Philosophy and Ideology. Z. A. Jordan 1963

The first distinguishing characteristic of the Warsaw School to appear, was the spectacular rise to prominence of formal and mathematical logic. This was due to Leśniewski and Łukasiewicz who jointly share the honour of being the founders of the Warsaw school. Its achievements have by now become widely known, even outside the narrow circle of professional logicians. Some of their works have been translated or rewritten in one of the world-languages, others have been competently summarised and reviewed, particularly in the columns of Journal of Symbolic Logic. The most important results have been incorporated, after the Second World War, in all contemporary textbooks of logic. There is no need, therefore, to give here another survey of its achievements. For the purpose of this study it is enough to indicate the most outstanding of them and to emphasise those of their features which were philosophically important and became a point at issue in the post-war period.

From this viewpoint Leśniewski’s system still holds its high rank of importance and philosophical significance^{[24]}. In the inter-war period he was a power second to none and he influenced deeply both the logicians, including Łukasiewicz, and the philosophers, among the latter Kotarbifiski, who ever since has emphatically recognised his indebtedness to Leśniewski^{[25]}. Tarski, through whom, one may almost say, Leśniewski became known internationally, has also repeatedly testified to Leśniewski’s inspiration in his own research^{[26]}. Kotarbiński’s and Tarski’s tribute could be extended to others. In fact, hardly anyone who spent his formative or mature years at that time in Warsaw escaped being influenced, in sonic way or other, by this extraordinary mind, extreme in whatever views he held, and whose combined power of criticism and of invention caused both admiration and fear.

Leśniewski’s position was based rather on personal contacts than on what he published. For he published little during his Warsaw period and what he did was written in a highly condensed manner; some parts of it could be followed only by those fully acquainted with Leśniewski’s ideas and methods. He was a perfectionist for whom nothing, either in his own or other people’s work, was good enough. Leśniewski died prematurely and during the war his manuscripts were destroyed. The efforts to reconstruct Leśniewski’s system are being made both in Poland and abroad^{[27]}.

Leśniewski seemed to have influenced the philosophy of the Warsaw school by some of his philosophical views and by his practice. His practice was an almost ruthless, radical formalism, to which he adhered not for its own sake, but as an instrument of conveying meanings and intuitions otherwise doomed, as he felt, to ambiguities and distortions. Formalism was to him the only effective method of making unequivocal philosophical statements and of consistent reasoning, free from the pitfalls of contradictions. He was thus one of the decisive influences in accepting the view, hotly contested by traditionalistic philosophers, that there is in formalism a creative power that liberates the mind from many evils which beset philosophy in the past.

Leśniewski was able to arouse and to establish respect for formalism by his unusual technical inventiveness and by his deep philosophical insight which he tried to express in a formalised language. During his philosophical apprenticeship in Twardowski’s school he was converted to the absolute conception of truth, and became a staunch opponent of any kind of relativism which, following Twardowski, he thought to be prompted by confusion of thought or terminological ambiguities. Formal theories which he constructed were to provide an exact and adequate description of the structure of the world. This prompted the utmost care which he took in every aspect of the construction – semantical categories, axioms, formation and transformation rules, definitions – and the attention which he paid to the deduction of theorems or to the perfecting of matters of detail. Mere formalism of the Hilbertian type or Russell’s logicism were of no interest to him. They could be and were interpreted as a play with meaningless symbols or a game of chess^{[28]}. A formalised deductive theory consisted for him of clearly meaningful, intuitively valid propositions. The most important of them were those involving the word ‘is’ (ontology) and ‘part of’ (mereology), or involving the concept of class in its distributive and collective sense. Leśniewski’s ontology and mereology are deductive theories of objects and relations that hold among them, and correspond in their content to the traditional ontology. They are not metaphysical theories in the sense that they do not make any assumptions about the nature of those objects. For what we know, they might be or might not be material. The axiom of ontology remains true irrespective of whether names of concrete entities, universal or empty names are substituted for its variables (on this account ontology is a more general theory than Aristotle’s syllogistic). But the meaning which the axiom ascribes to the term ‘is’ (‘∈’) in its main sense makes the propositional function ‘x ∈ y’ a true expression provided that ‘x’ does not admit empty or universal names as its value, or, affirmatively expressed, provided that only names of individual objects can be substituted for ‘x’. Similarly in mereology the name of a part of an object clearly cannot be objectless. Consequently, ontology and mereology are particularly suited for what is known as nominalism or rather the programme of a nominalistic reconstruction of logic and science. Leśniewski is a precursor, perhaps still unsurpassed, of what later was undertaken by Chwistek, Goodman and Quine.

Leśniewski’s examinations of antinomies were of high philosophical importance and exercised a powerful influence in the development of philosophy in Poland. Little of them has been left in writing and they constituted part of the ‘oral tradition’ of the Warsaw school, based on Leśniewski’s lectures and on personal contacts^{[29]}. They inspired the distinction between an object-language and metalanguage, a theory and metatheory, and prompted the emergence of philosophical and theoretical semantics.

In the examinations of antinomies, Leśniewski’s starting point was the recognition of the basic difference between an antinomy and an ordinary contradiction. The latter results from an error, from the failure to formulate fully and precisely all our premisses and directives of transformation; it can be, therefore, removed by purely technical means. This cannot be done with the former. Leśniewski criticised the way in which antinomies were eliminated in Principia Mathematica, because the theory of types was a kind of a police-theory or of prophylactic, as Tarski put it, to guard the deductive sciences against the known and other possible antinomies. It cured the symptoms without touching the cause of the disease.

An antinomy differs from a contradiction in that the former does not result from an error or imprecision, but follows in a valid manner from the premisses and rules of inference, which we believe to be true. The elimination of antinomies, of any possible antinomy, requires, therefore, a re-examination of these premisses and rules of inference. Leśniewski’s system of logic and foundations of mathematics was such a re-examination (it has been shown convincingly that the logical antinomies cannot be reproduced in Leśniewski’s ontology)^{[30]}. It included the theory of semantical categories which, technically speaking, corresponds to the role played by Russell’s theory of types, but otherwise differs from it. The theory of semantical categories was not added to the system of logic, but the system itself grew out of the theory and owes to it many of its peculiar characteristics. Moreover, the theory of semantical categories is not concerned with logical objects such as individuals, classes, relations; it classifies and orders logical constituents of language, hence its name and immediate philosophical relevance. In concrete applications the signs and expressions belonging to different semantical categories are made apparent by being enclosed in different kinds of brackets, the use of which is as much restricted and fixed as that of other expressions. In Leśniewski’s words, the theory of semantical categories did not replace the hierarchy established by the simplified theory of types, which for him lacked intuitive justification, but ‘intuitively undermined’ such assumptions and consequence relations that together lead to contradictions. As Leśniewski said, he would feel bound to accept the theory of semantical categories, if he wished to speak sense, were there no antinomies whatsoever in the world^{[31]}.

