marek lechniak...marek lechniak 24 other the late one, dating from 1940’s and 1950’s. basically...

SOME REMARKS ON JAN ŁUKASIEWICZ’S UNDERSTANDING OF NECESSITY*

MAREK LECHNIAK

Modal concepts were the object of J. Łukasiewicz’s interest for many years (from 1918 to the end of his life). Study of Aristotle’s logic supplied Łu-kasiewicz with basic inspiration for conducting research in that area. How-ever, along with the development of Łukasiewicz’s interests and formal skills, his interest in philosophical analyses decreased1. Łukasiewicz’s late works are perfect with respect to their formal features; however, philosophical justifica-tions that accompany the formal results are controversial, to say the least2. In the present article we show some of Łukasiewicz’s views on modal functors and especially on the necessity functor. In Łukasiewicz’s views we deal with sort of two conceptions of necessity (just like two conceptions of the whole modal logic) – one of them was the early one, whose crowning was the article Uwagi filozoficzne o wielowartościowych systemach rachunku zdań [Philoso-phical Remarks on Many-valued Systems of Propositional Calculus], and the

* Translated from: Kilka uwag o Jana Łukasiewicza rozumieniu konieczności, “Roczniki Filo-zoficzne”, 48 (2000), fasc. 1, pp. 195-221.

1 Which caused Twardowski’s criticism of Łukasiewicz in the article of 1921 Symbolo-mania i pragmatofobia [Symbolomania and Pragmatophoby], in: K. Twardowski, Wybrane pisma filozoficzne [Selected Philosophical Writings], Warszawa 1965, pp. 355-363. On the discussion between Łukasiewicz and Twardowski see: R. Jadczak, Logistyka a pogląd na świat. Przyczynek do biografii Jana Łukasiewicza [Logistic Versus the View of the World. A Contribution to Jan Łukasiewicz’s Biography], in: Fragmenty filozoficzne ofiarowane Henrykowi Hiżowi [Philosophical Fragments Presented to Henryk Hiż], ed. H. Zelnik, War-szawa 1992, pp. 40-48.

2 “Both Łukasiewicz and Leśniewski were very critical of the state into which philoso-phy had got itself, after centuries of unending discussions and argument. While Łu-kasiewicz, impressed by the successes of logical researches, advocated new methods in phi-losophizing, Leśniewski went so far as to declare himself an apostate from philosophy. However, those who have know both of them and have studied under them seem to agree in thinking that Leśniewski was in truth more philosophically minded than Łukasiewicz or indeed than any other of his fellow logicians” (C. Lejewski, A Handful of Reminiscences Related to Jan Łukasiewicz (typescript), p. 22).

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other the late one, dating from 1940’s and 1950’s. Basically Łukasiewicz al-ways focused on the possibility functor, whereas he took up the necessity functor either because of the former one, or because Łukasiewicz’s under-standing of this functor may be reconstructed from his analyses of the possi-bility functor3.

I. ARISTOTLE AND MEGAREANS’ UNDERSTANDING OF MODAL CONCEPTS

Since Łukasiewicz often refers to Aristotle’s works as well as to works by other ancient authors, let us remind the reader in several points what some recognized authors determined in this field.

1. The issue of sentences containing modal functors appeared mainly in two of Aristotle’s works: On Interpretation and Prior Analytics. One can speak of roughly4 two meanings of the term “possible” (Aristotle himself distinguishes these meanings in Prior Analytics (e.g. 25 and 37-40, 32 and 18-21):

– “possible” in the meaning of a one-sided possibility, where “possible” is equivalent to “not impossible”;

– “possible” in the meaning of a two-sided possibility, where “possible” is equivalent to “not-impossible” and “not-necessary”. In On Interpretation Aristotle devotes Chapters 12 and 13 to this issue5.

Basically, the first meaning of the term “possible” occurs6 in On Interpreta-

3 Generally it can be said that Łukasiewicz was not very keen on the necessity functor,

and was quite fond of the possibility (and it was the two-sided possibility that was dealt here with) functor. The origin of this emotional aura is probably that Łukasiewicz con-nected necessity with determination (and determinism), which view he opposed, and he connected possibility – with indeterminism (and human freedom). Cf. e.g. J. Łukasiewicz, Wykład pożegnalny [A Farewell Lecture], “Studia Filozoficzne”, 1988, vol. 5, pp. 127-129, or, on the other hand, the last sentences in his book Aristotle’s Sylogistic from the Stand-point of Modern Formal Logic (Oxford 19572, pp. 207-208).

4 Complications in Aristotle’s logical texts concerning this matter are pointed to by I.M. Bocheński (Z historii logiki zdań modalnych [From the History of Logic of Modal Propositions], Lwów 1938, pp. 23-44).

5 In On Interpretation certainly there are more fragments devoted to analysis of modal concepts; it is enough to mention Chapter IX, that has voluminous literature, concerning considerations of the problem of truth and logical necessity of sentences concerning future events. This multitude of meanings of modal concepts seems to be occasioned by the fact that in On Interpretation Aristotle starts analyses – as Ross says – starting from the way

A Few Remarks on Jan Łukasiewicz’s Understanding of Necessity

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tion, and in Prior Analytics – the other meaning (two-sided possibility). Mo-dal syllogistic contained in the work is built with the use of the functor “pos-sible” understood as the functor of two-sided possibility7.

2. Also Aristotle’s understanding of necessity is a rather complicated prob-lem; it can be pointed out that he distinguished: – necessity in the strict sense, that is necessity which is a feature of proposi-

tions, understood as “it is not possible (in the meaning of one-sided possi-

modal expressions are used in the natural language. This is why there we have a mixture of various meanings of modal concepts that is characteristic of the natural language. Cf. e.g. W.D. Ross, Aristotle, a Complete Exposition of His Life and Thought, New York 1959, p. 34. On the various meanings of modal concepts let us also quote here the following Ross’s distinctions: both 1) “necessary” and 2) “not-unnecessary”, as well as 3) “capable of being” are called possible. However, the first one of them meets only one of the conditions of being possible; it is not impossible. It does not meet the second condition (i.e. that it is not necessary) and that is why it is called possible only in the second meaning (i.e. of a one-sided possibility). What is actual may similarly be called possible in the same, im-proper meaning. However, when we turn to the difference between non-necessary and ca-pable of being, we can notice that Aristotle, speaking about the latter one, means the cases, in the world of possibilities (occasions) and change, of the usual (but not unchange-able) possession of a property by an object; and speaking about “not-necessary” Aristotle means the situations when either a rule that is used in most cases does not exist, or such a rule is violated by an exception” (ibid., pp. 34 ff.). Cf. also: J.L. Acrill, Aristotle’s Catego-ries and De Interpretatione, Oxford 1963, p. 149 ff.

6 Although in the list of relations between modal propositions in Chapter XIII (cf. On Interpretation, 22b, 10-28) mixing up of the meanings of concepts happens: implications given in Tab. I and III occur only for “possible” used as a two-sided possibility, whereas implications given in Tab. II and IV occur only for a one-sided possibility. By further transposition I 4 (does not have to be) from III 4 (does not have not to be) Aristotle made the whole table correct; “possibility” is understood then in the whole table as a one-sided possibility – Aristotle did not work out a table for the two-sided possibility. Cf. Acrill, op. cit., where a detailed commentary for Aristotle’s whole table can be found. The phrase “admissible of being” (treated in the Middle Ages as “is contingent”) is here a synonym of the phrase “may be” (“possible”) and that is why it may be omitted from those tables. Cf. A.N. Prior, Formal Logic, Oxford 1955, p. 187. Incidentally, it is a pity that in the Polish edition of On Interpretation there are no footnotes concerning logical interpretation of this fragment of the work.

