Logique & Analyse 146 (1994), 169-208 LOGICS OF REJECTION: TWO SYSTEMS OF NATURAL DEDUCTION O. Introduction Allard M. TAMMINGA This paper contains two systems of natural deduction for the rejection of non-tautologies of Classical Propositional Logic. The first system is correct and complete with respect to the body of all non-tautologies, the second system is correct and complete with respect to the body of all contradictions. The second system is a subsystem of the first. We begin with an historical synopsis o f the development o f the theories o f rejection fo r the classical logic of propositions, taking our starting-point from the theories of their 'founding father', Jan Lukasiewicz. Subsequently, the systems o f natural deduction are set forth and their correctness and completeness is showed. We shall conclude with an interesting 'Theorem of Inversion'. 1. Lukasiewicz's theories of rejection Jan Lukasiewicz (1878-1956) was among Kazimierz Twardowski's pupils at Lw6w University. In 1910 and 1911 Lukasiewicz participated in a seminar, presided by Alexius Meinong (1853-1921), at Graz University. Both Meinong and Twardowski, but also Edmund Husserl (1859-1938), studied with Franz Brentano (1838-1917). In 1911 Lukasiewicz was appointed as an extraordinary professor at Lw6w University. In 1915 he was offered a chair in philosophy at Warsaw University, where he lectured until 1939. After the Second World War Lukasiewicz held a position as professor of mathematical logic at the Royal Irish Academy o f Science in Dublin. Lukasiewicz owes his present-day fame to his research into multiple-valued logics, to his research into propositional logic, to the so-called 'Polish notation', and to his studies of the logic of Aristotle and of Stoic logic. In this paper Lukasiewicz's work is o f importance because he was the first to introduce the concept of 'rejection' into formal logic. According to Jerzy Stupeckil, Lukasiewicz's idea to introduce the concept of 'rejection' 1 Stupecki (1970); p. ix. 170 A L L A R D M. TAMM1NGA into formal logic, besides Frege's concept 'Anerkennung der Wahrheit', was first set forth in the paper 'Logika dwuwartoAciowa', which appeared in 1921. It will be recommendable to sweep the dust from this paper. In the introduction of the paper 'Two-valued Logic', the English translation o f 'Log ika dwuwartokiowa ', Lukasiewicz writes: ' I n adding 'rejection' to 'assertion' I have followed Brentano.' 2 B r e n t a n o i n h i s m a g num opus, his Psychologie vom empirischen Standpunkt which was published in 1874, defines the concept of 'Urteilen' as follows: Unter dem Urteilen verstehen wir [...] ein (als wahr) Annehmen oder (ais falsch) Verwerfen. 3 Brentano's lectures on logic, posthumously published as Die Lehre vom richtigen Urte il, a lso conta in some passages o n th e concept o f 'Verwerfung', that fa ll in with Lukasiewicz's ideas on the concept o f 'rejection'. Brentano here describes the notion of an 'Urte il' in the following way: wo immer etwas anerkannt oder verworfen, bejaht oder verneint wird, haben wir ein Urteil vor uns 4and Wer urteilt, stellt das, was er beurteilt, vor D o c h eine zweite neue Beziehung zum Gegenstand kommt zu de r im Vorstellen selbst gegebenen hinzu, sic ist charakterisiert durch das Anerkennen oder Verwerfen, Bejahen oder Vemeinen 5and ein Urteil [ist] immer dann gegeben [...], wenn etwas anerkannt oder verworfen, bejaht oder verneint wird und die Kategorien wahr und falsch anwendbar sind 6 2 Lukasiewicz (1921); P. 89. In (1951); p. 94. Lukasiewicz reports on the distinction between 'assertion' and 'rejection': '1 owe this distinction to Franz Brentano, who describes the acts of believing as anerkennen and verwerfen: 3 Brentano (1874); p. 34. 4 Brentano (1956); p. 32. 5 Brentano (1956); p. 33. 6 Brentano (1956); p. 97. LOGICS OF REJECTION: TWO SYSTEMS OF NATURAL DEDUCTION 1 7 1 The fo llowing passage most clearly presents Brentano's conceptions. Besides this, Brentano is the first to propound a pair of symbols to distinguish between the two kinds of judgment: Da jedem Urteil eine Vorstellung zugrunde liegt, so wird die Aussage ais Ausdmck des Urteils notwendig einen Namen enthalten. Dazu wird aber noch ein anderes Zeichen kommen mtissen, das demjenigen inneren Zustand entspricht, den wi r eben Urteilen nennen, d. h. e in Zeichen, das den bloBen Namen zum Satz ergdnzt. Und da dieses Urteilen von doppelter Art sein kann, ndmlich ein Anerkennen oder Verwerfen, so wird auch das Zeichen daftir ein doppeltes sein mtissen, eines f iir die Bejahung und eines ft ir die Verneinung. Far sich allein bedeuten diese Zeichen nichts [...], aber in Verbindung mit einem Namen sind sie Ausdruck eines Urteils. Das allgemeinste Schema der Aussage lautet daher: A ist (A +) und A ist nicht (A -). Au f dieses Schema muB sich jedes einfache Urteil zurtickftihren lassen, d. h. jedes Urteil, in welchem wirklich nur eine Bejahung oder eine Verneinung vorkommt, gleichgtiltig, ob die Materie einfach oder zusammengesetzt ist, denn diese Ausdrucksform enthdlt alles, was zu einem einfachen Urteil gehôrt: einen Namen, der das Beurteilte nennt und ein Zeichen, welches zu erkennen gibt, ob das Beurteilte anzuerkennen oder zu verwerfen sei. 7 Brentano also gives some examples o f an 'Anerkennung' and o f a 'Verwerfung'. The judgment 'Eine Million Menschen ist' must be classified as an 'Anerkennung', the judgment 'Ein weiBer Rabe 1st nicht' as a 'Verwerfung'. In this way, Brentano tries to build up a non-propositional theory of judgment, because that which is rejected or asserted in these examples is not a proposition, but the content of an idea ('Vorstellung'). In these examples, names o f the concerning contents are 'E ine Mil l io n Menschen' and 'Ein weiBer Rabe'. With respect to the concepts 'assertion' and 'rejection' Lukasiewicz, in the paper 'Two-valued Logic', writes: I do not define these terms, and by assertion and rejection I mean the ways of behaviour with respect to the logical values, the ways known to everyone from his own experience. I wish to assert truth and only truth, and to reject falsehood and only falsehood. The words 'I assert' are denoted by U, and the words 'I reject' by N. I consider the sentences: 7 Brentano (1956); p. 97-98. 172 A L L A R D M. TAMMINGA U: T, N : which are read: ' I assert truth' and ' I reject falsehood', respectively, to be the fundamental principles o f two-valued logic, although I do not quote them anywhere. These propositions are also read: ' I assert that truth is' and 'I reject that falsehood is' . 8 Although Woleriski 9 a n d S l u p e c k i l ° d i s r e g a r d t h e d i f f e r e n c e , t h i s c o n c e p t 'rejection' is not identical with the concept 'rejection' which comes to the fore in Lukasiewicz (1951), (1952) and (1953). While Lukasiewicz in (1921) explicitly demands the falsity of a proposition as a condition for its rejection, in (1951) a proposition must be rejected i f there is at least one distribution o f truth-values over the propositional variables in which the proposition under consideration is false. On the other hand, nothing changes with respect to the conditions under which a proposition can be asserted: Of two intellectual acts, to assert a proposition and to reject it, only the first has been taken into account in modern formal logic. Gottlob Frege introduced into logic the idea of assertion, and the sign of assertion (1-), accepted afterwards by the authors of Principia Mathematica. The idea of rejection, however, so far as I know, has been neglected up to the present day. We assert true propositions and reject false ones. Only true propositions can be asserted, for it would be an error to assert a proposition that was not true. An analogous property cannot be asserted of rejection: it is not only false propositions that have to be rejected. It is true, of course, that every proposition is either true or false, but there exist propositional expressions that are neither true nor false. Of this kind are the so-called propositional functions, i.e. expressions containing free variables and becoming true for some of their values, and false for others. Take, for instance, p, the propositional variable: it is neither true nor false, because for p/1 it becomes true, and for plO it becomes false. Now, of two contradictory propositions, a and --ice, one must be true and the other false, one therefore must be asserted and the other rejected. But neither of the two contradictory propositional functions, p 8 tukasiewicz (1921); p. 91. I have adapted the notation. LOGICS OF REJECTION: TWO SYSTEMS OF NATURAL DEDUCTION 1 7 3 and c a n be asserted, because neither of them is true: they both have