Expressions	Connotative	Non-connotative
with symbolic function	‘man’ ‘green’ ‘the possessing by every man of the property of mortality’	‘object’
without symbolic function	‘round square’ ‘centaur’ ‘the possessing by every man of the property of immortality’	‘for’ ‘abracadabra

 In particular, ‘object’ is a symbol of anything, and is non-connotative because it cannot be defined per genus proximum et differentiam specificam unless we fall into a regressus in infinitum, since ‘object’ is synonymous with ‘being’, and ‘being’ is summum genus. Expressions may or may not have the property of symbolic disposition: an expression which has this property only seems to symbolize something, no matter whether it does or not. Sentences are those expressions that all have disposition of symbolizing relations of inherence. A sentence is true when it has symbolic function, it is false when it has not; the symbolic function of a sentence depends on the symbolic functions of the component terms. True sentences of the form ‘a is b’ symbolize relations of inherence. The sentences

(1) ‘Every man is mortal’

(2) ‘A hippocentaur possesses the property of horseness’

are linguistic expressions with symbolic disposition, that is both have disposition of symbolizing a relation of inherence, i. e. the possessing by every object denoted by the subject of the properties connoted by the predicate, but (1) has symbolic function, while (2) has not.

S2. The Ontological Principle of Contradiction is the sentence OPC. We may substitute several synonymous formulations to OPC. In order to see if any two sentences are or are not synonyms we need to reduce them to the canonical form ‘a is b’. The convention re-written as follows is an example of such a reduction:

(3) Any sentence of the form ‘x is b ® x is c’ is synonymous with the sentence ‘xb is c’,

that is to say that a conditional sentence of the form ‘x is b ® x is c’ is synonymous with the sentence in canonical form ‘x-with-the-property/ies-connoted-by-b is c’.

S3. From S2 it stems that the sentence

(4) If x is an object, then x cannot have and not have at the same time the property c

is not synonymous with OPC. The proof follows from applying (3) to (4), where b is represented by ‘object’. Since ‘object’ is non-connotative, no object is denoted by ‘xb’, i. e. is the object with the properties connoted by ‘object’; for this reason (4) and OPC, whose subject denotes anything, are not synonyms. The sentences

(5) Every A is B

(6) If something is A, then it is B

are not synonyms, too.

S4. As philosophia prima, metaphysics - as Aristotle indicated - is the system of all the true sentences about all the objects in general.(14) From S3it follows that metaphysics can be built not as a system of conditional sentences, but as a system of categorical sentences. However, metaphysics has nothing in common with sentences about the so-called ‘general objects’. Conceptions about general objects lead to non-objectual speculations, and we might get rid of those conceptions once and for all by the following proof. Let a ‘general object’ be an object which is general with respect to a certain group of individual objects. Such an object may possess only those properties which are common to all the individual objects corresponding to it, for instance triangularity for the ‘triangle in general’ which does not possess equilaterality, or isoscelesness, etc.

Proof.

(Premise) A certain object Pk (15) is a general object corresponding to the individual objects P'1, P'2, P'3... P'n.

i. For every individual object P'k it is always possible to find a property pk not common to the individual objects P'1, P'2, P'3... P'n.

(I) The general object Pk has not the property pk.

ii. The individual object P'k having the property pk does not possess the property of not possessing the property pk; otherwise, if it were non-possessing pk, it would be a contradictory object, because it would be an object possessing and at the same time non-possessing the property pk.

iii. The property of not possessing the property pk is not common to all the individual objects P'1, P'2, P'3... P'n since the individual object P'k possesses the property pk.

(II) (For this reason) the general object Pk has not either the property of not possessing the property pk, i. e. it is not non-possessing the property pk; that is, it possesses the property pk.

From comparing (I) and (II) it turns out that the premise leads to a contradiction. Thus the sentence that a certain object Pk is a general object is false, that proves at the same time that no object is a general object.

S5. Establishing linguistic conventions has nothing in common with conventionalism: linguistic conventions are not indemonstrable sentences about objects and properties over which one has no power, but true sentences about states of affairs (stany rzeczy) created by whoever establishes them.

