## 1. Life



Jan Lukasiewicz's life was that of a career academic and
scholar, seriously disrupted by the upheavals of war in the twentieth
century. Born and educated in Polish Austria, he flourished in
Poland's Second Republic, endured the hardships of war, fled
ahead of the Red Army to Germany, and found a final haven in the
Republic of Ireland.



Jan Leopold Lukasiewicz was born on 21 December 1878 in
 Lwow[1],
 historically a Polish city, at that time the capital of Austrian
Galicia. Lukasiewicz's father Pawel was a captain in
the Austrian military, his mother Leopoldine, nee Holtzer, was
daughter of an Austrian civil servant. Jan was their only child. The
family spoke Polish. Lukasiewicz attended school (classical
*Gimnazjum* or grammar school, emphasizing classical languages)
from 1890, completing in 1897 and beginning the study of law at the
University of Lwow. Under Austrian rule the university
permitted instruction in Polish. In 1898 he switched to mathematics,
studying under Jozef Puzyna, and philosophy, studying under
Kazimierz Twardowski, who had been appointed Extraordinary (Associate)
Professor there in 1895, and also Wojciech Dzieduszycki. In 1902
Lukasiewicz was awarded his doctorate in philosophy under
Twardowski with a dissertation "On induction as the inverse of
deduction". Having achieved only top marks in all examinations
between his school leaving examinations and his dissertation, he was
awarded the doctorate *sub auspiciis Imperatoris*, a rare
distinction, and he received a diamond ring from Emperor Franz
Josef.



From 1902 he was employed as a private teacher and as a clerk in the
university library. In 1904 he obtained a scholarship from the
Galician autonomous government and went to study in Berlin then in
Louvain. In 1906 he obtained his *Habilitation* with a piece on
"Analysis and construction of the concept of cause". As a
*Privatdozent* in Philosophy, he was able to give lectures at
the university, becoming the first of Twardowski's students to
join him in doing so. His first course of lectures, delivered in
Autumn 1906, was on the algebra of logic, as formulated by Couturat.
In 1908 and 1909 he obtained a stipendium which enabled him to visit
Graz, where he made acquaintance with Alexius Meinong and his school.
In 1911 he was appointed Extraordinary Professor, and continued to
teach in Lwow until the outbreak of war in 1914. During this
time his students included Kazimierz Ajdukiewicz and Tadeusz
Kotarbinski, who would later become famous philosophers in their
own right. He also in 1912 got to know Stanislaw Lesniewski,
who had however come to Lwow after studying abroad and cannot
be counted as his pupil.



In 1915 the fortunes of war put Germany in control of Warsaw, and they
decided to re-open the university, which had not been allowed to
function as a Polish-speaking university under Russian rule.
Lukasiewicz became professor of philosophy there. In 1916 he was
dean of the Faculty of Arts, and in 1917 prorector of the university.
In 1918 he left the university, being appointed Head of the Department
of Higher Schools in the new Polish Ministry of Education, and after
Poland obtained full independence he became Minister of Education in
Paderewski's Cabinet, serving from January to December 1919.
From 1920 until 1939 he was, as was Lesniewski, a Professor in
the Faculty of Natural Sciences at the University of Warsaw. In
1922/23 and again in 1931/32 he served as rector of the university. In
1929 he married Regina Barwinska.



The interwar period was most fruitful for Lukasiewicz. He was a
leading figure, with Lesniewski and Tarski, in what became known
as the Warsaw School of Logic. He made a friend of the only German
professor of mathematical logic, Heinrich Scholz, and was awarded an
honorary doctorate by the latter's University of Munster in
1938. Other honors awarded him in this period were Grand Commander of
the Order of Polonia Restituta (1923), Grand Commander of the
Hungarian Order of Merit, a monetary award from the City of Warsaw
(1935) and memberships of the Polish Academy of Arts and Sciences in
Krakow, and the Polish Scientific Societies in Lwow and
Warsaw.



Students whom he supervised through their doctoral dissertations were:
Mordechaj Wajsberg, Zygmunt Kobrzynski, Stanislaw
Jaskowski, Boleslaw Sobocinski, and Jerzy
Slupecki.



At the outbreak of war in September 1939 the Lukasiewiczes'
home was bombed by the Luftwaffe: all his books, papers and
correspondence were destroyed, except for one volume of his bound
offprints. The Lukasiewiczes lived in provisional accommodation
for academics. The German occupiers closed the university and
Lukasiewicz found employment for a meagre salary in the Warsaw
city archives. Additional financial support came from Scholz.
Lukasiewicz taught in the underground university. From late 1943,
fearing the imminent arrival and occupation of Poland by the Red Army,
and under suspicion by some colleagues of being pro-German and
anti-Jewish, Lukasiewicz expressed the wish to Scholz that he and
his wife should leave Poland. As a first step to their going to
Switzerland, Scholz managed to obtain permission for the
Lukasiewiczes to travel to Munster. They left Warsaw on 17
July 1944, just two weeks before the outbreak of the Warsaw Rising.
Following the 20 July 1944 bomb plot against Hitler there was no hope
of them obtaining permission to leave for Switzerland. They stayed in
Munster, enduring allied bombing, until January 1945, when they
were offered accommodation by Jurgen von Kempski at his farm in
Hembsen (Kreis Hoxter, Westphalia), where they were liberated by
American troops on 4 April.



From the summer of 1945 Lukasiewicz taught logic at a Polish
secondary school set up at a former Polish POW camp in Dossel. In
October 1945 they were allowed to travel to Brussels. There
Lukasiewicz again taught logic at a provisional Polish Scientific
Institute. Being unwilling to return to a Poland under communist
control, Lukasiewicz looked for a post elsewhere. In February
1946 he received an offer to go to Ireland. On 4 March 1946 the
Lukasiewiczes arrived in Dublin, where they were received by the
Foreign Secretary and the Taoiseach Eamon de Valera. In autumn 1946
Lukasiewicz was appointed Professor of Mathematical Logic at the
Royal Irish Academy (RIA), where he gave lectures at first once and
then twice a week.



In his final years in Ireland Lukasiewicz resumed contacts with
colleagues abroad, particularly Scholz, with whom he was in constant
correspondence. He attended conferences in Britain, France and
Belgium, sent papers to Poland before being expelled (with 15 other
exiled Poles) from the Polish Academy in Krakow, lectured on
mathematical logic at Queen's University Belfast and on
Aristotle's syllogistic at University College Dublin. His health
deteriorated and he had several heart attacks: by 1953 he was no
longer able to lecture at the Academy. In 1955 he received an honorary
doctorate from Trinity College Dublin. On 13 February 1956 after an
operation to remove gallstones he suffered a third major coronary
thrombosis and died in hospital. He was buried in Mount Jerome
Cemetery in Dublin, "far from dear Lwow and
Poland", as his gravestone reads. Regina deposited most of his
scientific papers and correspondence with the RIA. In 1963 the Academy
transferred their holdings to the library of the University of
Manchester, where they remain, uncatalogued. The choice of Manchester
was due to the presence there as a lecturer of Czeslaw Lejewski,
who had studied with Lukasiewicz in Warsaw and twice been
examined by the latter for doctoral theses, once in 1939, when war
intervened, a second time in London in 1954. Lejewski had seen the
second edition of Lukasiewicz's book on Aristotle's
syllogistic through the press: it appeared posthumously in 1957. In
2022 on the initiative of the Polish government his remains were
repatriated and reinterred with military honors in Warsaw's
Powazki Cemetery.