to be rejected." Notwithstanding Lukasiewicz's remark in his 'A System of Modal Logic': The idea of rejection was introduced into logic by myself in 1951 12 the formal apparatus set forth in 'Two-valued Logic' suffices to formulate the later concept of 'rejection' adequately. The heart of the matter can be met in terms of Lukasiewicz's definition of (propositional) quantifiers. In his 'Two-valued Logic' Lukasiewicz supposes that implication (-) conforms to the following principles: Z 1 U : I - › Z 2 U : 1 - > T Z 3 N : Z 4 U : T - - - > T Further, he introduces the symbols 0, iv, x as variables, ranging over the logical values T and I Finally, Lukasiewicz introduces the quantifiers 11 and E with adjacent subscripts 0, tif, X. In this way, 11 0 means 'for every ' and E tit means 'for at least one t if F o r the universal quantifier Lukasiewicz formulates the following truth-condition: '1 assert any expression containing variables with universal quantifiers which yields only asserted expressions on the replacement of the variables by the values I and T . ' 1 3 L u k a s i e w i c z d o e s n o t f o r m u l a t e a t r u t h c o n d i t i o n f o r t h e i n t r o duction of an existential quantor. Nevertheless, I trust that the following truth-condition would be entirely in accordance with Lukasiewicz's ideas in 'Two-valued Logic': ' I assert any expression containing variables with existential quantifiers which yields at least one asserted expression on the replacement of the variables by the values a n d T.' This language enables tukasiewicz to formulate the following proposi-tion: 9 Wolefiski (1989); p.86. IC/ Slupecld (1970); p. ix. 11 tukasiewicz (1951); p. 94-95. 12 Lukasiewicz (1953); p. 353. 13 Lukasiewicz (1921); p. 99. 174 A L L A R D M. TAMM1NGA U : 1 1 0 [ 0 0 ] By substitution o f the logical values T and J.. fo r 0 , we obtained the aforementioned principles Z 1 e n Z 4 . We can also formulate the following: U: v i [ For, when v is replaced by T , we obtain U: T, which was accepted above. Using this terminology, the notion of 'satisfiability' can be formulated thus: 1.1 Definition Let op O n b e t h e c o n s t i t u e n t v a r i a b l e s o f a f o r m u l a E Then: E is satisfiable <=> U: E 0 1 . . . E 0 , ,Now, we are in a position to define the concept of 'x is rejected': 1.2 Definition Let 0 1 , o n b e t h e c o n s t i t u e n t v a r i a b l e s o f a f o r m u l a E Then: E is rejected <,> U : E 0 1 . . . E 0 , [ - a 7 1 • 1 4 As a consequence, the next theorem can easily be established: 1.3 Theorem L e t 0 1 , 0 , b e t h e c o n s t i t u e n t v a r i a b l e s o f a f o r m u l a E. Then: E is rejected <,> In short, the 1921 paper 'Two-valued Logic' already contains all the ingredients necessary for a formal treatment of the concept of 'rejection'. A concept that will reach its full growth in Lukasiewicz's later works, starting with the 1951 monograph Aristotle's Syllogistic from the Standpoint o f Modern Formal L o g ic 1 5 . Y e t L u k a s i e w i c z d o e s n o t f u l l y e x p l o i t t h e p o s s i bilities offered by the formal apparatus contained in 'Two-valued Logic'. Perhaps this can be explained by Lukasiewicz's strict adherence, in the 14 Compare Lukasiewicz (1951); p. 95: By introducing quantifiers into the system we could dispense with rejection.' In the following Lukasiewicz presents two applications ofDefinition 1.2. 15 Lukasiewicz (1951). LOGICS OF REJECTION: TWO SYSTEMS OF NATURAL DEDUCTION 1 7 5 thirties, to Brentano's demand that falsity be a necessary condition fo r rejection. Therefore it seems more appropriate to interpret the operator , as i t is used in the paper 'Two-valued Logic', in strict analogy to the interpretation o f 'U: ' , as 'Anerkennung der Falschheir. Th is wou ld conform to Brentano's use of his concept of 'Verwerfung'. Lukasiewicz's use of the verbs 'to assert' and 'to reject' in his lecture ' 0 Determinizmie' ('On Determinism') offers additional support for this interpretation. In an attempt to refute an argumentation in favor of determinism, derived from Aristotle, he writes: On the assumption of John's presence at or absence from home tomorrow noon not yet being decided, the sentence ' it is true at the present instant that John wil l be at home tomorrow noon' can neither be accepted nor rejected, i.e. we cannot consider it either true or fa lse . 16 In brief, in order to be justified in saying '11 assert 0 ', we first have to know that a certain sentence 0 is true, and likewise, in order to be justified in saying ' I reject 0 ', we first have to know that a certain sentence 0 is false. As long as we do not (yet) possess the required knowledge, 'we can neither accept nor reject the sentence but should suspend our judgement' 17 . I n o r -der to escape from determinism, Lukasiewicz sees no other possibility than to reject the principle of bivalence and to introduce a third truth-value, besides the truth-values 'true' and 'false': 'there are propositions which are neither true nor false but indeterminate.' 18 According to Lukasiewicz's later, more formal, concept of rejection f a l s i t y of the proposition concerned is no longer a necessary condition for its rejection; rather, the one and only condition for ' -I 0 ' is that 10 ' (' 0 is a tautology') does not hold. Thus, pace Stupecki and Wolefiski, i t is not quite correct to trace Lukasiewicz's later concept o f rejection to his 1921 paper 'Two-valued Logic'. For, clearly, the concept of 'rejection' underwent a considerable change between 1921 and 1951. Besides the operator decribed above, the paper 'Two-valued Logic' contains an interesting 'verbal rule'. Later on, we shall encounter a more precise fo rmu la t ion o f th is ru le kn o wn a s Lukasiewicz's 'Ru le o f 16 Lukasiewicz (1922); p. 125. 17 Lukasiewicz (1922); p. 124. 18 Lukasiewicz (1922); p. 126. 176 A L L A R D M. TAMMINGA Detachmene. 19 I n ' T w o v a l u e d L o g i c ' , L u k a s i e w i c z g i v e s a n e m b r y o n i c formulation of this rule: 'I reject any expression which means the same as some rejected express io n ' 2 0 In order to induce a better understanding o f this rule, Lukasiewicz offers the following example. 21 G i v e n t h e d e f i n i t i o n U: 1 1 0 1 1 1 4 0 v t v D f - * v l we obtain by the substitutions I for 0 and 1 for V: (1) U : 1 E D . , -Let it, moreover, be given that: (2) N : - - > 1 With the rule of rejection quoted above we finally obtain from (1) and (2): N: l v i These reflections on the paper 'Two-valued Logic' lead to the following conclusion: The paper does not contain the concept of rejection as formulated in Lukasiewicz's later works. I t does, however, contain some seeds which were developed into what may count as Lukasiewicz's mature theory of rejection. In my opinion, the most important of these seeds is not the operator , but the rule of rejection quoted above. Next we shall consider the mature concept of rejection as it is presented in Lukasiewicz's Aristotle's Syllogistic. In this monograph, Lukasiewicz dissociates h imse lf f rom traditional views on Aristo t le 's syllogistic. According to Lukasiewicz, Aristotle's syllogistic is a system of theorems, not a system of rules of inference, concerning propositions in which only variables fo r general terms (like 'tree', 'man ' and 'horse', but unlike 'Socrates' o r 'Plato') occur. Further on Lukasiewicz, following Aristotle, 19 Cf. tukasiewicz (1951); p. 71: ' I f the implication 'I f a, then /3 is asserted, but its consequent f i is rejected, then its antecedent a must be rejected too.' 20 tukasiewicz (1921); p. 99. 21 tukasiewicz (1921); p. 99-100. LOGICS OF REJECTION: TWO SYSTEMS OF NATURAL DEDUCTION 1 7 7 establishes all the rules of conversion and all the 'affirmative moods' (true syllogistic forms) on the basis of four axioms, two definitions and the resources of the classical propositional logic (CPL). Lukasiewicz writes: Aristotle in his systematic investigation o f syllogistic forms not only proves the true ones but also shows that all the others are false, and must be rejected. 22 When a proof is required to show that a certain syllogistic form is not a theorem, Aristotle usually presents a counter-example. Let us consider the following syllogistic form: If every B is (an) A and no C is (a) B, then there is at least one C which is no A. I f we substitute the term 'animal' for A, the term 'man' for B and the term 'horse' for C, we obtain a false proposition. Therefore, the syllogistic form, of which this proposition is an instance, is not a theorem of the Aristotelian syllogistic. Concerning this procedure Lukasiewicz writes: This procedure is correct, but it introduces into logic terms and propositions not germane to it. 