Storrs McCall
From The Development of Mathematical Logic in Poland between the Two Wars 1945 by Zbigniew Jordan
V. The Discovery of ManyValued Systems of Logic
Whatever value may be attached to the abovementioned results, the discovery by Łukasiewciz of manyvalued systems of logic stands out against all of them. Without any doubt it is a discovery of the first order, eclipsing everything done in the field of logical research in Poland.
It is well known from some remarks made by Łukasiewicz in his book The Principle of NonContradiction in Aristotle . . . as well as from some personal statements, that l'ideéeforce leading him to the discovery of manyvalued systems of logic was the conviction of the indeterminism of future events. The idea of manyvalued systems of logic, however, took shape only in 1920 with the introduction of truth matrices. In the same year the threevalued system was originated; that is to say, the truth matrices determined, the definitions of primitive ideas formulated, and the directions of interpretation indicated. The generalized manyvalued systems (of any denumerable set of values) came two years later (1922). In the course of time it became apparent that they form a general scheme . . . (etc)
(in McCall 1967, p. 389).
* * *
It seems safe to say that manyvalued logic has grown out of the stage during which it was just a new field of research, its results leaving now no room for doubts as to its validity. It seems also very likely that the implications of the discovery concern such widely different problems as the theory of probability, Brouwer's intuitionist logic, quantum mechanics, the discussion about universal strict determinism, etc. But it is difficult to fully realize now all the consequences of Łukasiewicz's discovery. Investigations in this direction have gone little beyond the first shocking conclusions that logical truth has got a multiform character and that there is a variety of ways in which it may be considered. In some respects the discovery and the foundation of manyvalued logic makes us think of the shattering blow dealt by the discovery of the NonEuclidean geometries to the deeplyroted conviction that there is one and only one way of constructing the spatial reference frame of our experiences. Similarly it was supposed that a consistent deductive system must follow the Aristotelian pattern, in accordance with his most general 'laws of thought'. In particular it was believed that a statement must be either true or false. In some circles there arose doubts concerning this principle when Brouwer constructed definite examples of mathematical theorems—dealing with 'the infinite' as it occurs in analysis—which are neither true nor false. The construction by Łukasiewicz of a selfconsistent deductive system in which the proposition 'a statement is
either true or false' no longer holds turned the balance definitely against the Aristotelian assumption.
VI. History of Logic
This sketch of the development of logic in Poland would not be complete without mentioning the awakening of a lively interest in the history of logic. Łukasiewicz once again was the energetic and influential protagonist. He himself for the firs time drew attention to many so far neglected or misunderstood discoveries made in ancient Greece (the School of Megara, the Stoics) and in the Middle Ages (Duns Scotus, Petrus Hispanus). (Etc.)
* * *
In his researches on the logic of the ancient world Łukasiewicz benefited by the collaboration of several classical scholars. Among them A. Krokiewicz should be especially mentioned ; and it may be added that the help he gave to Łukasiewicz influenced in turn his own work. (Etc.)
Łukasiewicz, whose own studies provided a crushing criticism of Prantl's Geschichte der Logik, demanded that the whole history of logic, till now distorted by involuntary ignorance of the subject and consequently limited to the Aristotelian syllogistic, should be rewritten by someone who had mastered equally modern logic and the technique of historical studies. H. Scholz's Geschichte der Logik (Berlin, 1931), the first step in this direction, was made thanks to the lead of, and in collaboration with, Łukasiewciz and his pupils.
(pages 3945 and 3967).
POLISH LOGIC 19201939,
editor Storrs McCall,
Oxford 1967.
From MANYVALUED LOGIC, 1969 by Nicholas Rescher The question of the semantical interpretation of manyvalued logics is a pivotal issue. For, as we have just argued, without an interpretation that gives some semantical rationale for its truthvalues—somehow connecting them with the conception of the truth status of propositions—a manyvalued system may be an interesting abstract mechanism, but is deficient in its claim to the rubric of a "logic." On our view the problem of semantical interpretation is absolutely fundamental in the study of manyvalued logics. One very able logician has written: The problem of finding an interpretation for the truthvalues is still far from a satisfactory solution, at least in the general nvalued case. This need not deter us. The abstract nature of truthvalues has been indicated already by Peirce. And in the formal development of manyvalued logic the semantical meaning of truthvalues is quite unessential.^{1} To be sure, if one is interested simply in the formal development of an abstract corpus of symbolic apparatus, the matter of the interpretation of the values involved is quite irrelevant. But if one is to consider such systems as systems of logic—and not as abstract games—then the issue of interpretation becomes absolutely crucial. Otherwise one is forced to take the position of the eminent German logician Heinrich Scholz,^{2} who insisted that one should speak of nvalued calculi rather than of nvalued logics (or logical calculi), or, similarly, to agree with Paul F. Linke,^{3} who holds that we have to do here not with logic, but with logiclike formalisms ("logoide Formalismen"). The critical considerations have been formulated with clarity and force by Z. Jordan: The difficulty with the nvalued systems does not consist so much in technical problems, considerable as these are, as in finding an interpretation of the n "truthvalues" involved in the system. Without an interpretation assigning a definite logical meaning to the n "truthvalues" any given nvalued calculus remains an abstract structure. The importance of studying such structures cannot be denied. But according to the accepted opinion, coming ultimately from Wittgenstein, the value of a logical system consists in providing a set of rules (possessing definite properties) for transforming expressions of a given meaning into other expressions, and thus in revealing their hidden properties and relations. this requirement is not satisfied by an abstract nvalued calculus.^{4} ^{1}Arto Salomaa, "on ManyValued Systems of Logic," Ajatus, vol. 22 (1950), pp. 115159 (see p. 121).
^{2}Heinrich Scholz, "In Memoriam Jan Łukasiewicz," Archiv für mathematische Logik und Grundlagenforschung, vol. 3 (1957), pp. 318.
^{3}"Die mehrwertigen Logiken und das Wahrheitsproblem," Zeitschrift für philosophische Forschung, vol. 3 (1948), pp, 378398, 530536. Also cf. his "Eigentliche und uneigentliche Logik," Methodos, vol. 4 (1952), pp. 165168. Z. Jordan, "The Development of Mathematical Logic in Poland between the Two Wars" in S. McCall. (ed.), Polish Logic : 19201939 (Oxford, 19670, pp. 347397 (see pp. 393394). New York, etc. : McGrawHill 1969.