7 “In the introduction to syllogisms about the premise or premises H (i.e. possible in the meaning that is not precisely defined) Aristotle gives the thesis EMpMNp (An. Priora A 13, 32 and 29 ff.) and then consistently uses it in the course of his exposition of the sub-ject” (Bocheński, op. cit., p. 28). Cf. also e.g. G. Patzig, Aristotle’s Theory of Syllogism, Dordrecht 1968; R. Patterson, Aristotle’s Modal Logic, Cambridge 1995.

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bility) that not…”; for this concept the scholastic principle holds: Ab oportere ad esse valet consequentia;

– logical necessity (according to the terminology proposed e.g. by Bocheński) that is a feature of a necessary connection between the premises and the conclusion in reasoning8;

– necessity (called by Bocheński) hypothetical necessity, that is a feature of any true proposition; when something is, it is necessary; for this concept the principle holds (called by Łukasiewicz “Aristotle’s theorem”) Unum-quodque, quando est, oportet esse; this use of the phrase “is necessary” may also be called temporal necessity, i.e. non-changeability of what has already happened – if something has come into being, it cannot be differ-ent9. 3. The Megarean logician Diodor (called Kronos) had a different view of

modalities. He understands them in the temporal way. He defines “possible” as what either is, or at a certain time will be true, “impossible” as what neither is nor will ever be true, and “necessary” as what both is and always be true. These definitions assume, as was accepted both in ancient and medieval logic, that the same proposition may be true at one and false at another time10.

II. GENERAL REMARKS ON ŁUKASIEWICZ’S UNDERSTANDING OF MODALITY

After these – of necessity selective – remarks concerning understanding of modality by ancient authors let us pass on to analysis of Łukasiewicz’s works. For a beginning let us make an obvious remark: Łukasiewicz’s views in this matter were changing, although till the end of his life he did not give up cer-tain findings he had arrived at in his first works on modalities (e.g. in the so-

8 It is in this meaning that Aristotle often says that something „is necessary of neces-

sity” or even „possible of necessity”. Cf. Bocheński, op. cit., p. 27. 9 This meaning of necessity occurs especially in Chapter IX of On Interpretation,

where Aristotle analyzes the possibility of undetermined future events. Cf. e.g. Acrill, op. cit., p. 133.

10 Cf. A.N. Prior, Diodoran Modality, “Philosophical Quarterly”, 1955, pp. 205-213; idem, Time and Modality, Oxford 1957, pp. 84-93 (Lecture IX).


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called Lvov lectures of 1920)11. These persisting views may be summarized in the following statements:

1. All functors, including modal ones, should be truth-functional. And since modal functors are monadic proposition-forming functors from proposi-tional arguments, and in two-valued logic there can be only four such func-tors, if they do not want to be a trivial repetition of classical functors they have to be functors characterized by more than two-valued matrices12; the difference between three-valued logic and the so-called Ł-modal system being that the four-valued matrices of the latter one are formed by multiplying two-valued matrices of the classical propositional calculus by themselves, so the matrices for equivalents of the classical functors N, C, K, A, E, verify the same theses as two-valued matrices.

2. The former statement is connected to the fact that in thinking about modal functors Łukasiewicz attached a great significance to the protothetics thesis CδpCδNpδq that is an expression of bivalency: if something is stated both about p and non-p, it is also stated about any sentence13.

3. Various meanings of modal concepts (e.g. various understandings of pos-sibility or necessity pointed out by Aristotle) may be united in one modal logic functor (respectively: possibility functor or necessity functor); it is an expression of some – I admit, difficult for me to understand – willingness to construe the most general modal concepts; this is so starting with the lecture O pojęciu możliwości [On the Concept of Possibility], through Uwagi filozo-ficzne o wielowartościowych systemach rachunku zdań [Philosophical Remarks on Many-valued Systems of Propositional Logic]14 until Łukasiewicz’s late works. This conviction lead Łukasiewicz from modal justification of three-valued logic to a decided criticism of the concept of necessity, ultimately

11 Cf. J. Łukasiewicz, O pojęciu możliwości [On the Concept of Possibility], “Ruch Fi-

lozoficzny”, 5 (1920), pp. 169 ff. or “Studia Filozoficzne”, 1988, vol. 5, pp. 129 ff. 12 Although the „early” Łukasiewicz did not know Lewis’s works yet, in his late works

he clearly criticizes the systems of modal logics in which modal functors are non-extensional functors – cf. Łukasiewicz, op. cit., p. 189.

13 On the grounds of protothetics it can be shown that the only axiom of propositional logic may be, among others, either a very general formulation of the principle of bivalence, or the principle of extensionality. This principle plays an important role in arguing in fa-vor of three-valued logic, although certainly it cannot be a thesis of the three-valued sys-tem. However, Łukasiewicz willingly used the principle, being aware of its great deductive possibilities.

14 In: J. Łukasiewicz, Selected Works, ed. by L. Borkowski Warszawa 1961, pp. 153-178.

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reaching a complete elimination of that concept from the scope of terms that are useful in logic.

After making these most general remarks let us pass on to outlining and critical analysis of Łukasiewicz’s understanding of necessity on the ground of the three-valued Ł3 system.

III. THE Ł3 SYSTEM

The system of three-valued logic construed about 1920 by Jan Łukasiewicz (in short called the “Ł3” system) may be presented by means of three-valued ma-trices characterizing the basic functors of this system15. In 1930 Łukasiewicz gave (in the article that has already been mentioned, namely, Uwagi filozo-ficzne) the so-called modal justification of three-valued logic. In the article he postulates joint validity of three groups (that he calls obvious) of modal theo-rems including: a) one-sided possibility the theorem: Ab esse ad posse valet consequentia, which, after transformations gives the formulation: I. If it is not possible that p, then non-p. b) temporal possibility, connected (on the ground of the rule, following from the Square of Opposition, and stating that the sentence “It is possible that p” is equivalent to “It is not necessary that non-p”), with temporal ne-cessity. The principle referred to by Łukasiewicz here is the so-called Aris-totle’s theorem saying that:

Unumquodque, quando est, oportet esse, which, if the temporal phrase quando is replaced by the implication functor16, assumes the following shape:

II. If it is assumed that non-p, then (with this assumption) it is not possi-ble that p.

15 Since these matrices are well-known we will not include them here; for a more de-

tailed characterization of these matrices and their philosophical interpretation cf. M. Lechniak, Interpretacje wartości matryc logik wielowartościowych [Interpretations of Val-ues of Matrices of Many-valued Logics], Lublin 1999.

16 And such a replacement – suggests Łukasiewicz (Philosophical Remarks on Many-valued Systems, in: Selected Works, p. 155) – is acceptable: “The word quando […] and the corresponding ὅταν of Aristotle, is not a conditional, but a temporal particle. Yet the temporal merges into the conditional, if the determination of time in the temporally con-nected propositions is included in the content of the propositions”.