S6. OPC is a true principle and it can be proved. The proof is divided in two parts:

I. Proof that every object is non-contradictory;

II. Proof that the sentence ”Not every object is non-contradictory” is a priori false.

Lesniewski versus Lukasiewicz

The real heart of Lesniewski’s Attempt2 is the semantic analyses presented and the theoretical tension associated by Lesniewski with OPC. Lesniewski requires that if a sentence ‘a is b’ has to be true, the subject a must have symbolic function (or must be not empty or non-denotative or non-objectual);(16) therefore the sentence

(7) A hippocentaur is a horse

is false, because ‘hippocentaur’ is an empty name. In a remark which advances the conclusions of the Attempt2, from this point of view the most important of the paper, Lesniewski proclaims his disagreement with Lukasiewicz’s statements, according to which ‘hippocentaur’

denotes truly something of non-existing, but it is not <expression devoid of meaning

and ‘the square built by rule and compasses and identical as regards the surface area to the circle of a radius of 1’ denotes ”an object with contradictory properties”. (17)

Classical examples of contradictory objects are ‘wooden irons’ [...] ‘square rounds’ or ‘round squares’. Some regard these funny combinations of words as empty sounds, sounds devoid of meaning. As to me, I deem that they are not simply empty sounds, like ‘abracadabra’ or ‘mohatra’, but yet they mean something. In fact it is possible to predicate about the round square that it is a round, that it is a square, a contradictory object, etc., while it is not possible to predicate something about ‘abracadabra’, because this word does not mean anything. [...] ‘the square built by rule and compasses and identical as regards the surface area to the circle of a radius of 1’ [...] is therefore a contradictory object, and yet it means something, is something, is an object.(18)

Although also for Lesniewski the meaning of ‘hippocentaur’ and of ‘object with contradictory properties’ is perfectly determined (for instance ‘hippocentaur’ connotes the property of humanity and the property of horseness),(19) neither ‘something of non-existing’ nor ‘object with contradictory properties’ denote any object, because no object is ‘something of non-existing’, since

no object has any property of ‘non-existence’ connoted by that expression <‘something of non-existing’.(20)

I should also remark that Lesniewski’s definition of synonymous sentences differs from Lukasiewicz’s in so far as the first requires the subjects to be not only denoting the same objects, but also connoting the same properties, so that the sentences

(8) Aristotle was the creator of logic

(9) The Stagirite was the creator of logic

are not synonyms in Lesniewski’s view (‘Aristotle’ connotes the property of having the name ‘Aristotle’, while ‘Stagirite’ does not).(21) Moreover, Lesniewski does not discuss the difference synonymousness/equivalence: all the transformations of sentences he deals with are salva significatione, and synonymousness seems to be the relation between sentences he wants to preserve in inferences. The argument in S3. is addressed directly against Lukasiewicz: Lukasiewicz asserts that OPC is synonymous with its conditional formulation (4),(22) since

every general judgement, positive or negative, presents a link between two judgements: ‘Every A is B’ means in fact that ‘if something is A, then it is B’ [...] There is not any doubt about the synonymousness of these forms.(23)

Lesniewski refutes the synonymousness of OPC with (4) for the non-connotativity of ‘object’. On (4) Lukasiewicz founds the sole formal proof of the Principle of Contradiction: (24) (4) is indeed the appendix’s theorem

T30. 1 ® Ø(a Ù Øa)

which is proved on the basis of definition of ‘object’ as ‘something which is non-contradictory’ and is

like all the laws of symbolic logic, only a hypothetical theorem which establishes that if P is an object, then P cannot at the same time have and not have c. But it does not follow from this that P is an object, i. e. is simply an object and is not at the same time a non-object.(25)

To obtain a concrete proof of PCit would be necessary to prove not (4) but

(10) If P is something and is not nothing, then P is a non-contradictory object. (26)

Lesniewski is not seeking for any concrete proof of OPC beside the formal one: on the contrary, to prove OPC means to prove that OPC is a true sentence in so far as accepted conventions and semantic premises allow. Well, for Lukasiewicz the impossibility to prove PC concretely (10) and not only formally (4) - where for ‘concrete proof’ is meant an answer to the question ‘Are there contradictory objects?’ - was opening new and fruitful perspectives to logic. We know actually that from this moment Lukasiewicz was driven to theorize a non-bivalent system of logic (‘non-chrysippean’ he was to christen it later):(27) reality does not prove nor deny PC, so if it is an ordinary theorem and not a principle and less than ever the supreme law of logic,(28) as Lukasiewicz tried to show, other logics are possible.

