## 1. The Origin and Development of the Lvov-Warsaw School



Kazimierz Twardowski (1866-1938) began his post as professor of
philosophy at Lvov University in 1895. He came to Lvov from Vienna,
where he had studied philosophy under Franz Brentano and Robert
Zimmermann. Twardowski belonged to the last group of Brentano's
students. His *Habilitationschrift* (1894) concerned the
concepts of the content and the object of presentations; it 
clarified and sharpened this important distinction. This work strongly
influenced Meinong and Husserl.



Twardowski appeared in Lvov with the ambitious plan of creating a
scientific philosophy (in Brentano's spirit) in Poland (at that
time, Poland was partitioned between Austro-Hungary, Germany and
Russia; Lvov belonged to the Austro-Hungarian Empire.) In fact, he
subordinated all his activities to achieving this task and
considerably limited his own scientific work. Twardowski was an
extraordinary and charismatic teacher. He very soon attracted many
young people to philosophy. After ten years of teaching he sometimes
had about 200 candidates for seminars and 2000 attendants of his
lectures. He propagated a clear style of writing and speaking about
philosophical matters, insisted upon justification of philosophical
theses and sharply distinguished philosophy as a science from
world-views. Following Brentano, he favoured problems on the
borderline of descriptive psychology, grammar and logic (he
supplemented his object/content distinction by that of
actions/products). A photo of Twardowski' seminar participants
taken during the 1925-1926 academic year is available (see the
 supplement),
 with most of the participants identified.



Although Twardowski was not a logician and did not consider himself as
such, his program formed a friendly environment for logic in all its
subdomains: formal logic, semantics and methodology of science. Jan
Lukasiewicz (1878-1956) was the first of Twardowski's
students to be interested in logic. He began his lectures in logic in
Lvov in 1906. Kazimierz Ajdukiewicz (1890-1963), Tadeusz
Czezowski (1889-1981), Tadeusz Kotarbinski
(1886-1981) and Zygmunt Zawirski (1882-1948) studied
mainly under Twardowski, but they also attended courses conducted by
Lukasiewicz. Stanislaw Lesniewski (1886-1939)
joined this circle in 1910. Warsaw appeared on the stage exactly in
1915, when the University of Warsaw was reopened. The academic staff
was mainly imported from Lvov; Lukasiewicz was appointed
professor of philosophy.



Poland recovered its independence in 1918 and Polish scholars began to
build national academic life. The program for the development of
mathematics elaborated by a mathematician Zygmunt Janiszewski (the
Janiszewski program) had a great importance for the subsequent
development of the LWS. According the Janiszewski program, Polish
mathematicians should concentrate on set theory, topology and their
applications to other branches of mathematics. In particular, the
Janiszewski program attached great importance to mathematical logic
and the foundations of mathematics. Two philosophers, namely
Lesniewski and Lukasiewicz, became professors of the
University of Warsaw at the Faculty of Mathematical and Natural
Sciences. Both began intensive teaching of mathematical logic, mostly
among mathematicians but also among philosophers. Thus, logic in the
LWS had two parents: mathematics and philosophy.



Alfred Tarski (1901-1983) opened the list of young
mathematicians and philosophers attracted by logic in Warsaw. The
logical community in this city included (in alphabetical order and
covering the whole period 1918-1939: Stanislaw
Jaskowski (1906-1965), Adolf Lindenbaum (1904-1941?),
Andrzej Mostowski (1913-1975), Moses Presburger (1904?-1943),
Jerzy Slupecki (1904-1987), Boleslaw Sobocinski
(1904-1980; a philosopher by training)) and Mordechaj Wajsberg
(1902-1942?). The names of three other logicians who graduated
shortly before 1939 or studied during War World II and began their
academic work after 1945 should be added, namely Jan Kalicki
(1922-1953; a mathematician), Czeslaw Lejewski
(1913-2001; a classicist and philosopher) and Henryk Hiz
(1917; a philosopher).



The development of logic in Warsaw had two subperiods in
1918-1939, namely 1918-1929 and 1929-1939. The first
decade consisted in intensive teaching and scientific work at the
seminars of Lesniewski and Lukasiewicz. Not many results
were published at that time. The explosion of publications took place
in 1929 and later. There are several factors which caused the
development of mathematical logic in Poland. The Warsaw school of
logic appears to be model case, but the power of this circle
influenced other places where the general environment was not so
favourable to logic. The fruitful co-operation of mathematicians and
philosophers in Warszawa had the utmost significance. The founders of
the Polish mathematical school made a brave experiment consisting in
inviting two philosophers with a modest mathematical background as
professors at the Faculty of Mathematics and Natural Sciences; this
did not happen in any other country. The gifts of Lesniewski and
Lukasiewicz as teachers and the abilities of the latter as an
organizer attracted young mathematicians. In Poland mathematical logic
was considered to be an autonomous science, not a part of mathematics
or philosophy. From the present-day point of view this might seem as
an exaggeration, but this ideology contributed essentially to the
strength of Polish logic. Their representatives were fairly conscious
of the fact that the propagation and defence of the autonomy of this
field had to be confirmed by important scientific results and
international recognition. Moreover, this view about logic motivated
various purely theoretical investigations on formal systems. On the
other hand, Polish logicians strongly insisted that logic should not
be restricted only to mathematics and required the co-operation of
representatives of all field in which logic might be used. Still
another factor played an important role, namely the conviction about
the social significance of logic as a weapon against all kinds of
irrationalism. Tarski once said "Religion [you can also say
'ideology' -- JW] divides people, logic brings them
together." According to Lukasiewicz, "Logic is
morality of thought and speech". Thus, Polish logicians doing
logic and teaching it were convinced that they were performing an
important social service.



