BETWEEN THE LOGIC OF PARMENIDES AND THE LOGIC OF LIAR

Transcription

1 Bulletin of the Section of Logic Volume 38:3/4 (2009), pp Kordula Świȩtorzecka BETWEEN THE LOGIC OF PARMENIDES AND THE LOGIC OF LIAR Abstract In the presented text we shall focus on some specific concept of changes that - in opposition to Parmenides - does not generate any contradiction, but allows to talk consistently about some objects usually considered as paradoxical - about so called Eubulidenian sentences. For this aim we will use a standard sentential language with one primitive operator C read: it changes that... which is intended to describe some sort of dichotomic changes - intuitionally speaking changes: from A to not-a or from not-a to A. This operator is axiomatizated in the logic of change LC. An epistemological realist usually places his opinions somewhere between two - in his opinion equally improbable - concepts of being: Parmenides theory of the impossibility of any change and variabilism by Heraclitus. Obviously, if we reject these extreme visions, we are still left with the wide range of opinions that may be assumed on the issue of the existence of various changes in the reality. In the presented text we shall focus on some specific concept of changes that - in opposition to Parmenides - does not generate any contradiction, what is more, it allows to talk consistently about some objects usually considered as paradoxical - about the so-called Eubulidenian sentences. The inspiration to take up this subject I owe to the late Professor Jerzy Perzanowski, to whom I dedicate the following considerations 1. 1 The suggestion to consider the changeability of Eubulidenian sentences has been included in [3]

2 124 Kordula Świȩtorzecka 1. Formal base - LC logic The framework of our analyses will be the LC logic which may be definitionally extended to the sentential logic of changes LCG formulated in [2]. The language of LC is intended to describe some sort of dichotomic changes intuitionally speaking changes: froma to not-a or from not-a to A. To this aim we will use a standard sentential language with one primitive operator C read: it changes that Language and axiomatics of LC The language of LC uses the following primitive symbols: (i) elementary propositions: z 1, z 2, z 3,..., (ii) truth connectives:,,,,,; (iii) one-argument (primitive) operator of change C to be read: it changes, that..., (iv) parentheses: (,). The set of LC formulas is defined in standard way. The axiomatics of LC is obtained from the same schemata as LCG. By the way, it is interesting notice that originally formulated list of these schemata may be shortened. From the original set of the accepted six schemas of axioms ([2]), it is enough to choose only the following: Ax1. Ax2. Ax3. CA C A C(A B) CA CB A B CA CB C(A B). We shall also use the following rules: (MP) A, A B B ( C-rule) A CA (Rep) A[B], B B A[B ], (A[B] symbolizes the formula A, in which B is subformula) 2 In LCG there are also considered changes which consist in occurring of new situations. Such changes are characterized with the use of the concept of level of a formula and definitions of some operators which enable to construct next levels. However, in the presented paper we are not going to speak about this sort of changes.

3 Between the Logic of Parmenides and the Logic of Liar 125 Definition 1. Calculus LC is characterized by: (i) axioms: all laws of the classical propositional logic and formulas of the following shapes: Ax1, Ax2, Ax3, (ii) primitive rules: MP and C-rule and Rep. 1.2 The sketch of LC semantics Let us consider the following semantics which is, in some sense, a simplified version of the semantics developed for LCG in [2]. Let N be the set of natural numbers and ϕ a function which assigns to every natural number n N a subset of the set of all elementary propositions. To give some intuitive interpretations, ϕ(n) may be understood as (i) the set of elementary propositions which are true at stage n in the development of the universe, or (ii) the set of elementary propositions which are considered to be true at stage n in the development of some agent s belief, or (iii) the set of elementary propositions which are assumed to be true at step n in the course of some argumentation or proof 3. A relation = between such ϕ s, natural numbers n and formulas A is defined inductively in a very common way: Definition 2. For any ϕ and n N we have: ϕ = n z i iff z i ϕ(n) ϕ = n A iff ϕ = n A (i.e. not ϕ = n A); ϕ = n (A B) iff ϕ = n A andϕ = n B ϕ = n (A B) iff ϕ = n A or ϕ = n B ϕ = n (A B) iff ϕ = n A or ϕ = n B ϕ = n (A B) iff (ϕ = n A or ϕ = n B) and (ϕ = n A or = n B), ϕ = n CA iff (ϕ = n A and ϕ = (n+1) A) or (ϕ = n A and ϕ = (n+1) A) (Therefore, ϕ = n CA means that A changes its truth value when proceeding from stage n to stage n + 1.) 3 This intuition we will use later cf. part 3.

