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Modal Logic before Kripke

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Abstract

100 years ago C.I. Lewis published A Survey of Symbolic Logic, which included an axiom system for a notion of implication which was ‘stricter’ than that found in Whitehead and Russell’s Principia Mathematica. As far as I can tell little notice was taken of this until 1930 when Oskar Becker provided some additional axioms which led Lewis in Symbolic Logic (written with C.H. Langford, 1932) to revise the system he had produced in 1918, and list five systems which could be obtained using Becker’s suggested formulae. The present paper reviews the development of modal logic both before and after 1932, up to 1959 looking at, among other work, Becker’s 1930 article and Robert Feys’s articles in 1937 and 1950. I will then make some comments on the completeness results for S5 found in Bayart and Kripke in 1959; and I will finally look at how modal logic reached New Zealand in the early 1950s in the work of Arthur Prior.
Organon F 26 (3) 2019: 323–339
https://doi.org/10.31577/orgf.2019.26302
research article
ISSN 2585–7150 (online)
ISSN 1335-0668 (print)
Modal Logic before Kripke
Max Cresswell
Received: 27 October 2018 / Accepted: 30 November 2018
Abstract: 100 years ago C.I. Lewis published A Survey of Symbolic
Logic, which included an axiom system for a notion of implication
which was ‘stricter’ than that found in Whitehead and Russell’s
Principia Mathematica. As far as I can tell little notice was taken
of this until 1930 when Oskar Becker provided some additional ax-
ioms which led Lewis in Symbolic Logic (written with C.H. Lang-
ford, 1932) to revise the system he had produced in 1918, and list
five systems which could be obtained using Becker’s suggested for-
mulae. The present paper reviews the development of modal logic
both before and after 1932, up to 1959 looking at, among other
work, Becker’s 1930 article and Robert Feys’s articles in 1937 and
1950. I will then make some comments on the completeness results
for S5 found in Bayart and Kripke in 1959; and I will finally look
at how modal logic reached New Zealand in the early 1950s in the
work of Arthur Prior.
Keywords: Modal logic; history of modal logic; C. I. Lewis; strict
implication; Saul Kripke
1 Introduction
In recent times modal logic has been seen as the logic of relational frames, a
development which took place in the early 1960s. In order to reinforce the
Victoria University of Wellington
Philosophy (HPPI), Victoria University of Wellington, PO Box 600, Wellington 6140,
New Zealand
max.cresswell@vuw.ac.nz
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324 Max Cresswell
importance of this change it is timely to reflect on what modal logic was like
up to that time. Such reflections are especially appropriate in that 2018 marks
the centenary of the publication of C.I. Lewis’s Survey of Symbolic Logic which
represents the first presentation of a formal axiomatic system of modal logic.
2Principia Mathematica (PM) 1910
Although modal logic was studied since the time of Aristotle, and was re-
vived in the late 19th century by Hugh McColl, the modern development of
the subject really begins with C.I. Lewis’s dissatisfaction with some of the
‘paradoxical’ sentences found in Whitehead and Russell 1910. These include:
(1) p(qp)
(2) p(pq)
(3) (pq)(qp)
(1) and (2) can be read as claiming that if pis true then it is implied by every
proposition, and if it is false it implies every proposition. (3) claims that of
any two propositions one implies the other. Beginning in 1912 C.I. Lewis
objected that these theorems conflict with the interpretation of as ‘implies’.
3 C.I. Lewis Mind 1912, 1914, Journal of
Philosophy, 1913, 1914
Lewis makes a distinction between an implication which holds materially and
one which holds necessarily or strictly. The first article in Mind (Lewis 1912),
after exhibiting (1)-(3) as what Lewis regards as defects in a theory of im-
plication, concentrates on pointing out that you can understand implication
as pqprovided that you understand the disjunction intensionally i.e. as
holding necessarily. (Lewis 1912, 523) On p. 526 he calls the implication used
in the “Algebra of logic”1material’ and contrasts it with his own ‘inferential’
1In footnote 1 on p. 522 he instances PM as ‘the most economical development of the calculus of proposi-
tions’ in particular with its definition of pqas pq.