After his Lwów period Leśniewski hardly ever said anything on philosophical subjects. Unlike Leśniewski, Łukasiewicz spoke often on these matters. Compared with Leśniewski, Łukasiewicz enjoyed the reputation of being a logician with pronounced philosophical interest. This opinion cannot be accepted without qualification.

łukasiewicz admired formalistic methods irrespective of what philosophical purpose and intentions they served. Formal logic was not to him a method or an instrument, but an autonomous discipline, to be studied for its own sake. The importance of this discipline was increased in his eyes by the fact that it carried philosophical implications and provided useful techniques for mathematics. But if that were not the case, formal logic would still possess an intrinsic value. Łukasiewicz was critical of Hilbert’s formalism, because it deprived formal logic of its independent status, subordinated it to mathematics and transformed it into some kind of mechanical device for getting results^{[32]}.

This criticism was prompted by the consideration that formal logic has achieved a degree of precision which mathematics cannot emulate and that by making of formal logic a maid-servant of mathematics the standards of precision achieved by formal logic are being debased. In fact, however, Łukasiewicz shared Hilbert’s approach to formalistic methods; he valued them because they did get results. He was fascinated by what can be done with symbols once the appropriate techniques are worked out and he did not spurn any of them, as Leśniewski did, since they were considered in some sense or other philosophically unsound. Łukasiewicz rejected the assumption of the Lwów school which required that basic logical concepts should be philosophically examined before they are accepted and made use of. In a discussion with Adam Żółtwski, representing the traditional trend in philosophy, he said plainly that this was an unfounded presumption of philosophy.

This attitude was probably responsible for the influence Łukasiewicz exercised on the mathematicians in Poland; he provided them with methods that they found useful in their own work. While Leśniewski, who became a mathematician and devoted his life to the task of securing a safe and consistent foundation for mathematics was more or less ignored by the mathematician, Łukasiewicz, in spite of not being a mathematician sensu stricto, did play a considerable role in the development of mathematical thought in Poland. He was particularly influential with the Warsaw mathematical school grouped round the editors of *Fundamenta Mathematicae*^{[33]}.

Within classical logic Łukasiewicz’s work was concentrated on the propositional calculus and the theory of A-I-E-O relations (Aristotle’s syllogistic), of which he was recognised an undisputed master and to which he gave their modern form, unsurpassed in simplicity, clarity and formal precision. The publication of Elements of Mathematical Logic established his authority in very wide circles in Poland and the appearance of Untersuchungen ber den Aussagenkalkl, written together with Tarski, marked the beginning of his international reputation. But the full recognition abroad came later, not until the years following the end of the Second World War, when the Republic of Eire generously offered him hospitality in Dublin. His creative power remained almost unaffected by the progress of age and he died like Euler, working and making plans for future work^{[34]}.

łukasiewicz became acquainted with the modern propositional calculus probably through Frege and there is no doubt whatsoever that Frege influenced him deeply, both by his logical and philosophical ideas. Under Frege’s influence he became a Platonist in logic, the position which he rejected as ‘mythology’ in his later years, and to which he again returned later still 35^{[35]}. He also thought Frege’s calculus to be superior to that of Russell-Whitehead and perfected it considerably, in particular by reducing the number of axioms from six to three (compared with four axioms of Principia Mathematica after one had been found deducible from the others by Bernays and Łukasiewicz working independently of each other). He devised a symbolism of his own, based upon a suggestion of Chwistek, the now widely known CN calculus, making it possible to dispense with dots and brackets, and, consequently, constituting an essential step forward in the strict formalisation of logic. He constructed other propositional calculi and their subsequent investigation greatly contributed to the understanding of the structure of formal theories. Łukasiewicz’s interest in the propositional calculus was based on the belief that the calculus of propositions is the fundamental logical discipline and the ‘deepest foundation’ of all deductive sciences. The whole structure of logic rests on it and mathematics rests on logic. If there are different logics of propositions, irreducible to each other – and manyvalued logics have established this fact – there must be different calculi of predicates, and, consequently, different theories of sets and different arithmetics^{[36]}.

Apart from the axiomatic method, he worked out with Tarski’s assistance the matrix method as a general method of constructing formal calculi, investigated the consistency, independence, and completeness of logistic system, and devised new methods of providing proofs to this effect. He initiated a vast body of investigations on the extended and restricted propositional calculi, as well as on calculi with single axioms (based on Sheffer’s stroke, equivalence or implication as primitive terms), and himself achieved important results in this field. While some of his discoveries should be credited to Łukasiewicz himself, others could have been made only because he worked with a team of logicians of great brilliance. One of them was Tarski. Others were Stanisław Jaśkowski (born 1906), Adolf Lindenbaum (1909-1941), Andrzej Mostowski (born 1913), Mojżesz Presburger, Jerzy Slupecki (born 1904), Bolesław Sobociński (born 1906), Mordchaj Wajsberg, all of whom left their mark and contributed to the achievements of the logical school as a whole. The team work was an essential and characteristic feature of the school and one of the secrets of its success. The collaboration was so close and intimate that it is often hard, if not impossible, to say, who should be credited with what. The team spirit without depreciating individual merits, those of Łukasiewicz and Tarski in particular, is a tribute to the disinterestedness and purposiveness of all^{[37]}.

The rich crop of particular and general results not only enriched our knowledge of formal structures but also constituted a prerequisite of the emergence of metalogic. This term occurs already in On the Principle of Contradiction in Aristotle (1910). Łukasiewicz used it to refer to investigations on the relations between logical principles which may lead, as he suggested, to various logical systems, including non-Aristotelian ones, in a similar fashion as the investigations on the parallel axiom had led to the construction of non-Euclidean geometries 38^{[38]}.

