Abstract
I focus on Leśniewski’s papers written before he published formal work on his systems. I begin with a discussion of what Leśniewski called linguistic conventions: various postulates introduced to elucidate the meaning and role of certain natural language devices. Then I show how he used them to draw conclusions about existential propositions, the principle of contradiction, the principle of excluded middle, the eternity of truth and the existence of abstract objects. I also explain his approach to paradoxes: Nelson–Grelling’s, Meinong’s, Epimenides’ (the Liar), and Russell’s. All Leśniewski’s arguments are carefully reconstructed and critically assessed.
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- 1.
For instance, Tarski-style hierarchization of natural language seems to be a regimentation of the same sort.
- 2.
Leśniewski’s father was a railroad engineer.
- 3.
“I divide all linguistic expressions into connoting and non-connoting; I adopt the expression ‘connoting expression’ to denote expressions that can be defined, and the expression ‘non-connoting expression’ to denote expressions that cannot be defined. The expressions ‘man’, ‘green’, ‘square circle’, ‘centaur’ are examples of connoting expressions; the expressions ‘to a man’, ‘well’, ‘at’, ‘abracadabra’, ‘object’, ‘every man is mortal’, etc. are examples of non-connoting expressions.” (1911, 31)
- 4.
The similarity between Leśniewski and Aristotle can be observed at least insofar as we speak of the form of a definition:
“Because there is nothing else in the definition besides the primary genus and the differentiae.” (Met., Z, 1037\(^{b}\), 29–30) Aristotle’s approach is more metaphysically involved, because he attaches special status to 
(the closest kind) and 
(specific differences). These notions are strongly connected with his metaphysics. - 5.
“I divide all linguistic expressions into those denoting something and those denoting nothing, in other words symbolizing something and symbolizing nothing or expressions which are symbols and those which are not. I call the relation of expressions to the objects denoted (in other words—symbolized) by these expressions, a symbolic relation. I call that property of an expression which consists in its symbolizing something, the symbolic function of that expression. An expression which denotes something, or which possesses the symbolic function, can be exemplified by the following: ‘man’, ‘green’, ‘object’, ‘the possessing by every man of the property of mortality’, ‘every man is mortal’, etc. The expressions which do not denote anything, or do not possess symbolic functions, can be exemplified by the following ones: ‘abracadabra’, ‘square circle’, ‘centaur’, ‘the possessing by every man of the property of immortality’, ‘every man is immortal’, etc. The expression ‘square circle’ does not possess a symbolic function because no object is a square circle, in other words there is no such object as could be symbolized by the expression ‘square circle’; thus the expression ‘square circle’ symbolizes no object, in other words symbolizes nothing. The expressions ‘possessing by every man of the property of immortality’, ‘every man is immortal’ do not possess a symbolic function because no man is immortal, in other words there is no object that could be symbolized by the aforementioned expressions. Therefore, these expressions symbolize no objects, or symbolize nothing.” (1911, 31–32) It is interesting that among denoting expressions Leśniewski included true sentences, but not false sentences. He was not too explicit about what sentences refer to, but it seems that rather than accepting Fregean Truth, he rather took them to refer to facts or relations of inherence between objects and properties.
- 6.
I add this reservation, because Leśniewski wouldn’t say that, for instance, ’how are you’ has a symbolic disposition just because someone can use this phrase to name an elephant.
- 7.
“I call the property of an expression which consists in that expression’s application or treatment (according to, or against, the adopted linguistic conventions) as one possessing the symbolic function, the symbolic disposition of that expression. Thus, e.g., I say that the expressions: ‘man’, ‘hippocentaur’, ‘every man is mortal’, ‘the possessing by a hippocentaur of the property of horseness’—possess a symbolic disposition when they are applied, or treated as expression-symbols. The first and third of these expressions possess a symbolic function, but the second and fourth do not because no object is a hippocentaur or the possession by a hippocentaur of the property of horsiness.” (1911, 33)
- 8.
“All connoting expressions possessing a symbolic function can be divided into two groups: expressions which correspond with any non-connoting expression symbolizing the same object, and expressions which correspond with no non-connoting expression symbolizing the same object. Thus, e.g., the connoting expression ‘the possessing by every man of the property of mortality’ corresponds with a non-connoting expression symbolizing the same object, namely the expression ‘every man is mortal’. The latter also symbolizes the possessing by every man of the property of mortality and also the expression ‘the possessing by every man of the property of mortality’. Whereas the connoting expression ‘man’ corresponds with no non-connoting expression which would symbolize the same objects as those symbolized by the word ‘man’.
