Jan Lukasiewicz. Selected works

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The two propositions are different from one another, and hence by the fact that Balzano did not know the concept of indefinite proposition it is not smprising that they have different truth values. One of the most error. Should it be possible to formulate and to solve problems occur- important is expressed by the following remark by Balzano: "It is self- ring in the theory of truth values without the concept of variable and evident that the validity of a sentence must depend on which ideas of indefinite proposition, then such a procedure would even have to be and how many are considered chanfoeable".

Entia non sunt multiplicanda praeter remark by the example: if in the proposition: "this triangle has three necessitatem. But if, along with the ical variable, play an important role not only in probability theory idea "this" also the idea "triangle", or instead of those two the idea but also in logic in general. All the laws of formal logic are formulated "side", is taken as changeable, then the degree of vilidity of the propo- and proved with the help of indefinite propositions. For instance, the sition turns out to be quite different, since in addition to true propo- law of the conversion of general negative propositions is: from the sitions we also obtain false ones.

In these propositions occm the logical validity and that of truth value, is strongly borne out if we select examples variables A and B, which may denote all possible objects; hence the propositions themselves are indefinite and can be neither true nor constructed by analogy to those which I have used in the "theory of truth values". According to Balzano, the true definite proposition: false, even though they have no :truth values, which can be calculated, "6-is diviS1ble by 3" and likewise the false proposition: "5 is divisible since the ranges of the values of A and B are not strictly outlined.

But the degree of validity of the these propositions "man" and "angel" are replaced by any other terms, 23 Ibid, Vol. II, p. A misprint seems to have crept into the text, which should then the propositions obtained in this way are always either both true read "mehrere" many instead of "wahre" true. Bolzano, like Aristotle, considers truth and falsehood as exclusive properties of sentences, but not of ideas. I, or both false. If we refuse to accept proposition. That the conceptual truth, e. Mathematics has developed only when former, can be proved conceptually.

But then the existence of the indefinite letters, that is variables, have been introduced instead of object A is not only possible, but also necessary. Hence if we want definite numbers, and the foundations of algebra have been laid in this to define pure possibility, which would be confused neither with neces- way. It need not be mentioned that at present mathematics would :riot sit-y-nor-with--aetttattty,-we-must assume that the second proposition, be possible without the concept of variable; but the concept of mathemat- "A exists", is also not a conceptual truth.

But it can never be proved ical variable falls under that of logical variable. Hence at least in mathe- a priori that out of the two contradictory propositions: "A exists" matics it is impossible to do without the concept of variable and that and "A does not exist" neither the former nor the latter is a.


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Bolzano's objects falling under the concept A some verify the one proposition procedure can be used as a striking testimony of the fact that the same and some the other, If a person wants to prove, for instance, that neither is also valid in logic: without realizing the lack of consistency in his of the two propositions: "man errs" and "man does not err" is a pure procedure, which he thus accepts, Balzano formulates all logical laws conceptual truth, he must find both human individuals who err and with the help of indefinite propositions, which he denotes sometimes who do not.

But then the existence of erring and non-erring men is not by single letters A, B, C, If we want to sort pure possibility out of etc. It is possible, and only these reasons can be eliminated neither from logic nor from science possible, that "the man x errs", but it is neither necessary nor real, in general, they prove even more necessary when it comes to the expla- since there are erring and non-erring men. It is obvious that Bolzano's nation of possibility and probability.

In my opinion, the essence of pos- definition of the concept of possibility does not lose its validity, since sibility cannot be grasped if it is not reduced to the concept of vari- if there are men who err then the sentence "there are no erring men" able.

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Balzano failed to fathom the essence of possibility, although is certainly not conceptual truth. But in addition to Bolzano's negative to explain that concept he chose the path which could have led him condition other, positive, conditions are necessary too in order to ex- to his goal. According to rum, the existence of an object is called possible plain the. These positive conditions consist if it is not impossible. But an object A is impossible if the sentence in the acceptance of logical variables and the assumption that there are "A does not exist" is a pure conceptual truth.

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For instance, we say indefinite propositions, which need not be either true or false. On the other hand, it is valid for the concept of probability. Balzano sensed clearly that there possible that a man errs, since there is no conceptual truth which denies is close relationship between the concept of the validity of a proposition the existence of an erring man.

Since, however, the concepts of logical variable tions to be found in Balzano concerning the concept of possibility. Had and of indefinite proposition were not known to him, he did not find he advanced his analysis of this concept somewhat further, he would the correct solution. Balzano defined the concept of -probability as undoubtedly have come across the concepts of variable and of indefinite follows: "We consider I take absolute probability is confused with relative probability, but also the liberty Let A and B stand existence of indefinite propositions.

Copsequently, definite propositions for the following definite propositions: "18 is divisible by 2" and "18 is must be treated as examples of probabilistic propositions. Moreover, divisible by 3". The proposition M might be: "18 is divisible by 5". I am convinced that Bolzano would have changed his mind in view In all these propositions the idea "18" is treated as changeable, and of this consequence, if he had thought.

Grelling's probability theory have sense when the definite numbers 18 or 10 are replaced by the var- More or less at the same time that I first presented my probability. Then the problem is: to find the relative truth value of the theory at the meetings of the Polish Philosophical Society in Lwow, indefinite proposition "x is divisible by 5" with respect to the indefinite the very interesting and valuable paper by Kurt Grelling was published propositions: "xis divisible by 2" and "xis divisible by 3'', under the under the title Die philosophischen Grundlagen der Wahrscheinlich- assumption that x ranges over all two-digit integers.