The choice of the term ‘metalogic’ might have been also prompted by the distinction between ‘mathematics’ and ‘metamathematics’ made a few years earlier in Germany to differentiate between two parts of Hilbert’s programme. Hilbert tried to show: firstly, that the whole of mathematics can be presented as a system of formulae derived from axioms according to some fixed rules; secondly, that this system is consistent. ‘Metamathematics’ was the name given to the investigations and arguments which were expected to secure the proof of consistency. As Hilbert conceived it, metamathematics was to make use – apart from other restrictions – only of the very simplest logical operations. Metalogic was conceived differently. Some of its seminal ideas can be traced to Leśniewski’s lectures at the University of Warsaw from the early ‘twenties onwards and to the stimulus which he provided in scientific discussions. Metalogic grew out of the critical examination of the principles and methods used in the construction of deductive systems, which also included investigations on the consistency of these systems as one of its important tasks. In view of the generality of these examinations they were known at first by the name of ‘methodology of deductive sciences’. In the course of time this conception turned out to be too narrow. On the one hand, there arose the need to differentiate between the language of the systems considered and the language in which the methodological investigations were carried out (the language and meta-language); on the other, various special concepts, created for the purpose of investigating the construction and structure of deductive formalised systems, called for critical examination and systematisation (the theory and the meta-theory). In the late ‘twenties these facts were widely recognised. The appearance of Łukasiewicz Elements of Mathematical Logic made them more pronounced and the publication of Łukasiewicz"s and Tarski Untersuchungen ber den Aussagenkalkl showed the metalogic in the process of its creation by the sheer weight and number of results and efforts at their systematisation.

The decisive step was taken by Tarski, who, from the evaluation of methods applied in the construction of deductive theories, passed to investigating them theories as wholes and to the elaboration of concepts necessary for this task. In his papers Tarski avoided the expression ‘metalogic’ and used instead that of ‘metatheory’ or ‘metamathematics’. This was prompted by his conviction that every deductive theory, and consequently also every logical system, is a mathematical discipline. ‘Metamathematics’ was, therefore, a more general term than ‘metalogic’. As to its meaning, ‘metamathematics’ in Tarski’s sense differed considerably from that in Hilbert’s^{[39]}.

łukasiewicz’s second major contribution to logic was his research into its history and the encouragement which he gave to others to follow suit. Łukasiewicz’s interest in the history of logic dated from the time when he was still a follower of the Lwów school. His acquaintance with modern formal logic constituted, however, a decisive turning point; only then did he realise that the existing works on the history of logic had no scientific value^{[40]}. His most important contributions concern the logic of the Stoics, which he rescued from contemptuous oblivion restoring it to its true greatness, and Aristotle’s syllogistic, which he liberated from its distorting accretions accumulated through centuries, and presented in its original formal and modern formalised form. The interest in Aristotle’s logic, which had occupied his mind since the early ‘twenties, was brought to its fruition in the first edition of Aristotle’s Syllogistic. He died before the second edition, supplemented by an exposition of Aristotle’s modal logic, appeared in print. His life-long concern with Aristotle’s logic was closely bound with many of his own discoveries and testified to the fact that historical research is fruitful in more than one respect.

łukasiewicz found followers and collaborators in Poland, among whom Jan Salamucha (1903-1944) and J. M. Bocheński (born 1902) were the most distinguished. The first examined the concept of deduction in Aristotle and Thomas Aquinas, St. Thomas’ proof ex motu for the existence of God, Ockham’s propositional calculus, and the problems of antinomies in medieval logic^{[41]}. Before the war the second published several contributions on ancient and medieval logic, to become in the post-war period an internationally recognised authority on the subject^{[42]}. Other contributions which were prompted or influenced by Łukasiewicz’s historical research, came from Zbigniew Jordan, Maria Kokoszyńska, and Konstanty Michalski^{[43]}.

łukasiewicz conceived the idea that means should be provided to equip an able young man, willing to undertake the task, with the necessary knowledge in classical philology, history and formal logic in order that he might devote his entire life to the rewriting of the history of logic. This idea was not and could not be implemented; the task clearly overreaches at present the strength of body and mind of any single man^{[44]}. But however unfeasible Łukasiewicz’s idea was, his insistence and efforts have achieved wonders. Since he spoke for the first time on the logic of the Stoics in 1923, a steady output of studies began to follow, first a trickle in Poland and Germany, later, in particular after the Second World War, a gathering stream all over the world. Many contributors recognised their indebtedness to Łukasiewicz and hardly anyone can ignore what he had to say on the subject^{[45]}. In Poland after the Second World War the interest in the history of logic revived, but little work, based on the study of original sources, has been done so far.

łukasiewicz’s greatest achievement, as is now widely recognised, is the discovery of many-valued logics. The fact that they do arouse considerable controversies -concisely and suggestively summarised in the Introduction to J. B. Rosser and A. R. Turquette Many-Valued Logics – perhaps only enhances the audacity not so much of the idea itself, entertained before Łukasiewicz, but of working it out and of establishing many-valued logics as consistent formal structures^{[46]}. The main facts concerning the logical discovery are easy to ascertain. The L_{3} was established by Łukasiewicz by the matrix method in 1920^{[47]}, that is, before E. L. Post’s celebrated paper was published. The L_{n} followed in 1922, according to Łukasiewicz’s own statement, but was not published until 1929, that is after Post’s generalisation^{[48]}. The mutual independence of Post and Łukasiewicz is evident from their respective publications, from the differences in the approach, formal in the case of Post, philosophical in that of Łukasiewicz, as well as in their respective treatment of the designated truth-values. In its abstract form the L_{n} system was presented in Łukasiewicz and Tarski *Untersuchungen fiber den Aussagenkalkl,* its philosophical origin and implications were examined in Łukasiewicz Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalkls. A primitive base of L_{3} was established by Wajsberg in 1929 and the axioms of a full L_{3} were given by Słupecki in 1936^{[49]}. Finally, in the ‘fifties Łukasiewicz examined L_{4} more closely by the matrix method in Aristotle’s Syllogistic and A System of Modal Logic^{[50]}.

The idea prompting Łukasiewicz to construct L_{3} originated from his studies of Aristotle’s logic and his interest in providing a consistent basis for the thesis of indeteminism. The latter interest is apparent from first to last in his attempts to formulate a theory of contingency which would allow the acceptance of the existence of true contingent propositions^{[51]}. Probably already in 1910, when he wrote On the Principle of Contradiction in Aristotle, he came to the conclusion that the thesis of indeteminism and the principle of bivalence, or rather that of the excluded middle, as he then saw it, were incompatible. He stated this view in an address delivered at Warsaw University in 1918 and again in a much fuller form in 1922; the former was recently reprinted, the latter was never published and perished during the Warsaw rising^{[52]}. Besides his indeterministic convictions, at the source of the discovery of L_{3} was Łukasiewicz’s concern with modal logic. He was prompted by the intuition, which has turned out to be right, that a system of modal logic cannot be accommodated within L_{3}.

