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Bulletin of the Section of Logic Volume 17:3/4 (1988), pp. 92–98 reedition 2005 [original edition, pp. 92–103] P. Bystrov TABLEAUX VARIANTS OF SOME MODAL AND RELEVANT SYSTEMS The tableaux-constructions have a number of properties which advantageously distinguish them from equivalent axiomatic systems (see [1]). The proofs in the form of tableaux-constructions have a full accordance with semantic interpretation and subformula property in the sense of Gentzen’s Hauptsatz. Method of tatleaux-construction gives a good substitute of Gentzen’s methods and thus opens a good perspective for the investigations of theoretical as well as applied aspects of logical calculi. It should be noted that application of tableau method in modal, tense, relevant and other non-classical logics is connected with serious difficulties. Tableaux variants (in Beth-Kripke style) are constructed only for a few normal modal systems. As to relevant and paraconsistent logic, the absence of its tableau variants may be considered as a question of special interest. We shall formulate the tableaux for propositional modal system S4.1, S4.2, S4.3, S4.4 and relevant R∗ (R0 ) and E∗ (E0 ) using Beth’s tableaux construction with indexed formulas. Let axiomatic propositional system S4 be given in usual way. Axiom schemes: A0. A1. A2. A3. Schemes of axioms of classical propositional logic 2A ⊃ A 2(A ⊃ B) ⊃ (2A ⊃ 2B) 2A ⊃ 22A Rules: modus ponens; ` A ⇒ ` 2A. If we have as additional axiom schemes: A4. 2(2(A ⊃ 2A) ⊃ A) ⊃ (32A ⊃ A) 93 Tableaux Variants of some Modal and Relevant Systems A5. 32A ⊃ 23A A6. 2(2A ⊃ B) ∨ 2(2B ⊃ A) A7. A ⊃ (32A ⊃ 2A) systems S4.1, S4.2, S4.3 and S4.4 can be obtained by means of adding to S4 one axiom scheme A4, A5, A6 and A7 accordingly. To construct the tableau variants T4.1, T4.2, T4.3 and T4.4 of axiomatic systems in question we define the following notions: Index is the sequence of natural numbers beginning with 0 in which no number is repeated. Let u, w, w0 , w00 , . . . be idexes and i, k, l, m, n, . . . - natural numbers. R is a binary relation such that uRw iff w = u, k or w = u or for some l ≥ 1 there exist such indexes w1 , w2 , . . . , wl that w1 = u, wl = w and wi Rwi+1 for any i (1 ≤ i < l). Index w is called subindex of index w0 iff wRw0 . If A, ¬A, A&B, A∨B, A ⊃ B are well formed formulae (in usual sense), w w w w w then A, ¬ A, A & B, A ∨ B, A ⊃ B will be indexed formulae. Now we introduce tableaux systems T4.1 - T4.4 by means of the following rules. Rules of construction: (BR) Usual rules for construction of Beth’s tableaux with the following addition: index of formula subjected to “non-modal” rule application is transferred without change from the main logical sign of this formula to its subformula(e) standing in the scope of this sign. For instance, the rule for implication are formulated in the following way. w w If A ⊃ B occurs on the right of (sub)tableau, write A on the left of it w w and B on its right. If A ⊃ B occurs on the left of (sub)tableau, split it into w w two subtableau writing A on the right of first subtableau and B on the left of the second. u,k u (2)T If formula 2 A occurs on the right of (sub)tableau, write A on the right of it, k being a number which does not occur in indexes described to formulae occurring in this (sub)tableau. (2)T 4.1 If formula A occurs on the left of (sub)tableau write A on the left of it, where w w0 94 P. Bystrov a) wRw0 or u,m b) w0 = u, k and w = u, m, n if formula A has already occurred on the left of this (sub)tableau. (2)T 4.2 – rule is obtained from (2)T.4.1 by replacement of point b) by c) w = u, n and w0 = u, m, n index w00 = u, n being subindex of at least one of indexes described to formulae which