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c) two-sided possibility; the concept is represented by the rule saying that: III. For a certain p it is possible that p, and it is possible that non-p. These theorems, taken together, lead either to trivialization of modal con-

cepts (modal functors would be reducible to one of one-argument functors of the classical propositional calculus or to contradiction. Therefore, in order to make it possible to introduce monadic modal functors, it is necessary – ac-cording to Łukasiewicz – to annul the principle of bivalency and to add to the division of sentences into false and true ones a new, third logical value. Modal functors may be then characterized in the following way:

p Mp Ip Lp Qp

1 1 0 1 0

½ 1 0 0 1

0 0 1 0 0

where “Lp” = “It is necessary that p”, “Mp” = “It is possible that p” (in the sense of one-sided possibility), “Qp” = “It is contingent that p”, that is “Qp” = “KMpMNp”, and “Ip” = “It is impossible that p”, where “Ip” = “NMp”.

The above table is a consequence of defining the possibility functor as “Mp” = “CNpp” (“it is possible that p” means the same as “if non-p, then p’), that is, so that “definition of the concept of possibility which would allow me to establish all the intuitive traditional theorems for modal propositions without contradiction”17. “One must grasp the intuitive meaning of this defi-nition. The expression ‘CNpp’ is according to the three-valued matrix false if and only if ‘p’ is false. Otherwise ‘CNpp’ is true”18. It is easy to notice that on the ground of classical logic the expression CNpp is equivalent to p; the expression is the antecedent of Clavius’ thesis whose counterpart on the

17 Cf. ibid., p. 166. The definition that is discussed in this sentence is not a definition

used by Łukasiewicz in his further analyses. The latter one comes from Tarski and is more general than the definition discovered by Łukasiewicz earlier, that has a less intuitive form:

“Mp = “AEpNp∏qNCpKqNq”, which means the same as: either the sentences p and non-p are equivalent, or there is not such a pair of contradictory expressions that would follow from the sentence p.

18 Ibid., p. 167.

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ground of the three-valued system is not a thesis. The equivalent of Clavius’ thesis, after applying this definition, has the following form: CMpp, that is, it corresponds to the formal formulation of theorem II and is not a thesis of the Ł3 system, as CM ½ ½ = C1 ½ = ½. The fact that applying the definition of the possibility functor formulated by Tarski to the definition of the necessity functor, gives the obvious result:

“Lp” = “NMNp” = “NCpNp”, also speaks in favor of the definition, that is, “freely speaking, we can then as-sert that a certain proposition ‘α’ is necessary if it does not contain its own negation”19.

Let us have a close look at the necessity functor in the Ł3 system now. In the above matrix we see that the sentence Lp has the value 1 only when p = 1, that is, it is necessary when p is true20. In turn, I0 = 1, that is, when p is “completely false” then it is true that it is impossible. These equations find justification in the fact that in the Ł3 system, like in Aristotle’s works, “proposition is counted definitely true (assigned the value „1”) only (a) when it is already determined that the event which it refers to shall turn out as it says, or (b) when the event referred to has passed from the future into the present or the past, and so has lost the potentiality it had, while it was still future, of turning out otherwise. It is only propositions which are in this sense ‘necessary’ which are definitely true, and only propositions which are in an analogous sense ‘impossible’ which are definitely false. A proposition is defi-

19 Ibid., p. 169. For modal functors characterized in this way equivalents of the follow-

ing rules are valid in Ł3: a) CpMp and CLpp – the laws of group I; b) CipLCpq and CLqLCpq – equivalents of paradoxes of strict implication; c) CQpQNp – the Aristotelian law of contingency that, when the above mentioned rule

of protothetic is absent (the theorem is absent, as its acceptance would have to lead to acceptance of the expression ApNp, and this is not a thesis of the Ł3 system; we will return to the thesis CδCδNpδq below), does not lead to paradoxes;

d) C*LpLLp and C*MMpMp – the reduction axiom for the S4 system, and e) C*MpLMp and C*MLpLp – the reduction axiom for the S5 system, where C*pq

may be defined as LCpq. Cf. Prior, Formal Logic, p. 247. 20 A discussion of how to interpret the values of Łukasiewicz’s tables can be found in:

Lechniak, op. cit., pp. 15-76; here we will only point to the fact that is seems that the truth understood in the classical way does not allow complementing the logical values with new values; Słupecki’s and Borkowski’s analyses have shown that the values “1” and “0” should be connected with sentences talking about determined events, and the value “½” with undetermined events.


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nitely false only when any chance it may have had of being true has gone”21. It seems that such a definition of necessity conveys well the meaning of what Aristotle had in mind when he stated the principle: whatever, that is, since it is, is necessary. Let us emphasize here that if the necessity functor is to be understood in a way that will not be trivial, that is if necessity is not to be a feature of all true sentences, then “1” in the table should really be interpreted as the value of a sentence about a present, past or determined event.

The concept of necessity characterized in this way is very broad. It not only comprises the necessity of sentences about completed and unavoidable events (that is temporal necessity), but also necessity of the laws of logic. However, “it is in any case important to distinguish the ‘necessity’ which is expressed by the monadic three-valued truth-operator (1, 0, 0) from the ne-cessity which consists in exemplifying a logical law. The latter, of course, can-not be a truth-function in any system at all – it is rather a consequential characteristic of certain truth-functions, consisting in the fact that all truth-functions formed in the same way, whatever their arguments, are true”. This can be seen in the following example. The sentence with a modal functor: “It is logically necessary that if Socrates died, then Socrates died” is true regard-less of the question whether its argument “Socrates died” is true or not; it is true because of the fact that the implication with the form p → p is a law of logic and as such it is logically necessary22. It is different with the sentence “L(If Socrates died then Socrates died)”. The sentence is true precisely be-cause the sentence “Socrates died” is true. Similarly, the sentence “L(Socrates died)” is true, although the sentence “Socrates died” is not a substitution of a law of logic23. Hence propositional formulas that are laws of logic are neces-sary in Łukasiewicz’s meaning, as they are always true. However, they are necessary in Łukasiewicz’s meaning for a reason that is different from logically necessary sentences. This is so since necessity in Łukasiewicz’s meaning is a feature of those sentences because of their being true at present, whereas logi-cal necessity is a feature of those sentences because of their form that guaran-tees them being laws of logic, or being true expressions regardless of the logi-cal value of the arguments of the proposition-forming functors that occur in

21 See: Prior, Formal Logic, p. 248. 22 In some standard systems of modal logic this logical necessity is ascribed to the the-

ses of logic by the so-called Gödel’s rule with the form α, hence It is logically necessary that α.

23 Cf. Prior, Formal Logic, p. 248.

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them. It seems that the meaning of the L functor is temporal (or temporal-causal); it is the unavoidability of what has already happened, is happening at present, or whose cause already exists.

From the above considerations certainly the conclusion cannot be drawn that e.g. CpLp is a thesis; it is true only when p assumes classical values as they express determination of an event to be or not to be. However, it is not true for p = ½, which value expresses an undetermined event. Then we have CpLp = ½. It may also be remarked that for necessity of sentences under-stood in this way ALpLNp and AMpMNp occur, although ApNp is not a the-sis of the system. Hence for sentences about present events the statement about occurrence of these events and the statement about their necessity are equivalent (for sentences about the present or about the past as well as for sentences about non-temporal facts, e.g. mathematical theorems, the third logical value can never be used). When the attitude is assumed that there are sentences that have the third logical value, all classical logic is acceptable, ac-cording to Prior, with respect to past, present and determined future events24.