4 126 Kordula Świȩtorzecka Theorem 1. means: LC is complete in respect to this semantics, which that LC A iff ϕ n N (ϕ = n A). The above can be inferred from a corresponding completeness theorem for LCG in respect to a richer and more sophisticated semantics which is based also on the idea of growing universe whose development is described by sets of elementary propositions extended from stage n to n + 1 by some new elementary proposition (cf. footnote 1). To speak about such a change from stage n to n + 1 we introduced in the language of LCG the distinction of levels of formulas. Intuitively speaking, each formula A has got its minimal level which is linked with the smallest universe in which A can be interpreted. In case of LC we do not distinguish between levels of formulas or in other words: all formulas have the same level. The definition of satisfability of formulas in the intended semantics of LCG differs from this which is described in Definition 1.2 by the condition that only such formulas are satisfied in stage n that have its minimal level not smaller than n. All inductive conditions for logical connectives are the same as in the Definition Of course, Theorem 1 can be proved independently of the completeness theorem of LCG. Theorem 2. The formulas of the following shapes are LC theses: (i) A B CA CB C(A B) (originally: Ax4) (ii) A CA B C(A B) (originally: Ax5) (iii) C(A B) CA CB (originally: Ax6, where CA CB (CA CB)) (iv) CA C A (v) A B CA CB C(A B) (vi) A CA B CB C(A B) (vii) C(A B) CA CB (viii) C(A B) CA CB 4 For more details and motivations, we have to refer the reader to [2] where he can find two different proofs of the completeness theorem for LCG (one with normal forms and one in Henkin s style).

5 Between the Logic of Parmenides and the Logic of Liar 127 To prove the above theorem is enough to say that LC characterized only by axioms Ax1-Ax3 is complete (Theorem 1) and formulas of shapes (i)-(viii) are valid in the described semantics and therefore they are drivable in LC. (To check that they are valid cf. also [2].) Let us also present a few schemas that do not generate LC theses: Theorem 3. theses: Not all instantiations of the following schemata are LC (i) C(A B) CA CB (ii) CA CB C(A B) (iii) CA CB C(A B) (iv) C(A B) (CA CB) (v) C(A B) (CB CA) (vi) C(A B) ( CA CB) (vii) CA C(A B). The above theorem can be easily proved by counterexamples. Consider, e.g. scheme (i): Let z, z be elementary propositions and choose ϕ in such a way that: z ϕ(1), z ϕ(1), z ϕ(2) and z ϕ(2). Then we will have: ϕ = 1 (z z ), but ϕ = 2 (z z ) and, therefore: ϕ = 1 C(z z ). On the other hand, there is: ϕ = 1 Cz and hence, not ϕ = 1 (Cz Cz ). About LC we also notice that: Theorem 4. (a) LC is not (i) a normal logic, although it is (ii) semi-normal, (b) LC is neither (i) monotonic nor (ii) anti-monotonic, because: ad (a) (i) formulas of the shape: C(A B) (CA CB) are not theses of LC logic; (ii) all tautologies of the classical logic are LC theses and the set of LC theses is closed under C-rule. ad (b) (i) LC is not closed under: (Mon) A B CA CB; (ii) LC is not closed under: (AMon) A B CB CA.