Organon F 26 (3) 2019: 323–339
Modal Logic before Kripke 325
or “strict” implication. In the second of the articles in Mind (Lewis 1914a)
he introduces two symbols for intensional () and extensional (+) disjunction
respectively, though he only uses one symbol for implication (), insisting on
its ambiguity. Although in the 1912 paper he speaks of an intensional dis-
junction as one which is necessarily true, he does not adopt any symbol for
necessity. (If is intensional you can of course define the necessity of pas
pp, but Lewis does not do so.) In neither of these articles does Lewis attempt
anything like an axiomatisation of the logic of strict implication, though on p.
243 of (Lewis 1914a) he does provide a list of formulae which are valid, using
for material implication and +for material disjunction; and contrasting
these formulae with some corresponding formulae in a language in which is
strict implication and both and +appear. The two articles from the Journal
of Philosophy (Lewis 1913, 1914b) have a slightly different focus. The 1913
article lists a large number (35) of valid formulae in PM which Lewis regards
as questionable when is interpreted as implication. As in his other articles
Lewis uses for strict implication and for strict disjunction. As in (Lewis
1914a), in (Lewis 1914b) he uses +for material disjunction, but in this article
he introduces <for material implication. He also uses for impossibility and
for negation. On p. 591 of (Lewis 1914b) he does produce an axiom set,
though two of the axioms are defective when is understood intensionally.
One is (pq)((qr)(pr)), which, when added to the other
axioms, reduces each strict operator to its material counterpart.2It’s clear
that what Lewis is trying to do is retain as much of PM as he can without
running into what he thinks of as its paradoxical consequences. He clearly
thinks that material implication does not reflect what he supposes ordinary
logicians (uncorrupted by truth-functional logic) think of as the relation of
implication.
4 Lewis 1918
(Lewis 1918) is the first axiomatic presentation of modern modal logic.3What
2Problems like this are discussed in (Parry 1968). Parry notes on p. 126 that in the 1912 paper Lewis
accepts the equivalence of (p(qr)) and (q(pr)) even when is a strict disjunction, and comments in
footnote 39: “These mistakes, long since corrected, show the fallibility of logical intuition. Parry’s chapter
provides a thorough survey of Lewis’s modal logic, and for that reason the present paper concentrates on the
work of others at that time, in particular works which are not available, or not easily accessible in English.
3For an elaboration of this claim see (Parry 1968).
Organon F 26 (3) 2019: 323–339
326 Max Cresswell
Lewis says is “Various studies toward this system have appeared in Mind and
the Journal of Philosophy. ... But the complete system has not previously
been printed. (Lewis 1918, 291, fn. 1).
The chapter on strict implication (J) is only a small part of the book. In
that chapter Lewis does not do any metalogic, and even definitions seem to
be regarded as object-language formulae.4
(Lewis 1918) takes impossibility () as primitive, and defines pJqto
mean that it is impossible that pshould be true without q’s being true too.
In 1918 is still used for ‘not’ and for impossibility, so − ∼ pmeans that p
is possible and ∼ −pmeans that pis necessary. Lewis uses juxtaposition for
conjunction, but I have used the now common and therefore write Lewis’s
definition as pJq=def (p∧ −q).5What you then find is a collection of
axioms (Lewis 1918, 291ff.) of which the first five are no more than strict but
conjunctive versions of axioms of PM. Specifically Lewis’s axioms are:6
1.1 (pq)J(qp)
1.2 (qp)Jp
1.3 pJ(pp)
1.4 (p(qr)) J(q(pr))
1.5 pJ− − p7
1.6 ((pJq)(qJr)) J(pJr)
1.7 pJp
4A criticism of these early papers is found in (Wiener 1916). Although Wiener does not put the point
in exactly this way his defence of Russell against Lewis could easily be seen as pointing out that when we
are giving an account in the metalanguage of implication we are claiming that for any formulae αand β,α
implies βwhen αβis valid, which is to say that it is true no matter what formulae αand βare. Or, if
you prefer, either αis false or βis true, no matter what values are given to its propositional variables. Lewis
seems unaware of this way of describing the situation. Here and elsewhere in the paper I use α,βetc. as
metalogical variables for well-formed formulae (wff) of the relevant object language. The need for making
such a distinction seems not to have been recognised by these earlier writers.
5In doing so it must be remembered that on p. 293 of (Lewis 1918), in 1.04, Lewis uses the symbol , not
for conjunction, but for strict disjunction. In 1.05, he uses +for material disjunction.
6In addition to using for conjunction in place of Lewis’s juxtaposition, I have inserted parentheses.
