be pointed out that his conception of the modal notions is utterly different from a conception of them which employs the ordinary, two-valued interpretation of the so-called truth-connectives and the terms ‘true’ and ‘false’. The difference being what it is, Łukasiewicz´s modal logic will, from the two-valued point of view, appear utterly absurd. Whether this is enough to ‘condemn’ it as a modal logic, I do not know—though I should feel inclined to think so. At any rate it is important that the absurdity should be clearly seen.
In its first version,^{12} Łukasiewicz´s system of modal logic
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made use of a principle to the effect that any proposition implies its own necessity. (Vide supra, p. 121.) This, in combination with the principle that any proposition implies its own possibility and the usual definition of necessity in terms of possibility, which were also accepted, lead to the consequence that any proposition is provably equivalent, in this system of modal logic, to the proposition that the first proposition is necessary. If we were allowed to speak ‘two-valuedly’ of propositions and of implication and equivalence this consequence means that truth and necessity of a proposition coincide. Łukasiewicz later abandoned the view that this early system of his were a satisfactory modal logic.
In publications from the end of his life, Łukasiewicz developed a new modal logic.^{13} This system, like ours, is based on (‘classical’) propositional logic. To this basis is being added a principle δ p → (δ ~p → δ q), where δ is a variable proposition-forming functor of one propositional variable—such as ‘~’ in propositional logic or ‘◊’ in modal logic. We shall refer to it as Leśniewski´s Principle. A further axiom of the Łukasiewicz system is the familiar ab esse ad posse-principle p → ◊ p. Finally, there are two ‘rejection-axioms’ (vide infra).
Łukasiewicz says of his system that it ‘fulfils, in my opinion, all our intuitions concering modalities without having the defects of the known modal systems’. Of what we have here called ‘classical’ modal logic he says that ‘the systems of Lewis are certainly very interesting and may have their own merits; I think, however, that they cannot be regarded as adequate systems of modal logic.’
Against this it must be pointed out, I think, that Łukasiewicz´s system could be said to ‘fulfil our intuitions concerning modalities’ only with a logic which is contrary to our ordinary, ‘two-valued’, intuitions of truth and falsehood and the truth-connectives. And within the framework of such a logic,
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I would contend, we have no ‘intuitions concerning modalities’ whatsoever.
The absurdities, from a two-valued point of view, come from the extension of Leśniewski´s Principle from propositional logic to modal logic. The principle is certainly true for the one monadic proposition-forming operation of ordinary propositional logic, viz. negation. ~p → (~ ~p → ~q) is a tautology of propositional logic. The principle, moreover, sometimes holds in modal logic too. The reader may easily verify that it holds both for necessity and impossibility. The status of possibility, however, with regard to Leśniewski´s Principle would seem to be quite different. If the proposition-forming operation δ be ‘possibilification’, then the principle says that the contingent character of an arbitrary proposition p, i.e. the fact that both p and ~p are possible, implies the possibility of any other arbitrary proposition q. In symbols: ◊p → (◊ ~p → ◊q). Could anything more absurd be said about the logic of the concept of possibility—granting the ordinary two-valued interpretation of negation and material implication?
Another example to illustrate the absurdity of Leśniewski´s Principle in modal logic will here be mentioned.
By means of this principle we can deduce (p ↔ q) → (◊p → ◊q), which Łukasiewicz calls a Law of Extensionality. He expresses the law in words as follows: ‘If p and q are equivalent to each other, then if p is possible, q is possible’. He says that the law is ‘perfectly evident’. But it is perfectly evident that the law is false, if negation, implication, and equivalence are understood in the usual two-valued way. For, consider two propositions, p and q, which are both false: p because it just so happens that it is false and q because it is impossible. Here the material equivalence p ↔ q is true and the material implication ◊p → ◊q false, and therefore the Law of Extensionality false. The law, in other words, requires for its truth, that it must not be the case that of two arbitrary, false propositions the first is contingent and the second impossible!
Łukasiewicz shows that the Law of Extensionality is inconsistent with the acceptance of true propositions as being necessary. (As can in fact be easily deduced from the example just
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discussed.) Faced with the choice between the acceptance of the Law of Extensionality and necessary truths, he chooses the former. The Law of Extensionality, he says, ‘should be taken as the corner-stone of any system of modal logic’. (The rejection of the alternative choice is laid down in a ‘rejection-axiom’.)
Thus Łukasiewicz is forced to banish the notion of necessary truths as inconsistent with modal logic. This is a very awkward position. His defence for it is not very strong. ‘It is certainly true’, he says, ‘that every a is an a, if a is not an empty form, but nothing is gained by saying that it is necessarily true.’ In other words: nothing is gained by distinguishing, among true propositions, the contingent truths from the necessary truths. This seems to me to be a strange view as to what constitutes ‘gains’ in the development of logic.
With the above remarks I have not wished to exclude that the Łukasiewicz modal logic, in its many-valued interpretation, may not be of some importance to the study of modality. I do not think that one can monopolize this study for so-called two-valued logic. There may be aspects of the possible and the necessary which are, so to say, out of the reach of ‘classical’ modal logic—both in its monadic and in its dyadic form. But Łukasiewicz´s formalism by itself does not establish that there are such aspects and therefore it is not clear whether it can appropriately be said to be a modal logic at all.
Note on Aristotle and the Law of Extensionality. Łukasiewicz seems to think that his Law of Extensionality can be found in Aristotle's modal logic.