6 128 Kordula Świȩtorzecka 2. Logic of Parmenides LC logic may be extended in various ways. LC extension that we are going to consider will be the smallest logic based on LC in which Parmenides thesis about non-occurrence of any change is expressed. Let us call a formula of the shape CA Parmenides formula. We have the following: Definition 3. Parmenides logic (LP ) is the smallest extension of LC, in which every formula of the shape: CA is a thesis (i.e. every Parmenides formula is thesis). About LP logic we should note that: Theorem 5. because: LP A iff ϕ n N (ϕ = n A iff ϕ = n+1 A) LC + [ CA] A iff ϕ n N (ϕ = n A iff ϕ = (n+1) A). The above equivalence follows directly from the fact that for every formula A in LC + [ CA] the formula CA is an axiom and from the Definition 2 (last item). Professor J. Perzanowski noticed that we can derive Parmenides thesis also in an extension of LC by the rule (Mon) or the rule (AMon). We have: Theorem 6. For any formula A: (a) CA LC + [Mon]; ad (a) 1. A (A A); 2. CA C(A A) [Mon]; 3. C(A A) [ C reg]; 4. CA; ad (b) 1. (A A) A; 2. C A C (A A) [AMon]; 3. CA C(A A) [Ax1]; 4. C(A A) [ C reg]; 5. CA. (b) CA LC + [AMon].

7 Between the Logic of Parmenides and the Logic of Liar 129 In addition to the considerations on logic realizing Parmenides concept, let us say that any attempt to find an extension of LC that would be a logical base of Heraclit s concept, immediately causes a contradiction. The addition of formulas of the shape: CA into the set of axioms is impossible because of C-rule. The situation does not change even when we take schema CA limited to contingent formulas - that is such formulas which are not LC theses, nevertheless they are not contradictory: Theorem 7. System LC + [CA for every contingent formula A] is inconsistent. Let us confine ourselves to the outline of the proof of Theorem 7: Consider two elementary propositions: z, z and their disjunction which are obviously contingent. Then: Cz, Cz and C(z z ) are axioms of this theory. Take any ϕ such that z ϕ(1) and z ϕ(1). Then, because of Cz and Cz, we will have ϕ = 1 (z z ) and also ϕ = 2 (z z ) which contradicts C(z z ). If ϕ(1) is the whole set of natural numbers or the empty set, then the contingent formula z z will not change. It means that there is no model for this theory. 3. The logic of Liar We expect that the logic of Liar should include as theses at least some socalled Eubulidenian sentences. Intuitively speaking, they are self-referential formulas which looks paradoxically, because they say about themselves that they are not true. The key to formulate the notion of such Eubulidenian sentences within the frame of LC calculus and to discus the possibility of using in without getting paradoxes is a certain interpretation of the following definition that we shall introduce to LC calculus: (N/C) NA (A CA) ( A CA) (NA (A CA)) and its consequence: (C/N) CA (A NA) ( A NA) (CA (A NA))

8 130 Kordula Świȩtorzecka To understand the operator N in our semantics we would extend Definition 2 by the condition: ( ) ϕ = n NA iff (ϕ = (n+1) A). Let us also say that LC + [N/C] calculus is equivalent to the system with the same distribution of modality N as it is in Prior s system F for the operator F that we read: next it is, that... ([1]). LC + [N/C] theses are: (N1) NA N A (N2) N(A B) (NA NB), and the derivable rule is the following: (N-reg) A NA. If we take as axioms formulas of schemas (N1), (N2) we add as the definition of C operator the equivalence (C/N), then with the use of (Nreg) and the laws of classical logic we obtain axioms LC: Ax1, Ax2, Ax3. ( C-reg) rule is derivable in such a system. Intended for Prior s system temporal interpretation connects operator N in the standard temporal semantics of Kripke type with the relation of the time succession which orders linearly a discrete set of moments. In such semantics Prior s system is complete. However, the given distribution of modality N does not force a temporal interpretation. We obtain an interesting perspective for our analysis when we connect operator N with the same meaning which is assigned to operator of truth T read: it is true, that... and is described by logic T F 2. [This very idea I owe to Professor Perzanowski.] Logic T F 2 characterizes operator T with the help of axioms of the same structure as axioms of system F describing operator of the time succession. However, this new meaning gives an opportunity to consider also a new semantics. An outline of this semantics - or rather: a description of the intuition that could enable its precise formulation - will be here a subject of our interest.