IV. THE FOUR-VALUED SYSTEM OF MODAL LOGIC

In the 1950’s Łukasiewicz gave a few versions of the system of modal logic25 that we will further call the Ł4 system (or the Ł-modal system). Łuka-siewicz bases the system on the so-called basic modal logic that should meet the following eight conditions: I. The implication If p, then it is possible that p is accepted; or using sym-

bols: 1.1. CpΔp (where “p” means “It is possible that p”). II. The implication If it is possible that p then p is rejected; or using symbols: 1.2. CΔpp (where is the sign of rejection).

24 Cf. ibid., p. 250. 25 Cf. J. Łukasiewicz, A System of Modal Logic, in: J. Łukasiewicz, Selected Works, pp.

252-390; J. Łukasiewicz, Aristotle’s Syllogistic. Cf. also many other articles from that pe-riod, e.g. J. Łukasiewicz, On Variable Functors of Propositional Arguments, in: J. Łu-kasiewicz, Selected Works, pp. 311-324, or the rather controversial article Arithmetic and Modal Logic, “The Journal of Computing Systems”, 1 (1954) 213-219.


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III. The sentence It is possible that p is rejected; or using symbols: 1.3. Δp. IV. The implication If it is necessary that p then p is accepted; or using sym-

bols: 1.4. CΓpp (where “p” means “It is necessary that p”). V. The implication If p then it is necessary that p is rejected; or using sym-

bols: 1.5. CpΓp. VI. The sentence It is not necessary that p is rejected: 1.6. NΓp. VII. The equivalence It is possible that p – if and only if it is not necessary

that non-p is accepted: 1.7. EΔpNΓNp. VIII. The equivalence It is necessary that p – if and only if it is not possible

that non-p is accepted: 1.8. EΓpNΔNp.

Condition I characterizes one-sided possibility, and Condition IV – neces-sity connected with one-sided possibility; together they correspond to group I of the so-called obvious modal theorems mentioned by Łukasiewicz in connec-tion with the Ł3 system; they are correlations following from the Modal Square of Opposition. Condition II corresponds to the saying A posse ad esse non valet consequentia; Condition V is its equivalent for necessity; and the last two conditions determine the connections between the concepts of neces-sity and possibility following from the Modal Square of Opposition. The six conditions are known in traditional logic; they present dependencies of the Modal Square of Opposition. Conditions III and VI have a special character. “The third condition states that not all formulae beginning with Δ are as-serted, because otherwise Δp would be equivalent to the function ‘verum of p’ which is not a modal function. [...] I accept throughout the paper that both Δ and Γ are proposition-forming functors of one propositional argument, and that Δp and Γp are truth-functions [...] As there exists in two-valued logic no functor of one argument which would satisfy the formulae 1.1, 1.2, and 1.3, or 1.4, 1.5, and 1.6, it is plain that the basic modal logic, and, consequently, every system of modal logic is a many-valued system”26.

For the basic modal logic Łukasiewicz suggests the following set of axioms (in which Δ is the primitive functor):

26 Łukasiewicz, A System of Modal Logic, p. 354.

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2.1. CpΔp. 2.2. CΔpp. 2.3. Δp. 2.4. EΔpΔNNp27. The basic modal logic may be generalized and in this way the Ł4 system is

obtained that, in the axiomatic form, may be presented in the following way: A) Axioms: 1. CδpCδNpδq, 2. CpΔp, 3. CΔpp, 4. Δp,

where δ is a variable propositional-forming functor from one propositional ar-gument.

B. Rules: a) the rule of substitution for accepted expressions (with an extension for δ-substitution)28, b) the rule of detachment for accepted expressions, c) the rule of substitution for rejected expressions: if α is rejected and α is a substitution of β, then β has to be rejected, d) the rule of detachment for rejected expressions: if Cαβ and β is rejected, then α has to be rejected.

If we accept the conditions 1.7 and 1.8, a parallel to the above axiomatic set of axioms can be obtained, axioms that contain the necessity functor as

27 The formula 2.3. is equivalent to the formula 1.7 (we will return to a discussion of

the relation between 2.4 and 1.7 in the further, critical part of this item), and it is used as an axiom for the reason of elegance, as 1.7 contains a term that has already been defined. Cf. ibid., p. 277.

28 On the rule of δ-substitution see: Łukasiewicz, On Variable Functors, pp. 311 sq. The rule often leads to very non-intuitive consequences – cf. e.g. the following remark by Lejewski (op. cit., p. 16): “Łukasiewicz allows for variable functors which require one pro-positional argument to form a proposition. Obviously, a calculus like that is part of proto-thetic. The most characteristic feature of the δ-Calculus are its rule of substitution for functorial variables and its rule of definition. The former is by far less intuitive that the corresponding rule in protothetic but among other things it makes the working of the lat-ter possible. Together with the traditional rule of detachment they form such a powerful instrument of deduction that Łukasiewicz was able to base a complete system of the δ-Calculus, i.e. the Calculus of Propositions with functorial variables, on the principle of bi-valence as the only axiom”.


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the primitive term, where beside Axiom 1 we have equivalents of expressions 1.4, 1.5, 1.6.

The following matrices for the basic functors (that are formed for functors corresponding to the classical ones by multiplying respective two-valued ma-trices by themselves) are the consequence of the assumptions accepted by Łu-kasiewicz (resulting in the above axiomatic)29:

C 1 2 3 4 N Δ Γ

*1 1 2 3 4 4 1 2

2 1 1 3 3 3 2 2

3 1 2 1 2 2 3 3

4 1 1 1 1 1 3 4

After this short presentation of the basic modal logic and the Ł-modal sys-

tem let us pass on to more detailed analyses of numerous intuitive problems offered by this system. As a starting point for these analyses let us remind the reader a few expressions that are accepted and a few ones that are rejected (connected with the necessity functor) in the system.

1. SOME ACCEPTED EXPRESSIONS OF THE Ł-4 SYSTEM

From the set of axioms given above the following formulas, among others, may be derived30:

29 In the system presented in the book Aristotle’s Syllogistic the matrices for the neces-sity functor and possibility functor are a little different from the above ones. There we have:

p Δ Γ

1 1 2

2 1 2

3 3 4

4 3 4 30 We use Łukasiewicz’s phrase “formulas may be derived”, and not “theses may be

proven”, because in the system some of its axioms are accepted (theses) and others are re-

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1. All the theses of the classical propositional calculus, including the law of extensionality: 73. CEpqCδpδq.

2. Formulas for the necessity functor: 130. CΓCpqCΓpΓq “If implication is necessary and its antecedent is neces-

sary, then its consequent is necessary; the equivalent of the T system axiom.