In Analytica priora 34^{a}5-7 Aristotle says: πρωτον δε λεκτεον οτι ει του Α οντος αναγκη το Β ειναι, και δυνατου οντος του Α δυνατον εσται και το Β εξ αναγκης. This invites two interpretations in symbols: ~◊ ~[~◊ ~(p →q) → (◊p → ◊q)] and ~◊ ~(p → q) → ~◊ ~(◊p → ◊q)—depending upon how one understands the reference of the etc. at the end of the sentence^{14}. (Łukasiewicz suggests the interpretation ~◊ ~(p
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→ q) → (◊p ↔ ◊q), from which, in ‘classical’ modal logic, the first of the above two formulae can immediately be derived.) The first formula and the one suggested by Łukasiewicz is indeed a truth of ‘classical’ modal logic. The second formula is a truth of the ‘classical’ systems M´ and M´´ (S4 and S5), but not of M. From Aristotle's argument in 34^{a}7-24 it would seem that he has the first interpretation in mind. This point, however, is here of minor importance only.
In 34^{a}29-31 Aristotle then refers to what has just been proved (επει γαρ δεδεικται) as being that ει του Α οντος το Β εστι, και δυνατου οντος του Α εσται το Β δυνατον. And this, obviously, suggests (p → q) → (◊p → ◊q) or (a case of) Łukasiewicz´s extensionality principle (at least if we accept the translation of ‘if-then’ by means of material implication^{15}). There is, however, as far as I am able to see, nothing to support that Aristotle really believed this principle to be true. Aristotle uses the words, just quoted, to refer to the principle mentioned in 34^{a}5-7. And, since 34^{a}29-31 does not mean the same as 34^{a}5–7, one would think that he is, in the later passage, simply guilty of inaccurate speech. In view of the clear-headedness and good common sense, characteristic of Aristotle´s genius, it is difficult to believe that he would have been willing to commit himself to holding—as one ought to on Łukasiewicz´s view—that of two arbitrary, false propositions, the first cannot truly be pronounced contingent and the second impossible.
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^{1} As an attempt in the same direction may be regarded E. J. Nelson´s treatment of consistency (compatibility) and entailment in the paper 'Intensional Relations' in Mind 39, 1930.
^{2} A logician, who holds the view that an impossible proposition ipso facto, i.e., by virtue of its impossibility alone, entails any proposition, may find the example unconvincing. For, he would say that our first proposition entails both the second and the third. And for this reason he may wish to say that the second and the third propositions are equally compatible with the first. But this view of entailment is, in my opinion, unsatisfactory. Anyone who denies that the first proposition entails the second but not the third, is either maintaining something which is not true or using the word ‘entails’ in some eccentric way. The concept of entailment and the view that impossible propositions entail anything and everything is discussed in the last essay of the present collection.
^{3} This view, of course, cannot be proved without further investigations into the way in which the concept of entailment is related to the dyadic modalities.
^{4 }It may be noted that the ‘Essay’ uses the symbol ‘M’, and not ‘◊’, for monadic possibility.
^{5} For the meaning of ‘elementary’ here, see my Treatise on Induction and Probability, p. 175.
^{6} Thus by Łukasiewicz in ‘Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalküls’ (Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Cl. III, fasc. 1-3, 1930).
^{7} The Oxford-translation (The Works of Aristotle, translated into English under the editorship of W. D. Ross, vol. 1) reads: ‘For there is a difference between saying that that which is, when it is, must needs be, and simply saying that all that is must needs be’. In view of the use which Aristotle elsewhere makes of απλος. in connexion with αναγκη. it seems to me certain that the translator has misplaced the word ‘simply’ here.
^{8} Thus by Ph. Boehner in ‘The Tractatus de Preadestinatione et de Praescientia Dei et de Futuris Contingentibus of William Ockham’ (Franciscan Institute Publications 2, St. Bonaventure, N. Y., 1945), p. 70 ff.
^{9} See Physica 199^{b}34-35; De generatione et corruptio 337^{b}25-27; De somno et vigilia 455^{b}26.
^{10} Cf. Analytica priora 30^{b}38-39 etc.
^{11} Cf. De interpretatione 18^{b}9-11. I shall not here discuss the further question whether, on Aristotle´s view, the necessity of that which is, when it is, can only belong to truths of the present and past, and not to truths of the future.
^{12} As presented in ‘Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalküls’.
^{13} ‘A System of Modal Logic’ in The Journal of Computing Systems, I, 1953; ‘Arithmetic and Modal Logic,’ ib. 1954; ‘On a Controversial Problem of Aristotle´s Modal Syllogistic’ in Dominican Studies, 7, 1954. Łukasiewicz´s views have been examined and compared with those of other modal logicians by A. N. Prior in ‘On Propositions neither Necessary nor Impossible’ in The Journal of Symbolic Logic, 18, 1953 and ‘The Interpretation of Two Systems of Modal Logic’ in The Journal of Computing Systems, I, 1954.
^{14} The Oxford translation of the quoted passage runs: ‘First we must state that if B´s being follows necessarily from A´s being, B´s possibility will follow necessarily from A´s possibility’. This supports the second symbolic interpretation. I do not think, however, that much weight can, on this point, be attached to the translation.
^{15} The Oxford translation of the quoted passage runs: ‘if B's being is the consequence of A´s being, then B´s possibility will follow from A´s possibility’. Here the translator has, in my opinion, at the expense of faithfulness to the text given a correct rendering of Aristotle's actual thought.