##### FORMAL AND MATHEMATICAL LOGIC

One of the most important and spectacular features of post-war development was the swift recovery of logic. Polish logicians have again been in the vanguard of logical research and as a team are second only to the American school of logic[232]. Closely associated with the recovery was the division of modern logic into two branches, formal (or philosophical) and mathematical logic (‘Philosophical logic’ is a term of pre-war origin and now not in use). The division was formally recognised in the early ‘fifties[233], but it became apparent much earlier. It came to light in the approach, selection of subjects and emphasis given to various logical theories in textbooks, including their titles, published by Mostowski and Czeżowski. One of its manifestations was the incorporation of mathematical logic, in the sense defined below, into the Polish Mathematical Institute set up towards the end of 1948. The Institute has the departments of theoretical and of applied mathematics, and mathematical logic constitutes a section of the former.

The division into formal and mathematical logic was brought about by the specialisation and division of labour. Some of the logicians with mathematical training have acquired highly specialised formal techniques which they apply in the solution of mathematical problems, in the investigations of new fields of mathematical research and in the inquiry on the foundations of mathematics. Gödel’s discovery that no formalised system, which is of interest to the mathematician, can be complete, radically changed the direction of the research on the foundations of mathematics and transformed it into a complex of clearly mathematical problems to be solved by methods technically different from those generally applied in formal logic, such as decision procedures, the theory of recursive functions, or algebraic methods. The foundation problems of mathematics have become still more specialised by their being diversified and considered separately for particular branches of mathematics (e.g. the arithmetic of natural numbers, the theory of sets or that of real numbers). The sequel of Gödel’s discovery was that logic applied to the study of the foundations of mathematics ceased to be a purely logical discipline and has grown into a branch of ‘applied mathematics’. This is the development against which Łukasiewicz once warned the logicians and which has been brought about by their discoveries. These discoveries turned out to be the rock on which the hopes of solving the problem of truth in the deductive sciences, once and for all, by means of constructing formalised logical and mathematical theories were wrecked. The concluding sentence of Mostowski Mathematical Logic emphasises this dramatic development and lets it be understood that the curtain has fallen on a whole era in the history of the foundation problem[234].

Thus, the division of logic into formal and mathematical became inevitable. The methods used in the study of the foundations of mathematics cannot be understood unless the logician turns into a mathematician. These methods are not, as a rule, applicable to the problems in which a formal logician is interested. On the other hand what a philosophically-minded formal logician is concerned with, namely a general study of formal structures and the application of formal logic to natural languages, no longer attracts much attention on the part of a mathematical logician. The latter lost interest in them once he realised that the understanding of the notion of truth in mathematics cannot be reached by the construction of formal systems embracing the whole of intuitive mathematics. Formal logic ceased to be the common ground on which a philosopher and mathematician used to meet to investigate deductive sciences and to search for the solution of the foundation problem of mathematics. The division of formal and mathematical logic is not absolute, but the difference in methods and interests is apparent. Modal logic, logic of imperatives, contrary-to-fact conditionals are examples of questions which hardly any mathematical logician would care to take up.

Mathematical logic has in Andrzej Mostowski an undisputed leader. With him works a group of logicians, some of whom, like Henryk Greniewski, Stanislaw Jaśkowski and Jerzy Słupecki had already made a name for themselves before the war, while others – Andrzej Grzegorczyk, Jerzy Łoś, Helena Rasiowa, Juliusz Reichbach, Czesław Ryll-Nardzewski, Roman Sikorski, Roman Suszko, Wanda Szmielew – belong to the younger generation. Some of them are clearly mathematical logicians, others work both in the field of mathematical and formal logic.