All connoting expressions possessing a symbolic disposition can be divided into two groups: expressions which correspond to any non-connoting expression possessing the same symbolic disposition, and expressions which correspond to no such non-connoting expression.”(1911, 34)
- 9.
Whenever I refer to someone’s work, it is my reading of the text, unless quotation marks are present.
- 10.
“...any proposition possessing a symbolic function symbolizes the possessing by the object, symbolized by the subject of that proposition, of properties connoted by its predicate. This convention implies that propositions can symbolize only the relations of inherence.”(1911, 36) What is somewhat interesting, Leśniewski, instead of saying that this refers only to simple (atomic) propositions adds a footnote which says: “...I speak of propositions in the sense of the ones reduced to the form of categorical propositions with positive copulas and predicates in the Nominative.” (1911, 36) which seems to suggest the claim that all propositions are reducible to propositions of this specific form. This claim however is not essential for further discussion; neither does Leśniewski give a complete set of directions describing how this reduction should proceed for any arbitrary proposition. Therefore, I decided to treat (2.9) as referring to simple propositions only, without assuming the claim about the reducibility of complex propositions (which is implausible anyway).
- 11.
Leśniewski in his early writings used ‘if’ in definitions in the sense of ‘iff’.
- 12.
Some conventions I just quote, if they were originally phrased in a concise manner. If I find a simpler and more accessible reformulation, I use it instead.
- 13.
Probably ‘alone’ should be read as ‘as the only extralogical assumptions.’ “I call false a priori all such propositions whose falseness can be demonstrated by means of linguistic conventions alone or the propositions which can be inferred from those conventions. ...On the analogy of the definition of the expression ‘proposition false a priori’, I define the expression ‘proposition true a priori’. I employ the latter expression to denote such propositions whose validity can be demonstrated by means of linguistic conventions alone or the propositions which can be inferred from these conventions.”
- 14.
“...a proposition with a negative copula can symbolize possessing, by the object denoted by the subject of that proposition, properties connoted by the expression consisting of the word ‘not’ and the predicate of the proposition with a negative copula in question (the negation ‘not’ is to apply to the whole expression that follows it). If the proposition with a negative copula has the form: ‘no etc ...’, then the word ‘no’ will be substituted, in the process of reduction, by the word ‘every’. Thus the proposition with a negative copula ‘no object can both possess and not possess one and the same property’ ...symbolizes the possessing by the object denoted by the subject of that sentences ...the possessing by every object of properties connoted by the expression consisting of the word ‘not’ and the predicate of the proposition with a negative copula—that is, the expression ‘able to both possess and not possess one and the same property’.” (1912, 23)
- 15.
See (Kripke 1980) regarding related issues.
- 16.
See also (Katz 1990).
- 17.
“I have said that the predicate of a positive existential proposition which has been brought to the form of a proposition with a positive copula, does not connote anything, except—at most—the property of being greater than one. I maintain this because such a predicate is synonymous with the words ‘being’ or ‘beings’ which connote nothing else, even though they denote (‘denotation’ as used by Mill) everything. This view conflicts with J.S. Mill’s theory which says the word ‘being’ connotes the property of existing. I consider Mill’s theory wrong because, should the word ‘being’ really connote the property of existing, we could define that word as ‘that which has the property of existing’, or in other words, as a ‘being which has the property of existing’ (since the definition must indicate not only differentiae specificae, but also the genus); this would, then, give rise to an inevitable regressus in infinitum. The word ‘being’ cannot be in fact defined at all; the statement that this word does not connote anything is fully in keeping with this fact.” (1911, 4–5)
- 18.
About the idea that some classifications are natural and some are artificial he says: “One hears, from time to time, of ‘natural’ and ‘artificial’ methods of classification. People who use this form of expression do not usually limit themselves in characterizing particular methods of classification, to the inclusion of these descriptively to either of the above categories; they usually combine such a descriptive characterization of methods of classification with the teleological element of valuation, and they value ‘natural’ classifications higher than artificial’ ones. The origins of the above characterizations of methods of classification, and the positive or negative estimations which accompany these characterizations, can vary immensely from case to case.
Some such cases are determined by various linguistic habits and traditions, others—by more or less well thought out and justified views concerning the problems of theoretical usefulness.
