T H » JOCBMAL or SYMBOLIC LOOIC
Volum* 3. Number 3, June 1938
ON FINITE AND INFINITE MODAL SYSTEMS
C. W E S T CHURCHMAN
In Oskar Becker's Zur Logik der Modalitdien1 four systems of modal logic are
considered. Two of these are mentioned in Appendix II of Lewis and Langford's Symbolic logic. The first system is based on Al-8* plus the postulate,
cii'.
0p-3~0~0p-
From A7: ~ 0 p ~ 3 ~ P we can prove the converse of Cll by writing ~ 0 p for p,
and hence derive
Cll.
Op = ~ 0 ~ 0 P -
The addition of this postulate to Al-8, as Becker points out, allows us to "reduce" all complex modal functions to six, and these six are precisely those
which Lewis mentions in his postulates and theorems: p, ~p, 0p» ~ 0 p >
~ 0 ~ P » an< l 0~P« This reduction is accomplished by showing
(1)
0»P=Op,
where On means that the modal operator 0 is repeated n times; e.g., 0* P =
0 0 OP- Then it is shown that
(2)
( ~ 0 ) » P = O P if n is even, and
(3)
( ~ 0 ) » P = ~ 0 P if n is odd.
By means of (1), (2), and (3) any complex modal function whatsoever may be
reduced to one of the six "simple" modals mentioned above.
It might be asked whether this reduction could be carried out still further,
i.e., whether two of the six "irreducible" modals could not be equated. But
such a reduction would have to be based on the fact that Op = P> which is
inconsistent with the set Bl-9 of Lewis and Langford's Symbolic logic and independent of the set Al-8. Hence for neither set would such a reduction be
possible.
It is interesting to note that if Cll is added to the set Bl-9, the two "zeroelements" Z and Z*' * become identical, as do the two "one-elements" Z' and Z*.
For Huntington has shown4 that Z**=Z. But by CI 1, when Z is written for p,
Z * ' = 0 Z = ~ 0 ~ 0 £ = Z * * , and hence Z=Z**=Z*'.
Received January 10, 1038.
1
Jahrbuchfiir Philotophie und phUnomenologische Forschung, vol. 11 (1930), pp. 496548.
* The postulate set of Lewis's Survey of symbolic logic.
' E. V. Huntington, Postulates for assertion, conjunction, negation, and equality, Proceed'
ings of the American Academy of Art* and Sciences, vol. 72 (1937), pp. 1-44.
* Ibid., Theorem 87.
77
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78
C. WEST CHTJBCHMAN
Next Becker mentions what he believes to be an analogous postulate:
C14.
~Q~p = ~ < 0 ~ 0 p .
Becker apparently thinks that C14 and Al-8 also leads to a similar reduction
to six simple modals. I believe that as a matter of fact this system contains
an infinite number of irreducible modals. The only way of proving this would
be by means of matrices; but an indication of its truth can be found in the fact
that while in the first system it was a fairly simple matter to show that
<>~<^~0p — ~ 0 ~ 0 p
and hence that
0P=
00P,
in this system we have
0 ~ 0 ~ 0 P = OOP,
and hence an analogous reduction is impossible.
Becker's third system is based on two additional postulates instead of one:
CIO'.
~ 0 ~ p -3 ~ 0 / ^ " v / 0 ~ P >
which, on the basis of A7, becomes the equivalence
CIO.
~<>/>^p =
~§~~Q~pf
and the famous Brouwerian Axiom
C12.
p-3~0~0pBecker believes that this system reduces all complex modals to ten and not six.
But on p. 498 of Lewis and Langford's Symbolic logic it is shown that CIO and
C12 imply Cll, and that Cll implies CIO and C12, and hence this system and
the first are identical in their properties.
Perhaps the most surprising thing of all, however, is the fact that Becker's
fourth system, which he believed to contain an infinite number of irreducible
modals, is also identical with the first system. Here no apparent equivalences
between modals were postulated, but, instead, what Becker calls a "generalized"
Brouwerian Axiom:5
C15.
p -3 ~ 0 » ~ 0 P There is a certain peculiarity concerning CI 5. For if it is true for n = 3,
then we can show that it is true for n = 2 and n = 1. For by A7: ~0p~3~P»
we can show that
~00~0P~3 ~ 0 ~ 0 P
by writing 0 ~ 0 p for p. Similarly, we can show that
~ 0 0 0 ~ 0 P -3 ~ 0 0 ~ 0 P ,
5
Cf. H. B. Smith's (identical) "induction formula," Abstract logic or the science of modality, Philosophy of science, vol. 1 (1934), pp. 369-397. This formula was discovered independently.
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ON FINITE AND INFINITE MODAL SYSTEMS
79
and in general
~ 0 » + I ~ O P -3 ~ 0 » ~ 0 P Hence, the form of the axiom as Becker puts it is stronger than that needed.'