'Man' and 'animal' are not logical terms, and the proposition 'A ll men are animals' is not a logical thesis. Logic cannot depend on concrete terms and statements. 23and Aristotle rejects most invalid syllogistic forms by exemplification through concrete terms. This is the only point where we cannot follow him, because we cannot introduce into logic such concrete terms as 'man' or , a n i m a l , . 2 4 Further on, Lukasiewicz even speaks of a ' f la w ' 2 5 i n A r i s t o t l e ' s e x p l a n a tion concerning the rejection of syllogistic forms by concrete te rms. 26 Y e t ,this is not the whole story: 22 Lukasiewicz (1951); p. 67. 23 Lukasiewicz (1951); p. 72. 24 Lukasiewicz (1951) p. 96. 25 Lukasiewicz (1951); p, 131. 178 A L L A R D M. TAMMINGA There are, however, cases where he [Aristotle] applies a more logical procedure, reducing one invalid form to another already rejected. On the basis of this remark a rule of rejection could be stated L.]; this can be regarded as the commencement of a new field of logical enquiries and of new problems that have to be solved. 27 What, then, is Aristotle's 'more logical procedure'? Lukasiewicz translates the crucial text in Aristotle's Analytica Priora (i. 5, 2 7 1 1 1 2 2 7 1 3 2 3 ) a s f o l lows: Let M belong to no N, and not to some X. It is possible then for N to belong either to all X or to no X. Terms of belonging to none: black, snow, animal. Terms of belonging to all cannot be found, if M belongs to some X, and does not belong to some X. For if N belonged to all X, and M to no N, then M would belong to no X; but it is assumed that it belongs to some X. In this way, then, it is not possible to take terms, and the proof must start from the indefinite nature of the particular premiss. For since it is true that M does not belong to some X, even if it belongs to no X, and since if it belongs to no X a syllogism is not possible, clearly it will not be possible either. 28 Here, Aristotle tries to show the falsity of the implications (a') and (pi) (consequently of ( a ) and ( ) ) : ( a) I f no N is (a) M and at least one X is not (a) M, then every X i s (a) N. ( a ') I f no N is (a) M and at least one X is not (a) M, then at least one X is (a) N. (s ) I f no N is (a) M and at least one X is not (a) M, then no X is (a) N. ( /3 ') I f no N is (a) M and at least one X is not (a) M, then at least one X is not (a) N. In order to refute (a') , and consequently ( a ), Aristotle finds the terms 'black' for M, 'snow' for N, and 'animal' for X. In order to refute ( S '), and consequently (0) , Aristotle looks for terms making true all categorical sentences of (a) . Further, Aristotle shows that no terms can be found 26 I would like to attend the reader to the fact that Tarski's logical semantics enable us to construe counter-examples without any reference to concrete terms. 27 Lukasiewicz (1951); p. 74. 28 Lukasiewicz (1951); p. 70. LOGICS OF REJECTION: TWO SYSTEMS OF NATURAL DEDUCTION 1 7 9 which satisfy both this demand and the dual condition that 'at least one X is (a) M' and 'at least one X is not (a) M' be true as well. In conclusion, Aristotle finds an other possibility: in order to refute (0 ) and (0 ') it is sufficient to refute the following syllogistic form: y) I f no N is (a) M and no X is (a) M, then at least one X is not (a) N. For ( 0) implies (p') and (0') implies ( y). If we know that ( y) is rejected, then we are justified in making the inference that ( 0 ) must be rejected. Therefore ( fi) must be rejected too. In short, these results entitle us to conclude that no single syllogistic form with the antecedent ' if no N is (a) M and at least one X is not (a) M' and a consequent consisting of a subject X and a predicate N, can be a theorem. This procedure, which Lukasiewicz characterized as 'more logical', can be summarized as follows: (c) I f the implication ' I f a , t h e n 1 3 ' i s a s s e r t e d , b u t i t s c o n s e q u e n t 1 3 is r jected, then its antecedent a must be rejected too. 29 Immediately, Lukasiewicz formulates a second rule of rejection: (d) I f a is a substitution for p, and a is rejected, then p must be rejected too. 30 In the chapter 'Aristotle's System in Symbolic Form' (op. cit.) Lukasiewicz constructs a formal system for Aristotelian syllogistic. He introduces four axiomatically asserted theses, two axiomatically rejected theses and the rules (c) and (d) mentioned above. This basis allows him to deduce all 232 false syllogistic forms. However, this fac t does not imply that Lukasiewicz's system would be inconsistent: all theorems of his system are preceded by a number, and all non-theorems of the system are preceded by a number and an asterisk. Of course, no single formula is both a theorem and a non-theorem of the system. Unfortunately, it is not possible to deduce every formula, constructed Out of the languages of propositional logic and syllogistic: not every formula of Aristotelian syllogistic can be either asserted or rejected. The system is L-undecidable 31 29 Lukasiewicz (1951); p. 71. 30 LAJkasiewicz (1951); p. 72. 180 A L L A R D M. TAMMINGA In 1948 Jerzy Stupecki showed that Lukasiewicz's system of Aristotelian syllogistic is powerful enough to deduce all true formulas of the system, but also that there is an infinity of false formulas that cannot be rejected within the system. In addition Stupecki showed that if a new rule of rejection were added to the system, all false formulas could be rejected within the sys tem. 3 2 T h u s t h e L d e c i d a b i l i t y o f t h e e x t e n d e d v e r s i o n o f Lukasiewicz's system for Aristotelian syllogistic was proved. When proving the L-decidability of the extended system (the system augmented with Stupecki's new rule of rejection), it turned out to be useful, in order to establish a certain lemma, to introduce 'rejection' for propositional logic as well. For CPL (Classical Propositional Logic) Lukasiewicz constructs the following system: We reject axiomatically the variable p, and accept the clear rules of rejection, (c) and ( 4 3 3 If we accept the symbols '1-' for 'assertion' and '-1' for 'rejection', symbols which Lukasiewicz for the first time uses in his paper 'A System of Modal Logic ' 34 , w e c a f o r m u l a t e t h i s t h e o r y o f r e j e c t i o n f o r C P L a s f o l l o w s : axiom H p ; detachment I f F Ø 4 t y a n d I v , t h e n d o ; substitution I f -1 i and tif can be obtained out of 0 by substitution, then -10 This system presupposes a complete system for the derivation of all theses of CPL. If 0 is a thesis of CPL, we shall write: 1-0 31 A system is L-undecidable iff there is at least one formula, formulated in the language of the system, which can not be asserted or rejected with the available techniques of the system. 32 Cf. Lukasiewicz (1951); p. 103ff for details. Slupecki's rule of rejection is of no use within propositional logic. 33 Lukasiewicz (1951); p. 109. Cf. Lukasiewicz's description of this system in (1952); p. 333-334. 34 Lukasiewicz writes: 'The idea of assertion and its sign w e r e introduced into logic by Frege in 1879, and afterwards accepted by the authors of the Principia Mathemanca. In my previous papers 1 always omitted this sign, but here 1 am bringing it in because, besides assertion, I introduce rejection. [...] 1 denote rejection by an inverted sign of assertion following a suggestion of Ivo Thomas.' Lukasiewicz (1953); p. 352-353. In 'A System of Modal Logic' Lukasiewicz also uses the concepts 'assertion' and 'rejection' f or a four-valued modal logic: the L-modal logic. LOGICS OF REJECTION: TWO SYSTEMS OF NATURAL DEDUCTION 1 8 1 The formula p--->q is not a thesis of CPL, for it is false if p is true and q is false. Accordingly, we may, using Lukasiewicz's theory o f rejection for CPL, show that: -Ip q 3 5 . (1) p ) -› p) -) p (2) d p (3) -I ((p p ) - > p) (4) d p -› ((p -> p) -› p) -› p is a thesis of CPL axiom detachment; by (1) and (2) substitution; take in (4) p -> p for p and p for q and we obtain (3) Therefore -Ip q . The system that results out o f a standard type of deductive system for CPL, augmented by this axiom and these rules of rejection, is both correct and complete with respect to the class of non-tautologies. In other words: exactly those formulas which are false according to at least one distribution of truth-values over the atoms can be rejected within this system. Klaus Hdrtig offers a short proof of this theorem. 36 B e s i d e s , H d r t i g s h o w s t h a t , i f , instead of the axiom dp a contradiction is axiomatically rejected and the rule o f substitution is cancelled, then the resulting system is correct and complete with respect to the class of contradictions. 