7This axiom is repeated in the list on p. 493 of (Lewis and Langford 1932), but was proved to be redundant
in (McKinsey 1934). Lewis notes McKinsey’s result in the 1959 Appendix (Appendix III) to (Lewis and
Langford 1932) on p. 503.
Organon F 26 (3) 2019: 323–339
Modal Logic before Kripke 327
1.8 (pJq) = (qJp)
It’s not difficult to see how these axioms arose. The idea is that PM’s theorems
also hold when the main operator is strict, in the sense that, for instance, when
is replaced by Jthe result will still be valid.81.6 states the transitivity of
J. 1.7 says that impossibility implies falsity. 1.8 is in many ways the most
interesting. In the first place it was shewn by E.L. Post to lead to p=p,
and was replaced, in Lewis 1920, by9
1.8(pJq)J(qJp)
But even 1.8was one which later caused Lewis some trouble. It’s not difficult
to see why. In the antecedent both variables are in the scope of a Jwhich is
itself only in the scope of the main operator, while in the consequent both the
variables are in the scope of (which for Lewis is a modal operator) which
in turn is in the scope of a Jwhich is itself in the scope of a modal operator.
Except for Lewis’s 1920 correction to 1.8 little notice appears to have been
taken of his work until a 1930 article by Oskar Becker.10
5 Becker 1930
Although in the Survey Lewis had taken the one-place impossibility operator
as the only modal primitive, for which he used , his principal interest was
in the defined operator J. I mentioned above that the pand qin the conse-
quent of 1.8are in the scope of three operators: one occurrence of and two
occurrences of J. (Just as pin ∼∼ pis in the scope of two modal operators,
which prevents its elimination by a double negation rule.). In more recent
notation, with (or L) for necessity,11 and (or M) for possibility, Lewis’s
∼∼ pis equivalent to □♢p,would be ,∼ − would be ,∼ − ∼ − would
8Thus, for instance, not only is p(pq)valid, so is pJ(pq), but not of course pJ(pJq).
9The label 1.8is given by (Becker 1930, 504) (but using <for strict implication). In (Lewis 1918) it is
2.2 on p. 297.
10There are a few exceptions. As well as (Wiener 1916) mentioned in footnote 4 above there is an article by
E. J. Nelson (1930) on what have come to be called the ‘paradoxes of strict implication’ – that an impossible
proposition strictly implies every proposition, and that a necessary proposition is implied by every proposition.
(See Lewis 1918, 506: 3.52 and 3.55.) Although this topic has caused philosophical controversy I am not
concerned with it in this paper.
11In the time in question modal notation was somewhat fluid, and, except when commenting specifically
on notational matters I have used Fitch’s symbol for necessity and Lewis’s symbol for possibility. The
first published use of is in (Barcan 1946).
Organon F 26 (3) 2019: 323–339
328 Max Cresswell
be □□,− ∼ − ∼ would be ♢♢, and so on. These sequences of operators are
what Becker called ‘modalities’, and in particular, where one modal operator
is inside the scope of another he called them ‘iterated’ or ‘complex’ modalities.
(Becker 1930, 502) comments that:
The more complicated modalities are not handled by Lewis. It is
striking that the iterations of impossibility are mentioned.12
By this he means that Lewis’s axioms do not guarantee the reduction of iter-
ated modalities. So (Becker 1930, 508) says:
We therefore add to the Lewis axioms the new axiom 1.9
1.9 − ∼ p <∼∼ p13
A few pages later he introduces two further axioms:
1.91 p <∼∼ p(p. 513)
1.92 ∼ −p <∼ − ∼ −p
In current notation 1.9 is equivalent to pJ□♢p, 1.91 is pJ□♢p, and 1.92 is
pJ□□p. These axioms are all ones considered by (Lewis 1932). 1.91 is the
proper axiom of S5 and 1.92 of S4. 1.91 is the Brouwerian axiom. Becker’s
principles led Lewis to write the appendix to the 1932 book with Langford, in
which he set out a number of modal systems of increasing strength.
6 Lewis and Langford 1932 (L&L)
I won’t say much about (Lewis and Langford 1932) since it is well known,
and surveys of the various Lewis systems exist all over the place. In this work
Lewis takes possibility, which he writes as , as the basic modal notion, and
defines pJqas (p∧ ∼q). (is now the regular negation, not impossibility
as in Lewis 1918.) By 1932 Lewis had been convinced that even axiom 1.8
was too strong, and opted for something weaker. In an appendix Lewis sets
out five systems which he calls S1S5. The system of (Lewis 1918) is the
12This translation is by Jacques Riche.