The disputation between Lesniewski and Lukasiewicz seems from these first remarks to centre in its fundamental features on pure ontology, if it is true that - according to Kotarbiñski’s 1966 definition - Lukasiewicz’s appendix was a treatise of general theory of objects.(29) If the crucial element to be noticed appears to be a purely ontological controversy between Lukasiewicz and Lesniewski , that is the possibility - admitted by the first, denied by the second - that in reality there are contradictory (and fictitious) objects, the controversy ends up by regarding different ways of understanding ‘object’. In this respect there are several matters to be considered from the historical point of view, that lay in the background of this discussion. One should not forget that Lukasiewicz’s position, according to which ‘object’ is that which is something and not nothing - distinct from the object that is something but also exists, so that there are objects which exist and objects which do not exist - recalls on one side immediately Twardowski’s ideas,(30) on the other side has the very redundant ontology of Meinong as background, with the distinction Sein-Sosein. It is not a mystery that Meinong had a considerable influence on the development of Lukasiewicz’s logical ideas. It should be noticed that - as regards the genesis of three-valued logic - Lukasiewicz launched his attack simultaneously against PC and the Principle of the Excluded Middle:(31) in the latter case a non-marginal role was played by meinongian incomplete objects,(32) more than twardowskian general objects, to which - however - the former owed very much. Lukasiewicz not only quotes several times Meinong’s name in the book and in the dense German abstract which he published,(33) but immediately after having drawn up the work on Aristotle he was as privatdozent in Graz, where at that time Meinong was teaching.(34) Meinong’s name appears for the first time in a note Lesniewski added to the Attempt2, where he cites the second edition of Meinong’s Über Annahmen, but presumably Lesniewski knew Meinong’s ideas much before, for one of his teachers, Hans Cornelius, had discussed them in his Versuch einer Theorie der Existentialurteile (1894).(35) Well, one could hardly conceive of a more distant position from Meinong’s ideas than Lesniewski’s. That no object is a contradictory object is a metaphysical claim - which semantically expresses itself as: there are empty names - that one meets in all the works by Lesniewski , and ‘to be something’ and ‘to be a non-contradictory object’ appear for the latter to be one and the same thing, besides the ‘square circle’ is not a non-existing object: it is not an object at all. In Lesniewski , contrary to what Meinong was thinking, the totality of objects coincides with the totality of what is real or existing, hence to be an object, to be existing, to be something, to be an individual object, to be real, to have defined spatio-temporal dimensions are in the final analysis the same thing. In this perspective it has no sense to ask for a concrete proof of PC as distinct from a formal one, as Lukasiewicz did. Moreover, it was not by chance that Lesniewski included his critique against general objects in the Attempt2 (see S4). The nearest target of the critique was Twardowski, guilty for having enriched his ontology of such unlikely objects.(36) If one thinks that in Twardowski general objects are nothing but special cases of contradictory objects, characterized both of them by the fact that they may be presented non-intuitively and indirectly and, furthermore, by their non-existence, (37) the critique finds its natural place in the Attempt2. Besides, the key passage in Lesniewski’s proof is S4. ii., in which Lesniewski shows that in order to build a general object from individual objects one should violate OPC, and since the latter is true, that construction is impossible. For this reason Lesniewski was not considering general objects to be objects violating the ontological tertium non datur. Undoubtedly it is true that - as Küng wrote - Lesniewski’s argument is applicable only to concrete objects, since an abstract object (a class, a universal idea) [...] cannot be defined as an object which possesses the properties of the concrete individuals subsumed under it, because an object must possess properties that are not assignable to any of the individual at issue”. (38) Anyway, on one hand ‘to be constructable from individual objects’ seems to be Lesniewski’s requirement for an object to be admitted in his universe, on the other hand Lesniewski’s aim, as a matter of fact, is precisely to exclude general objects from reality, just as there are not contradictory objects in the constructions of thought, of which - as to Lukasiewicz - Russell’s antinomy was a sample. But, according to Lesniewski , since there is only one ontological level and only one existence