Słupecki discovered the functionally-complete axiomatic system of the three-valued propositional calculus, solved the decision problem of the Aristotelian syllogistic, and gave an exposition of Leśniewski’s protothetic and ontology[235]. Jaśkowski formulated a system of the so-called logic of discussion and investigated the theory of modal and causal functions[236]. Łoś started as a formal logician. He investigated various axiomatic presentations of Aristotle’s syllogistic (those of Łukasiewicz, Śleszyński, and Słupecki), subjected the validity of Mill’s method of agreement and method of difference to scrutiny on the ground of an axiomatised partial language of physics, and dealt with the formalisation of intentional functions[237]. Suszko did research on the possibility of extending the idea of natural deduction to rich formalised languages, tried to apply some of his results obtained in mathematical logic (in which he follows the semantical line) to the theory of knowledge, and is interested in some purely philosophical problems of the foundations of mathematics[238]. Grzegorczyk, who has done some outstanding work in mathematical logic, came to it from philosophy and has retained a pronounced interest in the latter[239]. Ludwik Borkowski, Jerzy Kalinowski and Tadeusz Kubiński can be described as pure formal logicians. Their investigations are concerned with matters of direct philosophical significance, those of definitions, propositional calculus, modal logic and logic of imperatives[240]. But there seems to be a strong trend among the logicians towards mathematical logic in the above indicated sense. It attracts the ablest minds and clearly predominates over formal logic. This position provides a striking contrast to that before the war.

Unlike formal logic, mathematical logic suffered hardly any ideological and political interference even at the worst times of the Stalinist period in Poland. The main lines of its development can, therefore, be briefly described without reference to the evolution of the Marxist-Leninist doctrine. It is true that the results obtained in mathematical logic were sometimes presented as a confirmation of the materialistic views on mathematics, namely, that the latter is a natural science[241], but this pronouncement was an appendage to and not a guiding principle of research, which was free from any preconceived assumptions. What Mostowski and the others probably had in mind was the rejection of the neo-positivist doctrine which sharply separated the empirical and formal sciences. Carnap’s neat classification of truths into analytic and synthetic, the former being tautological and conventional in the sense that they are obtained by arbitrarily agreed upon transformation rules, was challenged by Łukasiewicz in 1936. It had been challenged for the same reason for which Mostowski and some other logicians rejected it, namely on the ground that logic and mathematics do say something about reality. Without any further additions there is nothing peculiarly materialistic about this perfectly sound, though somewhat constringent view.

The starting point of the mathematical investigations concerning the foundations of mathematics was the realisation that Gödel proved the futility of the great undertaking to make intuitive mathematics coincide with a formalised system of mathematics. If that had succeeded, the nature of mathematical concepts and the notion of mathematical proof would be definitely clarified. On the other hand, if the class of true mathematical theorems and that of provable ones within a formalised system have been shown never to be co-extensive, a new way of solving the basic foundation problems of mathematics must be found. This explains why Gödel’s discoveries occupy the central place in mathematical logic[242]. In the light of the incompleteness theorem the formalised logical and mathematical systems are only of a subsidiary and historical value.

One of the points in Hilbert’s programme was the contention that the scope and content of mathematical entities are defined in a unique fashion by the set of axioms established in the branch of mathematics in which these entities occur. The incompleteness of any sufficiently rich system of axioms makes it clear that this view is unacceptable any longer. Each such system has numerous mutually non-isomorphic models. Consequently, an axiomatic system does not define a single notion, but a whole class of them, let us say, not one but a whole class of notions of natural numbers. This consequence involves no contradiction; it is, however, hardly acceptable.

There arose the need, therefore, to ascertain the limits of the applicability of the axiomatic method as a means by which the nature of mathematical notions can be investigated. In this respect the concept of model has proved very fruitful and a general theory of models or structures determined by sets of axioms has been developed. In these investigations general algebraic notions are applied, and thus what is called the algebraisation of logic has been initiated. Finally, models themselves were conceived as some kind of algebraic entities and incorporated into modern abstract algebra to be investigated by methods appropriate to this branch of mathematics. The research on models led to unexpected results and established for the second time a close connection between logic and algebra. It applies both ways. On the one hand problems concerning formal structures (deductive theories) can be formulated as algebraical problems to be dealt with by algebraic methods (algebraic logic); on the other, some general algebraic results can be obtained from studies in the theory of models (logical algebra). This idea was compared with Descartes’ discovery of analytical geometry. In algebraic logic and logical algebra the correspondence between certain entities of abstract algebra and classes of all models of a certain kind takes the place of the relation between points and numbers in analytical geometry.

Algebraisation of logic is not by any means a trend restricted to Poland, although its inspiration was native[243]. Its origin goes back to Tarski’s calculus of systems and has been closely connected with the investigations on Gödel’s completeness theorem, which has become the subject of numerous studies (łoś, Rasiowa, Sikorski, Reichbach).