łukasiewicz’s conviction concerning the incompatibility of indeterminism and of the principle of bivalence was strengthened by one of Kotarbiński’s studies, first published in 1913. In fact Kotarbiński went further than that and came very close to the idea of a three-valued logic. If a man is free, Kotarbiński argued, i.e. is able to make choices and bring about what without his action would not materialise, there must be things in the Universe which may or may not happen, an ‘ambivalent possibility’, as he described it. A statement about an ambivalent possibility cannot, therefore, be either true or false; it constitutes a ‘third kind of propositions’, different from those which are true or false. Kotarbiński rejected expressly the principle of bivalence, that a ‘given proposition is either true or false’, which ‘unlawfully passes off for the principle of the excluded middle’. The latter is valid only in so far as it states that contradictory statements cannot be false together, from which it does not follow, however, that one of them must be true. Kotarbiński argued in a Brouwerian manner that if a proposition is proved not to be false, this does not entail that it is true. He concluded: ‘Every proposition is either true or false or neither true nor false and quartum non datur’ ^{[53]}. He was not certain, however, whether this assumption does not lead to contradictions and whether the quartum non datur principle is not absurd^{[54]}. Kotarbiński’s essay was known to Łukasiewicz and according to his own confession strongly influenced his thoughts on the subject^{[55]}. The seeds of L_{3} were sown in the Lwów school and more than one mind participated in their cultivation.

As is well known, these seeds go back to Aristotle. Łukasiewicz’s discovery carries, above all, the Aristotelian imprint and was prompted by his studies of Aristotle’s logic guided by his knowledge of formal modern logic. Aristotle accepted the view that particular statements referring to some future events, e.g. the statement ‘there will be a sea-battle to-morrow’, when pronounced to-day can be neither true nor false. Łukasiewicz drew the conclusion therefrom that they must have a third value different from truth and falsehood. Let us assume, in accordance with Aristotle’s suggestion, that: it is contingent that p (symbolically: Tp) if and only if it is possible that p and it is possible that not p. In Łukasiewicz’s notation this definition can be written as the equivalence (Q):

*Q T p K M p M N p.*

Now, if a statement is either true or false, there is no true value of p for which Tp is true. The right-hand side of the equivalence is a conjunction which for both truth-values of p must become falsehood. A consistent theory of contingency requires that the principle of bivalence of our intuitive and classic logic is replaced by a more general principle. There is nothing to prevent us from doing it in view of the fact that the statement: ‘for every p, p is either true or false’, is not a theorem but a principle of logic and can be replaced by a different principle, provided that certain conditions are satisfied. On this basis Łukasiewicz proceeded to the construction of L_{3}, an interpretation of which was to be modal logic, the third value standing for possibility. This attempt failed and Łukasiewicz was the first to see and to recognise it. L_{3} can be constructed and proved to be a consistent and complete formal system, but modal logic is not its interpretation. If we consider the theorems of L_{3} as those of modal logic, some theorems of L_{3} turn out to be inconsistent with the accepted meaning of modal functors, some others could not be interpreted in terms of modal logic. In particular, the theory of contingency cannot be accommodated within a three-valued modal system. If Tp is equivalent to KMpMNp and KMpMNp is true for some values of p, say ?, then we can assert both M? and MN?. However, from these assertions and a theorem due to Leśniewski follows that we have then to assert Mp, or, in other words, to admit that all problematic propositions are true. This destroys what Łukasiewicz called basic modal logic, a set of assumptions which must be included in any system of modal logic, if it is to make sense. The rejection of the expression Mp, which has to be asserted in a three-valued modal logic is a constituent of the basic modal logic^{[56]}.

Twenty years after this failure Łukasiewicz returned to his original idea and by defining certain four-valued matrices he succeeded in constructing a fourvalued system of modal logic that renders faithfully, in his opinion, our intuitions associated with modal functors and includes a consistent theory of contingent propositions^{[57]}. The theory of contingency is very ingenious but requires purely symbolical treatment. It assumes the distinction of two kinds of possibility and contingency, for which an appropriate terminology does not exist in ordinary speech (they can perhaps be described and expressed as different degrees of assertibility, associated with respective problematic and contingent statements). By means of a four-valued modal system and two kinds of possibility and contingency the existence of true contingent propositions in Aristotle’s sense:

Q T p K M N p,

can be asserted, with the functor ‘M’ denoting different possibilities.

There were some other attempts, all made outside Poland – in France, the United States, and the USSR – to find an interpretation or to make use of the L n in the calculus of probability, quantum mechanics, set theory, and the theory of electronic circuits. Neither was fully successful and neither contributed substantially to the development of the branch of science which was to benefit from it^{[58]}. Thus, at present, the interpretation of L_{4} in terms of modal logic is the only example of L n being something else but a purely formal structure. Moreover, this instance is not yet safely established. Hence some logicians look askance at the many-valued logics, as Leśniewski did. Leśniewski considered as useless such consistent deductive theories as enable us to prove an ever increasing number of theorems which are irrelevant in view of their being unrelated to reality. But the history of science provides numerous examples of abstract theories which had been developed long before any use was found for them. Moreover, there can be little doubt that Łukasiewicz’s performance of a ‘bold experiment’ has fundamentally changed our conception of logic and provided a new insight into the nature of formal structures, including those based on the intuitive bivalent logic.

łukasiewicz became a logician through search in philosophy for exactitude and precision in speech and thought. In the propositional calculus, greatly improved by his own achievements, he found the unsurpassed model of perfection which every science should strive for and try to emulate. Mathematics was no exception to this rule. As he saw it, modern formal logic was no branch of mathematics, but an autonomous science which set up a new ideal of scientific precision even for mathematicians. Compared with this function any other service that formal logic may render to mathematics, for instance by helping to solve otherwise very important questions of the consistency and completeness of mathematics (written 1929), has a secondary value. By raising still higher the standards of scientific procedure, formal logic makes every branch of knowledge exert itself to raise its own standards and to approximate the ideal model of formal logic. This was in Łukasiewicz’s opinion the highest contribution of formal logic to science and philosophy^{[59]}.

To train one’s mind in the methods of formal logic, wrote Łukasiewicz, is to allow scales to fall from one’s eyes. ‘One notices distinctions where there is none to the others and one sees nonsense where others look for deep mysteries’. Then comes the realisation that one has not learnt to think logically, precisely, consistently, thoroughly, neither in philosophy nor in science, neither in public nor in personal life^{[60]}. Certain conclusions follow inevitably. Much that has been done in the past in philosophy has no scientific value and to continue in the old habits of thought is simply a waste of time and mental energy.