(spatio-temporal), to exclude something from reality is to deny it tout court, for there is not any other world or realm where this something could be. It is easy to see how much Lukasiewicz does not agree with these ideas:

Logical and ontological principles are not only surer, but also more general than metaphysical principles; in fact they regard equally metaphysical beings, constituting the essence of the world (istotê) as the objects of experience and creations of human intellect which do not really exist, in general everything that is something and not nothing. If Aristotle’s principle of contradiction is only a metaphysical law, then it would not be improbable as of now the assertion that its logical and ontological meaning is not great. (39)

The distinction Lukasiewicz draws between ontology and metaphysics, which seems to be a difference between ”possible structures of beings [...] and the research on ontology as realized in ‘our world’”,(40) is clearly rejected by Lesniewski . Lesniewski accepts the distinction between logical and ontological principles, but certainly for him there is not one between metaphysical and ontological ones: ontology and metaphysics are interchangeable names to speak of the system of the sentences about all the objects in general, where ‘object’ always stands for ‘existing object’. Lesniewski does not claim LPC to be equivalent with OPC ”since they correspond to objective facts”. On the contrary, they are to be kept rigorously separated: an ontological principle is about all the objects in general, while a logical principle is about sentences, which are only some of the objects. For instance in the Critique it will be clear that the Ontological Principle of the Excluded Middle is true, but the Logical one is false. Lesniewski was to write that between ontological and logical principles there is a certain kind of ‘correspondence’, unfortunately not specified by Lesniewski better than it is

(thanks to this particular sharing-out of ontological/logical, Lesniewski could theorize the idea of a hierarchy of languages, which in principle seems to be infinite, that must have played a crucial role in the development of Tarski’s semantic inquiries).(41)

As Woleñski claims, a fundamental thing in Lukasiewicz’s book is his ontologism, that is a strongly ontological view of logic thanks to which logic always has an ontological interpretation.(42) It would be according to this feature that OPC and LPC are said to be equivalent, although one should accept the hypothesis that Lukasiewicz was not to believe in a ‘true’ logic. Lesniewski’s ontologism seems to have been stronger than Lukasiewicz’s, even at that period, and to have been kept on that way later, since actually Lesniewski’s research shows itself to have always been the pursuit of The True Logic. The issue recalls the large place the discussion about conventionalism has in the Attempt2 (see S5). Lesniewski introduces his logico-semantic ideas by means of what he calls ‘conventions’, pointing out carefully that they are true sentences about objects created by whoever establishes them, and not indemonstrable sentences about objects over which one has no power. The polemic against Poincaré and conventionalism as a matter of fact seems to be directed at Lukasiewicz, in accordance with whom the ethico-practical worth of PC is similar to that of the laws of Euclidean geometry:

Although the proof of the principle of contradiction is not complete, we should not despise it. Also for other principles we have not better proofs. The laws of geometrical figures, just as the principle of contradiction, base themselves on definitions, and we are right in doubting the truth of them in application to real figures as we doubt the principle of contradiction in application to the real world. We do not know in fact whether the definition of Euclidean space corresponds to real space, nor have we guarantees that the definition of object corresponds to real objects. But since in applying these laws to reality we do not meet any obstacle, we make use of them without scruple and we will act in this way as long as we succeed in doing it.(43)

It seems, however, that Lukasiewicz was not thinking at that time that the choice of one or other logic was a matter of convention (he neither had built a system itself of ‘non-Aristotelian logic’, yet).(44) And still in 1936 he was convinced that the choice at issue depended on experience.(45) But perhaps the accent he put on the practical worth of PC drove Lesniewski to stress his distance from conventionalism, in any case. While it is easy to become aware of the fascination the parallel - to which he was to be faithful for many years -(46) between non-Euclidean geometry and the ‘new logic’ exerted on Lukasiewicz, noticed among others by Kotarbiñski,(47) Lesniewski is not attracted by the new logic - be it ‘symbolic’ or ‘non-Aristotelian’ - nor would he still have been for years. Lesniewski’s efforts will always be directed to a modernized traditional logic,(48) for the moment conceived in a non-symbolic language. Lesniewski’s shift to symbolic logic - which was less dramatic than is commonly believed to be