132. CCpqCΓpΓq “If there is an (accepted?) implication and its necessary antecedent, then the consequent is necessary”.

133. CΓCpqCpΓq 134. CΓqCpΓp 135. CΓpΓΓp An equivalent of Axiom I of reduction – a counterpart of the

axiom of S4 system. 136. EΓΓpΓp 141. EΓNΓpΓNp II law of reduction; according to 136 and 141 a apodeictic

proposition (and also a apodeictic-problematic one “It is necessarily pos-sible that p”) is always equivalent to a categorical proposition. Respec-tively, in the S5 system we have, as a thesis, the expression EΓNΓpNΓp stating conversely that a apodeictic-problematic proposition (“It is neces-sarily possible that p”) is reducible to a problematic proposition.

149. ΓKpqKΓpΓq 154. EΓApqAΓpΓq The sentences are accepted: It is necessary that p and q

(p or q) when and only when it is necessary that p and (or) it is neces-sary that q.

2. SOME REJECTED EXPRESSIONS OF THE Ł-4 SYSTEM (FORMULAS FOR THE NECESSITY FUNCTOR)

156. CpΓp 157. NΓp 158. ΓCpp

jected (counter-theses?); discussions of the understanding of the operation (?) of acceptan-ce (rejection) will be presented below. To guide the reader through the analyses of Łuka-siewicz’s system, for expressions mentioned in the article A System of Modal Logic we use the numbers of the expressions that are used in it by Łukasiewicz; those expressions have been chosen that are interesting for intuitive reasons.


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3. THE QUESTION OF VALIDITY OF THE PRINCIPLE CδpCδNpδq

According to Łukasiewicz, the system of modal logic should be based on classical logic, and all the modal functors should be truth-functional ones. This basis of classical logic is to be ensured by the principle CδpCδNpδq. Along with the rule of δ-substitution and the usual rule of detachment it makes it possible to derive from it (at least) all the theses of classical logic. However, the rule leads to non-intuitive consequences, e.g. to the expression 160. ΓCpp, stating that the proposition “It is necessary that if p then p” (the proposition that the laws of logic are necessary is rejected) is rejected. In connection with this, let us consider two problems: 1) the question of validity of this principle in any four-valued calculus (based on classical logic); 2) the question of modal functors being truth-functional.

1. The Ł-modal system is based (like standard systems) on classical logic, but only this system is based on the principle CδpCδNpδq taken from proto-thetic. “In all these forms, and also with Leśniewski’s of protothetic the exten-sionality thesis CEpqCδpδq is contained – as long as we remain on the ground of the classical propositional calculus functors determine two-valued functions, and we do not have to care about this law or about its consequences. However, if we include functors Δ and Γ in proposition-forming functors, we should either explicite exclude modal functors from the domain of the functor δ, or annul CδpCδNpδq”31.

In systems of standard modal logics the former possibility occurs – it is de-clared that the modal functors are not truth-functional, so they are not in-cluded in the functor’s domain. In Ł4, on the other hand, modal functors can be substituted for δ, so both CΔpCΔNpΔq and CΓpCΓNpΓq. If any of these expressions were a thesis in Lewis’ systems, this would follow accepting the expression CΔpp (because CΔpCΔNpΔq leads to CΔpCΔNpΔKpNp, and this, taken together with NΔKpNp, leads to CΔpNΔNp, and hence to CΔpΓp). But in the Ł4 system no accepted expression may begin with NΔ, so NΔKpNp is not an accepted expression of the system. Similarly, neither CEpqCΓpΓq nor CEpqCΔpΔq is a thesis of standard systems (for, if e.g. p and q are false, then still with a possible p, q may be impossible – and such a situation may be

31 Prior, Time and Modality, p. 123.

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taken under consideration in these systems). According to Prior32, even in an Ł-modal system we have “something like the dilemma of dropping CδpCδNpδq or restricting the range of δ”. In a purely assertoric calculus, limiting the range of δ in the expression CδpCδNpδq is present, albeit it is not felt, as such a calculus does not contain any functors for which this law does not hold. This limitation is also assumed in Ł4, as the expression CδpCδNpδq keeps its validity either on condition that we will limit the range of δ to such functors as Δ or Γ, which have to be treated as variable functors changing their values only in a part of their ranges, or we will accept that a calculus is not functionally complete, i.e. that there are monadic functors that are not definable in Ł4. In a four-valued calculus there are 256 monadic functors, but only 16 of them are definable in Ł4. They are functors determined by compos-ing two-valued monadic functions of the classical propositional calculus; “The expression (Na, Nb) [by means of which a four-valued negation functor is de-fined – M.L.] is a particular case of the general formula (εa, ζb), where ε, ζ may take as their values the functors: V (verum), S (assertion), N (negation) and F (falsum) from the two-valued calculus. Since each of the four values of ε may be joined to each of the four values of ζ, we obtain 16 combinations that define 16 functors from one four-valued argument”33. So, functor of ne-gation is defined N(a, b) = (Na, Nb), functor of possibility is defined Δ (a, b) = (Sa, Vb) = (a, Cbb), and functor of necessity is defined V (a, b) = (Sa, Fb) = (a, Fb). Hence we have a limitation of functions of system that can be considered to those that can be treated as a composition of classical monadic functions (we will return to considering the intuitive meaning of such a con-struction of functors later on). If we go outside this domain and assume that we can define every functor, the expression CδpCδNpδq has to be left out. E.g. such an functor Xp can be defined that Xp = (respectively) 1, 1, 1, 4, when p = 1, 2, 3, 4. Then we have CX3XN3X4 = C1CX24 = C1C14 = C14 = 4. Or the functor Yp = 2, 3, 4, 1, when p = 1, 2, 3, 4, and then CY1CYN1Y3 = 3. Similarly the law of extensionality CEpqCδpδq is false for these functors. Prior sums this up in the following way: “In fact I do not know of any system containing this law which has not at least one of the fol-lowing three limitations: Either (i) the system is 2-valued, or (ii) it is func-tionally incomplete, or (iii) it does not contain all the classical laws C and

32 The analysis below is to a considerable degree based on Prior’s considerations in

Time and Modality, pp. 3-5, 123-132. 33 Łukasiewicz, Aristotle’s Syllogistic, p. 167.


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E”34. So it seems, as yet, that Łukasiewicz arbitrarily assumed that modal functors should behave like truth-functional functors.

2. In Formal Logic Prior presented an interesting argumentation support-ing the thesis that modal functors are not truth-functional ones35. The start-ing point for this argumentation is Aristotle’s thesis concerning two-sided possibility (contingency) stating that If it is contingent that p, then it is con-tingent that non-p (as Qp = KΔpΔNp = KΔNpΔp → KΔNpΔNNp = QNp). Łukasiewicz36 showed that with the use of the thesis CδpCδNpδq and Aris-totle’s thesis CQpQNp one can obtain paradoxical consequences, namely, that if a proposition is contingent, then every proposition is contingent37.