Kotarbinski was appointed professor of philosophy in Warsaw in
1919. His teaching activity resulted in a group of scholars working
mainly in the philosophy of science, including Janina Hosiasson (later
Mrs. Lindenbaum; 1899-1942), Edward Poznanski
(1901-1976), Dina Sztejnbarg (later Mrs. Kotarbinski)
(1901-1997) and Aleksander Wundheiler (1902-1957).



Twardowski and Ajdukiewicz (appointed professor in 1928) remained in
Lvov. They trained a group that included Izydora Dambska
(1904-1983), Maria Kokoszynska (1905-1981), Henryk
Mehlberg (1904-1978) and Zygmunt Schmierer (?-1943).
Although Twardowski's students also taught at other Polish
universities (Czezowski in Vilna, Zawirski in Poznan and
Krakow), Lvov and Warsawa were the main centres of the LWS. The
school was also joined by a group of catholic philosophers, including
Father Innocenty (Jozef) M. Bochenski (1902-1995)
and Father Jan Salamucha (1904-1944).



World War II had disastrous consequences for the LWS. Twardowski and
Lesniewski died before September 1, 1939. Of the people mentioned
above who lost their lives (mostly Jews murdered by the Nazis): the
Lindenbaums, Presburger, Salamucha, Schmierer and Wajsberg. Zawirski
died in 1947. Many emigrated from Poland during World War II or
shortly after it: Lukasiewicz (Dublin), Tarski (Berkeley),
Hiz (Philadelphia), Kalicki (Berkeley), Lejewski (Manchester),
Mehlberg (Toronto, Chicago), Sobocinski (Notre Dame) and
Wundheiler (New York); Bochenski (Fribourge) and Poznanski
(Jerusalem, before 1939).



The situation in Poland in 1945-1948 was similar as before 1939.
The Marxist ideological offensive against bourgeois philosophy started
in 1949. The policy became more liberal after 1956. Although many
scholars of the LWS actively taught and worked in the new political
reality, it would be difficult to say that the school continued its
former manner of existence. The tradition of the LWS was rather
preserved in individual hands, but not as an organized enterprise.



Note: the present essay focuses on the logical wing of the LWS. In
1939, the entire school comprised about 80 scholars actively working
in all branches of philosophy as well as in other academic fields,
like psychology, sociology, theoretical linguistics, history of art
and literary studies.


## 2. Metaphilosophy



Most philosophers of the LWS understood philosophy as a collection of
disciplines, including logic, ethics, aesthetics, metaphysics, and
epistemology. Philosophy is a science, like any other. All members of
the LWS inherited from Twardowski his main metaphilosophical claims
concerning clarity, justification and the separation of philosophy
from world-views. It also meant a radical rejection of all kinds of
irrationalism. A view, called anti-irrationalism by Ajdukiewicz,
demanded that every rationally accepted proposition be
intersubjectively communicable and testable. Although there was no
*a priori* list of meaningful questions subjected to
philosophical work, one should be sceptical about so-called great
metaphysical problems and their scientific status. Philosophical
activity must begin with a very careful linguistic analysis of
investigated problems and their meaning.



Twardowski himself favoured descriptive psychology as basic, but many
of his students found logic to be the most important source of
methodological criteria for philosophy. Perhaps Lukasiewicz was
the most radical in this respect. According to him, a reform of
philosophy was needed in order to avoid errors of the past. Philosophy
should proceed as logic does, axiomatically starting from clear
concepts and evident principles. Other philosophers of the LWS were
more modest and did not demand that philosophy should be axiomatized.
However, logical analysis of philosophical discourse became the
standard method of analysis. Yet the task of philosophy is not limited
to the analysis of language. Thus, philosophy, according to
methodological claims of the LWS, was analytic but not purely
linguistic. Philosophy is concerned with the world but it mainly
(though not 'only') performs its task also by the analysis
of language, which is used in speaking about reality. This view about
philosophy is to be contrasted with that of the Vienna Circle. In
particular, the LWS was not interested in a general metaphilosophical
scheme that sharply divided philosophy into good and bad, but rather
with an analysis of concrete problems. Thus, the LWS was united more
by a common methodological attitude and very general claims concerning
rationality than by a commonly accepted philosophical theory.



However, some general views were shared by most ('most' is
very important here) members of the LWS. These include:
anti-scepticism, anti-naturalism in humanities and axiology, realism
in epistemology and philosophy of science, absolutism in epistemology
and axiology, and empiricism. These views were characteristic for
Brentano and became implemented in Polish philosophy by
Twardowski.


## 3. Logic


### 3.1 Polish notation, the requirements for logical systems and metalogical concepts



Lukasiewicz invented a parenthesis-free logical notation. The
idea consisted in writing logical constants before their arguments.
Lukasiewicz replaced the usual signs for logical operations by
letters: *N* (negation), *K* (conjunction), *A*
(disjunction), *C* (implication) and *E* (equivalence).
Any well-formed formula (the present explanations are restricted to
propositional calculus) must begin with a capital letter
(propositional variables are symbolized by lower-case Latin letters)
which is the main functor of the entire formula. The main connective
has as its arguments variables or formulas consisting of variables and
constants. The following are examples: *Cpp* for (*p*
- *p*), *CCppNq* for ((*p* -
*p*) - !*q*). The structure of a formula (and
hence also its meaning) in Polish notation is uniquely determined by
the position of the letters. The parenthesis-free notation is
unambiguous in the sense that any finite sequence of symbols for
connectives and variables is interpretable in a unique way. This
implies that any wff coded in Polish notation has only one translation
into the standard symbolism. The main advantage of Polish notation is
its economy, because it avoids special punctuation devices such as
brackets or dots. When Lukasiewicz met Turing in 1949, the latter
remarked that Polish notation was much better for computers, because
formulas with function-symbols in front could be better elaborated by
mechanical devices.