Within the theory of axiomatic systems the investigations on many-valued logics have undergone a radical change. The discovery of many-valued logics was made on the ground of philosophical considerations and they also guided Łukasiewicz’s later research in this field. It was done by the matrix method, by which the number of truth-values and the manner for establishing values of compound sentences were fixed. On this basis the deduction of logical theorems was undertaken. It was discovered, however, that matrices constitute a special case of models, determined by a peculiar type of axioms, which might be represented in the form of algebraic systems. Thus the so-called non-classical calculi have been absorbed by algebra to be studied by algebraic methods (Mostowski, Rasiowa and Sikorski).

The second main line along which the problems of clarifying the nature of mathematical notions is approached is the constructivist trend. It is free from the philosophical assumptions of Brouwer’s intuitionism, and it is not associated with the revision of logic by bestowing a different meaning on the logical constants from that usually accepted. The constructivist trend is bound up with the problem of decision, i.e. with the method of effectively deciding whether a given sentence is a theorem or not, and with the efforts to distinguish constructive and non-constructive mathematical entities. The relevance of this distinction to the foundation problem of mathematics is obvious. On the other hand, the question how far the constructivist programme can be carried out and whether the classical results can be obtained in constructive mathematics, has an inherent interest. The theory of recursive functions constitutes the main development line of constructive mathematics. A particular field of application of this theory is the so-called computable analysis, studied by Grzegorczyk, Mazur and Mostowski.

So far as the clarification of the notion of mathematical proof is concerned, the investigations are more or less concentrated on decision problems and decision methods in various branches of mathematics (Grzegorczyk, Janiczak[244], Jaśkowski, Mostowski, Szmielew). They strengthen the constructivist trend in view of the fact that in proofs of undecidability for a class of problems the use of recursive functions is essential. On the other hand the importance attached to decision problems is not unrelated to the recognition of the failure to establish the foundations of mathematics by means of formalised mathematical systems.

The investigations on the foundations of mathematics are not concerned any longer with one problem and do not constitute a single undertaking. There are instead foundation problems approached from different directions and attacked by different, though inter-related, methods. The problems themselves have no direct philosophical relevance. They are essentially mathematical problems. The question, therefore, arises whether any philosophical question of the foundation of mathematics has actually remained, or, otherwise expressed, whether the mathematical investigations on the foundation problems have no philosophical assumptions to be further elucidated and critically assessed.

Mostowski has clearly recognised that it is not possible to re-establish the foundations of arithmetic and to explain the nature of mathematics in general by the exclusive means of mathematical methods[245]. Ultimately the problems involved have a philosophical character. Mostowski has also felt that what he calls the ‘materialistic philosophy of mathematics’ is, in principle, the right solution. This materialistic philosophy is, however, either purely negative or extremely vague. It says what mathematics is not. It is not a set of tautologies in the formalistic or neo-positivistic sense; it is not a mere game, consisting of manipulations with symbols, formation and transformation rules. On the positive side it says that mathematics either corresponds in some way to reality or is an ontological theory whose development is closely connected with the natural sciences and with what Marxist-Leninists prefer to call ‘the history of the relation between man and Nature’. These assumptions are clearly inadequate. They are not strengthened by an attempted revival of John Stuart Mill’s view, according to which mathematical concepts are empirical generalisations based on experiments and the accumulated experience of many generations[246]. This view is trivial, if it means that experience has played an important role in the development of mathematics, and it does not seem to provide much enlightenment when it is intended to state something more than that.

To close this short survey a few words should be added on the history of logic. The interest in this subject aroused by Łukasiewicz’s and his pupils’ investigations in the inter-war period has remained alive after the war. Both Czeżowski Logic and Greniewski Elements of Formal Logic include chapters on history of logic from the modern viewpoint, of which the first in particular is worth noting[247]. More specialised inquiry includes the general history of logic and the history of logic in Poland. In the ‘fifties a considerable number of studies dealing with various detailed problems of ancient logic and logic in modern times were published, but no more ambitious work, comparable with Łukasiewicz Aristotle’s Syllogistic or Bocheński Ancient Formal Logic and Formale Logik, appeared[248]. The history of logic in Poland is a new undertaking, started after the war. It has some achievements, though of limited interest, to its credit[249].

By far the highest rank should be accorded to Kotarbiński Lectures on the History of Logic[250]. It is not a continuous and exhaustive survey of the whole subject for which the preparatory work has yet not been accomplished. It is, however, a remarkable achievement, one of the most comprehensive accounts of the history of logic from the modern viewpoint in existence so far.