13Becker’s symbol for strict implication is <. I am omitting superfluous parentheses.
Organon F 26 (3) 2019: 323–339
Modal Logic before Kripke 329
system he calls S3, and it is the presence of 1.8in S3 which appears to have
worried him. The system he favoured was the one he called S2. In the 1932
book he includes a proof that S1 is weaker than S2, and he is clear that S3
is stronger than he wanted, but at the time of writing L&L he had no proof
that S3 was stronger than S2, and used S1 as a ‘fallback’ position in case
S2 should contain S3. (Parry 1934 proves that S3 is stronger than S2, and
Parry 1939 is a study of the modality patterns of S3.)
Although L&L was published in 1932 the next two articles appear to have
been written in ignorance of that work, and so do the papers they refer to.
They mostly use the notation of (Lewis 1918), and I will follow them in this
(though they sometimes use < instead of J). But I will refer to the systems
S3,S4 and S5 even though these names were only introduced in an appendix
to L&L.
7 Gödel 1933
In a one page plus three lines article, Gödel provides an axiomatisation of
the system now called S4. Although Gödel is aware of Lewis’s earlier work,
and also aware of Becker’s work, and of the work of W.T. Parry, he does not
yet seem to be familiar with L&L. From the perspective of modal logic the
main feature of Gödel’s paper is to axiomatise S4 by shortcutting the separate
postulation of strict versions of the axioms of PM, and replacing them with a
rule which says that if a formula is a theorem then so is that formula preceded
by a necessity operator which Gödel writes as B. This rule is added to an
axiomatisation of the classical (non-modal) propositional calculus, together
with the axioms:
Bp p
Bp ((B(pq)Bq)
Bp BBp
In these axioms is the symbol for material implication (Whitehead and
Russell’s ) used in (Hilbert and Ackermann 1928).
One might wonder how Gödel came to consider an axiomatisation with
such a rule. I owe the following conjecture to a discussion with Rob Goldblatt.
Organon F 26 (3) 2019: 323–339
330 Max Cresswell
Gödel’s paper is concerned to shew how to interpret intuitionistic logic with
the aid of Bas a ‘provability’ operator. In intuitionistic logic, mathematical
truths —represented by formulae which can be ‘asserted’— only exist because
they have been proved. It therefore seems natural to add a rule which explicitly
states this. As far as I understand, Gödel was not a modal logician, though
his article does make the link with Lewis’s system extended by an axiom of
Becker’s which is equivalent to Gödel’s Bp BBp. Nevertheless, although
Gödel at that stage may not have been familiar with Lewis and Langford’s
book he was familiar with (Becker 1930), since he had reviewed it in (Gödel
1931), and therefore he would have been familiar with (Lewis 1918), at least
to the extent that Becker represents Lewis.
Gödel cites (Parry 1933)14 as evidence that the system that he, Gödel,
has just presented is indeed the system of (Lewis 1918) provided that it is
extended by one of Becker’s axioms. Gödel is correct that Lewis’s system (S3)
with the addition of Np < NNp does axiomatise his (Gödel’s) system. This
axiom is Becker’s 1.92 on p. 514, which gets you S4. (Parry 1933) however
is concerned with Becker’s 1.9, which is what is one form of the proper S5
axiom. Parry’s purpose is to provide a decision procedure by conjunctive
normal form for first-degree modal formulae —formulae in which no modal
operator is within the scope of any other modal operator. Parry does not
himself prove that Becker’s 1.9 guarantees that every formula can be reduced
to first degree, though he does add a footnote acknowledging the appearance
of (Wajsberg 1933), in which the reduction of every formula to the relevant
normal form is established with the aid of 1.9. Since such a reduction is not
possible in S4, nothing in (Parry 1933) helps (Gödel 1933). So in all likelihood
it may be accidental that Gödel refers us to (Parry 1933), and probable that
the connection with modal logic is incidental to Gödel’s aim. It is perhaps
significant that the item immediately following (Parry 1933) is a paper by
Gödel, though unconnected with modal logic.15
8 Wajsberg 1933
Wajsberg concentrates on S5 rather than S4. He uses a reduction to a normal
form, and gives an interpretation in an ‘extended’ logic of classes which con-
14(Parry 1933) is Gödel’s Parry (1933a). del’s Parry (1933) is my (Parry 1934).