The parenthesis-free symbolism was closely associated with some ideas
of Polish logicians concerning the good properties of formal systems.
Of course, any correct logical system should be consistent and, if
possible, syntactically and semantically complete. It should also be
based on independent sets of primitive terms and axioms. The Warsaw
School of Logic strongly emphasized the last property, often
considered secondary. Thus, the dependence of primitive terms or
axioms was regarded as an essential defect. Moreover, some additional
structural properties of logical systems were recommended: (a) a
system with fewer primitive concepts is better; (b) a system with
fewer axioms is better; (c) if we define the length of an axiom system
as the number of symbols occurring in all of its axioms, the shortest
axiom system is the best; (d) a system with fewer different symbols is
better; (e) if we define an organic theorem as one which has no other
theorem inside it (for example, the formula *CpCqq* is not an
organic theorem), organic axioms are better than non-organic ones.
Thus, the ideal axiom system consists of a sole organic axiom of the
shortest possible length, provided that it is consistent. Requirements
(a)-(f) apply particularly well to propositional calculus. They became
the guiding principles of many logical investigations in the Warsaw
School of Logic. Logicians of this school also made precise many
important metalogical concepts, including those of the logical matrix,
consequence operation, deductive system and model.


### 3.2 Investigations of classical propositional calculus



Lukasiewicz formulated several axiomatic bases for the
functionally complete propositional calculus, that is PC in which all
16 binary connectives can be defined. The most popular is the
*N*-*C* system, having as axioms the formulas:
*CCpqCCqrCpr*, *CCNppp*, *CpCNpq*, and the usual
rules of inference (substitution, detachment). This system is
consistent, independent, Post-complete (= semantically complete):
Lukasiewicz and his collaborators invented new methods for
proving these properties. According to the criteria mentioned in the
preceding section, one should look for the simplest axiom bases.


### 3.3 Many-valued, modal and intuitionistic logic



The discovery of many-valued logic is commonly considered to be one of
the major achievements of Lukasiewicz. He did it in 1918, a
little earlier than Post. However, although Post's remarks were
parenthetical and extremely condensed, Lukasiewicz explained his
intuitions and motivations carefully and at length. He was guided by
considerations about future contingents and the concept of
possibility.



Lukasiewicz observed that no functor of classical propositional
calculus could be read as "it is possible that" and
provided that the formula *Mp* (it is possible that *p*)
is extensional (i.e., that its value depends solely on the value of
*p*). The difficulty can be solved if we admit a third value.
Sentences about future contingent states of affairs are natural
candidates for having the third value (1/2). For example, the
sentence "I will visit Warszawa next year", is neither
true nor false, it is merely possible and has the value 1/2. Its
negation has the same value. This idea led to three-valued logic. The
usual equalities for *N*, *A*, *K*and *C*
are supplemented by (I list only some cases) *p* = 1/2 =
*Np*, *K* 1/21/2 = 1/2,
*A*1/21/2 = 1/2. Easy calculations shows that
*ApNp* and *NKpNp* have the value 1/2 for
*p* = 1/2. This means that the laws of contradiction and
excluded middle do not hold in three-valued logic. Later,
Lukasiewicz generalized it to logics with an arbitrary finite
number of values and finally to an countably infinite number of
values. The sense of implication is given by the equations:



> 
> *Cpq* = 1, for *p*[?]*q*
>   
> 
> *Cpq* = 1[?](*p + q*), for *p*>*q*,
> 



and the sense of negation by the equation:



> 
> *Np* = 1[?]*p*, where 0[?]*p*[?]*1*.
> 



If we have only two values, these equations determine the usual
truth-tables for *C* and *N*.



Three problems arose after discovering many-valued logic. The first
concerned its axiomatization and metalogical properties, the second
its philosophical foundations and intuitive interpretation, and the
third its applications. Due to the work of Lukasiewicz himself,
Wajsberg and Slupecki, the first group of questions was largely
solved. Wajsberg showed that the formulas: *CpCqp*,
*CCpqCCqrCpr*, *CCNpNqCqp*, *CCCpNppp* axiomatize
L3 (three-valued propositional calculus). The same
author proved that a finite Ln is axiomatizable if it
includes the theorems: *CCpqCCqrCpr*, *CCCqrCCpqCpr*,
*CCqqCpp*, *CCpqCNqNp*, *CNqCCpqNq*. If
*n* = 0, Ln can be
axiomatized by (Lukasiewicz's conjecture, proved by
Wajsberg): *CpCqp*, *CCpqCCqrCpr*,
*CCCpqqqCCqpp*, *CCCpqCqpCqp*, *CCNpNqCqp*.
However, all of the above axiom-sets are functionally incomplete. The
problem was solved by Slupecki for L3. He
introduced the new functor *T* defined by *T*1 =
*T*1/2 = *T*0 = 1/2 and added the formulas
*CTpNTp*, *CNTpTp* to Wajsberg's axioms. All
Lukasiewicz's many-valued logics are consistent.
Slupecki proved that L3 is Post-complete. Every
Ln (*n*>2) is contained in two-valued logic,
although the converse does not hold; for example, the formulas
*CCNpNp*, *CCNppp,* *CCpqCCpNqNp*,
*CCpKNqNp*, *CcpEqNqNq* are theorems in the two-valued
system only. If *n* = 0, Ln
is contained in every finite Ln.