37 L u k a s i e w i c z c o n s i d ered his application of Aristotle's idea of 'rejection' to CPL as one of the most important contributions yielded by the formal part o f his investigations in Aristotle's non-modal syllogistic. 2. Theories of rejection for propositional logics after Lukasiewicz In the paper 'On Proofs of Re ject ion ' 38 W a l e n t y S t a s z e k c o m p a r e s t w o definitions of a proof of rejection and, under certain conditions, establishes their equipollence. Lukasiewicz's proofs of rejection satisfy the first definition. In his paper 'Z badati nad klasyczna logika n a zw' 3 9 S t a s z e k s e t sforth another kind of proofs of rejection. These proofs satisfy the second 35 Cf. Lukasiewicz (1951); P. 109. 36 Hartig (1960); p. 241. 37 Cf. Theorem 4.3.7 of this article. 38 Staszek (1971). 39 Staszek (1969) ('On the Classical Logic of Names'). The article is available only in Polish, therefore I was unable to consult it. 182 A L L A R D M. TAMM1NGA definition. However, the 1971 paper does not contain any example o f a proof of rejection of this second kind. I shall attempt, however, a reconstruction o f an example o f a proof o f a formula o f CPL to illustrate Staszek's second definition of proofs of rejection. Staszek notices that Lukasiewicz's rule o f detachment and his rule o f substitution have their analogues in proof theory. The rule of substitution in the usual proof theory can be formulated thus: SUBSTITUTION I f F-0 a n d t y can be obtained ou t o f 0 b y substitution, then F ty The rule of detachment in proof theory is the modus ponens rule: DETACHMENT I f F-0 v f and I 0 , t h e n 1 - t v Staszek maintains the usual notation for these BOLDFACE printed rules: I f the ru le o f SUBSTITUTION is applied, he writes OF iv; i f the ru le o f DETACHMENT is applied, he writes: 0 - ) t v , t v . T h e i t a l i c p r i n t e d r u l e s are submitted to the following convention: I f the rule o f substitution is applied, Staszek writes: ty-10 ; i f the rule o f detachment is applied, he writes: 0 -) Vt V/Ho. Thereupon, Staszek establishes the fo llowing elations between these pairs of rules of inference° : KO O F ! I f < = > t v 1 0 GO Ø -> t v , 01 11/ , # > 0 - > v f , V t H Ø Staszek defines a sequence ( a ) as an arbitrary sequence o f formulas 01 , 0 2 , • • • , O n • T h e d e f i n i t i o n o f t h e c o n c e p t o f i p r o o f , w h i c h e m b o d i e s t h e way Staszek himself conceives of a proof of rejection, runs as follows: The sequence ( a ) is an i-proof of an expression 0 i f 1' t h e sequence ( a ) is a usual proof and 2* 0 1 = 0 a n d 0 „ i s a r e j e c t e d a x i o m . 4 1 I f we, following Lukasiewicz, axiomatically reject the propositional variable p, we may, using Staszek's method, prove the formula p->q as fo llows: 40 Staszek (1971); p. 18. 41 Staszek (1971); p. 19. I have slightly changed the notation. LOGICS OF REJECTION: TWO SYSTEMS OF NATURAL DEDUCTION 1 8 3 (1) 1-(p - ) •->(--ip v q) q)-->(-,pv q) is a theorem of C'PL (2) H H q Axiom 1 ( 3 ) 1 1 -1 1 1 3 Axiom 2 (4) v Disjunction; by (2), (3) and (13) n (q) =0 (5) O p q Detachment; by (1) and (4) (1) p -3q (2) (P - ) P ) - ) P and p ( 3 ) ( ( P / 3 ) 4 P ) - > P (4) p Therefore -1p -› q. Further Staszek establishes, without any additional conditions, that for every Lukasiewicz-style proof of rejection of a formula 0 there exists an i-proof o f Ø. I f the logical systems under consideration satisfy certain conditions, the converse can be proved as well. Working with the rule o f substitution o f Lukasiewicz's system of rejection for CPL can be rather difficult. Errors wil l easily be made. In 1961 Klaus Hdrtig faced this shortcoming of Lukasiewicz's system in his paper 'Zur Axiomatisierung der Nicht-Identitdten des Aussagenkalkills'. He replaced Lukasiewicz's rule of substitution by a set of more convenient rules. I f we define ' (0)' as '1 vf: yi is an atomic subformula of 01', then Hdrtig's extended version of rejection for CPL can be formulated as follows: Axiom 1 Axiom 2 Detachment Disjunction Hdrtig established that this extended system is correct and complete with respect to the class of non-theorems o f CP L . 4 3 T h u s , H d r t i g ' s s y s t e m i s equipollent with Lukasiewicz's system, although the former does not contain Lukasiewicz's rule of substitution. Let us illustrate the system o f Hartig with the following example; a deduction of H p - > q : Therefore H l p • - > q H 1 0 , i f 0 E P R O P L 4 2 if 0 e PRON., ; I f 1-0 t i f and H I V , t h e n 1 0 f H d o , H 1 V / a n d ( 0 ) n ( v ) = 0 , t h e n H 1 0 v 42 A definition of PROPL can be found in section 3. 43 Hartig (1960); p. 244. premiss SUBSTITUTION; take in (1) p -› p fo r p for q and we obtain (2) ((pi p ) ---> p) p is a thesis of CP DETACHMENT; by (2) and (3) 184 A L L A R D M. TAMMINGA In the paper under consideration, Hiirtig further investigates what conditions an axiom system of the non-theorems of CPL must meet in order to be complete. Lastly, Hdrtig points out several open problems for further investigation. For instance: 'Was entspricht Satz 1 [the completenesstheorem for tukasiewicz's system o f rejection fo r the propositional calculus] speziell im intuition istischen Aussagenkalkill?'" Later, this problem was solved by Dutkiewicz (1989). Hdrtig concludes his paper as follows: Man kann mit der Einsetzungsregel [...] arbeiten oder sie ausschlieBen, und man kann auch versuchen, bei den Regeln den Prdmissentyp (a) - also die Bezugnahme auf die Eigenschaft d . h . auf die gegebene deduktiv abgeschlossene Menge - z u vermeiden. Schon f t i r d ie Spezialfdlle der klassischen Nicht-Identititten und der intuitionistisch-ungtiltigen A u s d r t i c k e w d r e n m i i g l i c h s t e i n f a c h e Axiomatisiemngen, in deren Regan n icht [...] au f die Identitdten zurtickgegriffen wird, von Interesse. 45 Further on, Hdrtig's system wi l l play the leading part in the completeness proof of the system to be presented in § 3 of this paper. Already in the paper 'On the Intuitionistic Theory o f Deduction', tukasiewicz has proposed to add a rule to his system of rejection for CPL in order to obtain a correct and complete system o f rejection fo r the Intuitionistic Propositional Calculus (1PL). After a short description of his system of rejection for CPL, tukasiewicz writes: I f we add to these general rules a special rule of rejection which is valid according to Gôdel in the intuitionistie system: (g) I f a and p are rejected, then a v 13 must be rejected, we get, as far as I see, a categorical system in which all the classical theses not accepted by the intuitionists can easily be disproved. 46 Thus, the system of rejection for IPL, envisaged by tukasiewicz, must be as follows: 44 Hartig (1960); p. 246. 45 Hartig (1960); p. 247. 46 tukasiewicz (1952); p. 334. 1 slightly changed the notation. LOGICS OF REJECTION: TWO SYSTEMS OF NATURAL DEDUCTION 1 8 5 Axiom i Det ach men t i-Substitution i-Disjunction A deduction of the following way: (1) i I P (2) i ( ( p - - - > p ) - > p ) - > p (3) i K P p ) > p (4) i d (5) (6) (7) F i - p - q p - > q ) i d p v Lukasiewicz's conjecture that this system is correct and complete with respect to the class of non-theorems of IPL proved to be false. Admittedly, the addition of rule (g) to Lukasiewicz's classical system of rejection does not lead to the inconsistency o f the resulting in tu it ion istic system. However, this resulting system is not complete with respect to the class of non-theorems of IPL. In his paper 'The Method of Axiomatic Rejection for the Intuitionistic Propositional L o g ic ' 4 7 R a f a l D u t k i e w i c z p r e s e n t s a n a p p r o p r i a t e r u l e o f rejection, which yields, added to Lukasiewicz's classical system of rejection, a correct and complete system to reject every non-theorem of IPL. However, the principal a im o f Dutkiewicz's paper is to establish the L-decidability 4 8 o f I P L . T h e a d d e d r u l e o n l y s e r v e s t o a l l o w t h i s p r o o f o f L-decidability; it is not meant for practical purposes. The application o f this added rule is subjected to two complex conditions. In turn, these conditions presuppose a considerable technical apparatus." Therefore, it will be desirable t o construct a more convenient system t o re ject the non-theorems of IPL, the more so as Dutkiewicz showed that a complete and correct system for this task is possible. 47 DutIdewicz (1989). 48 Cf. footnote 31. 