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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

© The Author. Journal compilation © The Editorial Board, Organon F.

This work is licensed under a Creative Commons “Attribution-NonCommercial

4.0 International” license.

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⊃(q⊃p)

(2) ∼p⊃(p⊃q)

(3) (p⊃q)∨(q⊃p)

(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 ∼p∨qprovided 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 p⊃qas ∼p∨q.

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

p∨p, 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 (p⊃q)⊃((q⊃r)⊃(p⊃r)), 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∨(q∨r)) and (q∨(p∨r)) 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 (p∧q)J(q∧p)

1.2 (q∧p)Jp

1.3 pJ(p∧p)

1.4 (p∧(q∧r)) J(q∧(p∧r))

1.5 pJ− − p7

1.6 ((pJq)∧(qJr)) J(pJr)

1.7 ∼pJ−p

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 α⊃βis 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) = (∼qJ∼p)

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(∼qJ∼p)

But even 1.8∗was 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 1.8∗are 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⊃(p⊃q)valid, so is −pJ(p⊃q), but not of course −pJ(pJq).

9The label 1.8∗is 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 S1–S5. 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 1.8∗in 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(p→q)→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). Gödel’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 ♢pJ∼♢∼♢p, 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

•p⊃♢p(23.11)

•□p≡ ∼♢∼p(24.1)

•(□(p⊃q)∧□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 p⊃qare 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⊃(p∨q)

•p⊃(p∧p)

•(p∧q)⊃(q∧p)

•(p∨q)⊃(q∨p)

•(p∧(p⊃q)) ⊃q

•((p⊃q)∧(q⊃r)) ⊃(p⊃r)

•p∨ ∼p

•((p⊃q)∧(p⊃ ∼q)) ⊃ ∼p

•q⊃(p⊃q)

•∼p⊃(p⊃q)

•(p⊃q)⊃((p∧r)⊃(q∧r))

•((p⊃q)∧(r⊃q)) ⊃((p∨r)⊃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 ♢♢p⊃♢p).

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 S1–S5.

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 pJ♢p.

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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