Actually, we shall show that for Becker, the postulate
C15'.
p-3~00~0p
is sufficient, i.e., C15 when n = 2.
Becker also introduces an important Rule (hereafter referred to as Becker's
Rule) which is actually the cause of the finiteness of the system:
Ride: If p-3g is established, i.e., is a theorem or postulate, then we may assert
OP-30?. 7
This rule is an important one in the modal calculus. The demonstration
given below will show that if we wish to have an infinite system and if we start
with C15, then we must deny that this Rule is true in general.8
Becker's Rule is quite distinct from A8 of the set of the Survey, which is
A8.
(p -3 g) -3 ( ~ 0 ? "3 ~ 0 p ) .
Becker's Rule merely asserts that A8 holds when the antecedent is verified,
and hence is weaker than A8. The importance of this distinction will become
obvious when we say that while, as Parry has shown,9 A8 is independent of the
set Bl-9, Becker's Rule follows from this set as a theorem.
Assume p-3g is an axiom or a theorem.10
Tang Tsao-Chen has shown" that
1.
(p -3 q) -3 (p = pq).
Therefore, by the Rule of Inference, since p-3q is true, we may assert
2.
p = pq-
Hence, by the Rule of Substitution, we may assert
3.
Op = On-
* P. C. Rosenbloom has been able to dispense with the concept of number altogether
in this postulate (Postulate set for Smith's theory of modal logic, as yet unpublished).
* Becker actually generalizes this Rule to the extent of saying that MpSMq can be
asserted, where M is any "affirmative" modal, that is, any modal with an even number of
negation signs or "curls." But for our purpose, the form here given is sufficient. As a
matter of fact, by means of this form and the set Al-8, the general form can be proved.
It is also worth noting that the implication p-3g in the Rule refers to an implication between
modal functions; i.e., p and q are modal functions of the same variable; p might be ~ 0 ~ » "
and j be Or. Put otherwise, pSq is thought of as being any theorem which follows from
A7 or the generalized Brouwerian Axiom.
* This assertion needs, of course, the qualification that other "traditional" laws be
observed; in particular, those assumptions which we set down below.
»Mind, n.s. vol. 43 (1934), pp. 7&-80.
" Actually, we prove something stronger than Becker's Rule, but for our purposes, the
assumption that pSq is a theorem or postulate is enough.
11
Bulletin of the American Mathematical Society, vol. 42 (1936), pp. 743-746.
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80
C. WEST CHXJKCHMAN
But a
4.
Opq -3 Oq.
Hence, by substitution, we may assert
5.
Op -3 Oq.
In view of this proof, I believe that the simplest, and perhaps the most economical method of establishing the implication-relation between any two modals in
the set Bl-9 would be to add the postulate
BIO.
p-3~00~Op,
which is C15 forn = 2; as was shown above, we can prove C15 for n = 1 from
BIO, and we shall show that by means of these propositions and Becker's Rule
we can prove the familiar theorem
Cll.
Op =
~ 0 ~ 0 P ,
and Cll completes the set by reducing all modals to six.1'
We may now set down Becker's assumptions thus:
PI.
p -3 OP-
P2.
p -3 ~ 0 » ~ 0 P ,
where the n below the diamond means that the diamond is repeated n times;
as mentioned above, the more economical assumption is p - 3 ~ 0 0 ~ 0 p > but
this is the axiom as given by Becker.
P3.
If p-3g is established, then Op"302 may be asserted.
P4.
p -3 q • = 5 ~ g -3 ~ p .
P5.
p -3 q . q -3 r : -3 : P -3 r.
(11.6)
P6.
p = ~~p.
(12.3)
(12.44)
M
It will suffice to show that from this set we can prove the
THEOREM.
OP
= ~0~Op>
since, as we have seen, with this assumption a finite modal calculus of six irreducible modals results.
Proof of the Theorem:
LEMMA
1. Op "3 0 ~ 0 0 ~ O p -
11
This is theorem 19.13 in Lewis and Langford's Symbolic logic.
" Cll hardly enables us to determine the truth or falsity of any "meaningful expression"
of the system, however. For instance, the truth of (p-3?)-3(<>p-30?) or 0(p?)""P7 is
still undetermined.
14
These numbers refer to the theorems as they appear in Lewis and Langford's Symbolic
logic.
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ON FINITE AND INFINITE MODAL SYSTEMS
(let n = 2 in P2)
p-3 ~ 0 0 ~ 0 p -
(1)
(2)
OP-3
81
((1) and P3)
0~00~0P-
0 ~ 0 0 ~ 0 P -3 ~ 0 ~ 0 P -
LEMMA 2.
(1)
p -3 ~ 0 ~ 0 p -
(let n = 1 in P2)
(2)
0~<>P -3 ~ p .