49 Cf. Dutkiewicz (1989); p. 455-456. I f k i 0 a n d I v , t h e n i H O ; If i t i f and i can be obtained out of 0 by substitution, then If i 1 0 a n d i l v t , t h e n i v non-theorem p o f IPL can be completed in the Axiom i ((p -› p) -> p) -4 p is a theorem of !PL i-Detachment; by (1) and (2) i-Substitution; take in (4) p -› p for p and p for q and we obtain (3) -› q) is a theorem of IPL i-Detachment; by (4) and (5) i-Disjunction; by (1) and (6) 186 A L L A R D M. TAMM1NGA Almost a ll contemporary studies in the logic o f rejection share the characteristic that they have been developed in order to prove some meta-logical theorems, for instance, the L-decidability of CPL or IPL. To the best of my knowledge, there have been so far only two attempts - those by Xavier Caicedo and by Valentin Goranko - to abandon axiomatic structure in order to obtain a more practicable system. Caicedo is the first to present a self-sufficient system o f rejection fo r CPL, i.e., a system which does not rely on the body of theorems of CPL. He does so in his paper 'A Formal System for the Non-Theorems of the Propositional Ca lcu lus' 5 () T h i s f o r m a l s y s t e m c o m p r i s e s t w o a x i o m s a n d eight rules. The rules ensure that every formula occurring in a linear rejection proof is a non-tautology. Caicedo's system can be formulated as follows: Axioms A I 0 - - - * i f 0 e P R O P L ; A 2 1 0 i f 0 e P R O P L . Rules R ( a ) -) if i ) 0 EPROPL-, ii) e ( V ) . (b) t if i f i ) E PROPL; --> i i ) e V )• R2 0 - > 111 R 3 0 - ) V 0 - › ( 0 - ) I Y ) - ) 0 ) V R4 1 0 R 5 1 0 V ( - > X ) - " V 0 R6 0 - " g R 7 0 - > ( V R8 0 E and -iv/ w h e r e E has the form indicated below. 50 Caicedo (1978). LOGICS OF REJECTION: TWO SYSTEMS OF NATURAL DEDUCTION 1 8 7 The formula E i n R8 must have the fo rm E = E l o rE = E l ( E 2 E , = x , o r E , x, E PROPL, x, f o r i j , a n d ( 0 -> vt) g VI , X 2 , • • • , X n ) • Further on, Caicedo shows that his system is 'perfectly unsound and completely antitautological' 51 , i . e . t h a t h i s s y s t e m i s c o r r e c t a n d c o m p l e t e w i t h respect to the class of non-theorems of CPI, with a n d ' ->' as the only logical connectives. 52According to the system described above, the following linear proof is a proof of c . i p q :(1) -hq -› q A 2(2) q R 5(3) P --- >q R I ( a ) Therefore ( H pAlthough Caicedo's system does not rely on the class of tautologies, its rules, especially R 8 , l a c k a n ' i n t u i t i v e p l a u s i b i l i t y ' . G o r a n k o ' s r e m a r k c o n cerning some of his own systems is also true of Caicedo's system of rejection: An obvious drawback in the sentential refutation systems [...] is that in most cases the specific refutation rules employed in them, although semantically well-formulated, are rather unhandy for practical purposes 53 By and large, the extant systems o f rejection ve rify th is assertion. Apparently, the unwieldiness of these systems explains their relative obscurity outside the specialists. Another objection, both theoretical and practical, which can be raised against mo t o f the systems described above, is their dependence on another system to provide the theses (i.e., the tautologies), which are necessary for the system to work. With respect to this point Goranko writes (he uses MT to denote tukasiewicz's Rule of Detachment): 51 caicedo (1978); p. 149. 52 Caicedo (1978); p. 149-150. 53 Goranko (1994); p. 313. 188 A L L A R D M. TAMM1NGA Another feature of these systems is that the inference of the refutable formulae involves inference of the acceptable, i.e. provable ones, because of the rule MT. This makes the refutation systems inferior to the orthodox ones. 54 Subsequent to his objections against the majority o f the existing rejection systems, Goranko indicates two ways to counter them: A way to abolish this inequality is to construct refutation systems for refutable sequents rather than formulae 55 Goranko opts for the construction o f a system of rejection for sequents. Goranko is the first to present a system of rejection for CPL which is both practicable and independent o f a system which provides the theses. His system, a system in the style of the sequent systems of Gerhard Gentzen, consists of an axiom scheme and seventeen rules. The system can be formulated as fo llo ws 5 6 : Axiom scheme: FHA, i f i ) F PIZO1= 1 1_, a n d A c P R O P L ii) T n A = 0 Structural rules: r, 0-IA F H A , 0 F, 0, 0-IA (Co n t r -1 )F, 0-1A FHA ( W e a k - ' ) 54 Goranko (1994); p. 313. 55 Goranko (1994); p. 314. 56 Goranko (1994); p. 315. FHA, 0, 0 (Co n t r -1 )F-0„ 0 FHA ( W e a k - ' ) LOGICS OF REJECTION: TWO SYSTEMS OF NATURAL DEDUCTION 1 8 9 (1) B-IA Axiom (2) B, B-IA Contr -1(3) A -4 B, B-1,4 I -) (4) A -› B-IA, -B (5) A B , /-, (6) (A -› B) A -AH-B I A(7) H((A -)B) A A ) - > r Logical rules: 1 - , 0 , t y I A I', 0 A IA (IA) F, 0, HA 1 - , 0 v t v I A ( i v ) FHA, 0 F, 0 -> tv-IA (I --)) FHA, 0 -10-IA (i-,) FHA F, THA ( T ) 57 Goranko (1994); p. 316-317. F-IA, 0 FHA, 0 A ig (rA) F, tg-IA F, 0 v vf-IA ( t v ) F, 0-IA, iff FHA, tit FHA, 0 A ( r A ) FHA, 0, tv FHA, 0 v ty ( r v ) IF, 0 -› tp lA (1 -3 ) F-IA, 0 -) ty ( r F, FHA, -10 F-IA FHA, ( 1 ) The system described above is correct and complete with respect to the class o f non-theses o f CP L . 5 7 T h u s , t h e f o r m u l a ( ( A - ) B ) A t i t - B can be derived with Goranko's system: Therefore -I((A -› B) A -,A) -) -B The reader has to keep in mind that Goranko's system is not a system of natural deduction: for Goranko's system consists exclusively of introduction rules, whereas a decent system of natural deduction should comprise both introduction rules and elimination rules. Besides, there is a close relationship between the semantic tableau method o f Evert Willem Beth and Goranko's system for the rejection o f non-theses o f CPL. I f we add to Beth's semantic tableau rules a new closure rule, which meets exactly a ll cases where the usual closure rule can not be applied, then we have found a rule which can directly be reformulated as Goranko's axiom scheme. Beth's method of semantical tableaus decomposes a given sequent in order 190 A L L A R D M. TAMM1NGA to apply a closure rule, Goranko (following Gentzen) starts with the closure rule and tries to build up the intended sequent by means o f his structural and logical rules. In this case, the distinction between a semantic and a syntactic treatment is not very clear indeed. Haskell B. Curry writes: In regard to the inferential methods, the ultimate source is Gentzen [...] Beth's semantic tableaux constitute, in some respects a refinement of the Gentzen mle s. 5 8 In the next section another system for the rejection o f non-theses of CPL shall be set forth. This system differs from the systems of Lukasiewicz, Staszek, and Hdrtig, for it is, in contrast to them, independent of the class of theses o f CPL. Next, Lukasiewicz's rule o f substitution or Staszek's RULE OF SUBSTITUTION do not play a role in the system. Besides, the system to be expounded is constructed in the style of Stanislaw JaAkowski's systems of natural deduction. This means that, instead of the sequents o f Goranko's system, formulas constitute the basic units o f deductions. The system to be expounded shares this characteristic with the systems o f Lukasiewicz, Staszek, Hdrtig and Caicedo. In contrast to the latter, I hope the reader wil l appreciate the practicability o f my system. Besides, the system comprises, in contrast to Goranko's system, both introduction and elimination rules. 3. A system of natural deduction Our syntaxis of classical propositional calculus will be defined according to the following definitions: The alphabet of CPL consists of: (i) Proposit iona l Constants (ii) L o g ica l Symbols (iii) A u x i l ia ry Symbols 58 Curry (1963); p. 25. P I , P 2 , A, V , T , )• The set of atomic formulas consists of all propositional constants and of T and I . The set of formulas is defined inductively as follows: 1) Eve ry atomic formula is a formula; 2) I f 0 and l i f are formulas, then (0 A ip), (0 v ( 0 -> ig) areformulas; LOGICS OF REJECTION: TWO SYSTEMS OF NATURAL DEDUCTION 1 9 1 3) I f i s a formula, then -0 is a formula; 4) Formulas are only those expressions that are obtainable by 1)-3). The following abbreviations are introduced for the sake of convenience: ATOM denotes the set of all atomic formulas; PROPL denotes the set of all propositional constants; FORM denotes the set of all formulas. In Example 3.1.4 the letters A en B are used as specific elements o f PROPL, instead of the indexed letters P I a n d P 2 , T h i s i s b e i n g d o n e f o r t h e sake of legibility. The rules of the system of natural deduction can be classified in groups, according to the starting-points which are required for the application of a rule. Group I consists of the rules At, -.At and _Li; Group H A c o n s i s t s o f t h e r u l e s v E I , y E 2 , E 2 , A / I , A 1 2 a n d T E Group B B c o n s i s t s o f t h e r u l e s T r i v e n E ; Group III consists of the rules --, / and CRV ('classical reductio ad verum'); Gr up IV consists of the rules v i 1 , y / 2 , - › 1 2 , a n d A E The following points determine the use, the scope and the limits o f the ru les: 59I. E ve ry natural deduction consists of a column o f consecutively numbered formulas, accompanied, to the right of the column, by a justification of each formula and, to the left of the column, by a structure o f vertical lines that indicate the scope of each hypothesis. This structure shows, for each formula, in which part of the deduction this formula is operative and in which part it is accessible (see point 6). 