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  4. This problem can keitsrechnung. They are propositions especially those announced in recent decades in the German literature which have been well confirmed by experience so that objective validity on the subject. For this purpose he selects three works which seem to must b. Yet the judgements of the calculus of proba- him to exceed all others in importance: the works of von Kries and bility cannot be referred to objective definite events; hence they must Stumpf, which I have quoted above, and A.

    Sentences which contain variables but are either falls upon a table, it turns heads up". He presents the relationship which should hold the consequent would have to be: "it turns heads up or tails up'', or between indefinite judgements and probabilistic propositions in a way else the antecendent would have to be: "when a coin falls upon a table which points to a lack of clarity of his basic idea. He states: "We may and its tail-side forms an angle of less than 90 degrees with the surface say that the state of things formulated in an indefinite judgement is of the table". Grelling's comment on this presentation of Fick's fundamental idea This is not far from saying that in all other cases it is more or less is that Fick in fact by incompletely formulated hypothetical judge- probable".

    In Grel- That this obscure point is not due to stylistic clumsiness, but has ling's opinion, Aristotelian terminology is very poorly suited to de- deeper reasons, can be seen quite clearly from the way in which Grelling scribe the concepts mentioned above.

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    We find the following formu- Without engaging in a discussion of whether Fick in fact did know lation: "The question is about the probability of the in. Let clearly, or whether terminological helplessness was in his case combined the following definite judgements be given: 'If A occurs, then there with a lack of knowledge of concepts, I shall now outline Grelling's occurs one and only one of N equiprobable cases, among which there opinions. Grelling thinks In this definition it is, first, unclear to me why Grelling calls the that we could come to terms with the subjective interpretation if the on- former judgement indefinite and the latter definite.

    In the former occur ly point were to justify probabilistic propositions which are concerned two indefinite terms, A and B, but the same terms reappear in the latter with every-day life and in fact are usually nothing but expressions 29 Ibid. Now if A and Bare vari- 1 Every correct scientific theory has its forerunners. It is only gradually ables in the first judgement, they must also be so in the second. Or is, I that the human mind forces its way through to a clear comprehension perhaps, the second judgement to be interpreted like an identity which contains variables but must nevertheless be true?

    The fact that the logical theory of probability also has its own forerunners, and such eminent ones at that, allows -! This obscure point is, however, far less important than the way in us to hope that the path toward the resolution of the problem of prob- which Grelling defines the :measure of probability. Although he is correct ability has ultimately been found.

    Since Grelling does not possess that concept, he :must revert to the old, obscure concept of "equiprobable cases'', a concept that is ridden with difficulties. Finally, there is for him no other way out than to accept Kries's theory.

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    Both concepts had been known previously, but they have not always been interpreted clearly, and never associated with one another. Representatives of :modern mathematical logic, such as Frege and Russell, knew the concept of indefinite proposition, even if they did not always treat it as a judgement; yet none of them has tried to apply.

    I I l: that concept to probabilistic propositions. Bolzano formulated the concept of validity, which corresponds to the concept of truth value, and used it in his own way, though not quite satisfactorily, to explain the concept of probability; but he did not know the concept of indefinite proposition.

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    On the other hand, Grelling knew the last-named concept I and applied it to probabilistic propositions, but the concept of truth value escaped his notice. This logical improvement of mathematics is, in my opinion, to be I. Professor Zaremba's definition of magnitude. On the formu- J ascribed first of all to the unprecedented development of formal logic. On theorems "devoid ofcontent". On the principle: "the inequal- j also mathematicians. Other logical relationships of logic; he was followed by De Morgan, Peirce, and SchrOder, who between the properties defining the formal properties of the relations "equal to", "less than" and "greater than".

    How the formal properties of those relationships I improved that algebra and formulated the theory of relations; finally, in recent years Russell, Frege, and Peano and his school applied the are to be defined. A criticism of Professor Zaremba's definition of magnitude. Owing How the concept of magnitude is to be defined. How to present the theory of magnitude to make it easily comprehensible and precise. The importance to the work of these scientists, traditional formal logic, which has had and the tasks of contemporary formal logic.

    I tyczna Theoretical Arithmetic. His work, published in Cracow in Only some of its concepts, often distorted, are penetrating the circles by the Academy of Learning, met with the fullest approval from mathema- of those scientists who are not professional logicians. Much time will ticians. But I can appraise that work from the standpoint of logic, the more I That is why I was not astonished when, in a book written by a learned professor of the Jagellonian University, I found none of the names so as some of the problems it raises are closely connected with logic.

    I mentioned above, but did find in many places opinions and methods The author himself, when referring, on page XV, to advances in the- I which, from the standpoint of contemporary logic, are inexact or even oretical arithmetic, says: "These advances consist partly in the intro- j erroneous. Hoborski, docent of the Jagellonian University, published To justify my last statement I shall analyse critically in the present in Wektor Vector , Vol.

    III, No. It is r paper the definition of the concept of magnitude as formulated by included here in a largely abbreviated form, which retains only those parts which Professor Zaremba, discuss some logical and methodological issues are of general theoretical importance. The sections concerned purely with the criticism of Zaremba's book have been left out as being now of merely historical interest for the Polish reader, who can read Zaremba's book in the Polish original. The table of I connected with that subject, and offer my own tentative definition of that concept.

    All the principles sitions P, R, S, Should s, In other words, all the values a principle be derivable from others, it could be proved on their basis, of the variables which verify the propositions P, R, S, Every theorem verify the proposition Z. It is always necessary to investigate most carefully which principles are necessary to the proof of a given theorem and which are l equal to C", for if we assume that the variables A, B, C range over the class of natural numbers, there are no values of these variables not.