## 2. The Influence of Twardowski



Lukasiewicz was one of Twardowski's first students in
Lwow, and was influenced in his attitudes and methods by his
teacher. Twardowski was born and studied in Vienna, where he became a
disciple of Franz Brentano, and was imbued with the latter's
passionate advocacy of philosophy as a rigorous discipline, to be
investigated with the same care and attention to detail as any
empirical science, and to be communicated with utmost transparency. In
1895 Twardowski was appointed Professor Extraordinary in Lwow.
He found Polish philosophical life dormant and third-rate, and set
about vitalising the subject and building its Polish institutions, at
the cost of his own academic output. Like Brentano, he believed that a
sound descriptive psychology was methodologically basic for
philosophy, and like Brentano he advocated modest reforms in formal
logic. Lukasiewicz, under the influence of Husserl, Russell and
Frege, rejected any foundational role for psychology, and inspired in
particular by the latter two, he carried the reform of logic far
beyond anything Twardowski envisaged. He read Russell's *The
Principles of Mathematics* in 1904 and it influenced him
considerably. The general attitude that philosophy could and should
aspire to be scientifically exact was one that stayed with
Lukasiewicz, though his estimation of the state of the subject
tended to become more pessimistic than optimistic, and he advocated
fundamentally reforming philosophy along logical lines.



Another respect in which Lukasiewicz continued the tradition of
the Brentano School was in his respect for the history of philosophy,
particularly that of Aristotle and the British empiricists. (He and
Twardowski translated Hume's first *Enquiry* into
Polish.) Twardowski, who knew Bolzano's work well, pointed out
similarities between concepts in Bolzano's and
Lukasiewicz's theories of probability. The respect for
history also lay behind Lukasiewicz's groundbreaking
studies in the history of logic, notably his accounts of Stoic
propositional logic and Aristotle's syllogistic.



Lukasiewicz emulated and indeed surpassed Twardowski in his
attention to clarity of expression. Qualified experts agree that
Lukasiewicz's scientific prose, in whichever of the three
languages in which he wrote, is of unmatched clarity and beauty.


## 3. Early Work



In the years before World War I, Lukasiewicz worked predominantly
on matters to do with the methodology of science. His doctorate,
published in 1903 as "On induction as the inverse of
deduction", investigated the relationship between the two forms
of reasoning, in the light of work by Jevons, Sigwart and Erdmann.
Inductive reasoning, proceeding from singular empirical statements,
attempts on his early view to reach a general conclusion to which a
certain probability can be ascribed. But he soon shifted to the view
that it is impossible to ascribe a determinate probability to a
general statement on the basis of induction. Rather the method of the
empirical sciences is to creatively hazard the thought that a certain
generalization is true, deduce singular conclusions from this, and
then see whether these are true. If one conclusion is not, then the
general statement is refuted. This, an early formulation of the
hypothetico-deductive method of science, anticipates the ideas of
Popper by more than two decades, though expressed less forcefully.
Lukasiewicz also anticipated Popper by stressing what he called
"creative elements in science", against the idea that the
scientist's task is to reproduce or replicate the facts.



In 1906 Lukasiewicz published a substantial piece,
"Analysis and construction of the concept of cause", which
gained him his *Habilitation* in Lwow. It is significant
both for its carefully argued method and for presaging themes that
were to occupy him later. Taking concepts platonistically as abstract
objects, Lukasiewicz rejects psychological, subjective and
regularity accounts of causality and accepts that cause and effect are
linked by necessity, which he identifies with logical necessity:
"a causal relation is a necessary relation, and the relative
feature [...] due to which we call some object a cause, is the
feature 'entailing or bringing-about with
necessity'." The piece's motto is *Arceo
psychologiam*, "Reject psychology", marking a clear
break with Twardowski and Brentano, while the logical analysis is
aimed at extracting the logical features of the notion of cause. It is
a clear example of what later came to be called 'analytic
philosophy', and shows Lukasiewicz bringing logical
concepts to bear on scientifically central concepts, as he was later
to do with his logical analysis of determinism.



An interest in probability lay behind one of Lukasiewicz's
two monographs published before the war, namely *Logical
Foundations of Probability Theory*, which was written and
published not in Polish but in German. In 1908 and 1909
Lukasiewicz visited Graz where both Alexius Meinong and Ernst
Mally were also working on probability theory at this time, so it is
likely that the book was written in German because their language of
discussion was German, and also in order to ensure a wider audience.
Lukasiewicz's theory makes constructive use of ideas
plucked from elsewhere: from Frege he took the idea of a truth-value,
from Whitehead and Russell the idea of an indefinite proposition, and
from Bolzano the idea of the ratio of true values to all values for a
proposition. Consider the classical urn example, where an urn contains
*m* black balls and *n* white balls. Let the indefinite
proposition '\(x\) is a black ball in this urn' be such
that the variable '\(x\)' may take as value any expression
naming a ball in the urn: the variable is then said to *range
over* the individual balls, and different expressions naming the
same ball to have the same value. (Note that Lukasiewicz indeed
uses the terminology, later associated with Quine, of a variable
taking values, here expressions, and ranging over objects designated
by said expressions.) An indefinite proposition is said to be true if
it yields a true proposition (Lukasiewicz says
'judgement' for a definite proposition) for all values of
its variables, it is false if it yields a false judgement for all
values, and is neither true nor false if it yields true judgements for
some values and false judgements for others. The ratio true values/all
values is then called by Lukasiewicz the *truth-value* of
the indefinite proposition. For true indefinites it is 1, for false
indefinites it is 0, and for others it is a rational number between 0
and 1 (rational because only finite domains are considered). In our
urn case the truth-value of the indefinite proposition
'*x* is a black ball in this urn' is
\(\frac{m}{m+n}\).



On this basis Lukasiewicz develops a calculus of truth-values in
which he can deal with logically complex propositions, conditional
probability, probabilistic independence, and derive Bayes'
Theorem. The calculus of truth-values is used as a *logical*
theory of probability, assisting us in our dealings with definite
reality: Lukasiewicz denies that there can be a theory either of
objective or of subjective probability as such. Two ideas from this
short but remarkable work are worth emphasizing because they resonate
with later ones of Lukasiewicz. Firstly, there is the idea of a
proposition (in this case an indefinite proposition) being neither
true nor false; secondly, and connected with this, of such a
proposition having a numerical truth-value properly between 0 (false)
and 1 (true). Lukasiewicz's theory deserves to be better
known: it continues and extends earlier ideas of Bolzano, his
probability corresponding to the latter's degree of validity of
a proposition (with respect to variable components). Its chief
drawback is that it is formulated for finite domains only.



Of all the works Lukasiewicz published before World War I, one
most clearly anticipated his later concerns. This was the 1910
monograph *On the Principle of Contradiction in Aristotle*. It
marked a crucial turning point in the development of the
Lwow-Warsaw school. For Lukasiewicz it represented the
first sustained questioning of the assumptions of traditional
Aristotelian logic.



Lukasiewicz introduces the project of his monograph, a critical
investigation of the legitimacy of the Principle of Contradiction (PC)
as variously formulated by Aristotle, in the context of its critique
by Hegel and the opportunity to re-examine the PC in the light of the
development of mathematical logic from Boole to Russell.
Lukasiewicz's sources for the post-Hegelian discussion of
the 'logical question' are Ueberweg, Trendelenburg and
Sigwart. A more local background was probably Twardowski's
account of the absolute and timeless nature of truth.



Lukasiewicz distinguishes three different, non-equivalent
versions of PC in Aristotle: an ontological version, a logical
version, and a psychological version, as follows:



*Ontological* (OPC): No object may at the same time possess and
not possess the same property.



*Logical* (LPC): Contradictory statements are not
simultaneously true.



*Psychological* (PPC): No one can simultaneously believe
contradictory things.