At first, Lukasiewicz called his three-valued logic
"Non-Aristotelian", but later he preferred the
qualification "Non-Chrysippean". According to
Lukasiewicz, the Stagirite himself doubted the validity of the
principle of excluded middle in the domain of future contingents. On
the other hand, the Stoics believed that every proposition is true or
false, independently of its temporal reference. Thus, the Stoics
accepted the principle of bivalence in its unrestricted form. Now, the
foundation of two or many-valued logic lies not in this or that
logical theorem, but in metalogic; in particular, it is determined by
accepting or rejecting the principle of bivalence. Whoever, as
Chrisippus did, accepts the validity of the principle of bivalence,
opts for two-valued logic; whoever even partially rejects this
principle, as Aristotle did, thereby opens the door to many-valued
logic. Lukasiewicz took the side of Aristotle. However, this did
not close the problem of the interpretation of other logical values.
Lukasiewicz tried to go though indeterminism and causality. A
typical difficulty is the following. Take *p* as valued by
1/2. Its negation also has the value 1/2. The same holds for
*KpNp*, contrary to the firm intuition that any pair of
contradictory sentences is false. Difficulties with interpretation
changed Lukasiewicz's primary view concerning the relation
of many-valued logic to reality. At first, guided by a realistic
epistemology of logic, he maintained that one of the rival logics
could be proved to be the correct description of the physical world.
Later, he was rather inclined to look at logical systems as formalisms
having their own problems deserving research and as useful devices for
solving various questions but not as something leading to the only
"true" ontological scheme. Yet he believed that
many-valued logic would play a considerable role in the foundations of
mathematics.


#### 3.3.1 Lesniewski's systems



Lesniewski intended to a formulate the full logical system that
would serve as the basis for the whole of science, and in particular
for mathematics. This system consists of three parts (a) protothetic
(a generalized propositional calculus); (b) ontology (a logic of
terms); (c) mereology (a theory of parts and wholes). Protothetic is a
calculus in which quantifiers bind propositional variables and
variables referr to arbitrary functors constructible over the usual
functors: that is, functors of propositional variables, functors of
functors, etc. In general, if we start with the category of sentences
alone, protothetical quantifiers bind variables of all further
definable categories. The shortest axiom of protothetic (written in
the Russell-like symbolism) is the formula



> 
> [*pq*] :: *p* - . *q* -
> :.[*f*]:. *f* (*pf* (*p* [*u*].
> *u*)). - : [*r*]: *f* (*qr*). -
> . *q* - *p*
> 



(Sobocinski). Protothetic is an absolute propositional calculus
in the sense that the principle of bivalence is its theorem. In fact,
protothetic inspired Lukasiewicz's system with variable
functors, another absolute propositional logic.



If we add the functor e (read as "is") forming
sentences from two names, we obtain Lesniewski's ontology
(LO). The meaning of the constant e is perhaps the most
important matter for a proper understanding of LO. The epsilon
corresponds well to the sense of the copula "*est*"
in Latin sentences of the type "*Socrates est
homo*". The epsilon has no spatio-temporal connotations and
does not indicate the membership relation or identity. The rendering
of the epsilon by the English "is" may be misleading,
because the latter is modified by articles. The axiomatic
characterization of the meaning of the epsilon is given by



(O)
[*Aa*]:: (*A* e *a*) -
:.[S*B*]. (*B* e *a*):.
[*BC*]: (*B* e *A*). (*C*
e *A*). -
  

(*B* e *C*):. [*B*]: (*B*
e *A*) -. *B* e *a*.


Its simplified form (discovered by Sobocinski) is:



(O')
[*Aa*] *A* e*a*:
-.[S*B*]. (*A* e*B*).
(*B* e*a*).


The right sides of (O) and (O') are conjunctions. The intuitive
content of (O) is simple in spite of its formal complexity. It
establishes that the sentence "*A* is *a*"
is equivalent to the following conditions (a) *A* is not an
empty term; (b) there is only one *A*; (c) whatever is
*A*, is also *a*. Thus, "*A* is
*a*" is a singular sentence which is true if and only if
(a)-(c) hold. In particular, such a sentence is false if *A* is
a general or an empty term. On the other hand, (O) (or O') is
valid for all terms, even general or empty ones. Thus, LO is valid in
all domains, including the empty one and can be regarded as the first
system of free logic. In LO we can define two important concepts,
namely that of existence and that of being an object. This is done by
(I use non-symbolic forms): (1) for any *A*, *A* exists
= for some *x*, *x* is *A*; (2) for any
*A*, *A* is an object = for some *x*, *A*
is *x*. LO performs functions usually provided by predicate
logic. The meaning of the constant e is sufficiently general
to define identity and the inclusion of classes. Since these concepts
are definable in elementary ontology, it is stronger than first-order
logic.