occur in this (sub)tableau. (2)T 4.3 – rule is obtained from (2)T 4.1 by replacement of point b) with w00 d) w0 = w00 (where w00 6= 0) if some formula of the form 2 B w occurs on the left and formula A – on the right of this (sub)tableau. (2)T 4.4 – rule is obtained from (2)T 4.1 by replacement of point b) with u e) if w = u, k and formula A occurs on the left of this (sub)tableau, then w0 is an arbitrary index such that w0 6= 0. Rule of closure: (CR) If some formula with one and the same index occurs on the right and on the left of one and the same (sub)tableau, then such (sub)tableau is closed. (Sub)tableau is closed iff all its sub-tableaux are closed. System T4.1 is constituted of (CR), (BR), (2)T , (2)T 4.1 . Other systems T4.2, T4.3 and T4.4 are obtained from T4.1 by means of replacement of the rule (2)T 4.1 with rule (2)T 4.2 , (2)T 4.3 , (2)T 4.4 , respectively. Proof of equivalency between SM and TM can be given semantically, in Kripke style, using the notion of equivalency of tableaux and corresponding models for SM1) . But more constructive ”syntactical” proof can be given also. Let indexed sequent be an expression of the form Γ → Θ where Γ, Θ are lists (may be empty) of indexed formulas. If all occurrences of formulae in indexed sequent Γ → Θ have index 0, it is called pure sequent. So pure sequent is just the same as sequent in usual (Gentzen) sense. We introduce the calculus of indexed sequents GJ in the following way: 95 Tableaux Variants of some Modal and Relevant Systems w w Basic sequent (axiom): Γ1 , A, Γ2 → Θ1 , A, Θ2 . Rules:2) w ∨→ w A, A ∨ B, Γ → Θ w w B , A ∨ B, Γ → Θ w ; A ∨ B, Γ → Θ w w →∨ Γ → Θ, A ∨ B, A w ; Γ → Θ, A ∨ B w →¬ w A, Γ → Θ, ¬ A w Γ → Θ, ¬ A ; w ¬→ ¬ A, Γ → Θ, A w ¬ A, Γ → Θ ; Calculus GJ 4.1 is obtained from GJ by addition of the rules: u u,k Γ → Θ, 2 A, A Γ → Θ, 2A , where k does not occur in indexes described to any formula occurrence in conclusion sequent; w0 w A , 2 A, Γ → Θ , w 2 A, Γ → Θ where wRw0 or (∗) w0 = u, m, n and w = u, k if u,m formula A occurs in Γ. We can obtain calculi GJ 4.2, GJ 4.3, GJ 4.4 from GJ 4.1 by replacing condition (∗) by one of the following conditions respectively: (∗)4.2 w0 = u, m, n and w = u, n index w00 = u, m being subindex of at least one of indexes described to formulae in conclusion sequent; w00 (∗)4.3 w0 6= w00 (w0 6= 0) if some formula of the form 2 B occurs in Γ and w (∗)4.4 formula A occurs in Θ; u w0 is an arbitrary index such that w0 6= 0 if w = u, k and formula A occurs in Γ. Cut is eliminable in each of those systems. So, if we define in appropriate way the notion of representing formula of the indexed sequent Γ → Θ, it can be proved that representing formula of the axiom of GJ M is theorem 96 P. Bystrov of SM and all rules of GJ M are rules derived from SM in the following sense: if the representing formula of the premis (representing formulae of premises) can be proved in SM, then the representing formula of conclusion is theorem of SM also. On the other hand, if formula α is a theorem of SM, then pure sequent → α can be proved in GJ M because we can prove → α in GJ M if α is an axiom of SM and rules of SM are derivable in GJ M. Then the following proposition is valid for GJ M and SM. Proposition 1. A. formula α can be proved in SM iff pure sequent → α can be proved in GJ M. Now let us consider the set of indexed sequents – the expression of the form S1 ; S2 ; . . . ; Sm , where for each i (i = 1, 2, . . . , m) Si is an indexed sequent. Then calculus of the sets of indexed sequents B J M we obtain by adding to the axiom-schemata S1 ; S2 ; . . . ; Sn , where n > 0 and for each i (i = 1, 2, . . . , n) Si is a basic sequent of GJ M two following rules: U ; S0; W ; U ; S; W U ; S 0 ; W ; U ; S 00 ; W ; U ; S; W where U, W