This evaluation of formal logic made Łukasiewicz despair of philosophy. In an address delivered at the Second Polish Congress of Philosophy (Warsaw, 1927) Łukasiewicz gave the advice to the audience that they should forget the past and start everything again from scratch. If all human disciplines were ordered according to the scientific precision of their methods, philosophy would have to be placed at the bottom. Philosophical systems of the past might have some aesthetic and moral value, they might occasionally have made a true and intuitively justifiable observation, but scientific value they have none. The philosophers’ failure to make of philosophy a science results from their neglect of logic. They do not adhere to and do not follow the logical procedure or they base their views on wrong theories of logic. Philosophy has thus fallen into the abyss of vain speculations from which only formal logic, and the axiomatic method in particular, can rescue it.

Not all philosophical problems could be examined in the suggested manner, but not all philosophical problems have a definite sense. Those which are concerned with the essences of the world, with the mythological entities like Plato’s ideas or Kant *Dinge an sich*, cannot be formulated in a comprehensible, clear, and unambiguous manner. Questions concerning the structure of the world – time, space, causality, determinism, indeterminism, teleology – are a different matter; to these questions the axiomatic method can be applied. While the axiomatic method provides an instrument by means of which scientific philosophical theories can be constructed, experience and natural sciences would be used to verify them and to revise continually the basic concepts and assumptions of these theories. The latter would pave the way for a philosophical synthesis, a truly scientific and formally sound view of the world, to guide us in our efforts to improve ourselves and the world we live in^{[61]}.

This makes it clear that Łukasiewicz’s conception of philosophy substantially differed from that advocated by the Vienna Circle. Łukasiewicz spoke approvingly of Moritz Schlick’s and his followers’ efforts to revise philosophical method by making use of formal logic, but he emphatically rejected Carnap’s reduction of philosophy to logic of science. He was inclined to agree with Carnap that metaphysical propositions are meaningless, if they claim to convey knowledge about something which is over and beyond all experience (the essence of things, or things in themselves). But this is the Kantian understanding of metaphysics; there remain factual problems concerning the structure of the world which are metaphysical, whether they are so termed or not. The latter are not syntactical questions, as Carnap suggested. It is hard to see at all, how by investigating the structure of language, its formation and transformation rules, the problems as to whether the world is or is not finite in space can be solved. The same applies to the question of causality, determinism, and many others. These are questions of fact and their solution is not a matter of language. One can again agree with Carnap that there is a formal mode of speech correlated to a material mode and that some errors might be avoided by translating sentences expressed in the material mode into those clearly syntactical. But to say that by such a translation we get rid of the misleading impression, inherent in the material mode of speech, that a ‘material’ sentence refers to some extra-linguistic reality where no reference of this kind is in fact involved, is a dogmatic, unjustifiable statement. The sentence: ‘the evening star and the morning star are identical’ is correlated with: “the words ‘evening star’ and ‘morning star’ are synonymous”; their meaning, however, is quite different. The former refers to the extra-linguistic, the latter to the linguistic reality. The matter of fact involved in the former was resolved after long years of observations and could not have been decided upon by an examination concerned with the usage or by the reflections upon the meaning of the words referred to in the latter.

Finally, Carnap’s view that logical and mathematical sentences are tautologies which do not say anything about the world cannot be accepted either. There are various systems of geometry and logic but one and only one of them does apply to the outside world, irrespective of the fact whether we are at present able to state which of them does. When a formal a priori structure is applied to reality it must be treated as any other hypothesis, i.e. to be verified by experience. When this is achieved, a geometrical or logical system does convey some knowledge about the outside world. Carnap’s logic of science and the Wiener Kreis doctrine in general, Łukasiewicz stated, were risky philosophical speculations which would soon become obsolete^{[62]}.

The critical attitude to the logical positivism of the Vienna Circle was presumably reinforced by Łukasiewicz’s return to Platonism in logic. Łukasiewicz discovered that formal logic displays certain paradoxical features. On the one hand, formal logic is neutral; it does not commit anyone to any particular view in ontology or in the theory of knowledge, and it can be combined with empiricism or rationalism, with realism or idealism. On the other hand, a formal logician accepts, in practice, the nominalist viewpoint. The expressions and sentences with which he deals, are considered to be names of man-made inscriptions. This should commit him to finitism; only a finite number of inscriptions can be given at any time. Finitism makes, however, the validity of logic dependent on certain empirical facts, which is unacceptable, and is actually incompatible with the logician’s practice. There is no longest logical thesis, as there is no largest natural number. Moreover, it can be easily shown that in the two-valued propositional calculus the class of all theorems is infinite in Dedekind’s sense. The nominalistic outward appearances of formal logic mask metaphysical problems concealed in the foundations of logic^{[63]}. How should they be solved?

The answer belongs to philosophy and not to logic. Łukasiewicz never said that it is either too complex or impossible to give a theory which would do without postulated abstract entities, but he was inclined to accept their existence on the ground of what is implied by logical constructions and by the insight they provide. We believe that there are shortest possible axioms of various calculi which are to be found. In general, the logician only discovers formal structures. They appear to his mind ‘as if they were a concrete and palpable object, made of the strongest material, a hundred times stronger than concrete and steel’. Nothing can be changed, created, arbitrarily decided by the logician, who by his efforts gains the knowledge of ‘permanent and lasting truths’. Personally Łukasiewicz believed that they were the thoughts of God^{[64]}.

łukasiewicz was a man of few philosophical ideas and those which he held in one period of his life seem to have differed sharply from those expounded by him in another. The one exception was his unshakeable conviction in the truth of the thesis of indeterminism. In other respects he oscillated between the extremes. Philosophically Łukasiewicz was influential inconsiderably strengthening certain tendencies. He has done much to inspire the respect for the use of formal logic in philosophical investigations, implying some restrictions on the range of problems discussed and a sharp distinction between what can be scientifically examined and what does not lend itself to such treatment. He also strengthened the methodological orientation of Polish philosophy and its absorption in the problems of science. He was a moderate supporter of scientism; he believed and inspired trust in the methods and achievements of science, both in their theoretical and practical aspects. In general, he seems to have been a power in so far as some intellectual trends were concerned and to have exercised little influence in matters concerning the philosophical programme and methodological procedures. What he wished philosophers to accomplish was not feasible. He underrated the difficulties of reducing philosophical problems to the form which would provide a possibility of their solution by a purely deductive method, and also those connected with the verification of axiomatised philosophical systems. Practically nobody tried to put these ideas into effect. What was widely accepted was Łukasiewicz’s criticism of the state of philosophy, his diagnosis of its causes and his belief that much could be improved by sharpening the philosopher’s tools with the assistance of formal logic.