15Much of the information in this section, and indeed the whole paper, is based on material collected by
Jacques Riche, to whom I am deeply indebted.
Organon F 26 (3) 2019: 323–339
Modal Logic before Kripke 331
tains an operator on a class Awhich gives the universe class Uwhen A=U
and the empty class when A̸=U. Wajsberg axiomatises this calculus in
a way which can be seen as a set of axioms for S5 (and he notes this). In a
sense then this can be said to constitute the first completeness proof in modal
logic. It does however provide completeness only for S5. One feature that
Wajsberg’s work does illustrate is that there is a difference between reducing
modalities, strings of ,and , and modal functions, formulae with modal
operators in them. Though he does not use the name 1.9 or attribute the
axiom to Becker Wajsberg’s work shews that with Becker’s 1.9 you can not
only reduce all modalities to six, but you can reduce all modal functions to
formulae of first degree. This contrasts with S4 where, although there are only
finitely many distinct modalities, modal functions are not always reducible to
first degree.
Axiom 1.9 was first published in (Becker 1930), but from a footnote on p.
492 of L&L it seems that Wajsberg had seen its importance before Becker’s
work had appeared. Lewis in this footnote is commenting on some matrices
which can be used to distinguish the various systems he (Lewis) is putting
forward. Lewis writes:
Groups II and III, below, were transmitted to Mr. Lewis by Dr. M.
Wajsberg, of the University of Warsaw, in 1927. Dr. Wajsberg’s
letter also contained the first proof ever given that the System of
Strict Implication is not reducible to Material Implication, as well
as the outline of a system which is equivalent to that deducible
from the postulates of Strict Implication with the addition of the
postulate later suggested in Becker’s paper and cited below as
C11.16 It is to be hoped that this and other important work of Dr.
Wajsberg will be published shortly.
9 Feys 1937
In 1937 Robert Feys produced a survey of modal systems. Much of Feys’s
article is in the spirit of Lewis in looking at which laws of the non-modal
propositional calculus still hold when the symbols are interpreted ‘strictly’
16C11, in the notation of (Lewis and Langford 1932) is pJp, and is the characteristic axiom of the
system Lewis called S5. In the notation of (Lewis 1918), it is − ∼ pJ∼∼p. In Becker and in Wajsberg it is
− ∼ p < ∼∼p.
Organon F 26 (3) 2019: 323–339
332 Max Cresswell
and which do not. There is however an important difference. Feys seems to
be the first to notice that you can treat modal logic as a collection of sys-
tems, all based on classical logic, but in which the modal operators can be
given a variety of interpretations, and each system reflects some particular
interpretation. Unlike Lewis, Feys seems to feel that it is not at all an easy
matter to decide on just which logic is the ‘correct’ one, and points out that
mainstream logic concentrates on non-modal operators. Feys sees the need to
justify considering modality against those who think that ‘the logic of truth
and falsity is enough’. Here are some extracts from §1 of Feys’s paper.17
The idea of a logic of modalities is as old as logic itself.
When, two thousand years after Aristotle, logistics resumes its
work to give it mathematical rigor, the form it will adopt is that of
a logic of truth and falsity, almost such as the Stoics had conceived
it. In this restricted form it will prove susceptible of a tangible and
adequate expression by symbols of which Principia Mathematica
remains the model.
But such a methodical limitation does not eliminate the problems
of philosophy, and even of the sciences, which are stated in terms
of modalities. It is impossible to reason about causality, about the
very value of deductions, without resorting to the idea of necessity
and to those correlative to it, the idea of possible, of contingent.
Feys then sets out the nature of the axiomatic method, contrasting the situa-
tion in modal logic with that in non-modal propositional logic which he calls
‘the logic of true and false’. In the case of the logic of true and false there is
another method available that is to say we can study the logic of material
implication by the method of truth-tables, and therefore such logic may be
held not to require the axiomatic method. By contrast, in 1937 this method
seemed the only one available for modal logic.
An axiomatic presentation does not require you to attach any meaning to
the formulae. Feys first lists a set of axioms known to be sufficient for the
17I am using a translation by Jacques Riche, but have altered Feys’s own rather idiosyncratic notation to
a notation with and for the possibility and necessity operators.