Lukasiewicz criticises Aristotle for on the one hand claiming PC
cannot be proved, and on the other hand attempting an indirect or
pragmatic "proof". In partial agreement with the tradition
according to which PC is not the cornerstone or basic principle of
logic, Lukasiewicz claims that its status is less secure than
some other logical propositions, and that its function is principally
to serve as a pragmatic norm. Nevertheless, in an Appendix to the book
he gives a formal derivation of one version of PC from other
assumptions. This shows that PC is as it were just one logical theorem
among others, a statement that would raise few eyebrows today but was
fairly radical in its day. Among the assumptions used in the
derivation is a version of the Principle of Bivalence, that every
proposition is either true or false and none is both, so the
derivation of PC is not after all such a surprise.



Lukasiewicz described himself later as attempting in the
monograph to devise a "non-Aristotelian logic" but admits
that he did not succeed, principally because at this stage he was not
prepared to reject the principle of bivalence. It may well be
Meinong's influence at work when Lukasiewicz comes to give
his natural-language renderings of the symbolism of Couturat's
algebra of logic in the Appendix. There is little or no trace of the
propositional logic that Lukasiewicz was to make very much his
own: the renderings are clumsily object-theoretic: the constant
'0' for example, which might be naturally construed as a
constant false proposition (and is so in later Lukasiewicz) is
rendered as "the object that does not exist". This is one
reason why Lukasiewicz's formal work in the Appendix to the
1910 work appears relatively archaic. While the variable letters like
\(a, b\) etc. "signify affirmative statements" and their
negations \(a', b'\) etc. "signify negative statements",
and in practice do work like propositional variables and their
negations in modern propositional logic, Lukasiewicz's
renderings of them are curiously hybrid: '\(a\)' is
rendered as '\(X\) contains \(a\)' and
'\(a'\)' as '\(X\) contains no \(a\)', while
'1' signifies '\(X\) is an object' and
'0' signifies '\(X\) is not an object'. This
is all very confused, and by no means a classical sentential logic in
intent, even if it works like one in practice.



While not in itself a success, the book shows Lukasiewicz on the
threshold of his later logical breakthroughs. It was read in 1911 by
the young Lesniewski, who sought against Lukasiewicz to
prove OPC, and who first introduced himself in 1912 on
Lukasiewicz's doorstep with the words, "I am
Lesniewski, and I have come to show you the proofs of an article
I have written against you." The book also contains a brief
discussion of Russell's Paradox, and it was reading this that
inspired Lesniewski to become a logician, intent on providing a
paradox-free logical foundation for mathematics. The book promoted
further discussion in Lwow: Kotarbinski wrote in defense
of Aristotle's idea, discussed by Lukasiewicz, that a
statement about future contingent events may lack a truth-value before
the event and only gain one afterwards, while Lesniewski wrote in
opposition to this and brought Kotarbinski round to his own view
(which agreed with earlier views of Twardowski and later ones of
Tarski) that truth is timeless, or as Lesniewski expressed it,
both eternal and sempiternal. Lukasiewicz was soon to side with
the earlier Kotarbinski, and in so doing to make his most famous
discovery, that of many-valued logic.


## 4. Propositional Logic


### 4.1 Discoveries in Propositional Logic



Lukasiewicz came across propositional logic, which he originally
followed Whitehead and Russell in calling 'theory of
deduction', in their work and also in that of Frege. In 1921
Lukasiewicz published a ground-clearing article,
"Two-Valued Logic", in which he brought together results
in the algebra of logic governing the two truth-values true and false,
which, like Frege, Lukasiewicz construed as what sentences or
propositions *denoted,* but for which, unlike Frege, he
introduced constant propositional symbols '1' and
'0'. He intended it as the first part of a monograph on
three-valued logic, which however was never completed, probably
because Lukasiewicz became dissatisfied with the rather hybrid
approach which was already being outdated by his rapid development.
The article is notable for several innovations. Using a symbolism
derived from those of Couturat and Peirce, it introduces the idea of
axiomatic rejection alongside that of axiomatic assertion, which
latter was of course familiar from Frege, Whitehead and Russell. The
constants '0' and '1' also occur in asserted
and rejected formulas, in effect setting up an object-language version
of truth-tables. To show this we use Lukasiewicz's later
parenthesis-free notation (see the supplementary document
 (Lukasiewicz's Parenthesis-Free or Polish Notation)
 and his symbols '\(\vdash\)' for assertion and
'\(\dashv\)' for rejection, to be read respectively as
"I assert" and "I reject". The first
principles of logic are simply \({\vdash}1\) and \({\dashv}0\) but to
indicate the tabulation for implication the following principles must
be adhered to: \({\vdash}C00, {\vdash}C01, {\dashv}C10, {\vdash}C11\).
When Lukasiewicz employed propositional variables, he quantified
them in the manner of Peirce, using '\(\Pi\)' for the
universal and '\(\Sigma\)' for the particular
quantifier.



Lukasiewicz and his students made the study of propositional
calculi very much their own: the results obtained between 1920 and
1930 were published in a 1930 joint paper of Lukasiewicz and
Tarski, "Untersuchungen uber den
Aussagenkalkul". Work proceeded on both classical
(bivalent) and many-valued calculi. The clearest and most complete
demonstration of how Lukasiewicz in his maturity treated
classical propositional calculus comes in his 1929 student textbook,
based on lecture notes, *Elements of Mathematical Logic*. The
system, following Frege, is based on implication \((C)\) and negation
\((N)\) alone, with the elegant axiom set

 
\[ \begin{align}
&CCpqCCqrCpr \\
 &CCNppp \\
 &CpCNpq 
\end{align} \]


and three rules of inference: modus ponens, a rule of uniform
substitution of formulas for propositional variables, and a rule of
definitional replacement. On this basis, and using an extremely
compressed linear notation for proofs which is at the opposite extreme
of Frege's space-occupying proofs, Lukasiewicz proves
around 140 theorems in a mere 19 pages.



Lukasiewicz, aided and abetted by students and colleagues, not
just Tarski but also Adolf Lindenbaum, Jerzy Slupecki*,*
Boleslaw Sobocinski, Mordechaj Wajsberg and others,
investigated not only the full (functionally complete) propositional
calculus, with different sets of connectives as basic, including the
Sheffer functor *D*, but also partial calculi, in particular
the pure implicational calculus (based on *C* alone) and the
pure equivalential calculus (based on *E* alone). They strove
to find axioms sets satisfying a number of normative criteria: axioms
should be as few as possible, as short as possible, independent, with
as few primitives as possible. Undoubtedly there was a competitive
element to the search for ever better axiom systems, in particular in
the attempt to find single axioms for various systems, and the
exercise has been smiled upon or even belittled as a mere
"sport", but the Polish preoccupation with improving axiom
systems was a search for logical perfection, an illustration of what
Jan Wolenski has termed "logic for logic's
sake". At one time it was thought, not without some
justification, that only Poles could compete. When Tarski once
congratulated the American logician Emil Post on being the only
non-Pole to make fundamental contributions to propositional logic,
Post replied that he had been born in Augustow and his mother
came from Bialystok. Later, Lukasiewicz was to find in the
Irish mathematician Carew Meredith a worthy non-Pole who could outdo
even the Poles in the brevity of his axioms (see Meredith 1953).



Lukasiewicz used many-valued matrices to establish the
independence of logical axioms in systems of Frege, Russell and
others. He proved the completeness of full, implicational and
equivalential calculi, and proved that the equivalential calculus
could be based on the single axiom \(EEpqErqEpr\), with substitution
and detachment for equivalence, and showed further that no shorter
axiom could be the sole axiom of the system. Tarski showed in 1925
that the pure implicational calculus could be based on a single axiom,
but a series of improvements by Wajsberg and Lukasiewicz led to
the latter's discovering in 1936 that the formula
\(CCCpqrCCrpCsp\) could serve as single axiom and that no shorter
axiom would suffice, though the publication of this result had to wait
until 1948.