Tarski (born 1902) from the beginning enjoyed the advantages and disadvantages of his intellectual background. Unlike his teachers he was by training a mathematician and logician, and only afterwards a philosopher^{[65]}. In his philosophical views he was influenced by Kotarbiński, to whom the collection of his pro-war papers, translated into English, is dedicated. But what he took from philosophy he repaid with interest.

Already his first contribution^{[66]}, in which he showed that, granted the use of functions with propositional variables and of the universal quantifier, all the propositional functors can be defined in terms of the equivalence (Leśniewski adopted Tarski’s discovery in his protothetic), secured him recognition and his reputation has been rising ever since. While for Leśniewski logic and metalogic constituted an organic whole, and Łukasiewicz associated them closely together, Tarski early realised that their respective domains differ not only by the degree of abstraction but also by methods appropriate to each of them. The rise in volume and importance of metalogical research, as well as construction of the conceptual framework and methodological tools in metalogic, are bound indissolubly in Poland with Tarski’s name.

The shift of interest towards metalogic, which took place everywhere after the publication of Gödel’s celebrated paper, in Poland only enhanced the tendency previously initiated by Tarski. Tarski moved through the unexploited land of metalogic, improved whatever he touched, framed new notions, discovered new methods and ideas, gathered a rich harvest of ‘incidental’ results from the application of his analysis to a variety of subjects belonging to logic, mathematics and applied mathematics. In the late ’twenties he was the first, or one of the first, to introduce an informal axiomatic method to metalogical investigations. Those on the concept of truth in formalised languages, first published in 1933 but dating back to 1931, are based on an axiomatised metatheory. The power of the new methods was made apparent and the importance of a general metatheory of deductive systems enhanced when Tarski showed that some mathematical problems might be solved by metamathematical methods and that the examination of classes of deductive systems (calculus of systems) opens new vistas to the knowledge of formal structures^{[67]}.

Metalogic passed through several stages of development. At first, it was not clearly differentiated from logic itself and under the name of methodology of deductive systems was mainly concerned with the investigations of the consistency, completeness and independence of axioms. When it emerged from this stage to the status of an autonomous discipline, it was conceived as the study of the formal aspects of formalised deductive systems (formalised theories). After the publication of Carnap *Logische Syntax der Sprache* (1934). these investigations used to be called ‘logical syntax’.

Outside Poland, in particular in the Vienna Circle, metalogic and any metatheoretical examination were identified with logical syntax. This carried certain important philosophical implications. They were connected with the construction of various artificial formalised languages by means of which philosophical problems were to be examined and elucidated. The construction of artificial formalised languages was based on the assumption, prompted by the identification of metalogic with logical syntax, that its object-language (in this case – the ordinary speech) can be specified by its formal characteristics. In other words, it disregarded the fact that the expressions of everyday language and scientific discourse have a meaning, denote something or refer to some state of affairs, and that they are true or false. Artificial formalised languages which eliminated these semantical notions were consequently bound to be fragmentary and to constitute an inadequate medium for the examination of philosophical problems. As this was not at first realised by the inventors of artificial languages, certain serious philosophical errors became inevitable.

Tarski’s important contribution to philosophy in Poland, and later also to philosophy in general, was his early recognition of the fact that logical syntax (he then called it ‘morphology of language’) constitutes only a part, though an important one, of metatheoretical investigations. The decisive step was taken in 1931 when he supplemented the concept of syntax by that of semantics. While logical syntax investigates formal characteristics of formalised theories, the subject matter of semantics is the relation between the expressions of language and the objects which they denote or to which they refer, that is, between language and extra-linguistic reality. In Tarski’s words, the ‘semantical concepts (i.e. of theoretical semantics) serve to set up the correlation between the names of expressions and the expressions themselves.’ ^{[68]} The idea of semantics is general; it applies equally to the ordinary speech and to formalised theories. Theoretical semantics (semantics of the formalised theories) was in fact established after the need of semantical concepts was revealed by the examination of ordinary speech.

The term ‘semantics’ was taken in Poland from grammar or rather from that part of it which is called descriptive or historical semantics and which deals with the questions how it came about that words mean something and designate something, how their meanings change in the course of time and how they can be divided into classes according to what they mean or designate^{[69]}. The term was adopted from grammar, but its use was prompted by extra-linguistic considerations. These emerged from the examination of antinomies, particularly those of the liar, the Grelling and Nelson antinomy of heterological terms and Richard’s antinomy of definability, all of them resulting from the usage of ‘self-referring’ terms. They drew attention to the logical analysis of language and to what Kotarbiński called ‘semantical relations’. Already in the late ‘twenties a rudimentary logical theory of language (philosophical semantics) was worked out; it included the examination of the concept of meaning, designation, name, expression, definition and semantical categories as its main elements^{[70]}. These studies were greatly influenced by Husserl, and, above all, by Leöniewski’s penetrating criticism and original ideas. In particular Leöniewski devoted to the examination of antinomies much of his inventive energy and Tarski took advantage of Leöniewski’s views on this matter^{[71]}. Once again antinomies turned out to be a blessing in disguise. While antinomies of classes, and, above all, Russell’s antinomy, inspired efforts to construct a consistent formalised system of logic and mathematics, the antinomy of the liar was the starting point for the rise of philosophical and theoretical semantics.

The important conclusion which emerged from the examination of semantical antinomies stated that a semantically closed language and one within which the laws of logic hold, must be inconsistent and provide a breeding ground for antinomies. It should be understood that a semantically closed language is any natural or artificial language which does not differentiate between expressions about which we speak from those in which we speak, e.g. between object- and meta-language. In a semantically closed language the object- and meta-language coincide; the former thus contains its own semantics.

Semantics is a relative concept. While we can speak of semantical concepts in general – such as meaning, denotation, reference, definition, truth – semantics is always related to a particular language and each language has its own semantics, not to be confused, under the penalty of inconsistencies or errors, either with the language itself or with its structure (syntax), the form, arrangement and composition of the expressions of this language.