The classification of propositions into analytic and synthetic which I have carried out ...can be, in view of at least one of its consequences, characterized as regarded by some as ‘artificial’. Such an ‘objection’ can in the first place originate from the fact that one of the two classification labels, i.e., that of analytic propositions, comprises two ‘very’ or ‘too’ heterogenous groups of propositions: (1) propositions whose predicates connote any properties but not those connoted by the subjects’, and (2) propositions whose predicates do not connote the properties connoted by the subjects only because they do not connote any features.
I do not consider it my task to tone down all such dissonances if they arise solely from deeply rooted emotional impulses resulting from some linguistic habits, yet I cannot miss the opportunity to provide my classification of propositions with a ‘safety valve’ against objections supported by arguments of theoretical usefulness of my classification.” (1911, 9)
- 19.
For the sake of simplicity, I am putting well-known general concerns about the notion of analyticity aside. See however (Russell 2008) for a defence of analyticity.
- 20.
So, for example, it is a theorem of one of Leśniewski’s systems that \(\exists {a}\,\lnot ex(a)\), i.e. ‘For some \(a\), \(a\) does not exist’.
- 21.
- 22.
The diary is unpublished. The translation from Polish is due to Owen LeBlanc and Arianna Betti. Accessed on Agust 31, 2010 at http://www.segr-did2.fmag.unict.it/polphil/PolPhil/Lesnie/LesnieDoc.html#lukdiary
- 23.
Although Leśniewski did not give a definition of the contradictory of a sentence, from his discussion it seems that this was a syntactic notion: if \(S\) is taken to be a singular term, the sentence contradictory with ‘S is P’ would be ‘S is not P’, and if \(S\) is taken to be a general term, then the sentence contradictory to ‘Some \(S\) is \(P\)’ is ‘Every \(S\) is not \(P\)’.
- 24.
“Let us suppose that I am to answer the question of whether the following propositions are true: ‘every centaur has a tail’, ‘a certain centaur does not have a tail’, ‘every square circle is a circle’, ‘a certain square circle is not a circle’. If we take into account the above analysis, the answer to this question becomes quite easy. Each of the four mentioned propositions is obviously false because the subject of each denotes nothing. The word ‘centaur’ which is the subject in the first two propositions, and the expression ‘square circle’ being the subject in the remaining two—denote nothing because no object is a centaur and no object is a square circle. Thus, no object is such that it could be denoted only by the word ‘centaur’ or by the expression ‘square circle’. These expressions denote no objects, that is to say—they denote nothing.” (1913b, 59)
- 25.
“The points raised ...throw some light on the ‘problem’ of negative propositions. They demonstrate the falsehood of the theory of negative propositions, developed in considerable detail by Sigwart in his Logic and defended by some other modern logicians. According to this theory—the negative proposition ‘A is not B’ is equivalent to the affirmative proposition ‘the proposition ‘A is B’ is false’...Given the propositions ‘the centaur has no tail’, ‘a square circle is not a circle’ ...the respective propositions ...are ‘the proposition ‘the centaur has a tail’ is false’ and ‘the proposition ‘the square circle is a circle’ is false’. The propositions of the type ‘A is not B’ ...are in this case false because ...they have subjects which denote nothing ...For the same reason, the respective propositions of the type ‘A is B’, i.e. ‘the centaur has a tail’, ‘a square circle is a circle’—are also false. If, however, the last two propositions are false, then the propositions stating their falsehood must be true.” (1913b, 59–60)
- 26.
Even though Leśniewski rejected one of the forms of the PEM, he did not divide sentences into true, false and indeterminate—according to him, if a PEM failure takes place, it is because both a sentence and its negation are false. Gelber (2004) suggested that “on the basis of his interpretation of Aristotle, for instance, Leśniewski developed a three-valued classification with true, false and indeterminate as the values” (p. 231). This is not Leśniewski’s view. The charitable reading of Gelber is that Leśniewski is there mistaken with Łukasiewicz).
- 27.
The introduction of such a third value would invalidate Leśniewski’s reasoning. There are some other difficulties to Łukasiewicz account of futura contingentia, though. Alas, they lie beyond the scope of this book.
- 28.
“Każda prawda jest konieczności a̧ , każdy fałsz—niemożliwości a̧ . Gdy co jest bowiem, być musi, bo nie być nie może; gdy coś gotowe jest, a nie jest, to być nie może, bo już s a̧ d o nim twierdz a̧ cy jest fałszem, a wiȩc nie może siȩ stać prawd a̧ . Można wiȩc prościej sobie zapamiȩtać: s a̧ d prawdziwy jest, gdy rzecz jest, a wiȩc gdy jest konieczna, fałszywy—gdy niemożliwa, sprzeczna z czymś, co jest.” (Kotarbiński 1913: 88–89)
- 29.