### 4.2 Variable Propositional Functors



Standard propositional calculus employs neither quantifiers nor
variable functors, that is, functors of one or more places taking
propositional arguments, but which unlike such constant functors as
\(N\) or \(C\) do not have a fixed meaning. Such variable functors act
like the predicates of first-order predicate logic except for taking
propositional rather than nominal arguments. They thus add to the
expressive power of the logic. Lesniewski added both quantifiers
and bound propositional and functorial variables to propositional
logic, and called the resulting theory *protothetic*. Leaving
prefixed universal quantifiers tacit, it is a thesis of protothetic
that

 
\[ \begin{align}
& CEpqC\delta p \delta q 
\end{align} \]


where \(\delta\) is a one-place propositional functor, out of the same
syntactic stable as negation or necessity. This thesis is an
expression of the law of extensionality for propositional expressions.
If \(p\) and \(q\) are replaced by complex expressions \(x\) and \(y\)
the thesis can be used to enable definitions to be given in the
implicational form \(C\delta x\delta y\).



If \(\delta\) is replaced by the first part of a complex expression,
e.g., \(Cq\) or \(CCq0\), then simply adjoining a variable such as
\(p\) to give \(Cqp\), \(CCq0p\), is straightforward. But if the
"gap" where the variable is to go is not at the end, such
as \(Cpq\), or if the variable is to be inserted more than once, as
\(CCp0p\), this simple replacement procedure will not work.
Lesniewski got around the problem by introducing auxiliary
definitions which maneuvered the required variable slot into the right
position with only one occurrence. But Lukasiewicz found this
procedure unintuitive and wasteful. His preference--which in fact
echoes the practice of Frege--was to allow any context in which a
single propositional variable is free to serve as a substituend for a
functor like \(\delta\), and mark the places into which the argument
of \(\delta\) was to be slotted with an apostrophe, so in our examples
\(C \apos q\), \(CC\apos 0 \apos\). This more liberal
"substitution with apostrophe" allows definitions to be
given a satisfyingly simple implicational form. For example, in
propositional calculus based on implication and the propositional
constant 0, negation can be defined simply by \(C\delta Np\delta
Cp0\). The use of variable functors with liberal substitution enables
a number of principles of propositional logic to be given startlingly
compressed and elegant formulations, for example the principle of
bivalence in the form

 
\[ \begin{align}
& C\delta 0C\delta C00 \delta p 
\end{align} \]


which can be read as "if something is true of a false
proposition then if it true of a true proposition, it is true of any
proposition" (*C*00 is a true proposition). The supreme
achievements of compression using variable functors were made by
Meredith, who showed (as quoted by Lukasiewicz, referring to an
apparently unpublished paper) that the whole of classical
propositional logic with variable functors can be based on the single
axiom

 
\[ \begin{align}
& C\delta pC\delta Np\delta q. 
\end{align} \]


More astonishingly, Meredith (1951) showed that the whole of bivalent
propositional calculus with quantifiers and variable functors can be
deduced, using substitution, detachment and quantifier rules, from the
single axiomatic formula

 
\[ \begin{align}
& C\delta \delta 0\delta p. 
\end{align} \]


Lukasiewicz admiringly described this feat as "a
masterpiece of the art of deduction".


### 4.3 Intuitionistic Logic



Lukasiewicz was interested in intuitionistic logic, not least
because, like his own, it rejected the law of excluded middle. In a
late article published in 1952, he gave an elegant axiomatization with
ten axioms, using the letters \(F\), \(T\) and \(O\) for the
intuitionist connectives of implication, conjunction and disjunction
respectively, in order to obviate clashes caused by
"competition" for the connectives, though interestingly he
kept the usual negation for both systems. He then showed how to define
classical implication as \(NTpNq\), formulated this definition using a
variable functor as the implication

 
\[ \begin{align}
& F\delta NTpNq\delta Cpq 
\end{align} \]


and proved that in this version classical bivalent logic based on
\(C\) and \(N\) is contained in intuitionistic logic, provided that
detachment is confined to \(C\)-\(N\) formulas only. Classical
conjunction and disjunction can be defined in the usual way as
\(NCpNq\) and \(CNpq\) respectively. By differentiating intuitionistic
from classical connectives his perspective reverses the usual one that
intuitionistic propositional calculus is poorer in theorems than
classical: in Lukasiewicz's formulation it is the other way
around.


## 5. Many-Valued Logic


### 5.1 Possibility and the Third Value



Lukasiewicz's most celebrated achievement was his
development of many-valued logics. This revolutionary development came
in the context of discussing modality, in particular possibility. To
modern logicians, used to the idea of modal logic being grafted onto
classical bivalent logic, this may seem odd. But let us consider how
Lukasiewicz arrived at the idea. If \(p\) be any proposition, let
\(Lp\) notate that it is necessary that \(p\) and \(Mp\) that it is
possible that \(p\). The two modal operators are connected by the
usual equivalence \(ENLpMNp\). Everyone accepts the implications
\(CLpp\) and \(CpMp\). Lukasiewicz supposes one accepts also the
converse implications \(CpLp\) and \(CMpp\), as one would from a
deterministic point of view. That gives the equivalences \(EpLp\) and
\(EpMp\), which effectively collapse modal distinctions. Now add in
the idea that possibility is two-sided: if something is possible, then
so is its negation: \(EMpMNp\). From these it immediately follows that
\(EpNp\), and this is paradoxical in two-valued logic. The way out, as
Lukasiewicz portrays it, is to uncollapse the modal distinctions,
not by rejecting any of the principles above but by finding a case
where \(EpNp\) is *true*. We entertain the idea of the
proposition \(Mp\) being true when \(p\) is neither true nor false. In
addition to the truth-values **true** (1) and
**false** (0), allow then a third value,
**possible**, which we write
'\(\tfrac{1}{2}\)', so that when \(p\) is neither true nor
false, it is possible, and so is its negation \(Np\), for if \(Np\)
were true, \(p\) would be false, and vice versa. If \(Epq\) is true
when \(p\) and \(q\) have the same truth-value, then when \(p\) is
possible (we write '\(\tval{p}\)' for the truth-value of
\(p\), so \(\tval{p}=\tfrac{1}{2}\)) we have

 
\[ \begin{align}
& \tval{EpNp} = \tval{E \tfrac{1}{2} \tfrac{1}{2}} = 1 
\end{align} \]


This is, with minor changes, the way in which Lukasiewicz
introduces the third value in his first published paper on the
subject, which bears the title "On the Concept of
Possibility". This short paper is based on a talk given on 5
June 1920 in Lwow. Two weeks later a second talk in the same
place was more transparently titled "On Three-Valued
Logic". In this, Lukasiewicz sets out principles governing
implication and equivalence involving the third value. These in effect
determine the
 truth-tables[2]
 for these connectives:




|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 

|  |  |  |  |
| --- | --- | --- | --- |
| \(C\) | 1 | 1/2 | 0 |
| 1 | 1 | 1/2 | 0 |
| 1/2 | 1 | 1 | 1/2 |
| 0 | 1 | 1 | 1 |

 |  | 

|  |  |  |  |
| --- | --- | --- | --- |
| \(E\) | 1 | 1/2 | 0 |
| 1 | 1 | 1/2 | 0 |
| 1/2 | 1/2 | 1 | 1/2 |
| 0 | 0 | 1/2 | 1 |

 |



Together with the assumed definitions of negation, conjunction and
disjunction as, respectively

 
\[ \begin{align}
Np & = Cp0 \\
 Apq & = CCpqq \\
 Kpq & = NANpNq 
\end{align} \]


this yields truth-tables for these connectives as




|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 

|  |  |
| --- | --- |
| \(N\) |  |
| 1 | 0 |
| 1/2 | 1/2 |
| 0 | 1 |

 |  | 

|  |  |  |  |
| --- | --- | --- | --- |
| \(A\) | 1 | 1/2 | 0 |
| 1 | 1 | 1 | 1 |
| 1/2 | 1 | 1/2 | 1/2 |
| 0 | 1 | 1/2 | 0 |

 |  | 

|  |  |  |  |
| --- | --- | --- | --- |
| \(K\) | 1 | 1/2 | 0 |
| 1 | 1 | 1/2 | 0 |
| 1/2 | 1/2 | 1/2 | 0 |
| 0 | 0 | 0 | 0 |

 |



Lukasiewicz proudly declares "that three-valued logic has,
above all, theoretical significance as the first attempt to create a
non-Aristotelian logic" (*PL*, 18; *SW*, 88). What
its practical significance is, he thinks awaits to be seen, and for
this we need "to compare with experience the consequences of the
indeterministic view which is the metaphysical basis of the new
logic" (ibid.).