2. T h e hypothesis which was introduced last must be the first retracted. Thus, verticals indicating the scope of an hypothesis must not intersect. 3. A part of a natural deduction that belongs to one individual vertical indicating the scope of an hypothesis, will be called a •subdeduction'. 4. E ve ry rule of inference stipulates that i f the column contains certain starting-points, then a certain formula may be written down on a newly introduced line at the bottom of the column. Starting-points can be both occurrences of formulas and occurrences of subdeductions. 5. T h e position and the order in which the starting-points appear is of the greatest importance. An application of one of the rules of Group III or 59 I partly owe these points to Kradavis (1988); pp. 42-43. 192 A L L A R D M. TAMMINGA Group IV requires that the conlusion that can be reached through application of the rule under consideration is written down on the very next line after the termination of the (last) relevant subdeduction. Besides, an application of one of the rules of Group IV requires that the hypothesis of the (first) relevant subdeduction is introduced on the very next line after the formula which serves as starting-point for the application of the rule under consideration. An application of the rule A E requires that the hypothesis of the second relevant subdeduction is introduced on the very next line after the termination of the first relevant subdeduction. The order of the relevant starting-points is irrelevant only whenever the rule i s applied. 6. A rule can be applied only if the required starting-points are attainable. We need some definitions. A formula 0„, on line in is operative at line n with In n , if (i) O m o c c u r s w i t h i n t h e s c o p e o f a n h y p o t h e s i s w h i c h h a s n o t b e e n retracted at line n; (ii) for any hypothesis, if line m is in its scope, so is line n. 6 ° Thus, outside the scopes of all hypotheses no formula is operative. A formula O n , o n l i n e i n i s a c c e s s i b l e a t l i n e n w i t h i n < n , i f (i) both line in and line n lie outside the scope of all hypotheses. A formula O n , o n l i n e i n i s a t t a i n a b l e a t l i n e n w i t h n i < n , i f (i) O n , i s o p e r a t i v e a t l i n e ; o r (ii) O m i s a c c e s s i b l t l i . If a line n occurs under at least one hypothesis, then a formula O n o nline n can be justified only by one of the rules of Group H A o r G r o u pI I B b y r e f r e n c e t o a f o r m u l a O n , o n l i n e i n ( a n d , i n c a s e o f t h e r u l e -1E, a formula 0 / o n l i n e 0 , i f O n , ( a n d o i ) i s ( a r e ) o p e r a t i v e t l i n e n . If line n lies outside the scope of all hypotheses, then a formula 0, on lin n can be justified only by one of the rules of Group H A y r e f e r -ence to a formula O n , n l i n e i n , i f O n , i s a c c e s s i b l e o n l i n e n . If a line n occurs within the scope an hypothesis, then a formula O n o nline n can be justified only by one of the rules of Group IV by reference to a formula O m o n l i n e i n a n d o n e ( t w o ) s u b d e d u c t i o n ( s ) ( 5 1 ( , 8 2 ) , i f 0„, is operative at line n. If line n lies outside the scope of all hypotheses, then a formula O n o nline n can be justified only by one of the rules of Group IV by reference to a formula O n , o n l i n e m a n d o n e ( t w o ) s u b d e d u c t i o n ( s ) 8 ( 5 i f 60 A hypothesis O m o n l i n e m i s w i t h i n t h e s c o p e o f i t s e l f a n d , t h e r e f o r e , i s o p e r a t i v e a t line tn. 0„, is accessible on line n and if O rn , 1 o n l i n e m + 1 i s t h e h y p o t h e s i s o f the (first) relevant subdeduction. The rules of Group III do not need such stipulations. 7. T h e double tilde d e n o t e s parts of the deduction in which no hypothesis is retracted that was not introduced in the very same part of the deduction. This part may be empty as well. 8. T h e introduction and retraction of hypotheses is always permitted (but keep point 2 in mind). Note that the introduction o f an hypothesis is useful only if we intend to apply a rule which requires a certain subdeduction as a starting-point. 3.1.1 Definition ( 0 ) = ty: iv is a subformula of 0 and ig E PROPL1. 3.1.2 Definition T h e system o f natural deduction fo r the rejection o f non-theses o f CPL exactly consists of At, -dit, Triv, y E 1 , y E 2 , v i 1 , y / 2 , - > E , - › E 2 , - › - - > / 2 , A E , A / 1 , A / 2 , C R V , T E e n I t . LOGICS OF REJECTION: TWO SYSTEMS OF NATURAL DEDUCTION 1 9 3 At, i f i ) 0 E PROPL; ii) f o r every l wh ich is operative at line n: 0 ( -lilt, i f i ) 0 E PROPL; ii) f o r every ig which is operative at line n: 0 ( ty). Triv, i f i ) 0 on line m is operative on line n. 0 -› v f- -4 ) - › E t 194 ALLARD M. TAMM1NGA k m - , 0 n T -, E, if 0 on line k and -, operative on line n. 0 on line m are •, 0 V t i t 0 V 111._ _ i i+1 0 0 r 0 --°1 1 1 vE l V i k i f i+1 1 4 1 / _ y E 2 n -1 n -1 0 n 0 v iv v / i n v tit v1 2 i - , 0 i+1 - , 0n -1 LII/ n 0 --+ tit - › / I i i+1 n - 1 , 1 1 0 - ) V , V E 2 LOGICS OF REJECTION: TWO SYSTEMS OF NATURAL DEDUCTION 1 9 5 3.1.3 Definition 0 <,> T h e r e is at least one formal deduction of rejection fo r 0 (according to the system o f Definition 3.1.2). 3.1.4 Example 1. --, B -, At 2. --, B hyp._ 3. A At 4. - (A -› B)A -,A hyp. 5. A --> B hyp. 6. -› E 2 : 5 7. T -,E: 2, 6 8. r _ 1 A hyp. 9. I T -, E. 3, 8 10. T A E: 4, 5-7, 8-9 11. --1((A -4 B) A --À) -,I: 4-10 12. V I -4 B) A -,A)---> -,B -> 1 2 : 1 , 2 1 1 i 0 A V 0 i+10 F 0 A y i / I X l-11 r :n -1 V I n X A E 0 A l i f = = r 0 1 0n -1 LI_L n _lrT n --,0 -I n 0 0 TE A // A 1 2 CRV 196 A L L A R D M. TAMMINGA 4. Soundness and completeness of the system 4.1 Semantics for CPL 4.1.1 Definition A classical model of CPL is an ordered pair (S, / ), with S= (1, 0), and with / a function defined on a subset of PROPL with values in 11, 01. 4.1.2 Definition L e t M be a classical model o f CPL. The language o f M = (S r , 1 3 4 ) i s t h e f o l l o w i n g s e t 2 m : 2 m = {0 : 1 m (P i ) i s d e f i n e d f o r e v e r y P i E ( 0 ) 1 . 4.1.3 Definition L e t M be a classical model of CPL. The classical valuation o f CPL based on M is the following function, V m , defined for every formula o f 2 m and with values in {1,01: Sem m V m ( 0 ) , 1 3 4 ( 0 ) , i f 0 E P R O P L ; Sem V m (0 - › i g ) = < = > V 3 4 0 ) = 1 a n d V m ( t v ) = 0 Sem, V m (0 A t i l f ) = 0 < = > V m ( 0 ) = 0 a n d / o r V m ( t i f ) = 0 ; Sem, V m 0 v 0 = 0 <=> V m ( 0 ) = 0 a n d V m ( t y ) = 0 ; Se m V A I - 1 0 = 0 < = > V A , / ( 0 ) = I ; Sem, V m T)=1; Sem i 4 ( ) = . 4.2 Soundness of the natural deduction for the logic of rejection 4.2.1 Definition L e t I c F O R M . T h e n <=> 3 1 ( V o E F : ( V m ( 0 ) = 0 ) ) . The reader can easily prove the following: LOGICS OF REJECTION: TWO SYSTEMS OF NATURAL DEDUCTION 1 9 7 4.2.2 Lemma 4.2.3 Definition 4.2.4 Definition Let IF cFORM, F c FORM for i=1, 2 and 0, tp" EFORM. Then A F 1 a n d r 2 c F 1 A F 2 A r a r l d 0 E r A T , 0 A Fs, A l ' , 0, T• O vv f F, v r , v tv, I - , 0 - › I " , 0 - - - > t Y , • 0 - > t y Ail', OA Iv 1 F , O A t i f , 0 a n d i o r A F , O A l l f , AI ' , 0 , i A I ' , 0 , V , 0 V t i f r, v f , -•> A 1E, 0 A I ' , 0 A tv 1 9 1 1 1 r , l i f p Ø A V • T A F,T, A A F, Let F cFORM. Then (F)=T0 :0 e (ig) and yt e Let IF c FORM. Then a model M = (S, I ) is -min ima l <=> i ) A , f ( x ) = 0 f o r e v e r y x E i l ; ii) Fo r every ig u (F): /( vt) is defined; iii) For every VI E PROM, 4 ) : i( ig) is undefined. 4.2.5 Fact F •=> there is a -min ima l model. 4.2.6 Lemma L e t Fc FORM, 0 E PROPL. Then (i) A F and 0 ( F ) F , (ii) a n d 0 O (F) I ' , Proof: ( i i ) S u p p o s e A I" and Ø ( F ) a n d 0 e P R O P L . F a c t 4 . 2 . 