(P4 applied to (1), and P6)
(3)
LEMMA
0' '00~0P-3~0~0P3.
OP ~3 ~ 0 ~ 0 P -
(write 0 ~ 0 p for p in (2))
(P5 applied to Lemmas 1 and 2)
~ 0 ~ 0 P ~ 3 OP-
LEMMA 4.
Op~p -3 0~P~0~p~3 p.
~ 0 ~ 0 P ~3
(PI)
P-3
(1)
(2)
(3)
(4)
(write ~ p for p in (1))
(P4 applied to (2), and P6)
OP-
(write O P for P in (3))
Therefore 0 p = ~ 0 ~ 0 p , by Lemmas 3 and 4 and Lewis's 11.03 (j>=qi —
:p-3?.?-3p), Q-E.D.
The question might arise whether this last system is not really stronger than
the first, i.e., whether the reduction could not be carried out even further.
But it is a comparatively simple matter to show that the axiom of the first
system implies the axiom of the last, i.e., Cll implies C15 (for n=2), and the
two systems are identical in their properties:
Cii.
OP =
1.
~0p = 0~0P-
2.
0 ~ 0 P
3.
~0~0P
4.
OP
5.
p-3 OPP -3 ~ 0 0 ~ 0 p -
6.
~ 0 ~ 0 P -
= 00~0P-
=
00
0 0~0POp-
(The law p = g . - 3 . ~ p = ~ g applied to Cll.)
(Rule of Substitution and 1)
(same reason as 1)
(Cll and 3 and the Rule of Substitution)
(A7)
(5 and 4 and 11.6)
The importance of the principal theorem is that it demonstrates that if we
are to have a system which contains infinitely many distinct modal functions,
certain principles which at first sight seem "self-evident" must be denied in
general, particularly if the generalized Brouwerian axiom is admitted." Thus
11
When the converse of this axiom is postulated, ~0~Op~3p> then the system is not
necessarily finite; cf. the system based on C14 above, which is apparently infinite, though
it includes A8 and hence Becker's Rule. But the converse of the Brouwerian Axiom is
somewhat paradoxical, at least when verbally interpreted. If p — ~ 0 ~ O p is the axiom,
an infinite system is possible also.
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82
C. WEST CHURCHMAN
in the system of H. B. Smith, where all modal distinctions are maintained (i.e.,
no modal function of a variable is equatable to another modal function of that
variable), apparently paradoxical results are necessitated. Besides Becker's
Rule, certain other traditionally accepted laws must be denied in general. For
instance, since Becker's Rule was demonstrated chiefly on the basis of B8:
0 P ? " 3 0 P > this principle cannot be true in Smith's system. That is, there are
cases of its failure, though there are, of course, many cases where it is true: e.g.,
if p=l or 0 and q—1 or 0. Perhaps the most significant case of failure of a
traditionally "self-evident" law is the following "Law of Transitivity" (Syllogism):
(p -3 q . q -3 r) -3 (p -3 r).
Now in Smith's system, this law implies B8,1' and in view of the fact that B8
implies Becker's Rule, and that Becker's Rule plus the generalized Brouwerian
Axiom reduces the system to one having but six irreducible modals, the Law of
Transitivity in this general form must be denied if an infinite number of modal
distinctions are to be maintained. Some might think that the necessity of this
denial was a sufficient reason for denying that there are an infinite number of
modal distinctions, since the above law seems to be as much the life-blood of
logic as the Law of Contradiction. But such objections would miss the significant aspect of the matter. It is true that it would indeed be paradoxical,
and as a matter of fact (pragmatically) false, to assume that this law as usually
meant fails. But the usual meaning states the law in a restricted sense: "If p-3g
is a theorem (or postulate) and q-3r is a theorem (or postulate) [or, more generally, if p-3g is actually true, etc.], then we may assert pSr." But the symbolized form as given above asserts considerably more, and, indeed, too much:
"If we have pSq and q-%r as hypotheses, where p, q, and r are any expressions
whatsoever, then pSr is a consequence of this hypothesis." The only valid
objection to denying the law in this general form would rest on a point of convenience, I think. But Smith's reasons for insisting on a complete modal distinction in my opinion far outweigh any objection of convenience.
TJNIVSRSITT OF PENNSYLVANIA
M
Let j—0 ("0" is any statement satisfying the proposition pS~p, i.e., any impossible
statement). Then the law becomes (p-30.0-3r)-3(p-3r). The premiss 0-3r is true in
Smith's system and may be suppressed. Hence the law reduces to (p-30)-3(p-3r). We
may express the premiss as ~ O p and the conclusion as ~<>(p T ) by the definition of implication; so that ~C>P "3 ~ 0 ( p ~ r ) . By contradiction and transposition we get B8:
0(P ~ r ) -3 OP-
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