### 5.2 Indeterminism and the Third Value



This final remark reveals the motivation of Lukasiewicz's
drive to replace the old bivalent logic with the new trivalent one. It
was in order to defend indeterminism and freedom. In fact the idea had
come to fruition some three years earlier. Having been appointed to an
administrative position in the Ministry of Education in 1918, and
being about to leave academic life for an indefinite period,
Lukasiewicz delivered a "farewell lecture" to the
University of Warsaw on 17 March, in which he announced dramatically,
"I have declared a spiritual war upon all coercion that
restricts man's free creative activity." The logical form
of this coercion, in Lukasiewicz's view, was Aristotelian
logic, which restricted propositions to true or false. His own weapon
in this war was three-valued logic. Recalling his 1910 monograph, he
notes that:



> 
> Even then I strove to construct non-Aristotelian logic, but in vain.
> Now I believe I have succeeded in this. My path was indicated to me by
> antinomies, which prove that there is a gap in Aristotle's
> logic. Filling that gap led me to a transformation of the traditional
> principles of logic. Examination of that issue was the subject-matter
> of my last lectures. I have proved that in addition to true and false
> propositions there are *possible* propositions, to which
> objective possibility corresponds as a third in addition to being and
> non-being. This gave rise to a system of three-valued logic, which I
> worked out in detail last summer. That system is as coherent and
> self-consistent as Aristotle's logic, and is much richer in laws
> and formulae. That new logic, by introducing the concept of objective
> possibility, destroys the former concept of science, based on
> necessity. Possible phenomena have no causes, although they themselves
> can be the beginning of a causal sequence. An act of a creative
> individual can be free and at the same time affect the course of the
> world. (*SW*, 86)
> 



Because Lukasiewicz was involved in government until the end of
1919, it took until 1920 for his discoveries of 1917 to be revealed to
a wider academic public. Lukasiewicz returned to the subject of
determinism for his inaugural lecture as Rector of the University of
Warsaw on 16 October 1922. That lecture, delivered without notes but
later written down, was reworked, though not in essentials, up until
1946. It was published only posthumously in 1961 as "On
Determinism". Distinguishing logical from causal determinism,
Lukasiewicz claims that if a prediction of a future contingent
event such as an action is true at the time the prediction is made,
the event must occur, so the only way to rescue the agent's
freedom of action is to deny that the prediction is true, and assign
it instead the third truth-value of possibility.



Here is not the place to go into the problems with
Lukasiewicz's argumentation. Suffice it to say that the
principle *EpLp* need not be accepted by determinists, and that
other logicians who have considered adding a third value to logic,
such as (unbeknown to Lukasiewicz) William of Ockham, concluded
that there was no reason to reject bivalence while upholding freedom.
This is without even considering compatibilist views.


### 5.3 More than Three Values



Once the spell of bivalence was broken, a natural next step was to
consider logic with more than three values. In 1922 Lukasiewicz
indicated how to give truth-tables for the standard connectives in
systems with finitely or infinitely many truth values, according to
the following principles, where the truth-values are numbers in the
interval [0,1]:

 
\[ \begin{aligned}
\tval{Cpq} & = \begin{cases}
1, & \text{if } \tval{p} \le \tval{q} \\
 (1 - \tval{p}) + \tval{q}, & \text{if } \tval{p} \gt \tval{q} 
\end{cases} \\
 \tval{Np} & = 1 - \tval{p} 
\end{aligned} \]


In proposing logics with infinitely many values, Lukasiewicz was
thus the inventor of what was much later (43 years later, to be exact)
to be called 'fuzzy logic'. Commenting on these systems in
1930, Lukasiewicz wrote



> 
> it was clear to me from the outset that among all the many-valued
> systems only two can claim any philosophical significance: the
> three-valued one and the infinite-valued ones. For if values other
> than "0" and "1" are interpreted as "the
> possible", only two cases can reasonably be distinguished:
> either one assumes that there are no variations in degrees of the
> possible and consequently arrives at the three-valued system; or one
> assumes the opposite, in which case it would be most natural to
> suppose, as in the theory of probabilities, that there are infinitely
> many degrees of possibility, which leads to the infinite-valued
> propositional calculus. I believe that the latter system is preferable
> to all others. Unfortunately this system has not yet been investigated
> sufficiently; in particular the relations of the infinite-valued
> system to the calculus of probabilities awaits further inquiry.
> (*SW*, 173)
> 



We will discuss this philosophical attitude below.


### 5.4 Axioms and Definitions



Once the truth-tabular or matrix approach to many-valued logics was
established, it was natural to consider their axiomatization.
Lukasiewicz's students assisted in this. In 1931 Wajsberg
axiomatized the three-valued system L\(\_3\)
by the theses

 
\[ \begin{align}
& CpCqp \\
 & CCpqCCqrCpr \\
 & CCNpNqCqp \\
 & CCCpNppp 
\end{align} \]


Wajsberg also proved a conjecture of Lukasiewicz that the
denumerably infinite-valued system L\(\_{\aleph\_0}\)
can be axiomatized by



\[ \begin{align}
& CpCqp \\
 & CCpqCCqrCpr \\
 & CCCpqqCCqpp \\
 & CCCpqCqpCqp \\
 & CCNpNqCqp 
\end{align} \]


None of these systems is functionally complete: there are connectives
not definable on the basis of *C* and *N* alone. Among
those that are definable is possibility *M*: as early as 1921
Tarski showed that it could be defined as *CNpp*. In 1936
Slupecki showed that by adding a functor \(T\) specifiable as
\(\tval{Tp} = \tfrac{1}{2}\) for all values of *p*, all
connectives can be defined in L3. To axiomatize these
functionally complete system the formulas

 
\[ \begin{align}
& CTpNTp \\
 & CNTpTp 
\end{align} \]


have to be added to Wajsberg's axioms.



Adolf Lindenbaum showed that L\(\_n\) is
contained in L\(\_m\) \((n \lt m)\) if and
only if \(n - 1\) is a divisor of \(m - 1\), so if neither divides the
other their respective tautologies properly overlap but neither set is
contained in the other. Tautologies of the infinite-valued system
L\(\_{\aleph\_0}\) are contained in those
of all the finite-valued systems.