Organon F 26 (3) 2019: 323–339
Modal Logic before Kripke 333
propositional calculus followed by three modal axioms. With for possibility
and for necessity these are:18
pp(23.11)
p≡ ∼p(24.1)
((pq)p)q(25.3)
together with the rules:
A logical law remains true if one substitutes for a variable p,q,r, wher-
ever it appears, the same function containing modalities. (22.3)
If pand pqare logical laws, qis a logical law.19 (13.42)
From a logical law pone can conclude to a logical law p. (What is
tautologically true is tautologically necessary).20 (25.2)
In §20 Feys shows awareness of the difference between Lewis’s way of pro-
ceeding, in which the axioms are all stated in terms of strict implication, and
Gödel’s way of proceeding:
18The axiom set (for classical logic) is attributed to Heyting. It takes as primitive symbols more than are
customary today. In the notation of the present paper the axioms adopted are:
p(pq)
p(pp)
(pq)(qp)
(pq)(qp)
(p(pq)) q
((pq)(qr)) (pr)
p∨ ∼p
((pq)(p⊃ ∼q)) ⊃ ∼p
q(pq)
p(pq)
(pq)((pr)(qr))
((pq)(rq)) ((pr)q)
19Feys states this rule in terms of the letters pand q. For comments on this see footnote 4 above.
20It is the presence of this rule which distinguishes Feys’s logic from Lewis’s. None of Lewis’s logics weaker
than S4 contains this rule, and Lewis himself seems not to have appreciated the difference between this rule
and the S4 theorem p□□p(Alternatively ♢♢pp).
Organon F 26 (3) 2019: 323–339
334 Max Cresswell
Two methods make it possible to relate these kinds of logics to a
system of postulates (it seems to us obvious to call them logics of
the Aristotelian modalities). The logics of Strict Implication avoid
presupposing the logic of truth and falsity; their postulates intro-
duce, simultaneously with the true and the false, other modalities.
We will follow another path, indicated in particular by Gödel. We
will start from the logic of true and false and we will complete it
with postulates that will introduce the other Aristotelian modali-
ties.
Feys dropped Gödel’s S4 axiom p□□pto give the system now called T.
In the 1937 paper Feys seems unaware that he has produced a modal system
which is none of Lewis’s S1S5.
10 McKinsey and Carnap
In 1945/46 both J.C.C McKinsey, and Rudolf Carnap produced accounts of
modality which justify particular Lewis systems. (I have discussed these in
Cresswell 2013.) McKinsey’s characterised S4 and Carnap’s S5.21 They were
however not really flexible enough to provide the generality that we have come
to expect for the multiplicity of systems which have been studied after the de-
velopment of the relational semantics for modal logic.22
11 Feys 1950
In 1950 Feys explicitly presents a survey of the systems in (Lewis and Langford
1932), together with two systems S10and S20, which are like S1 and S2 except
that they lack the axiom pJp.
As in the 1937 paper a large amount of the 1950 paper is to list the various
respects in which the theorems of the various systems of strict implication do
or do not match up with the theorems of the ordinary non-modal proposi-
tional calculus. In footnote 13 of this article Feys makes some remarks about
21Carnap’s 1946 study is in fact very similar to Wajsberg’s and on p. 41 in footnote 8 Carnap acknowledges
his debt to Wajsberg’s 1933 article.
22Hans Reichenbach also discussed modalities (for instance in Reichenbach 1954), but his work stands
somewhat outside the development of mainstream modal logic.
Organon F 26 (3) 2019: 323–339
Modal Logic before Kripke 335
the system now called T, which is the system presented in (Feys 1937). The
footnote refers to von Wright, who mentions material from his then forthcom-
ing (von Wright 1951b), in which the system is called M. As Feys notes, his
system (T) is taken from (Gödel 1933) by omitting the S4 axiom. Gödel’s
own paper only discusses a system which is (deductively equivalent to) one of
Lewis’s, and so may have caused Feys to miss the crucial fact that his own
system is not one of Lewis’s.