Once this is recognised, semantical antinomies can be, in principle, eliminated and semantics of any particular language constructed in a logically unobjectionable manner. There is, however, an important difference in this respect between natural and artificial languages. The logically unobjectionable and materially adequate use of semantical concepts depends on the structure and the vocabulary of the object language. We can construct the semantics of a language by exact methods if and only if this language is exactly specified, and, as a matter of fact, only formalised languages are exactly specifiable^{[72]}. Strictly speaking only languages of logical systems are at present formalised. But these languages have been successfully applied to mathematics, theoretical physics, and also to some branches of biology. Outside this field, and this includes, of course, everyday language which provides means of scientific and philosophical discourse, the use of semantical concepts can have only an approximate character. Both the vocabulary and the structure of everyday language are not formally specifiable and depend on extra-linguistic factors. Apart from the structure, it has, as it were, ‘depth’. On this account Tarski doubted at first (1933) whether a consistent and fruitful use of semantical concepts, in particular that of a true sentence, can be made in everyday language^{[73]}.

This feeling was not shared by philosophers. Neither a scientist nor a scholar nor a philosopher can do without semantical concepts. They would have to keep silent if they were to dispense with terms such as ‘meaning’, ‘denotation’, ‘definition’, ‘true’ and ‘false statement’. The realisation that they might get involved in contradictions did not prevent them from using these terms. To be provided with a general theory of semantical concepts, their exact definitions and general characteristics, gave the scientist and philosopher a firmer hold of these concepts in the examination of their own problems. Theoretical semantics was, therefore, accepted as a discipline of considerable philosophical significance. In particular, as has already been mentioned, theoretical semantics spared philosophy in Poland committing some errors to which other philosophers of language fell victim. The question of the correct usage of language was never dissociated from its semantical aspects and absorbed by the syntax of language^{[74]}.

The philosophical significance of theoretical semantics was the more obvious owing to the fact that Tarski’s demonstration of the uses of semantics was made in connection with his efforts to provide a satisfactory definition of truth. His semantic conception of truth is universally known and widely accepted; only some of its salient features will, therefore, be considered.

As his starting point Tarski accepted the classical conception of truth, also known under the name of the correspondence theory of truth, which was in agreement with the tradition of the Warsaw school. The Warsaw school, however, went further and accepted what used to be called the ‘absolute conception of truth’ ^{[75]}. By this qualifying term several things were meant. The expression ‘every true proposition is absolutely true’ meant the same as ‘no proposition is relatively true’. Since the relativistic conception of truth implies that in the sentence ‘’p’ is true’, the term ‘true’ is not a full predicate, the absolute conception of truth can also be described as the view which holds the contrary opinion, namely that in the expression ‘’p’ is true’ the term ‘true’ is a full predicate. Finally, the absolute conception of truth claims that a proposition is true if and only if it states what is the fact (and not what somebody says or thinks to be the fact). The explication of this last meaning inherent in the absolute conception of truth might involve its supporters in considerable difficulties, and this is the point where Leöniewski’s stimulating and original ideas on the matter and Tarski’s partial definition of truth – the (T) schema – prove their usefulness. Tarski’s partial definition of truth states in a clearer and unambiguous manner what the supporters of the classical conception of truth (in its Aristotelian version) in general, and those of the absolute conception in particular, had in mind^{[76]}.

In the summary of main results obtained in his celebrated essay The Concept of Truth in Formalized Languages, Tarski expressed the view that the only way of overcoming confusions and contradictions resulting from the application of semantical concepts to everyday language is to reform this language. He wanted this reform to be done in a manner which would finally make the language of everyday life closely resemble formalised languages. In a later essay he approached this subject from a slightly different angle. Perhaps, he said, nonformalised languages with an exactly specified structure could be constructed to replace everyday language in scientific discourse or at least in a comprehensive branch of empirical science^{[77]}.

Tarski’s suggestion was not followed in Poland, where analytical trends prevailed over the tendency to construct formal artificial languages. The philosophers did not share Tarski’s gloomy evaluation of the position prevailing in colloquial speech. In their opinion the worst confusions and the more obvious antinomies can be avoided by fully accepting and carefully adhering to Tarski’s distinctions between the object- and meta-language, as well as by applying selfimposed restrictions in the use of semantical concepts. Subsequently, partial reform and not a total reconstruction was undertaken. With the differentiation of various levels of discourse and the distinction of the syntactical structure and semantical aspect of the ordinary speech, philosophical problems appeared in a new light and had to be re-examined. The impact of theoretical semantics on philosophy was lasting but of necessity limited in scope. To use Tarski’s own words, theoretical semantics could be applied ‘only with certain approximation’. The situation with which philosophy was confronted after the publication of The Concept of Truth in Formalized Languages did not differ essentially ‘from that which arises when we apply laws of logic to arguments in everyday life’ ^{[78]}.

Leon Chwistek (1884-1944) was the most colourful and the least influential logician of the period between the two wars. His unconventional, if not eccentric, personality set him apart and made him tread a path of his own. His intellectual background differed from and his interests seemed to be only marginally connected with what was being done in the Warsaw school.

Chwistek spent most of his life in Cracow which, with Warsaw and Lwów linked by the ties of a common school of philosophical thinking, was the third centre of logical research. While in Warsaw logic, being an autonomous discipline, kept close to philosophy and philosophy to logic, this was not the case in Cracow. The philosophers there adhered to more traditional ways and only mathematicians showed interest in modern formal logic. Among the latter the most distinguished was Stanisław Zaremba (1863-1942), one of Chwistek’s teachers^{[79]}. The greatest credit for spreading knowledge of modern logic in Cracow was due to Jan Śleszyński (1854-1931), also a mathematician and Chwistek’s teacher, whose lectures on logic published in book form by S. K. Zaremba, the son of the eminent mathematician, were widely read in Poland even when it was supplanted by better and more comprehensive textbooks. Being the concern of mathematicians, logic became a branch or an instrument of mathematics. The conviction that formal logic provides no useful knowledge but only a method was one of the ideas firmly established in Chwistek’s mind.

Chwistek’s personality is very apparent in everything he wrote. Although he was likeable and entertaining in personal relations, as a writer he excelled in a style that was both malicious and combative. He uttered forceful and not always just opinions of other people and their views. His knowledge was wide but often superficial. He did not take enough care to present the views which he wished to criticise in a scrupulously objective manner. He used the names of trends with which he disagreed, e.g. that of Cantorism, not as descriptive but as qualifying, invective-like terms. There are pages upon pages in his books and articles in which he came close to journalism, which are geistreich but rambling. Chwistek indulged sometimes in modes of expression and habits of thought that the Warsaw school set out to eradicate from academic philosophy.