### 5.5 Second Thoughts on Modality: System L



From 1917, Lukasiewicz had been happy with the three-valued logic
as formulating adequate notions of modality, with the noted preference
for the infinite-valued system as optimally precise. At some time,
probably around 1951-52, when he was working on
Aristotle's modal logic, Lukasiewicz changed his mind.
There are a number of reasons behind the change of mind but the
easiest to identify is Lukasiewicz's concern that in L\(\_3\)
there are theorems of the form
\(L\alpha\), for example \(LCpp\). Why should this be a concern, given
that most "standard" modal logics recognize the principle
that if \(\alpha\) is a theorem, so is \(L\alpha\)? Lukasiewicz
gives two examples to justify the worry. If \({=}ab\) is the
proposition that \(a\) is identical with \(b\), then basing identity
on the two axioms of self-identity and extensionality

 
\[ \begin{align}
& {=}aa \\
 & C{=}abC{\phi}a{\phi}b 
\end{align} \]


then instantiating \(L{=}a\apos\) for \(\phi\) gives

 
\[ \begin{align}
& C{=}abCL{=}aaL{=}ab 
\end{align} \]


and if we accept \(L{=}aa\) we are forced to conclude that \(L{=}ab\),
which Lukasiewicz thinks is false (*SW* 392, *AS*
171), citing Quine's (1953) example (now outdated because the
number has changed) that while it's true that 9 = the number of
planets, this is not necessarily true, although necessarily 9 = 9.
Dually, we have

 
\[ \begin{align}
& CMN{=}abN{=}ab 
\end{align} \]


that is, if \(MN{=}ab\) then \(N{=}ab\). But suppose *a* is
replaced by "the number thrown on this throw of this die"
and *b* by "the number thrown on the next throw of this
die" the antecedent can be true and the consequent false.



In the wake of much subsequent discussion of such examples by Quine,
Kripke and others, these examples are hardly convincing, but there is
another more general reason why Lukasiewicz rejects necessities
as theorems:



> 
> it is commonly held that apodeictic propositions have a higher dignity
> and are more reliable than corresponding assertoric ones. This
> consequence is for me by no means evident. [...] I am inclined to
> think that all systems of modal logic which accept asserted apodeictic
> propositions are wrong. (*SW* 395-6).
> 



Since \(LCpp\) is a theorem of all the systems of many-valued logic to
date, Lukasiewicz needed to come up with something new. This he
did in his 1953 paper "A System of Modal Logic".



Lukasiewicz begins the paper by laying out the conditions that a
modal logic needs to satisfy. These include axiomatic rejections as
well as assertions, as follows:

 
\[ \begin{align}
& \vdash CpMp \\
 & \dashv CMpp \\
 & \dashv Mp \\
 & \vdash CLpp \\
 & \dashv CpLp \\
 & \dashv NLp \\
 & \vdash EMpNLNp \\
 & \vdash ELpNMNp 
\end{align} \]


To obtain a system of modal logic respecting extensionality for
propositional functors, Lukasiewicz takes Meredith's axiom
for \(C\)-\(N\)-\(\delta\) propositional calculus

 
\[ \begin{align}
& \vdash C\delta pC\delta Np\delta q 
\end{align} \]


and adds one more axiomatic assertion and two axiomatic rejections



\[ \begin{align}
& \vdash CpMp \\
 & \dashv CMpp \\
 & \dashv Mp 
\end{align} \]


together with rules of substitution and detachment for both assertion
and rejection, to obtain his logic. The principles for assertion are
as usual, while those for rejection are:



\(\dashv\) Substitution: Any formula which has a rejected substitution
instance is rejected.



\(\dashv\) Detachment: If \(Cab\) is asserted and \(b\) is rejected
then \(a\) is rejected.



From these he can derive all the desired principles and
extensionality.



This is the logic L. Unlike standard modal logics, it has a
finite characteristic matrix, as follows, where like Lukasiewicz
we now replace '\(M\)' by a new symbol
'\(\Delta\)', with 1 as the designated (true) value and 4
the antidesignated (false) value:




|  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- |
| \(C\) | 1 | 2 | 3 | 4 | \(N\) | \({\Delta}\) |
| 1 | 1 | 2 | 3 | 4 | 4 | 1 |
| 2 | 1 | 1 | 3 | 3 | 3 | 1 |
| 3 | 1 | 2 | 1 | 2 | 2 | 3 |
| 4 | 1 | 1 | 1 | 1 | 1 | 3 |



The matrix was proved characteristic by Smiley in 1961. Functors of
necessity (\(\Gamma\)) and conjunction are definable in the standard
way. More intriguingly, Lukasiewicz notes that there is another
possibility operator \(\nabla\) with the truth-table also given
below:




|  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- |
| \(K\) | 1 | 2 | 3 | 4 | \(\Gamma\) | \({\nabla}\) |
| 1 | 1 | 2 | 3 | 4 | 2 | 1 |
| 2 | 2 | 2 | 4 | 4 | 2 | 2 |
| 3 | 3 | 4 | 3 | 4 | 4 | 1 |
| 4 | 4 | 4 | 4 | 4 | 4 | 2 |



Taken in isolation, this is indistinguishable from \(\Delta\), but the
two operators together interact differently, for while \(\dashv
\Delta\Delta p\) and \(\dashv \nabla\nabla p\), both \(\vdash
\Delta\nabla p\) and \(\vdash \nabla\Delta p\). Lukasiewicz
compares them to twins which are indistinguishable separately but
distinguishable together. Similar twins are the necessity operator
\(\Gamma\) and its counterpart (with values 3434), and indeed the two
intermediate truth-values 2 and 3.



The logic is very unlike Lukasiewicz's earlier multivalent
systems and also very unlike other modal systems. It is unlike his own
systems in that it is an extension of classical bivalent logic and
includes all bivalent tautologies. This is less surprising when we
note that the four-valued matrices for the standard connectives are
simply the Cartesian product of the standard bivalent matrices with
themselves. It is the modal operators that make the difference.
Several features make this very unlike standard modal systems. One is
the complete lack of any truths, let alone theorems, of the form
\(\Gamma a\), in line with Lukasiewicz's rejection of
truths of "higher dignity". Other odd theorems are:



\(\vdash CK{\Delta}p{\Delta}q{\Delta}Kpq\)
  

all possible propositions are compossible



\(\vdash CEpqC{\Delta}p{\Delta}q\)
  

materially equivalent propositions are both possible if one is



\(\vdash C{\Delta}pC{\Delta}Np{\Delta}q\)
  

if a proposition and its negation are both possible, anything is



Lukasiewicz was aware of many of these odd consequences, but
continued to uphold his system. Despite a number of attempts to make
sense of the system, it has generally been concluded that because of
these oddities it is not really a system of modal logic. If there is
one dominant reason for this it is Lukasiewicz's adherence
to the principle of extensionality (truth-functionality) even for
modal operators, which forced his account of modality to go
multivalent in the first place.


## 6. History of Logic


### 6.1 Stoic Propositional Logic



Lukasiewicz's third signal achievement, along with his
investigations of many-valued and propositional logics, is his work in
the history of logic. Indeed he can reasonably be considered the
father of the modern way of doing the history of logic, which is
pursued, to quote the subtitle of his book on Aristotle's
syllogistic, "from the standpoint of modern formal logic."
We saw that his early book on the principle of contradiction in
Aristotle was relatively unsuccessful in its own terms, though it
demonstrated his ability to go to the heart of the ancient Greek
texts.



A decisive event in Lukasiewicz's development as an
historian of logic was his discovery of ancient Stoic logic. It seems
he was examining a dissertation on the Stoics and to prepare for it he
read original texts. He thereupon discovered that Stoic logic,
contrary to the then standard opinion, voiced by Prantl, Zeller and
others, was not a bowdlerized and defective Aristotelian syllogistic,
but an early propositional logic, so that for example the first Stoic
indemonstrable, "if the first, then the second; but the first,
therefore the second" is simply modus ponens or detachment for
the conditional 'if', and the variables, represented not
by letters but by ordinal numerals, are propositional variables, not
term variables. He first voiced this view, which is now of course
standard, at a meeting in Lwow in 1923. A more systematic
treatment of 1934, "On the History of the Logic of
Propositions", is a delightful vignette taking in the broad
sweep from the Stoics, ancient disputes about the meaning of the
conditional, Petrus Hispanus and Ockham on the De Morgan laws, the
medieval theory of consequences, and culminating with Frege and modern
propositional calculi. Modern appreciation of the achievements of
Stoic logic dates from Lukasiewicz's clarification and his
unstinting praise of the Stoics, especially Chrysippus.
Lukasiewicz appreciated that Prantl did not have the benefit of
knowing post-Fregean logic, and despite Prantl's erroneous
dismissal of the "stupidity" of much Stoic logic did at
least provide helpful sources. Nevertheless, Lukasiewicz's
judgement on past historians of logic is scathing:



> 
> *The history of logic must be written anew*, and by an
> historian who has a thorough command of modern mathematical logic.
> Valuable as Prantl's work is as a compilation of sources and
> materials, from a logical point of view it is practically worthless
> [...] Nowadays it does not suffice to be merely a philosopher in
> order to voice one's opinion on logic. (*SW*, 198)
> 


### 6.2 Aristotle



In Lukasiewicz's logic textbook of 1929, after treating
propositional calculus he does not go on, as one would nowadays, to
expound predicate logic, but gives a brief formal account of
Aristotle's categorical (non-modal) syllogistic, presupposing
twelve theorems of propositional calculus. This foreshadowed his 1951
book, *Aristotle's Syllogistic*, by 22 years. This book,
which revolutionized the study of Aristotle's logic, had a long
and interrupted genesis. A talk on the subject given at Krakow
in 1939 was published in Polish only in 1946. In 1939 Lukasiewicz
prepared a Polish monograph, but the partial proofs and manuscript
were destroyed in the bombing of Warsaw. In 1949 he was invited to
lecture on Aristotle's syllogistic at University College Dublin,
and those lectures formed the basis of the book, completed in 1950 and
published the following year, his first in English. The first edition
dealt only with categorical syllogistic. For the second edition,
completed in 1955, less than a year before his death, Lukasiewicz
added three chapters on the modal syllogistic, employing the modal
logic L that he had developed in the meantime. The second edition
was proof-read and indexed by Lejewski, and appeared in 1957.



Lukasiewicz's understanding of Aristotle's
syllogistic is based on two specific interpretative principles and a
general attitude. The first principle is that Aristotle's
syllogisms are not, as had traditionally been supposed, inference
schemata, of the form '*p,* *q*, therefore
*r*', but conditional propositions of the form 'if
*p* and *q*, then *r*'*.* This leads
directly into the second principle, which is that there is behind the
syllogistic treatment of term logic a deeper logic, that of
propositions, and in particular a logic of opposition,
'and' and 'if', as well as (in modal
syllogistic) 'necessarily' and 'possibly'.
Lukasiewicz takes this propositional basis to be occasionally
invoked by Aristotle, for instance in the treatment of indirect
proofs, but for the most part left as tacit, and he therefore regards
it as legitimate to criticise Aristotle (unlike the Stoics) for not
explicitly formulating the underlying propositional logic.
Lukasiewicz's trenchant and controversial views sparked a
controversy over how to interpret the syllogistic. While the
principles did win an early adherent in Patzig (1968), subsequent
criticisms by Corcoran (1972, 1974) and, independently, Smiley (1974)
established clearly that syllogisms are not propositions but
inferences, and that Aristotle had no need of a prior logic of
propositions. That view is now universal among scholars of
Aristotle's logic. In retrospect, it appears that
Lukasiewicz was keen to wish onto Aristotle his own (Fregean)
view of logic as a system of theorems based on a propositional
logic.



The general attitude, present throughout Lukasiewicz's
treatment, is that Aristotle's work is of sufficient precision
and stature to warrant and withstand exposition using the most
rigorous modern logical methods and concepts. In other words, the
development of modern logic, while it may highlight lacunae and
deficits of Aristotle's logic, in fact brings out its merits,
innovations and genius more clearly than previous traditional or
philological studies. Lukasiewicz's attitude has prevailed
and is now pervasive among those studying Aristotle's logic,
whether or not they agree with his specific interpretative
principles.



After an exposition of the basics of Aristotle's treatment of
syllogistic, in which he criticises earlier commentators and notes
that Aristotle originated the method of rejected forms in order to
show not just which are the valid syllogisms but also to prove the
invalid forms to be such, Lukasiewicz presents his formalization
of categorical syllogistic, based on the following logical
expressions




|  |  |
| --- | --- |
| Expression | Meaning |
| \(Aab\) | All \(a\) is \(b\) (or \(b\) belongs to all \(a\)) |
| \(Eab\) | No \(a\) is \(b\) (or \(b\) belongs to no \(a\)) |
| \(Iab\) | Some \(a\) is \(b\) (or \(b\) belongs to some \(a\)) |
| \(Oab\) | Some \(a\) is not \(b\) (or \(b\) does not belong to some
\(a\)) |



Taking \(A\) and \(I\) as primitive and defining \(E = NI\) and \(O =
NA\), the axioms, added to propositional calculus, are




|  |  |
| --- | --- |
| \(\vdash Aaa\) |  |
| \(\vdash Iaa\) |  |
| \(\vdash CKAbcAabAac\) | (Barbara in the first figure) |
| \(\vdash CKAbcIbaIac\) | (Datisi in the second figure) |



together with modus ponens and a substitution rule for term variables.
This in fact was the system that Lukasiewicz had put forward in
his 1929 textbook. As the second axiom indicates, Lukasiewicz is
here following Aristotle in assuming that all terms denote. Rejected
forms can be added: Lukasiewicz gives from the second figure



\[ \begin{align}
& \dashv CKAcbAabIac & \text{and}\\
 & \dashv CKEcbEabIAc & 
\end{align} \]


which together with detachment and substitution for rejection deliver
all Aristotle's 232 rejected moods. Lukasiewicz's
verdict on Aristotle's categorical syllogistic is, that despite
its narrowness, it is "a system the exactness of which surpasses
even the exactness of a mathematical theory, and this is its
everlasting merit." (*AS*, 131)



The modal syllogistic on the other hand is little studied, according
to Lukasiewicz, both because it falls well below the standards of
perfection of the categorical, and for lack of a "universally
acceptable system of modal logic", which Lukasiewicz takes
himself, with L, now to have provided. Lukasiewicz's
own treatment falls short of being definitive, though it provides
material for later studies, and we shall not pursue it here.
Interestingly, in Aristotle's attempts in Book I, Chapter 15 of
the *Prior Analytics*, to establish the theses

 
\[ \begin{align}
& CCpqCLpLq \\
 & CCpqCMpMq 
\end{align} \]


Lukasiewicz sees an Aristotelian endorsement for the idea of a
principle of extensionality for modal operators as well as for
categorical ones.


## 7. Philosophical Positions



In his early philosophy, the most significant and influential position
adopted by Lukasiewicz is his anti-psychologism in logic. This
was influenced by Frege, Husserl and Russell. It manifested itself
terminologically in Lukasiewicz's replacement of the
traditional term *sad* (judgement), used by Twardowski, by
the term *zdanie* (sentence). This change of perspective and
terminology was adopted by subsequent Polish logicians *en
masse*. After 1920, Lukasiewicz is very sparing in his
statements regarding philosophy and philosophical problems. His
abiding commitment to indeterminism we have noted. His main comments
and indeed ire are reserved for those who criticise the place of
mathematical logic (or logistic, as it was then known) in philosophy
and thought in general. He noted certain convergences in method and
style between the Lwow-Warsaw School and the Vienna
Circle, but criticised the latter for their conventionalism and
rejection of all metaphysics and for their attempt to turn substantive
problems into linguistic ones. Despite its abstractness, logic is no
more detached from reality than any other science, and it is
constrained to conform to aspects of the world. It was his conviction
that determinism was false that drove his rejection of bivalent logic.
While maintaining the metaphysical neutrality of logic, he admitted
later in the 1930s that whereas he had earlier been a nominalist, he
was now a platonist. The source of this conviction is stated at the
end of his 1937 polemic "In defence of logistic":



> 
> whenever I work even on the least significant logistic problem, for
> instance, when I search for the shortest axiom of the propositional
> calculus, I always have the impression that I am facing a powerful,
> most coherent and most resistant structure. I sense that structure as
> if it were a concrete, tangible object, made of the hardest metal, a
> hundred times stronger than steel and concrete. I cannot change
> anything in it; I do not create anything of my own will, but by
> strenuous work I discover in it ever new details and arrive at
> unshakable and eternal truths. (*SW*, 249)
> 



Rarely has the motivation for platonism been so eloquently stated.