12 A.N. Prior
One of the clearest introductions to modal logic in the 1950s occurs in (Prior
1955). Prior was interested in modal logic from the early 1950s.23 In the Craft
of Formal Logic, a manuscript from 1952, Prior shews an awareness both of
(Lewis and Langford 1932) and of (Feys 1950), though he is more concerned
to discuss how problems which can be expressed in ‘Professor Feys’s notation’
had already been discussed by earlier thinkers, particularly in the 18th cen-
tury. So although he is becoming familiar with modal logic Prior does not
yet seem to have made it his own. He does adopt Feys’s symbol Lfor neces-
sity, which goes well with the modal operator M, as well as the Łukasiewicz
symbols for the truth-functional operators which he almost certainly got from
Bochenski. He was familiar with the work of G.H. von Wright, who, in (von
Wright 1951a), produces a system of deontic logic which Prior refers to in an
article published in the Australasian Journal of Philosophy later in that same
year.24 In that article Prior appears to suggest that his own interests do not
yet run to an appreciation of modern Modal/Deontic logic. His familiarity
with modal logic certainly emerges in (Prior 1952) where he imagines what
amounts to a two world model in which four truth values correspond to true
in both, true in the first and false in the second, false in the first and true in
the second, and false in both. He then points out that there will be formulae
whose validity in that model depends on there being only two worlds. But
perhaps Prior’s strongest contribution to modal logic was in his John Locke
Lectures at Oxford in 1956 (published as Prior 1957) in which he shewed how
to interpret modal logic as a logic of time. For he established in that work that
different assumptions about temporal ordering could lead to different systems
23Mary Prior stresses the importance of Prior’s teacher at the University of Otago, J.N. Findlay, whom she
reports as using (Lewis and Langford 1932) in a course in 1940.
24(Prior 1951). Page references are to the reprint in (Prior 1976).
Organon F 26 (3) 2019: 323–339
336 Max Cresswell
of modality. (Prior 1967, 27) speaks of a communication from Saul Kripke
pointing out that S4 was too weak to be the logic of futurity because it al-
lowed branching futures. This insight led inexorably to the semantic study
of modal logic in terms of a set of indices ordered by a relation which could
satisfy different conditions. Prior returned to New Zealand after his year at
Oxford in 1956, but at the end of 1958 took up a chair in Manchester.
13 Bayart 1958, 1959
In 1959 Saul Kripke produced a definition of validity, in terms of classes of
models, for quantificational S5.25 This was published in The Journal of Sym-
bolic Logic for that year. What was less noticed was that in Logique et Analyse
in 1958 Arnould Bayart produced a definition of validity for quantificational
S5 in which he used a set of ‘possible worlds’, acknowledging Leibniz but say-
ing that, for the purposes of logic worlds could be anything at all. Kripke of
course by 1963 had realised that worlds did not have to be models, as work
by authors like Kanger had supposed, but (Bayart 1958) explicitly denies that
they should be models. In 1959 Bayart produced a Henkin completeness proof
for quantificational S5. An English translation of Bayart’s two papers, using
more familiar notation and with an historical introduction and a commentary
is found in Cresswell (2015). (See also Cresswell 2016.)
14 Conclusion
Until the work of Kripke and others in the late 1950s and early 1960s modal
logic was frequently faulted through not having a viable semantic theory com-
parable to that produced for standard propositional and predicate logic. We
have seen that Feys invokes the axiomatic method in modal logic precisely to
compensate for this lack. What the possible-worlds semantics does reveal is
that it is the fact that modal logic has such a simple and intuitive metatheory
which goes a long way to explaining just why it survived for so long even when
its critics felt that it was uninterpretable.26
25It should be noted that the present survey has concerned itself solely with modal propositional logic. The
early developments in modal predicate logic begin with those found in (Barcan 1946; 1947) and elsewhere.
26The following list of references contains all the articles referred to in the text, together with other early
articles. It does not claim to be a complete bibliography of work in modal logic before the late 1950s.
Organon F 26 (3) 2019: 323–339
Modal Logic before Kripke 337
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Article
In the early days of the semantics for modal logic the `possible worlds' were thought of as models or interpretations. This was particularly so when the interpretation was of \emph{logical} necessity or possibility, where this was understood in terms of validity. Arnould Bayart in 1958 may have been the first modal logician to argue explicitly against the identification of necessity and validity. This note contrasts his semantics with that provided by Rudolf Carnap in 1946, and examines Bayart's proof that if you identify necessity with validity then certain theorems of S5 are not valid. The proof is then examined using Carnap's semantics.
Conference Paper
Carnap's 1946 semantics for S5 modal logic depends on the claim that α is true iff α is L-true, i.e., valid. I investigate some problems with this definition, and then consider the relation between this early attempt, and work in the late 1950s by Kanger, Hintikka and Kripke, which led to the development of the current possible-worlds semantics for modal logic. Finally I look at J.C.C. McKinsey's ‘syntactical’ account of validity for a modal logic between S4 and S5.