It would be wrong, however, to draw the conclusion that Chwistek exercised only harmful influences. His merits were as numerous as his demerits; those who saw clearly the latter did not disregard the former^{[80]}. The ultimate reason why he did not play a role commensurable with his achievements in Polish philosophy was the fact that he attached greater significance to certain philosophical problems than did the Warsaw school and other Polish philosophers under the influence of this school.

Chwistek was probably the only philosopher in Poland who not only watched and followed the dispute concerning the foundations of mathematics between intuitionism, formalism, and logicism, but also took a serious interest and part in it. When a young man he attended a lecture given by Henri Poincaré in Göttingen after which he remained under the spell of ‘constructivist’ ideas^{[81]}. This was the starting point of a long and sustained effort recorded in a number of contributions, the final result of which is the system presented in English in The Limits of Science.

Chwistek was the first Polish logician to gain recognition abroad by his criticism of Principia Mathematica, in particular of its theory of types and its extralogical axioms (those of infinity and reducibility). In this connection he worked out the theory of constructive types, based on the reduction of the notion of class to that of propositional function (Chwistek was the first to see that this can be done), and the simplified theory of types. The latter was very important as a working semi-intuitive tool, particularly in relation to the theory of sets. To-day they belong to the history of logic, the second having been incorporated into the body of common knowledge. From the point of view reached by Chwistek some years later, the simplified theory of types has outlived its usefulness. The principle involved was to be included in a more comprehensive theory which replaced the hierarchy of types within a language by a hierarchy of languages^{[82]}.

As Chwistek saw it, the problem of establishing consistent foundations for mathematics required the reconciliation of several not easily reconcilable principles. First, the principle of nominalism: to recognise the existence of only such objects as are given in immediate experience and with respect to which no serious question or doubt can be raised. These are clearly some objects of the physical world. The implementation of the principle of nominalism is a prerequisite for showing that in the impressive structure of mathematics no more complicated and less obvious operations are ultimately involved than those of laying brick on brick. Second, the principle of constructivism or la règle de Poincaré: never to consider any objects or a class of them which are not capable of construction. It is true that a great majority of mathematicians has become reconciled to un-constructive objects and that the assumption of the existence of such objects does not need to lead to contradiction (e.g. Russell’s universal class of real numbers), but the presence of these metaphysical, idealistic suppositions is a fundamental flaw in the whole structure^{[83]}. Third, the principle of formalism: without formalisation there is no construction of deductive theories by exact methods. The only alternative to formalisation is the use of everyday language with its confusions and pitfalls. Everyday language can never be made accurate and Poincaré was wrong and Hilbert was right when he wished to transform the entire content of mathematics into symbolic formulas derived from axioms in a mechanical manner. Fourth, the principle of ‘completeness’: every part of classical mathematics, irrespective of the way in which it has been acquired, should be included in the system. Such a hypothesis as Cantor’s actually infinite has its source in fantasy. It should be shown that what is really of value in it can be absorbed in a system based on the above enumerate principles.

This was the plan on whose implementation Chwistek was engaged since 1928 and in this undertaking he was assisted by three collaborators, Jerzy Herzberg, Władysław Hetper, and Stanisław Skarżeński, none of whom survived the war. The first step was the creation of what Chwistek called ‘elementary semantics’, which, besides its name, has nothing in common with semantics in Tarski’s sense. Chwistek claimed that in Hilbert’s metamathematics there is contained intuitive semantics, i.e. the rules for the construction of the simplest possible expressions from given elements (letters or signs). This intuitive semantics is formalised and expanded into a system of syntax in terms of which the propositional calculus and the theory of classes are constructed. On this basis the axiomatisation of classical mathematics which assumes no non-constructive objects is finally undertaken. If successful, and this matter must be left to the mathematician to judge, it would provide a proof of the consistency of mathematics. In this, more than in anything else, lies the importance of Chwistek’s system. The opinion has been expressed that Chwistek’s work represents the most important attempt ever made to establish such a proof^{[84]}.

The question might be asked why Chwistek undertook this enormous task and what philosophical significance he attached to it. He himself answered this question at length. The proof of consistency would put it beyond any possible doubt that a rationalistic view of the world can prevail against confused metaphysics and stem the rising wave of anti-rationalism. He traced the origins of modern anti-rationalism down to Hegel and among its representatives counted Nietzsche, Bergson, Husserl, the pragmatists, and many eminent scientists – such as Weyl and Eddington – who became affected by the intellectual atmosphere which he considered hostile to rational thought. A rational view of the world is based on simple and clear truths derived from experience and exact reasoning, and ultimately on what Chwistek called ‘sound reason’, not to be confused with common sense. While common sense accepts what seems ‘obvious and inevitable in a given society’, sound reason prompts both criticism and constructive thought. Hume, Comte, Marx, and Mach were its great exponents. Chwistek’s work, as he conceived it, was to turn the scale against anti-rationalism and restore sound reason to its place in human affairs. From this viewpoint he considered everything else done in philosophy in Poland as unimportant and ultimately irrelevant. This verdict was neither just not right. In many respects he was closer to the Warsaw school than he cared to recognise.

This review of the work accomplished by the four leading Polish logicians in the period between the two wars cannot close without a short mention of the mathematicians. The period under discussion witnessed a considerable expansion of mathematics in Poland, particularly in the domain of the theory of sets and topology^{[85]}. The theory of sets is perhaps the most ‘philosophical’ branch of mathematics. The concept of class and its obscurities fall within the region where the logician, the philosopher and the mathematician meet to clarify their ideas, to revise the premisses, and to make more precise the modes of reasoning they use in common. The concept of class is not the only one of this kind; the theory of relations or the notion of probability provide other examples. The wide application of statistical methods in more or less exact sciences drew attention to the mathematical theories of probability, which, with the use of statistics, come into play sooner or later.

Among the pure mathematicians, who either directly contributed to logic or whose works were widely read by students of philosophy, the names of Sierpiński, Kuratowski, Mazurkiewicz, Wilkosz (1891-1941) and Mostowski should be mentioned. As the first three were the leaders of the Warsaw mathematical school there is little wonder that the philosophers, the logicians, and the mathematicians were often lumped together into a single group to be known in the world outside as the Polish or the Warsaw school. Although confusing, this single name has the merit of throwing into relief the close and fruitful collaboration that existed at that time between the three sciences and in various ways contributed to the development of each of them.