Mereology, assumes protothetic and ontology as logically prior
theories, and has the term "part of" as its sole primitive
concept. Being a part is a non-reflexive and transitive relation.
There are no empty classes. Moreover, the class consisting of a single
element is identical with it. In general, mereology is a theory of
sets in the collective (mereological) sense, contrary to ordinary set
theory, which describes sets in the distributive sense. The main
difference between the two interpretations of the term
"set" consists in the fact that the membership relation is
transitive under the mereological reading, but non-transitive under
the distributive one. Lesniewski believed that his theory classes
would perform all the tasks of ordinary set theory without generating
paradoxes. In fact, he invented mereology when he tried to solved the
Russell paradox. Since there are no mereological classes which are not
their own elements, the question which led to the Russell paradox
simply makes no sense in Lesniewski's systems. On the other
hand, mereology is weaker than set theory.



Lesniewski's systems have some formal features, even very
peculiar ones in some respects. All are axiomatic. According to his
nominalistic preferences, they are concrete physical objects.
Expressions are always understood as sequences of concrete
inscriptions. There are as many expressions as have been written; no
expression exists merely potentially. This view is called constructive
nominalism. According to it, two intuitively equivalent systems, for
example, protothetic based on equivalence and protothetic based on
implication, are different systems. Every logical system, on
Lesniewski's view, is not finished at any time because
there is always the possibility of adding new elements to it. Hence,
the rules for constructing and developing formal systems are of
crucial importance for Lesniewski's logics. He understood
this very well and devoted much attention to explaining the details of
his formalization. Lesniewski formulated his procedural
directives purely syntactically and completely. Due to the role of
equivalence, he was able to treat definitions as theorems. In general,
Lesniewski's systems are commonly considered to be perfect
from the point of view of the requirements of correct formalization.
The Lesniewski project is a version of logicism.
Lesniewski's three systems form a grand logic and provide a
universal language to capture the whole of knowledge. It is certainly
not an orthodox system and lies on the margin of contemporary research
in logic. Yet it continues to attract many logicians and philosophers.
In spite of their marginality, Lesniewski's systems are
investigated in all parts of the world.



Lesniewski suggested the theory of syntactic categories, later
developed by Ajdukiewicz in the early 1930s. This theory takes the
categories of sentences and names (for Ajdukiewicz, following
Lesniewski, there is no syntactic difference between proper names
and common nouns) as fundamental and assigns the pointer *s* to
sentences and *n* to names. Now functors have fractions as
pointers. For example, "is" has *s/nn* as its
categorial index; it says that "is" is a two-placed
functor of two nominal arguments which forms a sentence. The
conjunction as a propositional connective forming sentences from two
other sentences has *s*/*ss* as its index. Consider now
the expression "*p* and *q*". Write the
categorial indexes of its parts. We thus obtain the sequence:
*s* *s*/*ss* *s.* Perform simplifications
by divisions similar to dividing algebraic fractions. The letter
*s* is the result. A simple algorithm says that an expression
is syntactically coherent if and only if *s* or *n* is
its index after performing all simplifications. Ajdukiewicz's
quasi-arithmetical notation was the first system of categorial
grammar.


### 3.4 Semantics and truth



Due to semantic paradoxes, Hilbert's formalistic metamathematics
and the syntacticism of the Vienna Circle, the concept of truth was
expelled from the domain of logic. It was Tarski who changed this
attitude. He was inspired by the Aristotelian tradition in philosophy,
as well as the non-constructive style of working on the foundations of
mathematics that was prevailed in Poland. In 1933, he published a book
on the concept of truth (in Polish), translated into German in 1936
and English in 1956.



Tarski's theory of truth (the semantic conception of truth) has
two aspects: philosophical and formal. Philosophically, it is a
version of Aristotle's idea that truth consists in saying that
what is, is and what is not, is not (it is related to the idea of
correspondence). However, the main problem was formal. Tarski had to
offer a construction free of semantic paradoxes, in particular the
Liar. He achieved this goal by postulating that the concept of truth
must be defined for a definite, well-constructed formalized language
**L**. However, the definition itself should be
formulated in the metalanguage **ML**. The definition is
to be formally correct, that is, it cannot lead to contradictions and
has to satisfy the usual conditions of correctness (non-circularity,
etc.). It should also be materially adequate. According to Tarski, the
basic intuition is captured by the T-scheme: *s* is true if and
only if *P*, where the letter *s* represents the name of
a sentence and *P* is a translation of this sentence into the
metalanguage **ML**. Now the condition of material
adequacy (the Convention *T*) says that a truth definition
**TD** is materially adequate if and only if all
equivalences (that is, for all sentences in **L**)
stemming from the *T*-scheme by appropriate replacements are
provable from the definition. The conditions are satisfied by the
following definition:



> 
> A sentence *A* of a language **L** is true if and
> only if it is satisfied by all infinite sequences of objects taken
> from the universe of discourse.
> 



A more refined version is model-theoretical:



> 
> A sentence *A* is true in a model **M** if and
> only if *A* is satisfied by all infinite sequences of objects
> taken from the carrier of **M**.
> 



This definition implies the metalogical principles of excluded middle
and contradiction, which both are equivalent to the principle of
bivalence.