In the philosophy of logic, one of Lukasiewicz's most
deep-seated convictions, one that he shared with the other logicians
of the Warsaw School, was that logic has to be extensional, that it is
the study of the calculi not of linguistic meanings or psychological
judgements but of the truth-values, whether just the classical two or
more besides. His view is that sentences denote truth-values, and that
logic is the science of such logical values, not of sentences (which
is grammar) or of judgements (which is psychology), or of contents
expressed by propositions, or of objects in general (ontology). He
does not justify this position, but simply accepts and assumes it. As
we saw, it has far-reaching consequences for his treatment of modal
logic, forcing it to be many-valued.



In addition to the general attitude to scientific philosophizing that
he derived from Twardowski, there is one identifiable source of some
other of Lukasiewicz's philosophical stances regarding
logic, or if not a source, at least a point of convergent convictions.
One is the rejection of a "supertruth" above ordinary
truth. This comes out especially clearly in the modal logic L.
The other is his liking for degrees of possibility intermediate
between truth (1) and falsity (0), by contrast with the
non-quantitative third case of possibility (or in L twin third
cases). An exactly similar distinction between two kinds of
possibility, "unincreasable", without degrees, and
"increasable", with infinite degrees, can be found in
Meinong's massive 1915 treatise *Uber Moglichkeit
und Wahrscheinlichkeit.* Like Lukasiewicz, Meinong does not
accord propositions a dignity of necessity higher than truth, and
despite having the most ample ontology known to philosophy,
Meinong's object theory lacks objects described as necessary: he
never mentions God, and ideal objects such as numbers are taken by him
to subsist, not to exist or subsist necessarily. It is perhaps not
incidental that on returning to Lwow after his visit to Graz,
Lukasiewicz spoke in 1910 about the law of excluded middle,
concluding that like the principle of contradiction it is not
fundamental, and has practical rather than logical significance. He
conjectured that it failed for general objects like the triangle in
general, which is neither equilateral nor not equilateral. Meinong
accepted such objects, which he called "incomplete", and
had in fact adopted the idea from Lukasiewicz's teacher
Twardowski. Lukasiewicz also regarded the application of the
principle to real objects as "connected with the universal
determinism of phenomena, not only present and past ones but also
future ones. Were someone to deny that all future phenomena are today
already predetermined in all respects, he would probably not be able
to accept the principle in question." The seeds of three-valued
logic were already germinating in 1910, after the visit to Graz.



Meinong employed the many values of increasable possibility to give an
account of probability. While Lukasiewicz's procedure in
his 1913 monograph was based on a different idea, he continued to be
pulled towards the idea that infinite-valued logic might be able to
shed light on probability. At the very latest by 1935, with the
publication of a short article on probability and many-valued logic by
Tarski, he knew that the most straightforward approach, that of
identifying probabilities with truth-values between 0 and 1, would not
work. The reason is that because of probabilistic dependence,
probability is not extensional: if \(p\) is the proposition that it
will rain in Dublin tomorrow and \(Np\) is its negation, the
probability of the contradictory conjunction \(KpNp\) is 0, but if
\(p\) has degree of truth \(\tfrac{1}{2}\), so does \(Np\), and so
\(\tval{KpNp} = \tfrac{1}{2}\) in both L\(\_3\)
and L\(\_{\aleph\_0}\).
Despite this, as late as 1955
Lukasiewicz could still muse,



> 
> I have always thought that only two modal systems are of possible
> philosophic and scientific importance: the simplest modal system, in
> which possibility is regarded as having no degrees at all, that is our
> four-valued model system, and the 0-valued system
> in which there exist infinitely many degrees of possibility. It would
> be interesting to investigate this problem further, as we may find
> here a link between modal logic and the theory of probability.
> (*AS*, 180)
> 


## 8. Legacy



Lukasiewicz once declared somewhat immodestly that the discovery
of many-valued logics was comparable to that of non-Euclidean
geometries (*SW* 176). Whatever their significance,
Lukasiewicz's hopes for such logics have not been realized
in the way he anticipated. The semantics and pure mathematics of
multivalued logics have flourished, leading to the development of
MV-algebras in use for the algebraic semantics of
Lukasiewicz's logics. Infinite-valued or fuzzy logic has
its own mathematics, and prominent among its developers is the Czech
mathematical logician Petr Hajek, whose work is influenced by
that of Lukasiewicz. Fuzzy logic is found in many practical
applications, where it is used to deal with vagueness, inexactness, or
lack of knowledge, whether these are the same or different. But
Lukasiewicz's championing of multivalence in the analysis
of modality has been almost universally rejected, and the logic of
modality has inexorably followed other paths, mostly bivalent,
non-extensional ones. His final logic L has resisted consensual
interpretation, and is regarded as at best an oddity and at worst a
dead end.



The outstanding work that Lukasiewicz and his students
accomplished in the logic and metalogic of propositional calculus, the
Polish speciality of ever-shorter axioms and so on, now belongs to the
bygone heroic age of logistic. Its results have indeed only been
bettered occasionally by automated theorem-provers. On the other hand
the emphasis on logical semantics, Lukasiewicz's abundant
use of truth-values notwithstanding, has shifted interest away from
axiomatic virtuosity.



In the history of logic, Lukasiewicz's pioneering studies
opened up a new and more fruitful interaction between the past and the
present, and the rediscovery and new appreciation of figures from
logic's past "in the light of modern formal logic"
has continued to this day, though not all of Lukasiewicz's
own views on how to approach Aristotle or the Stoics have stood the
test of time. His work also helped to inspire those historians of
logic from the Catholic tradition in Krakow, most notably Jan
Salamucha and Jozef Bochenski, who applied modern methods
to the investigation of logical problems and arguments from the
history of philosophy.



During the heyday of the Warsaw School, 1920-1939,
Lukasiewicz played a key role in educating the next generation of
logical researchers and inspiring them with methods, results and
problems. Even ideas he tossed off as exercises have changed logic,
for example a 1929 suggestion to formalize the informal procedure of
proof from assumptions led to Stanislaw Jaskowski's
1934 system of natural deduction, in essentials the way logic is
mainly taught to students today. The war irrevocably interrupted their
work. Several of Lukasiewicz's best students were Jewish,
and were killed in Nazi death camps. In his exile from Poland after
1944, Lukasiewicz had scant opportunity to continue this
pedagogical work, holding a research position in a non-teaching
institution in a country with no logical tradition. His interactions
with contemporaries were much more sparse, and those chiefly through
correspondence. The one notable logician who interacted with
Lukasiewicz at this time and whose work intersects with his in
both interests (time, modality, many-valuedness) and attitudes (the
importance of logic for philosophy) is Arthur Prior, who was the only
major logician to adopt Polish notation, and who also expended more
effort than anyone in the attempt to find a plausible interpretation
for the system L. It is also fair to say that of the major
figures among the Warsaw logicians, Lukasiewicz has received the
least attention from commentators and historians. There are relatively
fewer monographs and papers on Lukasiewicz than on other major
figures of the Lwow-Warsaw School.



Despite such disappointments, Lukasiewicz's achievements
and inventions ensure him a permanent and honorable place in the
history of mathematical and philosophical logic. Lukasiewicz was
justly proud of the prominence achieved by Polish logicians between
the wars, and fully deserves his commemoration by one of Adam
Myjak's four statues of prominent Lwow-Warsaw
School members at the entrance to Warsaw University Library.