-(49) did not signify anyway a shift to ‘non-Aristotelian logic’, too. Lesniewski in this sense was very conservative, and that sort of old-fashioned flavour his logic emanates - entangled with remarkably modern and far-seeing peculiarities - is due to his faith to traditional logic. As clear from chapter XV of On the Principle of Contradiction in Aristotle, Lukasiewicz started in 1910 to get interested in the theoretical meaning of a system of non-Aristotelian logic, whose possibility was connected, as said previously, with the dethronement of PC from the royal chair of the Supreme Principle of Logic. One cannot evoke such matters without remembering that ‘non-aristotelian’ logic is linked not only with the debate on PC, but equally with that on the Principle of the Excluded Middle and the Principle of Bivalence, already noticed previously. Nevertheless, since the issue would deserve an entire paper, which should include at least Kotarbiñski’s contribution,(50) I will not consider it in detail. I will just recall some philosophical traits connected with non-bivalent logic which are strictly related to the points presented here. When Lukasiewicz announced the discovery of the three-valued system of propositional calculus, he emphasized that this system ”has, above all, a theoretical significance as a first attempt to construct a system of non-Aristotelian logic”.(51) As Jordan wrote, ”whether it may be shown to have also a ‘practical significance’ cannot be decided until the consequences of the principle of trivalence are investigated in their relation to empirical knowledge [...] The question of the application of the trivalent system of logic, of finding a set of objects in which the axioms of this system are satisfied, is a distinct problem and independent of the theoretical discovery which should be judged by itself, irrespective of its application”.(52) On this point Lukasiewicz and Lesniewski had the opportunity of showing in the clearest way their very different opinions. In 1938 Lukasiewicz delivered a lecture to the Circle of Scientists in Warsaw, ”Genesis of three-valued logic”. Lesniewski took part in the discussion and his words are the sole evidence we have of his ideas on many-valued logics.(53) Lukasiewicz outlined the discovery of trivalent logic saying among other things that the importance of polyvalent logic was overcoming that of non-Euclidean geometry, and that it showed that ”non equivalent ways to speak of reality” were possible. The fundamental idea in the birth of three-valued logic was adding a third value to the matrix of bivalent logic, with the proviso of finding an intuitive interpretation of it. Without this,

if there had not existed at least a shadow of possibility to interpret intuitively this third value, then trivalent logic would not have been born. The author would have been accused of having had a thought devoid of sense.(54)

The interpretation Lukasiewicz had in mind was linked with Aristotle’s Perihermeneias and sentences on future contingent facts, that were in his view neither true nor false. Lesniewski contrasted this position as strongly as he contrasted Lukasiewicz’s non-existent objects in the Attempt2. For him the third value had no sense, because ”no one had been able until now to give to the symbol ‘2’ introduced in trivalent logic’s matrix any intelligible sense, which may ground this or that ‘realistic’ (rzeczywisto¶ciowej) interpretation of this ‘logic’”. Lesniewski declared never to have met in science any situation such as had required an integration of ordinary calculus of propositions that followed from the introduction of any third logical value in argumentations. Lesniewski was arguing that any ‘intensional function’ such as, for instance, ‘is possible that P’ had to be ‘de-intensionalized’ in order to be examined on the basis of extensional and bivalent logic, since he did not know any system of intensional logic that on his opinion was satisfactory. Lukasiewicz’s answer was particularly meaningful: he explained his end had been to build a system of pure logic without consideration for the applications it could have, although remarking his feeling obliged of giving an intuitive interpretation of it. Finally, Lukasiewicz disclosed the real issue of the disagreement with Lesniewski , that is indeterminism and Principle of Causality:

If there existed in the world an omniscient man [...] he could not infer, basing himself on the laws of nature, that tomorrow there will or not will be a sea battle, if it were not conditional already now; besides, he could not state if such a battle took place in the past, if its consequences had notlasted till now. At that moment, thus, the sea battle passes into the ‘realm of possibilities’, and this is not because we do not know anything about it, but because this is just the structure of the world.(55)