5 implies that there is a I" -minimal model. As 0 is not a subformula of F, we can, since M is m i n i m a l and PROPL, define a model M ' . ( S , I ' ) b y e x t e n d i n g M's interpretation function as follows: T = / u t (0 , 1)1. Sem A , i m p l i e s V m . ( 0 ) = I F u r t h e r , S e i n i m p l i e s t h a t M 1 i s I r , - , 0 1 m i m i m a l . F i n a l l y , F a c t 4 . 2 . 5 i m p l i e s : F, (i) c a n be proved analogously. • 198 A L L A R D M. TAMM1NGA 4.2.7 Lemma L e t A be a formal rejection deduction, running from line l to line k, in which a subdeduction (5 occurs. This subdeduction runs from line h with a hypothesis O h t o l i n e jwith a formula 0, The hypothesis O h h a s b e e n r e t r a c t e d at line j-1-1. Let 91 b e defined for all n with 1 n k as follows: 91, = {0:0 is a formula occuring on a line m with in < n which is operative at line n or 0 is a hypothesis which is introduced at line n ) . 6 1Suppose A 91, A 91„ 0, fo r every i with i Further, suppose A 91 h . Then A 91 ) , 0 ) . Proof: A s every hypothesis that was introduced between O h a n d 0 ) must have been retracted before line j, we may conclude that exactly the same hypotheses are operative at line j as those that are operative on line h. Since we are, in the course of this proof, interested only in line j, we can leave out of consideration any formula between lines h and j that occurs within the scope o f one ore more additional hypotheses. Accordingly, we renumber the lines between h and j, skipping each formula that occurs within the scope o f an additional hypothesis (introduced after line h). Line h gets number l and line j gets number r. We shall show that for every renumbered line q with I r we have: A 91 M q • Induction hypothesis: Let the proposition to be proved be correct for every line p with 1 p < P' P Ø .Induction step: Consider Op• If op is a hypothesis, then q =1 and o q = O h . W e a l ready know that A 91 h A s O h E 9 , L e m m a 4 . 2 . 2 ( i i ) implies: A 91 , O h . T h e r e f o r e A 9 1 q , 0 q . I f 0 i s not a hypothesis, then, according to the induction hypothesis, fo r q -1 i t holds that A 91,,_ 1 , 0 1 _ 1 . Also 91q--1 L.) frq--11= 9 q 1 T h e r e f o r e A . T h e r e f o r e , • using the imp 'cation that is supposed to hold, A 91 q 7 0 q • This implies: A 91,, O r . T h e r e f o r e 9 J ' 1 . 0 • J . 61 On EIR , o n l y i f y i s a hypothesis which is introduced on line n. LOGICS OF REJECTION: TWO SYSTEMS OF NATURAL DEDUCTION 1 9 9 4.2.8 Soundnesstheorem Let 0 E FORM, then: -I 0 A 0 . Proof S u p p o s e that Ø . Then there must be a formal deduction of rejection, say A , with a final conclusion Ø . A consists o f a column of formulas 01 , 0 2 , • • • , O k ( w i t h O k = 0 ) , p r o v i d e d w i t h verticals indicating the scope of each hypothesis and a correct annotation for every occurrence of a formula. The verticals and the annotation uniquely stipulate which formulas are operative at a certain line and which formulas are accessible at a certain line. Now we define for every n with I n 5 _ k. 91, =10:0 is a formula occurring on a line m with m<ri which is operative at line n or 0 is a hypothesis which is introduced at line n 1 . 6 2We shall show for every n with I On This will be sufficient, as we have for the case n4c: 91 k , O k Because on the last line of a natural deduction for the logic of rejection all hypotheses have been retracted, we have 91 k =Therefore we have A 0 O k B e c a u s e e v e r y m o d e l f a l s i f i e s every fo rmu la o f 0 th is imp lies A Ø. Further, O k = 0Therefore A Ø. The proof for * can be carried out by a course-of-values induction. Induction hypothesis: Let the proposition to be proved be correct for every line m with m < n: O n t • Induction step: We distinguish cases according to the annotation on line n. Suppose that A 91„. Case I - o n i s t h e f o r m u l a T o n l i n e n w h i c h h a s b e e n d e r i v e d with E . Then there must be lines i and j with i<J1 and je„.12 with formulas 0, and o f s u c h t h a t 0 , = A o t , w h i l s t b o t h f o r m u l a s 0 , a n d O f a r e operative a t l in e n . There fore 0 , e 91,, a n d 0 . 1 E 9 1 „ T h e r e f o r eA 91,, 0 „ 0,, and also A g i n , - O f , 0 1 . W i t h L e m m a s 4 . 2 . 2 ( i i i ) a n d ( i ) w e obtain: A 91 T Therefore A 9i,,, 0„. Case 2 - yE1 Ø i s a formula on line n which has been derived with vE l . T h e n w e m u s t h a v e o n e o f t h e t w o f o l l o w i n g c a s e s : i) T h e r e is a line m with M<I1 with a formula 0 „ 1 s u c h t h a t= Ø , y I v , w h i l e 9 i „ T h e n 0 „ c a n b e d e r i v e d out of 0 , only i f O m i s o p e r a t i v e a t l i n e n . T h e r e f o r e 62 c 91,, only if 0 , is a hypothesis which is introduced on line ns 200 A L L A R D M. TAMMINGA 0„ E W i t h Lemmas 4.2.2(1), (ii) and (iv) we obtain: H I N „ , 0 „ • ii) T h e r e is a line m with m<n with a formula o„, such that = 0„ t i f , while 91, = 0 . Then 0 „ can be derived out o f 0 „ „ o n ly i f 9 i n , = a W i t h o u r i n d u c t i o n hypothesis we have: H 9t„, a t „ „ o„,•, As 91„, = 0 this implies H Ø . With Lemmas 4.2.2(i) and (iv) we obtain: H O n . A s g i n = 0 , w e a l s o h a v e : A 9 i n , Ø . The cases v E 2 , - ) E 1 , - ) E 2 , A I , A 1 2 a n d T E c a n b e p r o v e d a n a l o g o u s l y . Case 3 y 1 1 O n i s a f o r m u l a o n l i n e n w h i c h h a s b e e n d e r i v e d w i t h v i 1 , w h i l e 0 „ = 0 , y 0 „ . T h e n t h e r e m u s t b e a l i n e i w i t h i < n w i t h a formula Ø. and a subdeduction (5, running from line i + I to n -1 . This subdeduction starts with the hypothesis 0 , such that 0, = 0 , ±1 , a n d e n d s on line n -1 with formula O I mme d ia t e ly after line n -1 the hypothesis O i + 1 i s r e t r a c t e d . I f 9 I „ 0 , t h e n 0 , i s o p e r a t i v e a t l i n e n , t h e r e f o r e 0, e 9/,,. Therefore, using Lemma 4.2.2(ii), we obtain: = 1 9 Ç 0,. I f 91„ = 0 , then surely 9t, = 0 , hence H 91, A s i<n, we can apply the induction hypothesis to get: H Ø. Therefore H 9 , 0,. Both cases lead to the same result: H 9 , Ø. Because it is impossible to introduce any formula between lines n -1 and n, we have for every 0 e 91, that there must be a line in with in < i+ l with a formula 0 „„ such that 0 = o„, and 0„, is operative at line n. Therefore, every formula 0 E 9 i n i s o p e r a t i v e a t l i n e 4 1 . H e n c e , f o r e v e r y 0 e 9 I „ , w e have 0 E 9 . Therefore g i n 9 1 , ± 1 . A s 0 , = 0 1 + 1 a n d a s 0 1 + 1 ( 0 , + I is a hypothesis), we have: 91„ u {0, g 91 , +1 . Conversely, suppose that 0 e 9 i, +1 \ t 0 , + 1 1 . T h e n t h e r e m u s t b e a l i n e m with m<i+1 and a formula 0 „„ such that 0 = 0,„ and O n , i s o p e r a t i v e a t l i n e 41. Th is fo rmula O n , m u s t a l s o b e o p e r a t i v e a t l i n e n . T h e r e f o r e 9 1 i +1 \ { 0 i 1 } ç 9 1 n - A s 0 , = 0 , 4 _ , , w h a v e : g i 9 t , , u 1 0 1 1 . C o m b i n i n g this w i t h the conclusion o f the preceding paragraph w e obta in: 91 i , 1 = 9 , H e n c e H It may now be checked that, i f we read number h for number i+1, and number j for number n -1 , every condition of Lemma 4.2.7 holds. For one thing, the induction hypothesis holds for every line in the subdeduction under consideration. Using Lemma 4.2.7, we obtain: H 9 1 , o n _ l . A s i t i s impossible to introduce any formula between lines n -1 and n, we have 91 n 9 1 i • W e a l s o h a v e 0 = 0 1 + 1 E 9 3 / 2 I • T h e r e f o r e 93 n 1 0 1 , O n i / 9 1 n I 1 • • • • ' 1 0 n 1 1 W i t h L e m m a 4 . 2 . 2 ( i ) w e o b t a i n : 9t 0 „ O n 1 . U s i n g L e m m a s 4 . 2 . 2 ( i ) e n ( i x ) w e o b t a i n : H 9 , 0 i v O n I • As O n = 0 , v 0 „ _ 1 , w e h a v e : H 9 i „ , O n • LOGICS OF REJECTION: TWO SYSTEMS OF NATURAL DEDUCTION 2 0 1 The cases y / 2 , - › / 1 a n d - ) 1 2 c a n b e p r o v e d a n a l o g o u s l y . The remaining cases are left as exercises (Use Lemma 4.2.6!).• 4.3 Completeness of the natural deduction for the logic of rejection In this section we shall prove the completeness of our system of natural deduction for the logic of rejection, by showing that we can simulate the axioms and the rules of derivation of the system of Klaus Hdrtig within our system of natural deduction. Hdrtig's system (Hdrtig, 1960) is complete. As we have seen in section 2, 1-15.rtig's system can be formulated thus: Axiom 1 Axiom 2 Rule of Detachment Rule of Disjunction 4.3.1 Definition a H 0 < = > H -i 0 , i f 0 e P R O N . , H H ' 0 , i f 1 0 E R O P L I-0 v i a n d 1 0 0 and 11 -1 W a n d ( 0 ) n ( 1 1 1 ) = 0 H 1 0 v t i f There is at least one formal deduction of rejection fo r 0 (according to the system o f Definition 3.1.2) in which no occurrence of a formula is justified with one of the atomic rules At or -,At. Remark A deduction o f rejection A I w i t h a c o n c l u s i o n 0 i n which no occurrence o f a formula is justified with an atomic rule, has a pleasant quality, namely, this deduction o f rejection A I w i t h c o n c l u s i o n 0 c a n , w i t h o u t causing any problems, be fitted in within the scope o f hypotheses of an arbitrary deduction of rejection A 2 .4.3.2 Lemma L e t 0, VEFORM, then a d ( 0 - - > t y ) a n d a d d 202 A L L A R D M. TAMMINGA Proof 1 . j. A I 4 0 4.3.3 Lemma L e t 0 e FORM, then: 10 Proof 6. 