Tarski's truth-definition is one of the most debated
contemporary philosophical and logical ideas. It strongly influenced
semantics, philosophy of language, philosophy of science and
epistemology. In particular, it became the first step toward
model-theory, a central branch of mathematical logic. Two applications
of this definition are worth mentioning. Firstly, Tarski succeeded in
formulating an exact definition of "logical consequence"
("following from" or "logical
entailment"):



> 
> A sentence *A* logically follows from the set
> **X** of sentences if and only if every model of
> **X** is a model of *A*.
> 



Secondly, Tarski proved the following limitative theorem:



> 
> If a formal system **S** captures Peano arithmetic, the
> truth-predicate (or: the set of **S**-truth) is not
> definable in it.
> 


### 3.5 The history of logic



Lukasiewicz revolutionized the history of logic. He proposed to
look at the history of logical ideas through the glasses of
mathematical logic. The reason was that he was convinced about the
continuity of formal logic from Aristotle to modern mathematical
logic, perhaps with a break from the 16th century to Boole
and Frege (of course, with the exception of Leibniz). Thus, old sound
logical theories should be considered as anticipations of ideas
advanced in the 19th and the 20th centuries.
Guided by this assumption, Lukasiewicz showed that the Stoics
invented propositional calculus, contrary to the prevailing view that
Stoic logic was a part of Aristotle's logic. In particular,
Lukasiewicz demonstrated that the Stoic logic of propositions was
a system of rules, not theorems. Another of Lukasiewicz's
historical discoveries consisted in the rehabilitation of medieval
logic, commonly neglected as fruitless scholasticism. He was joined in
these investigations by Bochenski and Salamucha.



Historical work inspired logicians of the LWS toward modern
interpretations of traditional logical doctrines. The most famous is
Lukasiewicz's formalization of the Aristotelian logic of
categorical sentences (the syllogistic plus conversion and other rules
of so-called direct inference). Lukasiewicz interpreted this
logic as a specific formal theory, not as a fragment of predicate
logic, as it was usually done (for example by Frege or Russell). Yet
the logic of categorical sentences assumes propositional logic as
prior. The logic of assertoric sentences (Lukasiewicz also
considered its modal extension) has the following form. Let the
formulas (lower-case letters are term variables) *Uab*,
*Iab*, *Yab*, *Oab* stand for the sentences
"every *a* is *b*", "some *a*
are *b*", "no *a* is *b*" and
"some *a* are not *b*". We can define
*Yab* as *NIab* and *Oab* as *NUab*. The
axioms are as follows: (a) *Uaa*; (b) *Iaa*; (c)
*CKUmbUamUab* (the Barbara modus); (d) *CKUmbImaIab*
(the Datisi mode); the rules are: all the rules of propositional
calculus, substitution for term variables, the definitional
replacement according to the definitions of *Yab* and
*Oab*.


### 3.6 The philosophy of logic and mathematics



There was no official philosophy of logic and mathematics in the LWS.
Most Polish logicians treated logical studies as independent of
philosophical commitments. Only Lesniewski had explicit
philosophical views which influenced the form of his systems. This
does not mean that concrete works were not influenced by philosophical
ideas. Lukasiewicz's many-valued logic and Tarski's
theory of truth are perhaps model cases. The former had the problem of
determinism as its background and the second was strongly inspired by
the Aristotelian tradition in thinking about truth. It was also the
case that Polish logicians had inclinations to empiricism as a general
epistemological attitude and this philosophy often resulted in
sympathies to nominalism (Tarski), constructivism (Mostowski) and
scepticism concerning the sharp distinction between logical and
extralogical truth (Tarski). However, the technical side of logical
problems decided about investigations and sometimes forced changes in
philosophical standpoints. The example of Lukasiewicz is
instructive once again. Although he at first thought of logic as a
true or false description of reality, he later adopted a more
conventionalist and instrumentalistic standpoint. This attitude
allowed him to accommodate various ideas coming from rival
foundational directions, that is, logicism, formalism and
intuitionism. In fact, Lesniewski and Tarski contributed to the
theory of logical types and combined it with the theory of syntactic
categories; Tarski's version in his work on truth is
particularly important. Tarski also showed new perspectives for
logicism by defining logical concepts as invariants under one-to-one
transformations. He also contributed to general metamathematics (the
theory of consequence operation) and intuitionistic logic. However, a
very special feature of logical investigations performed in the LWS
consisted in the free admission of all fruitful mathematical methods,
including non-constructive ones. This was the main point of the set
theoretical approach to the foundations of mathematics which replaced
logicism.


### 3.7 Supplementary and concluding remarks



The above survey does not do justice to many of the logical studies
carried out in the LWS. Let me only mention some of them: particular
historical studies of Bochenski and Salamucha, several
interpretations of traditional logic (Ajdukiewicz, Czezowski),
partial propositional calculi (all Warsaw logicians), propositional
calculus with variable functors (Lukasiewicz), paraconsistency
(Jaskowski), L-modal systems (Lukasiewicz), rejection
rules, natural deduction (Jaskowski), intuitionistic logic
(Jaskowski, Tarski, Wajsberg), free logic (Jaskowski,
Mostowski), the elimination of quantifiers (Tarski, Presburger),
undecidability (Tarski, Mostowski), the foundations of geometry
(Tarski), the elementary theory of real numbers, the calculus of
systems (Tarski), the Kleene-Mostowski hierarchy, generalized
quantifiers (Mostowski), as well as several particular results: the
deduction theorem (Tarski), the upward Lowenheim-Skolem theorem
(Tarski), the separation theorem for intuitionistic logic (Wajsberg)
or the Lindenbaum maximalization lemma.