According to Prior, the difficulty in this reasoning is in line 4: “it is sim-ply not true that if a proposition is contingent, then if non-p is also contin-gent, everything is contingent”. The functor “It is contingent that […]” is not such an functor that can be substituted for δ. Analogy with quantifiers can be here referred to. If for any predicate φ it is true that KΣxφxΣxNφx, then this will also be true for Nφ. Hence we have:

C(KΣxφxΣxNφx)(KΣxNφxΣxNNφx), (because K(Σxφx)(ΣxNφx) → K(ΣxNφx) (Σxφx) → K(ΣxN x)(ΣxNNφx)). If we use the abbreviation Oxφx for the ex-pression KΣxφxΣxNφx, we can note our thesis like this: CoxφxOxNφx and de-rive from it an absurd conclusion:

1. CδpCδNpδq 2. CCpqCCpCqrCpr (Frege’s syllogism) 3. COxφxOxNφx 1 p/φx, q/ψx – 4 4. CδφxCδNφxδψx

34 Prior, Time and Modality, p. 129. Further on Prior construes systems that have the

third one of these limitations. 35 See: pp. 190-192. 36 On Variable Functors, pp. 250 ff. 37 Łukasiewicz’s reasoning is the following: 1. CδpCδNpδq 2. CCpqCCpCqrCpr (Frege’s syllogism) 3. CQpQNp (Aristotle’s thesis) 1 δ/Q - 4 4. CQpCQNpQq 2 p/Qp, q/QNp, r/Qq – C3 – C4 –5 5. CQpQq 5 – Σp – Πq – 6 6. CΣpQpΠqQq.

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4 δ/Ox – 5 5. COxφxCOxNφxOxψx 2 p/Oxφx, q/OxNφx, r/Oxψx – C3 – C5 – 6 6. CoxφxOxψx 6 Σφ – Πψ – 7 7. CΣφOxφxΠψOψxψx So, if there is such a predicate φ about which we can say that something

has a property represented by it, and something does not have that property (e.g. the property of being a student in a set of people), then we can say the same about every predicate, which means that there are no true universal propositions. The problem lies here in treating the compound quantifier “Ox” as one that can be substituted for δ in 4. We can see that the function corres-ponding to the quantifier “Ox” corresponds to a contingent proposition and the function is not a truth-functional one (for in order to determine if the propositions “Oxφx” and “OxNφx” are true or not one has to know the pre-dicate represented by φ; for some predicates it is so, and for other ones it is not). So the functor of contingency (of two-sided possibility) is not a truth-functional one. And since it is so, then considering the (mentioned above) correlations between modal functors, no modal functor is a truth-functional one.

The above argumentation seems quite convincing. Of course, Łukasiewicz draws different conclusions from the expression CΣpQpΠqQq; his conclusion is that Aristotle’s principle should be rejected as it leads to paradoxical consequences. From the above presented argumentation and from reasoning concerning the principle CδpCδNpδq it follows that it is the latter rule that is responsible for non-intuitive consequences; modal functors cannot be substi-tuted for a variable δ.

4. AN ATTEMPT AT INTUITIVE INTERPRETATION OF MODAL FUNCTORS IN THE Ł4 SYSTEM

Prior, however, made an attempt at giving some understanding of Ł-modal functors, so that the way of defining these functors could seem more rational. “The form ‘Possibly p’ has many meanings, but there is as it were an upper and a lower limit to what it may mean. It never asserts more than that p is actually true, and it never asserts less than that p is true if it is true. Simi-larly ‘Necessarily p’ never asserts less than that p is actually true, and never


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asserts more than that p is at once true and false (for this last is a kind of upper limit to all assertion […] In all modal systems except this one, it is as-sumed that these limits lie outside, even if just outside, the range of permissi-ble meanings of ‘Possibly’ and ‘Necessarily’. That is, ‘Possibly p’ is taken to assert not only no more than but definitely less than the plain p (being true in cases in which p itself is not true) and not only not less but definitely more than “If p then p” (since this is true even for p’s which would ordinarily be called not possible but impossible. It is true, for example, that if 2 and 2 both are and are not 4 then 2 and 2 both are and are not 4). But the formulae which occur in the Ł-modal system as axioms and theorems are not those which hold when ‘possibly p’ is given as meaning something between the up-per and lower limit, but those which hold whether it be given the upper-limit meaning or the lower-limit meaning. That is, in the formulae of this system the operator M [i.e. Δ in our notation – M.L.] behaves as if it were not a con-stant but a variable operator, capable of standing either for the plain ‘It is the case that’ or for ‘If it is the case that (so-and-so) then it is’. CpMp [i.e. CpΔp in our notation – M.L.], for example, is a law because it holds whether you replace Mp [i.e. Δp in our notation – M.L.] by plain p or by Cpp; the converse CMpp [i.e. CΔpp in our notation – M.L.] is not a law because it does not hold when you replace Mp [i.e. Δp in our notation – M.L.] by Cpp; and the simple Mp [i.e. Δp in our notation – M.L.] is not a law because it does not hold when you replace it with the plain p. The L [i.e. Γ in our notation – M.L.] of the Ł-modal system may be interpreted similarly – the theses of the system containing L [i.e. Γ in our notation – M.L.] are those formulae con-taining L [i.e. Γ in our notation – M.L.] which will be laws of the ordinary propositional calculus whether Lp [i.e. Γp in our notation – M.L.] be replaced by the plain p or by KpNp. The odd formula CLCpqCpLq [i.e. CΓpqCpΓq in our notation – M.L.]38 is a law because if you simply drop L [i.e. Γ in our no-tation – M.L.] you obtain CCpqCpq, which is an obvious enough tautology, and if you replace Lα [i.e. Γα in our notation – M.L.] by KαKα it becomes CKCpqNCpqCpKqNq, which is always true because its antecedent is always

38 It is “odd” in the meaning that – as Prior remarks – it corresponds to the mistake

medieval authors often mentioned that it consists in mistaking necessitas consequentiae for necessitas consequentis. See distinguishing of various kinds of necessity in Aristotle’s works at the beginning of the present article.

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false”39. Let us have a closer look at Łukasiewicz’s considerations in order to examine legitimacy of Prior’s conclusions.

Łukasiewicz writes: “The formulae with Δ are obviously a product of for-mulae verified by S and V. CpΔp is asserted because it is asserted for Δ = S and Δ = V. CΔpp and Δp are rejected because the first formula is rejected for Δ = V and the second one for Δ = S. Now we can obtain a product of S and V by multiplying S by V, getting thus the function Δ (a, b) = (Sa, Vb) = (a, Cbb)”40. Also “the classical propositional calculus, to which all the formulae of our modal logic are matrically reducible, is ‘saturated’, i.e., any formula must be either asserted on the ground of its asserted axioms, or rejected on the ground of the axiom of rejection p, which easily follows from our axiom three or four”41. Here we can see the same property Prior writes about, one of skipping the possibility function (and the same may be said about the neces-sity functor, for which, respectively, it may be stated that the formulas are a product of formulas verified by S and by F. The formula CΓpp is accepted because it is accepted for Γ = S and for Γ = F, the formula CpΓp is rejected because it is rejected for Γ = F, and the formula NΓp is rejected because it is rejected for Γ = S) from the lower limit to the upper one, throughout the range that is in fact assigned for this functor, where standard modal functors “work” without obtaining those limits. Here we reach a place that is impor-tant for the present considerations – the role played in Łukasiewicz’s system by the scheme: accepted expressions – rejected expressions.