9 Between the Logic of Parmenides and the Logic of Liar 131 Let us take the already announced way of understanding the symbol = (by: ϕ = n A we mean that A is assumed to be true at step n in the course of some argumentation or proof cf. p. 1.2, (iii)). Within the frame of the proposed interpretation we will consider the succeeding one after another steps (stage) of an argumentation or a proof formulated by someone. Following this intuition, we will paraphrase a non-temporal version of the condition ( ) as follows: ( T ) formula NA is accepted (assumed) in a given proof in its step n iff formula A is accepted (assumed) in step n + 1 in this proof. Let us consider a situation now, when in step n of the constructed proof there occurs a sentence of the shape: A NA. We will name the sentences of the shape: A NA Eubulidenian sentences, because of the accepted way of reading N. With the use of the introduced semantic terminology, let us notice that: (1) If, in step n of an argumentation, the sentence A is assumed to be true, that means, ϕ = n A, then, if we have ϕ = n A NA, then ϕ = (n+1) A, i.e. A will be considered as false in the next step. (2) If, in step n of an argumentation, the sentence A is assumed to be true, then, by the same reason, A will be considered as true in the next step. In that sense, Eubulidenian sentences have oscillating truth values. The reasoning presented above does not lead to contradiction as long as we do not accept as theses the expressions of the following shape: (N ) NA A. If we accept schema (N ), then the considered situation can occur in no step of the constructed proof: a formula of the shape: A NA cannot occur in any step n. In our reasoning we obtain a contradiction with every assumption accepted in (1) and (2): ad 1) If sentence A is false in step n + 1, then it cannot be accepted in the proof in step n, and this is in contradiction to the assumption that A has been accepted in the proof in step n;

10 132 Kordula Świȩtorzecka ad 2) However, if sentence A is accepted in the proof in step n it is true in step n + 1 of the proof, then it must be also true in step n - that is, it has been accepted in the proof, which is in contradiction to the assumption that A has not been accepted in the proof. In terms of logic LC + [N/C] extended by (N ), it turns out that no sentence can be submitted to a change, because the theses of LC + [N/C] is the equivalence: (N/C) CA (A NA). In this way, logic LC + [N/C] extended by (N ) is Parmenides logic: LC + [N/C] + [N ] = LP. The proposed interpretation of operator N leads us to logic LC +[N/C] which controls the behavior of the Liar: In the successive steps of the constructed argumentation, there can be stated any (non-contradictory) sentences as long as they are not Eubulidenian sentences. When somebody says the Eubulidenian sentence, next he must change the value of this sentence - in the next step of his argumentation he must take a value contrary to the one he has taken before (and next: again and again). The behavior of Liar may have various rhythm of using Eubulidenian sentences. Because of Theorem 7 it is certainly impossible for the Liar to say only Eubulidenian sentences and on every step of his argumentation apart from the logical theses - he would also have to say some contingent non-eubulidenian sentences. Between the impossibility of the using only Eubulidenian sentences and the impossibility of not using them at all (such situation is forced in LP ) there is still the whole range of different behaviors of the Liar, controlled by (adequately interpreted) extensions LC + [N/C]. Let us consider only four exemplary possibilities: (P1) When the Liar uses logic LC + [N/C] + [CA] for any contingent formula A, in every step of his argumentation he can use Eubulidenian sentence of the shape: A NA. In such a case, in every step he will have to change value of A. (P2) For LC + [N/C] + [CCA], for any contingent formula A, the periods in which value of A does not change, will get longer to two steps than in case (1).

11 Between the Logic of Parmenides and the Logic of Liar 133 (P3) Calculus LC + [N/C] + [CCCCA] will describe the same rhythm as (1) and (2) - the periods of value of A changes will get longer up to n+4. (P4) For n + 3 the rhythm will be different than for other cases - the period of constancy alternately with one-time change will be happening with the frequency: 3 to 1. However, the generalization of the above mentioned examples are the subject of the future researche. References [1] Artur Prior, Time and modality, Oxford [2] Kordula Świȩtorzecka, Classical Conceptions of the Changeability of Situations and Things Represented in Formalized Languages, The Publishing Company of the University of Cardinal St. Wyszyński in Warsaw, 2008, pp. 240 (published English version of habilitation thesis). [3] Jerzy Perzanowski, The opinion on dr Kordula Swietorzecka s scientific achievements and habilitation thesis presented to the Board of the Faculty of Christian Philosophy of the University of Cardinal St. Wyszyński in Warsaw (manuscript). Institute of Philosophy University of Cardinal St. Wyszyński in Warsaw