## 4. Philosophy of Science



The philosophy of science was a favourite field of the LWS. Since
science is the most rational human activity, it was important to
explain its rationality and unity. Since most philosophers of the LWS
rejected naturalism in the humanities and social sciences, the way
through the unity of language (as in the case of the Vienna Circle)
was excluded. The answer was simple: science qua science is rational
and is unified by its logical structure and by definite logical tools
used in scientific justifications. Thus, the analysis of the
inferential machinery of science is the most fundamental task of
philosophers science. Inductivism was a prevailing view about
justification in empirical science. Hosiasson formulated an axiomatic
system of inductive logic, anticipating Carnap's later work.
Other attempts to establish the foundations of inductive inference
were undertaken by Ajdukiewicz (via statistics, decision theory and
game theory (he mainly investigated the problem of the rationality of
modes of fallible inference), Czezowski (via probability logic in
the sense of Reichenbach) and Zawirski (via a combination of
many-valued logic and probability theory).



Lukasiewicz worked on problems of the methodology of the
empirical sciences in 1902-1910. At first, he tried to develop
the inverse theory of induction (induction as inversed deduction)
proposed by Jevons and Sigwart. However, he abandoned this project
very soon and adopted a radical deductionist standpoint. For him,
induction plays no significant role in science. Deduction remains the
only credible mode of reasoning in all spheres of science. As applied
in empirical science, it leads to negative results; that is, it can
show that some hypotheses are false in the face of empirical data.
Lukasiewicz also offered a formal argument against induction
derived from probability theory. Assume that *H* is a universal
hypothesis. Its a priori probability is equal (or close) to zero and
no further empirical data can increase it. These ideas contain the
main points of Popper's philosophy of empirical science.



Tarski's semantic ideas converted most members of the LWS to
scientific realism. Formerly, under the influence of conventionalism,
instrumentalism concerning scientific theories had adherents
(Ajdukiewicz, Lukasiewicz). A radical form of anti-realism was
developed by Poznanski and Wundheiler in the 1930s. They pointed
out that verification in empirical science is cyclic and principally
anti-fundamentalistic. In particular, it is not possible to identify
any data without a reference to theories. Hence, truth in science
cannot consist in a correspondence with facts.



Of many investigations concerning special problems, let me only
mention Mehlberg's version of the causal theory of time and some
works on the causality problem in quantum mechanics. He admitted a
universal time as a synthesis of physical (intersubjective) and
psychical (subjective) time. The causal theory does not lead to the
anisotropy of time. It can be that universal time is symmetric, but
locally asymmetry is possible. Mehlberg and Zawirski defended a
moderate causalism in quantum mechanics. In particular, Zawirski
argued that the unpredictability of the future (Heisenberg) does not
entail that the principle of causality fails.


## 5. Ontology and Epistemology Inspired by Logic


### 5.1 Reism



Kotarbinski developed a general doctrine, called reism. It has
two aspects, ontological and semantic. Hence, we can speak about
semantic reism and ontological reism, although this distinction was
clarified by Kotarbinski later. In general, reism goes against
the acceptance of the existence of general (abstract) objects, that
is, facts, properties, states of affairs, relations, etc. The main
ontological thesis of reism is as follows (it is subdivided into two
subtheses): (R1) any object is a material, spatio-temporal, concrete
thing; (R2) no object is a state of affairs, property of relation
(according to Kotarbinski, these three categories exhaust the
domain of alleged abstract objects). Now (R1), that is, the positive
thesis of reism has a rich content. Firstly, it marks a formal feature
of existing objects, namely their concrete character.



Secondly, it characterizes things as material and spatio-temporal
entities, that is, as physical objects. Leibniz conceived monads as
spiritual concreta. For the later Brentano, every object is concrete,
but there are souls and bodies. Thus, Leibniz's reism was
monistic and spiritualistic, Brentano's was dualistic, and
Kotarbinski's monistic and materialistic. Although the
terminology varies (one can equivalently speak about reism, concretism
or nominalism), two claims of any theory going against general
(abstract) objects should be very sharply distinguished. The first is
formal-ontological and points out the formal feature of existents,
namely that they are individuals; but the second is
material-ontological or metaphysical and focuses on their nature as
physical or psychical entities.



Semantic reism is parallel to the ontological aspect of this doctrine.
The key idea consists in the distinction of genuine names and apparent
names (onomatoids). A name is genuine if and only if refers to things,
that is, to concrete physical things. By contrast, onomatoids are
words that allegedly refer to abstract entities,
"allegedly" because their referents do not exist. At first
glance, apparent names are similar to empty terms. However, this
resemblance is merely apparent, because empty nouns are genuine names
and can always be decomposed into non-empty genuine names (e.g.,
"round square"). This becomes evident when we try to
formulate the conditions of meaningfulness for sentences. In general,
a sentence is meaningful if and only if it consists (except for
logical constants) only of genuine names or is reducible to such
sentences. For example, the sentence "all cats are
animals" is reistically meaningful, but "properties are
abstract objects" is not. Furthermore, "a square triangle
is rectangular" is good, but "sets exist outside of time
and space" is not. The sentence "whiteness is a property
of snow" can be reduced to "snow is white". This
example shows how to translate some sentences with apparent names into
purely reistic statements.



Matters become clearer if one remembers that Lesniewski's
calculus of names is the underlying logic of reism. The copula
"is" in "snow is white" functions with its
meaning defined by the axiom of LO (see above). Thus, this sentence is
true if its subject refers to an individual object. The traditional
interpretation of common nouns and adjectives, consistent with LO, as
general terms referring to many objects, saves their reistic
character. Thus, one can say that the formal-ontological aspect of
reism is adequately displayed by LO. Of course, reism as a
metaphysical doctrine is an addition to LO.