5. THE SCHEME: ACCEPTED EXPRESSIONS – REJECTED EXPRESSIONS

The scheme: accepted expressions – rejected expressions requires separate analyses. In the “usual” system of logic we have one category of expressions – accepted expressions, the set of which coincides with the set of theses. All other expressions built in the language of the system are left out from consid-erations – they are not theses since they cannot be derived from axioms by means of rules leading from theses to theses. Hence in such a system we are dealing with only one logical value (truth?) being the upper limit of the set of

39 Prior, Time and Modality, p. 4. 40 Cf. Łukasiewicz, A System of Modal Logic, p. 369. 41 Ibid.


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logical values; values of all other expressions are under that limit. The fact that the propositional variable is not a thesis in the system follows from the fact that such a substitution of the variable can be given that is a false proposition. In the Ł4 system, if one takes into account Łukasiewicz’s expla-nations, this scheme should also be valid – “a rejected formula” means the same as “not every substitution of it is accepted” (the third condition (1.3) states that not all expressions starting with Δ are accepted42), so, speaking strictly, the symbols α, α should be read as (respectively): all substitutions of α are accepted, not all substitutions of α are accepted. Here we come to what must be striking when we are confronted with the Ł4 system – if we were limited to accepted axioms only and also if we suspended validity of the rule CδpδNpδq in relation to modal functors (because of the doubts men-tioned above), the “positive” knowledge about modalities contained in the system would be very poor (only the conditions 1.1, 1.4, 1.7 and 1.8 would remain in the system, that is the principle Ab oportere ad esse… and the rela-tion between Δ and Γ following from the modal square but not infringing the “healthy” intuitions concerning modal concepts). These intuitions are in-fringed by consequences of extending validity of the principle CδpδNpδq to modal functors and by rejection axioms. The role of the latter ones is such that (as Łukasiewicz points himself) they defend the Ł4 system against trivi-alization. Axioms 1.2 and 1.5 defend functors Δ and Γ against trivial reduc-tion to the assertion functor, axiom 1.3. defends functor Δ against reduction to the verum functor, and axiom 1.6 defends the necessity functor against re-duction to the falsum functor. If we assumed that modal functors are truth-functional functors, and there are no “rejected” propositions, the situation would have to be like in Ł3 – all sentences with the best value (true, about determined events, already past ones) would be necessary, and all the non-false propositions (with all values apart from the worst one) – possible; how-ever, Łukasiewicz wanted to avoid that (e.g. the principle Unumquodque, quando est, oportet esse is simply rejected). On the other hand, construing a saturated calculus (owing to the pattern: accepted expressions – rejected ex-pressions) resulted in a paradoxical situation, namely, no necessary proposi-

42 If the sign “ ” was read in the strong sense, i.e. as “a rejected formula” is the same

as “its every substitution is false”, then modal logic would not be possible at all, as no true problematic proposition would be allowed; however, this was not Łukasiewicz’s intention (because, as follows from the matrix Ł4, if p has designated value, then also Mp has such value).

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tion may be accepted, as in the Ł4 system there is no possibility of introduc-ing “the first” accepted necessary proposition into the system (in standard system introduction of necessary propositions into the system is made possible e.g. by the Gödel rule, which ascribes necessity to expressions on the basis of the fact that these expressions are e.g. theses of the classical propositional calculus); in Ł4 the rejection rules effectively eliminate all expressions starting with the necessity functor. This is so since if p (as its substitution conse-quence is the rejected expression Δp), then from the rule of detachment for rejection and the expression CΓpp we have the expression Γp. Since no proposition may be necessary, it is the same with the laws of logic, e.g. 160. ΓCpp or Γ(x = x)43. Although, strictly speaking, in accordance with the

above mentioned Łukasiewicz’s findings, expression Γp should be read not as: “any proposition is not necessary” but “we cannot assume that all apo-deictic propositions are true”, which is trivially true (as there is at least one contingent true proposition).

Possibly another way in which necessary propositions could “enter” the system could also be theses of reduction. In the S5 system we have, e.g. the thesis: It is possibly necessary that p if and only if it is necessary that p (so problematic-apodeictic propositions are reducible to apodeictic propositions). However, in the Ł4 system we have 141. EΓNΓpΓNp stating that apodeictic propositions are reducible to apodeictic propositions. These consequences make numerous authors consider the Ł4 system as a strange or peculiar one44. Also e.g. thesis 154. EΓApqAΓpΓq is peculiar. It concerns distribution the necessity functor into the arguments of the alternative. For contradictory events the alternative of propositions describing the events is true and, one would like to say, logically necessary (the rule of the excluded middle), whereas it is obvious that none of these propositions has to be necessary (the-sis 154 is a consequence of the rule CδpCδNpδq).

From the above results Łukasiewicz derives philosophical conclusions con-cerning the possibility of all apodeictic knowledge. He emphasizes the impor-tance of the discovery pointing to the fact that no apodeictic propositions can

43 Łukasiewicz offers a peculiar argument to prove that Γ(x = x), which is a response to difficulties connected with necessitation of analytical propositions that were pointed out by Quine. Cf. Łukasiewicz, Aristotle’s Syllogistic, pp. 228-231; idem, Arithmetic and Modal Logic, in: Łukasiewicz, Selected Works, pp. 391 ff.

44 It is probably these peculiarities, why the Ł4 system is not quoted in the work: R. Freys, J. Dopp, Modal Logic, Paris–Leuven 1965. Cf. J. Porte, The Ω-System and the Ł-System of Modal Logic, “Notre Dame Journal of Formal Logic”, 20 (1979), pp. 915-920.


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be true, i.e. to impossibility of apodeictic knowledge. It is worth noting here that this result of the Ł4 system is not actually a discovery, but an assump-tion made by Łukasiewicz: what in fact allows Łukasiewicz to assume the axi-oms of the system and its rules? Why are the rejection rules removing neces-sary propositions from the set of propositions among these rules? Hence the following quotation from the last chapter of Aristotle’s Syllogistic should be treated rather as a source of the Ł4 system, as its philosophical motivation, but not a philosophical consequence of this system45. Łukasiewicz writes: “There are no true apodeictic propositions, and from the standpoint of logic there is no difference between a mathematical and an empirical truth. Modal logic can be described as an extension of the customary logic by introducing the ‘stronger’ and ‘weaker’ affirmation; the apodeictic affirmation Lp [in the work Aristotle’s Syllogistic the functor L corresponds to our Γ, and M to our Δ – M.L.] is stronger, and the problematic Mp weaker than the assertoric af-firmation p. If we use the non-committal expression ‘stronger’ and ‘weaker’ instead of ‘necessary’ and ‘contingent’46, we get rid of some dangerous asso-ciations connected with modal terms. Necessity implies compulsion, contin-gency implies chance. We assert the necessary, for we feel compelled to do so. But if Lα is merely a stronger affirmation than α, and α is true, why should we assert Lα? Truth is strong enough, there is no need to have a ‘supertruth’ stronger than truth”47. In the quoted considerations one can detect instru-mentalism that the “late” Łukasiewicz openly promoted, and empiricism that issues from it. “I am fully aware that other systems of modal logic are possi-ble based on different concepts of necessity and possibility. I am firmly be-lieve that we shall never be able to decide which of them is true. Systems of logic are instruments of thought, and the more useful a logical system is, the

45 Analysis of all Łukasiewicz’s works leads to the conclusion that at the foundations of

the systems he created there are certain philosophical presuppositions, or, in other words, that Łukasiewicz actually tried to carry out his program of logical philosophy, treating the systems he created as ones being at service of his own views. What is misleading in Łu-kasiewicz’s works is that many of those logical-philosophical works are connected with stud-ies of Aristotle, which is the cause why they are treated as an attempt at interpreting the Stagirite’s views, whereas they are Łukasiewicz’s own works, that are an expression of his views of the world, only supported by certain theses taken from Aristotle (just like Aristotle himself uses theses formulated by his predecessors, when he argues for his own theorems).