The word comes from Latin in-(English: in) and -haerere (English: to hang, to stick). The relation of inherence is the relation between a property and an object that has it.
- 30.
“One might say that my proof would be quite valid but for the fault that it does not ‘accord with reality’. To support this objection one might cite a ‘random’, it seems, judgment which is originally true and then ceases to be true, e.g., the judgment ‘Stanisław Leśniewski will die’. This judgment is true for as long as I am alive; when I die it will become false because, when I shall not be here anymore, I shall not be able to die again. By becoming false at the time of my death, the judgment ‘Stanisław Leśniewski will die’ will give way to the true judgment ‘Stanisław Leśniewski died’ which in its turn is false until I shall die.” (1913a, 97–98)
- 31.
Grelling’s paradox is sometimes associated with quite a different reasoning pertaining to heterologicality. However, ‘Nelson-Grelling paradox’ is a term which Leśniewski originally used. Indeed, this is similar to a formulation to be found in Grelling and Nelson (1908).
- 32.
“...if it were true that there are no ‘contradictory objects’, in other words, no objects are contradictory, then it would be true that ‘a contradictory object is not an object’. It can be, however, true that ‘a contradictory object’ is not an object only in the case when a certain object is ‘contradictory’. If no object were ‘contradictory’, then no proposition about the ‘contradictory object’ could be true, including the proposition ‘a contradictory object is not an object’. Thus, if it were true: ‘a contradictory object is not an object’, then it must be also true that a certain object is contradictory. This being so, the assumption made at the beginning that no object is ‘contradictory’ entails the conclusion that a certain object is ‘contradictory’.” (1913b, 62–63)
- 33.
See for instance Simmons (1993) for a critical survey.
- 34.
Another problem is that this does not provide a way out of the Yablo’s paradox. Consider an infinite sequence of sentences \(s_0, s_1, s_2, \ldots \) such that:
$$\begin{aligned} s_0&= \text {`}\forall {x}\,(P_1(x) \rightarrow \lnot Tr(x))', \\ s_1&= \text {`}\forall {x}\,(P_2(x) \rightarrow \lnot Tr(x))', \nonumber \\ s_2&= \text {`}\forall {x}\,(P_3(x) \rightarrow \lnot Tr(x))', \ldots \nonumber \end{aligned}$$(2.65)Assume that the extension of every \(P_n\), for \(n=1, 2, 3, \ldots \), is \(s_n, s_{n+1}, s_{n+2}, \ldots \). So every \(s_i\) says that all \(s_j\)’s with \(j>i\) are not true. Now ask yourself: is \(s_0\) true? If yes, then for any \(k>0\) the sentence \(s_k\) is false. But this also means that for any \(k>1\) the sentence \(s_k\) is false. But this is exactly what \(s_1\) says and hence \(s_1\) is true, which falsifies \(s_0\). Suppose then that \(s_0\) is false. This means that there is a \(k>0\) such that \(s_k\) is true. But we can repeat the reasoning we led about \(s_0\), this time about this \(s_k\) to show that \(s_k\) can’t be true. Hence the paradox. However, no circularity in reference in Leśniewski’s sense takes place: the subject of each sentence refers to all sentences below it. The question whether circularity is involved is sensitive to the notion of circularity involved. See for instance Leitgeb (2002) and Urbaniak (2009a).
- 35.
- 36.
- 37.
This notion of parthood differs from the notion of proper parthood which requires the part to be “smaller” than the object of which it is a part.
- 38.
I translate using the indefinite article: ‘a class of objects’, because the assumption is not taken to imply the uniqueness of a class of objects \(a\).
- 39.
This is a streamlined version of Leśniewski’s argument, the original is less accessible.
“Because there is nothing else in the definition besides the primary genus and the differentiae.” (Met., Z, 1037
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Urbaniak, R. (2014). Leśniewski’s Early Philosophical Views. In: Leśniewski's Systems of Logic and Foundations of Mathematics. Trends in Logic, vol 37. Springer, Cham. https://doi.org/10.1007/978-3-319-00482-2_2
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DOI: https://doi.org/10.1007/978-3-319-00482-2_2
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