2j. j+1. j+2. r _ v i j+3. L . 7 2j + 4 . ø - * v i j-1-5. j+6. - 1 V 1 11 hyp. Given As in the course of the deduction of -1 yt we have not made an appeal to one o f the atomic rules, we have tv• • First, we prove the negations of every axiom of an axiom system for CPL, for instance the system KT l a x 6 4These deductions are left as exercises to the reader. a d - , - - > ( I V - * 0 ) ) (ii) a ( 0 -)(tv ) ( ) ) - ) 0 V I ) - > ( 0 - * X ) ) ) (iii) H ( 0 A kg) -> 0) (iv) - - , (0 A iy) i v ) (v) - > ( i i t -4 (0 A iv))) (vi) ( 0 -› X ) - > ( ( i l l - > X ) - > ( ( 0 y I g ) - ) X ) ) ) (vii) 0 -> (0 v iv)) (viii) a 1 1 ( l i t ( 0 v v f ) ) (ix) a 1 - , ( - , 0 - > ( 0 - > x) (xi) a H - > 0 ) (xii) a 1 - , ( ( ( 0 - > 0 ) - > 0 ) 63 Cf. HUrtig's less restrictive rule: I0 I -4. Hartig (1960); 244. Given Triv: 1 hyp. Triv: j ->1 2 : j + 1 , j + 2 - j + 3 -E. j j + 5 64 KT lax is the system K A ax in Barth and Krabbe (1982). pp. 211-227. extended with one extra axiom (Ax T ); T. The system KT lax is complete for CPL in languages with both vérum (T) and fa/sum (I ). LOGICS OF REJECTION: TWO SYSTEMS OF NATURAL DEDUCTION 2 0 3 Next to this finite set o f axioms, this system has one rule of inference: modus ponens. We complete the proof with induction over the number of lines occurring in the axiomatic derivation. Let i t be given that IØ. Then there must be a derivation in K T lax, say A, with a conclusion 0 A consists o f a column o f formulas 01 , 0 2 , • • • O k ( w i t h 0 k - = 0 ) . E v e r y o c c u r r e n c e o f a f o r mula is justified either with an appeal to one of the axioms of KT l a x „ or with an appeal to modus ponens. Now, we prove by induction that for every line n with I k we have: a I - 1 O n • Induction hypothesis: Let the proposition to be proved be correct for every line m with m < n: a d Ø .Induction step: Consider O n . O n i s j u s t i f i e d e i t h e r a s a n axiom, or by modus ponews. i) Suppose O i s a n a x i o m . T h e n w e h a v e f r o m o u r l i s t of proofs of negated axioms a HO n . ii) Suppose 0 „ is justified by modus ponens. Then there must be two lines i and j with i < n and j < n with formulas O i a n d 0 ) • s u c h t h a t O f = 0 , - - - > O n Our induction hypothesis implies: a H 0 , 0 , a n d • Using Lemma 4.3.2 we have: a d O n . Therefore, for every n with 1 n k we have: 0 „-ITherefore a -1 0 •4.3.4 Lemma L e t 0 e FORM, then 0 H 1 0 • Proof: C f . Eldrtig (1960). 4.3.5 Lemma L e t 0 E FORM, then 11 1 dProof: I t will be sufficient to show that the two axioms and the two rules of inference of 1 1 d r t i g ' s s y s t e m a n b e m i m icked with in our system o f natural deduction fo r the logic of rejection. Hdrtig's axioms can easily, without any need fo r adaptation, be simulated by the two atomic rules At en -1At of our system of natural deduction. I-Idrtig' s Rule o f Detachment can be proved as follows: Suppose: I0 -› tv and t i t . 204 A L L A R D M. TAMMINGA By Lemma 4.3.3 we obtain: a H t y ) a n d H t i t . .i• j+1. j+2. j+3. i + 4 • j+5. j+6. j+7. j+8. 1 1 Given hyp. Given hyp. hyp. Triv: i+1 j + 1 , j + 2 j +3 j , j+4 CRV: j+1-j+5 E 2 : j + 7 Therefore H 1 1 5 . r t i g ' s R u l e o f D i s j u n c t i o n c a n b e p r o v e d a s f o l l o w s : Suppose H 0 en H tif en (Ø )n ( t j i )= ø . Then there are two natural deductions o f rejection A I a n d A 2 , s u c h that A I ' s c o n c l u s i o n i s t h e f o r m u l a 0 a n d A 2 ' s c o n clusion is the formula t it . The formulas 0 and t y do not share any common propositional letter. I t st ill remains possible that 0 and the formulas occurring in the deduction A 2 h a v e c o m m o n p r o p o s i t i o n a l c o n s t a n t s . Let A = A 1 the set o f propositional constants of Ø. L t B B „ 1 the set o f all propositional constants of formulas occurring in deduction A 2 . A a n d B are finite sets o f propositional constants. Therefore there is an infinity o f propositional constants which are neither in A nor in B. Let C I , C 2 , b e a n e n u m e r a t i o n of th is in f in ite set o f propositional constants. L e t CE A n B . Replace very occurrence of C in A 2 b y C I . The resulting deduction will be called 1/4 2 N o t e t h a t t h e conclusion ty c a n n o t c h a n g e b y t h e s e p r o c e d u r e . L e t B i be the set o f propositional constants o f formulas occurri g i n deduction 1 A 2 . W e n o w r e p e a t t h e procedure described above until A n B -= 0 . Then 0 and 0 , 2 d o n o t h a v e a n y c o m m o n p r o p o s i t i o n a l constant. Deduction ' A 2 c a n , w i t h o u t c a u s i n g n y problems, be fitted in within the scope of hypotheses of another deduction if, within this scope, only formulas LOGICS OF REJECTION: TWO SYSTEMS OF NATURAL DEDUCTION 2 0 5 i+I. j. A2 Ø hyp. Given j+1. j+2. T j+ 1 j+3. T v E 2 : j + 2 consisting o f propositional constants f ro m A a re operative. Therefore: Ai j. 0 i+I. 0 r j . 0 V t i f Therefore H Ø y V. • 4.3.6 Completenesstheorem Let 0 e FORM, then: (i) A O -1 ( / ) (ii) 0 <=> 0 • Proof: ( i ) D i re c t ly from Lemma 4.3.4 and Lemma 4.3.5. (ii) Dire ct ly from (i) and Theorem 4.2.8. • 4.3.7 Theorem L e t 0 eFORM, then: a d 0 < = > 0 i s a c o n t r a d i c t i o n . Given hYP• Given Proof. ( ) I f a -I 0 , t h e n t h e r e i s a d e d u c t i o n A 2 w i t h a c o n clusion 0 such that no single occurrence of a formula is justified by one of the atomic rules. Suppose: 0 is not a contradiction, i.e. there is at least one model M such that V m ( 0 ) = I , B y S e m _ w e o b t a i n t h a t t h e r e i s a t l e a s t o n e odel M such that V m ( 0 ) = 0 , i n o t h e r s y m b o l s -0. B y the Completenesstheorem we have -1 -4 . Therefore there must be a deduction A I w i t h c o n c l u s i o n -0. We can, taking this deduction as our starting-point, proceed as follows: A I 1 Given 206 A L L A R D M. TAMMINGA ( I f 0 is a contradiction, then i s a tautology. By Lemma 4.3.3 we obtain: a -I1. hyp. A Ij. - 1 1 0 [ T Given i+1. I, i CRV: 1-i+ I Therefore H T. Using the Correctnesstheorem, we obtain: T , t h e r e f o r e , t h e r e m u s t a t l e a s t b e o n e m o d e l M such that V m T ) = O . T h i s i s i n c o n t r a d i c t i o n w i t h Sem i . h e r e f o r e 0 i s a c o n t r a d i c t i o n . Therefore a H 0 . •4.3.8 Definition L e t 0 , ty E FORM, then: a I 0 < = > There is, taking premiss t y as our starting-point, a t least o n e deduction o f rejection o f 0 , i n wh ich n o s in g le occurrence of a formula is justified by one of the atomic rules At or -1At. 4.3.9 Theorem of Inversion Let 0 , iii c FORM, then: OF V'=> V a 1 0 •Proof: W e already know that: 0 [-• V < = > F 0 -› V ( D e d u c t i o n t h e o r e m ) a I - 4 0 - 4 l i f ) ( T h e o r e m 4 . 3 . 7 ) . Therefore, it will be sufficient to show: a I - , ( 0 V ) < = > V a H O . ( ) Let it be given that: a H - - - , ( 0 - - > t g ) 1. V p r e m . 2. V h y p . A Ii. - 1 ( 0 - - V ) 1 G i v e n i+1. - - - , 0 h y p . 42. - , 0 h y p . i+3. L i f T r i v : 2 i+4. 0 -› V - ) / 1 : i + 1 , i + 2 i + 3 i+5. T - 1 E : i, 44 46. C R V : i+l-i+5 47 . 1 / / V 0 v 1 1 : 1 , 2 i + 6 i+8. 0 v E 2 : i + 7 Therefore t i r a -I O . LOGICS OF REJECTION: TWO SYSTEMS OF NATURAL DEDUCTION 2 0 7 ( ) Given: t if a -I O .1. - 0 -> lit h y p . 2 . 1 / 1 - ) E 2 : 1 A 2i. 0 1 Given i-1-1. - , 0 - ) E I : 1 i+2. T - 1 E : i , i + 1 i+3. - ( 0 - ) I l l ) - 4 : 1 i + 2 Therefore a i - , ( 0 - ) t i t ) . • 6 5 REFERENCES University of Groningen Else M. Barth Sz. Erik C.W. Krabbe (1982), From Axiom to Dialogue. A Philosophical Study of Logics and Argumentation, Walter de Gruyter, Berlin 1982. Franz C H A T Brentano (1874), Psychologie vom empirischen Standpunkt - zweiter Band - Von der Klassifikation der psychischen Phiinomene, Felix Meiner Verlag, Hamburg 1971. Franz C.H.H.J. Brentano (1956), Die Lehre vom richtigen Urteil, Francke Verlag, Bern 1956. Ludwik Borkowski (ed .) (1970), Jan Lukasiewicz. Selected Works, North-Holland Publishing Company, Amsterdam 1970. 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