46 It is interesting that Łukasiewicz used the term “contingent”, and not “possible” here. It seems that thinking about possibility, he understood two-sided possibility by it.

47 Łukasiewicz, Aristotle’s Syllogistic, p. 205-206.

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more valuable it is”48. Let us analyze these two quotations, treating them as a motivation for the Ł-4 system – maybe then the system will stop being strange and peculiar to us?

The first thing that is striking here is the reduction of the necessity func-tor to the role of an epistemic concept. Can necessity, which Aristotle attrib-uted to the relation between the antecedent and the consequent of a correct syllogism, be associated with the concept of acceptance?49 This necessity clearly means as much, as is convincingly argued by e.g. Patzig, as universal validity of the syllogism – “[…] necessarily true means nothing more than true in all possible cases, that is for any substitution of concrete terms for vari-ables A, B, C, occurring in the given syllogistic mood”50. Another kind of propositions Aristotle considered necessary are propositions based on defini-tions, e.g. “It is necessary that man be an animal”; in such propositions a connection between the subject and the predicate is necessary. Łukasiewicz treats necessary propositions based on definitions as examples of analytical propositions, since in analytical propositions it is so that the predicate is con-tained in the subject. However, for Aristotle propositions based on definitions derive their necessity from necessity contained in the real definition, and this necessity is derived from intellectual obviousness that is the source of the real definition. Hence propositions based on definitions are absolutely necessary, i.e. they meet two conditions:

48 Łukasiewicz, A System of Modal Logic, p. 378. 49 Such an approach was characteristic only of the late Middle Ages with their epis-

temic considerations connected with the concept of consequence (cf. my review of the book: I. Boh, Epistemic Logic in the Late Middle Ages, London – New York 1993, included in “Roczniki Filozoficzne”, 48(2000), fasc. 1, pp. 253-261). Impropriety of such an ap-proach was emphasized by Łukasiewicz himself (Aristotle’s Syllogistic, p. 146): “But how should we interpret necessity, when we have a necessary proposition without free variables, and in particular, when this proposition is an implication consisting of false antecedents and of a false consequent […] I see only one reasonable answer: we could say that whoever accepts the premises of this syllogism is necessarily compelled to accept its conclusion. But this would be a kind of psychological character [exactly like in the medieval treatises on consequences – M.L] which is quite alien from logic. Besides it is extremely doubtful that anybody would accept evidently false propositions true. I know no better remedy for re-moving this difficulty than to drop everywhere the L-functor standing in front of an as-serted implication”.

50 Patzig, op. cit., p. 27; also cf. the whole of Chapter II, entitled: “Logical Necessity” (pp. 16-42).


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– the predicate belongs to all objects falling under the term that is the sub-ject; – the predicate belongs to these objects as such, i.e. on the virtue of the defi-nition51. So not all analytical propositions have such absolute necessity, e.g. it is not a feature of such propositions as “Every man is able to laugh”, “Every man sleeps (at a certain time), and propositions like these, which state e.g. proprium about species. Łukasiewicz seems not to notice this at all. The third kind of necessary propositions for Aristotle that Łukasiewicz takes up, are sen-tences expressing basic metaphysical principles. Again, Łukasiewicz argues against them as if he did not know about Aristotle’s metaphysics. He writes: “We read in De Interpretatione: ‘If it is true to say that something is white or not white, it is necessary that it should be white or not white’. It seems that here a necessary connection is stated between a ‘thing’ as subject and ‘white’ as predicate. Using a propositional variable instead of the sentence ‘Something is white’ we get the formula: ‘If it is true that p, it is necessary that p’”52. Cer-tainly the fact that Łukasiewicz “did not notice” the law of the excluded mid-dle and treated it as a simple proposition that is true or false, points to a de-cidedly empirical attitude that Łukasiewicz assumes. Let us notice: in the first of the quoted examples concerning the syllogism the possibility of its argu-ments being false was the argument for denying the syllogism of necessity; and it is exactly this fact that is the source of necessity of the syllogism! It is so because the laws of logic are propositions concerning not facts but general and necessary relations between facts. Only extreme empiricism may explain the fact that in all the examples quoted here Łukasiewicz reduced these rela-tions to general propositions. Certainly the concept of necessity is not needed then: the laws of logic, metaphysics and the truth concerning the relations be-tween species and genus or differentia specifica in the real definition are only empirical generalizations, and the concept of necessity may be reduced, at most, to the unnecessary concept of acceptance.

On the ground of the above considerations we are not very surprised by Łukasiewicz’s statement that there are no true apodeictic propositions. We can see that Łukasiewicz went here further than Hume who allowed for at least necessity of relations between ideas (e.g. mathematical theorems). Ac-cording to Łukasiewicz theorems of mathematics do not deserve some special treatment: “The true a priori is always synthetic. It does not arise, however,

51 Cf. ibid., p. 34. 52 Łukasiewicz, Aristotle’s Syllogistic, pp. 205-206.

Marek Lechniak

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from some mysterious faculty of the mind53, but from very simple experi-ments which can be repeated at any time. […] On such experiments the axi-oms of logic and mathematics are based; there is no fundamental difference between a priori and a posteriori sciences”54. If construction of the Ł4 system was indeed preceded by the above philosophical decisions, the construction becomes comprehensible. If the system was conceived – as Łukasiewicz sug-gests – as one serving analysis of the contingency concept that is a feature of propositions about facts (that is of all theorems of the science), and the neces-sity functor is related to the two-sided possibility functor by the laws of the modal square, then construction of the Ł-4 system must have gone in this di-rection in order to make it impossible for any apodeictic propositions to as-sume a designated value. Hence the scheme: acceptance – rejection, the rules of rejection for substitution and detachment, no counterpart for the Gödel rule that would indeed trivialize the system if it were reduced to the laws of propositional calculus, forcing one to accept as necessary any true proposition and, finally, non-standard reduction theses. The system had to be tight enough to avoid accepting even one apodeictic proposition as true, at the same time allowing for existence of such propositions. In conclusion it has to be stated that the concept of necessity that Łukasiewicz propagated, over the years changed in the same way as his conception of logic did. The “late” Łukasiewicz, propagating instrumentalism in the question of the obligatory character of logic systems, and empiricism in the sphere of the origin of logic truths, could not allow for categorical propositions. So in the battle he fought throughout his life against the concept of necessity and determinism that he considered to be connected with it, Łukasiewicz finally won a victory.

Translated from Polish by Tadeusz Karłowicz

53 All this quotation is directed against Aristotle and his conception of apodictic know-

ledge based on definitions as the ultimate foundations. Intellectual obviousness giving defi-nitions necessity guaranteed for Aristotle truthfulness of all knowledge based on the syllo-gism that made it possible for that necessity to inherit the ultimate premises by proposi-tions proven in the apodictic way.

54 Łukasiewicz, Aristotle’s Syllogistic, p. 206.

marek lechniak...marek lechniak 24 other the late one, dating from 1940’s and 1950’s. basically...

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