Kotarbinski recommended reism as a sound view. In particular, it
defends philosophy and ordinary thinking before hypostases, that is,
accepting the existence of abstract objects on the basis of using
apparent names. Thus, reism defends us against *idola fori* in
Bacon's sense. Kotarbinski's reism is perhaps the
most radical materialistic nominalism in the history of philosophy.
Reism is exceptional to the main tendency in the LWS in that it
proposes a uniform language, proper everywhere, including the
humanities, sociology and psychology (Kotarbinski supplemented
reism by radical realism, that is, the view that there are no mental
contents). In this respect, reism resembles physicalism. The troubles
of reism are typical those which are in the case of any reductive
materialism and nominalism, and concern the interpretation of
mathematics, semantics, psychology, the humanities and social
sciences.


### 5.2 Radical conventionalism and semantic epistemology



Radical conventionalism is an epistemological theory developed by
Ajdukiewicz in the early 1930s. It is based on a conception of
language and meaning. The concept of meaning is taken as a primitive.
Now, the meaning of expressions in a language **L**
induces the rules for accepting its sentences. Ajdukiewicz lists three
kinds of meaning-rules (or sense-rules): (a) axiomatic (they demand
the unconditional acceptance of sentences, for example,
"*A* is *A*"; (b) deductive (they demand the
acceptance of a sentence relatively to the prior acceptance of other
sentences, for example, !*A* follows from *A* -
*B* and !*B*), (c) empirical (they demand the
acceptance of a sentence in a definite empirical situation, for
example "it is raining" when it rains).



The special significance of meaning-rules and their relation to the
meanings of expressions appears when special languages are taken into
account, namely closed and connected. A language **L** is
open if it can be extended to a new language **L'**
without changes in the meanings of other expressions; otherwise
**L** is closed. A language is disconnected if there is a
non-empty subset **X** of **L** such that no
element of **X** is linked by the meaning-rules to other
elements of **L**; otherwise, **L** is
connected. It follows from the above definitions that if
**L** is closed and connected, it cannot be enriched
without changing the meanings of the original expressions.



According to Ajdukiewicz, natural languages are open and disconnected.
By contrast, scientific languages are closed and disconnected. Let
**L** be closed and connected. The set of meanings of
**L** is its conceptual apparatus. If **A**
and **A'** are two conceptual apparatuses, they are
either identical or mutually non-translatable. Since the acceptance
and rejection of sentences is always related to a language
**L**, empirical data do no force us to accept or reject
any sentences, because there always remains the possibility of
changing a given conceptual apparatus. This is a considerable
radicalization of the conventionalism of Poincare. The
difference is the following. For Poincare, since theoretical
principles are conventions, we are free to modify them, but
experiential reports are perfectly stable. Ajdukiewicz extended
conventionalism to all sentences, because any sentence, no matter
whether experiential or theoretical, depends on a conceptual
apparatus. It is why Ajdukiewicz called this conventionalism
radical.



In the middle of the 1930s Ajdukiewicz changed his view. He came to
the conclusion that closed and connected languages are fictions. He
was influenced by the semantic ideas of Tarski. Tarski also argued
that, contrary to Ajdukiewicz's hopes, the invariance of
meaning-rules over permutations of expressions influences their
meaning-relations. Gradually, Ajdukiewicz developed the program of
semantic epistemology, mainly directed toward the defence of realism
against various forms of idealism. In particular, he criticized
Rickert's transcendental idealism and Berkeley's
subjectivism. For Rickert, reality is only a correlate of the
Transcendental Subject. Now the Transcendental Subject can be
identified with the set **T** of true propositions
obtainable on the basis of axiomatic and deductive rules. However, due
to incompleteness phenomena, **T** cannot be generated in
this way. For Ajdukiewicz, that was a justification that
transcendental idealism fails. Ajdukiewicz compared the language used
by Berkeley to the language of syntax, because the former reduces
relations of the mind to its objects to relations between thoughts. On
the other hand, the ordinary way of speaking about objects employs
semantic relations. Berkeley's claim *esse* =
*percipi* is similar to an attempt to define semantics in a
purely syntactic language. However, due to Tarski's results
about the relation between syntax and semantics, this is impossible.
Finally, Ajdukiewicz argued that any idealistic language is
understandable only if it is associated with a realistic language.
Hence, any attempt to consider idealistic language as self-sufficient
cannot be successful.


## 6. The Significance of the Lvov-Warsaw School



The LWS acted in a country which never belonged to the philosophical
superpowers. This circumstance is important for any assessment of the
significance of the LWS. One can measure it on a national or an
international scale. The importance of the LWS for Polish
philosophical culture was enormous. Twardowski fully realized his
task. He introduced scientific philosophy in his sense into Poland and
created a powerful philosophical school. It did very much for the
subsequent development of philosophy in the country. In particular, it
popularized very high standards of doing philosophy. This was
important in the difficult times after 1945, when Marxism started an
ideological and political offensive against bourgeois philosophy. In
fact, due to the strong methodological tradition related to the LWS,
Polish philosophy did not lose its academic quality in
1945-1989.



As far as the matter concerns international importance, one thing is
clear. The logical achievements of the LWS became the most famous.
Doubtless, the Warsaw school of logic contributed very much to the
development of logic in the 20th century. Other
contributions are known but rather marginally. This is partially due
to the fact that most philosophical writings of the LWS appeared in
Polish. However, this factor does not explain everything. Many
writings of the LWS were originally published in English, French or
German. However, their influence was very moderate, considerably
lesser than that of similar writings of philosophers from the leading
countries. This is a pity, because radical conventionalism, reism or
semantic epistemology are the real philosophical pearls. But perhaps
this is the fate of results achieved in cultural provinces.