Here I am obliged to leave out of my account a lot of things, Lesniewski’s views on the subject included, which are chiefly contained in his ”Is Truth only Eternal or is it also Sempiternal?”; I limit myself to remark that the most important point in this respect is once more an exclusively ontological controversy: for Lesniewski there are not indeterminate sentences (56) in the structure of the world which symbolize undecided facts as future contingent ones - which, furthermore, are not contingent at all. In Lesniewski objects seem to be set up ab aeterno in space and in time:

it is already now true that [...] I shall choose this rather than another profession, that of two crossroads I shall take the right rather than the left one, that at a given moment a certain thought will cross my mind as summoned by my attention, that at times I will give, refuse, keep or break my word of honour.(57)

Lukasiewicz’s opposition ontology/metaphysics kept on remaining still valid when he passed from a local (the inquiry on logical laws) to a global understanding of logic

(the study of logical systems). Ontological pictures of the world vary according to the system of logic one chooses: yet the world itself, from a metaphysical point of view, is as it is, and in the real choice one should be guided by experience, not by logic.(58) Lesniewski’s post 1920 logical systems were built with a completely different theoretical attitude: for him there was just Our World and The True Logic, which was just Lesniewski an, constructed on axioms which he believed firmly to be true, although being uncapable to explain why it was he believed so.(59)

Russell, Chwistek and the beginnings of Mereology

Lesniewski was right in ascribing to On the Principle of Contradiction in Aristotle an importance that largely overcame his declared dislike for Lukasiewicz’s positions. In fact the work did not cause a sudden shift of his thought, but left a long-lasting mark in the development of his ideas: he met Russell’s antinomy of the class of the classes which are not subordinated to themselves for the first time in Lukasiewicz’s book. Everyone who knows even very little about Lesniewski is aware that Mereology was born in consequence of the attempt of solving the antinomies of set theory. But maybe few know the whole story, which starts with the Attempt1. As already seen, Russell’s antinomy was regarded by Lukasiewicz as a contradictory object, but Lesniewski at that time was deeply convinced that there were not such objects, and probably was not really interested in Russell’s antinomy until he actually tried to analyze it. Lesniewski wrote his paper on Russell’s antinomy, ”Is the Class of Classes not Subordinated to Themselves Subordinated to Itself?” (1914) after the Critique. The latter criticizes the logical principle of excluded middle on the basis of the convention which he called ‘The Restricted Principle of the Excluded Middle’, i. e.

RTND A sentence with denotative subject and connotative predicate is true if and only if its singular contradictory is false,

which was already elaborated in the Attempt1. In the Critique Lesniewski quotes a paper by Leon Chwistek, ”The principle of contradiction in the light of Bertrand Russell’s more recent inquiries” (1912), and, indeed, in the Critique seems to take most of the materials from Chwistek’s paper as a starting point of his analysis. The important issue for the present ends in the Critique is the treatment of sentences with empty subjects: given Lesniewski’s theory of truth from which RTND stems, antinomies like Nelson-Grelling’s or Meinong’s Paradox could be solved simply by showing that the sentences which contradict themselves are both false, that means by showing that their subjects are empty. Lesniewski considers a series of antinomies which not only are exactly the antinomies Chwistek presents in his paper, of which the last and the most important is Russell’s one, but even the pages of the works quoted by Lesniewski are the same quoted by Chwistek.(60) Although there would be a lot of important things to notice about the remarkably pioneering solution of Epimenides’ Paradox, I should notice only that Lesniewski solved it more or less in a similar way, putting in addition a restriction to connotative self-referential names.(61) Well, the impression one has in reading Lesniewski’s Class of Classes is that it is actually the last chapter of the Critique published separately. Lesniewski’s approach to Russell’s Antinomy is the same as all the others solved in the Critique: he tried to show that the Antinomy was based on sentences with empty subjects. The brand new fact was that in this case it was not enough to restrict the expressive power of language, as in the solution of Epimenides’ paradox, though very brilliant. To show that ‘the class of classes not subordinated to themselves’ was an empty expression required to specify which kind of object was understood by ‘class’. And that was the birth of Mereology. So Lukasiewicz was an important source for Lesniewski’s approach to formal logic, but one should also consider the importance Chwistek’s paper had in this respect, and first of all one should not despise the idea that it was Lesniewski’s conviction that there were not contradictory objects in the one world there was that gave rise to Mereology.(62)