are sets of indexed sequents (may be empty), S 0 (S 0 , S 00 ) premis(es) and S is a conclusion of some rule of GJ M. It is easy to prove that calculi BJ M and GJ M are deductively equivalent. On the other hand, any inference of BJ M can be transformed into correct TM-inference. So, the GJ M and TM are deductively equivalent and on the ground of Proposition 1 we have the following: Proposition 2. The system TM is deductively equivalent to the system SM. To construct the tableaux variants of relevant systems the rule of closure must be modified. We shall say that it is closure (of some tableauconstruction) with respect to elementary3) formula α iff α occurs on the right and on the left of each subtableau of the construction in question. Rule of closure (CR)∗ results from (CR) when the last sentence in formulation of (CR) is replaced by the following: “(Sub)tableau is closed iff 97 Tableaux Variants of some Modal and Relevant Systems all its subtableaux are and there is a closure with respect to each elementary subformula of the formula occurring on the right of initial tableau”. Moreover, we replace the rules for conditional from the set of rules (BR) by the following two rules. (⊃r )∗ w,k w If formula A ⊃ B occurs on the right of (sub)tableau, write A w,k on the left and B on the right of this (sub)tableau (k being a number not occurring in indexes described to formulae in this (sub)tableau). (⊃l )∗ w If formula A ⊃ B occurs on the left of (sub)tableau, split this w00 (sub)tableau into two alternative subtableaux and write A w0 on the right of the first and B on the left of the second subtableau, w0 being index such that wRw0 and w00 graphically coincides with w or w0 . The resulting set of rules (BR)∗ together with (CR)∗ gives the propositional system R∗ . It may be said that R∗ is a strongly relevant system because all the so-called “paradoxical” formulae of classical propositional logic and formulae of the form (1) A ⊃ (A ∨ B) and (2) (A&B) ⊃ A are not theorems of R∗ . Let the occurrence of the formula A in the formulae of the form (1), (2) be called essential occurrence. Then the rule of closure (CR)0 results from (CR)∗ if we modify the last sentence of its formulation in the following way: “(Sub)tableau is closed iff all its subtableaux are and there is a closure with respect to each essential occurrence of each elementary subformula of the formula occurring on the right of initial tableau.” The set of rules (BR)∗ together with (CR)0 gives the relevant system 0 R. If we add to the formulation of (CR)∗ the condition: “All elementary subformulae with respect to which closure takes place have one and the same index”; the rule (CE)∗ results. Accordingly, if the condition: “All essential occurrences of each elementary subformula with respect to which 98 P. Bystrov closure takes place have one and the same index” is added to (CR)0 , we have the rule (CE)0 . (BR)∗ +(CE)∗ gives “strongly relevant” system E∗ . (BR)∗ + (CE)0 gives relevant (in usual sense) system E0 . As for interrelation between R0 , E0 and well known relevant systems R, E it can be proved that a class of theorems of the system R0 (E0 ) includes one of the R (E) but more probably not vice versa. Detailed consideration of such interrelation is a question of special interest. Notes 1) In “SM” and “TM” the letter M can be replaced by a number of some system 4.1, 4.2, etc. 2) Rules for ⊃ and & can be written just in the same way preserving indexes in premis(es) without change. w w w w 3) A formula is elementary iff it has the form A, ¬ A ∨ B, A & B or w A ⊃ B, where A and B are propositional variables. References [1] E. W. Beth, The foundations of mathematics, Amsterdam 1959. [2] P. I. Bystrov, Calculi of indexed sequents and tableaux constructions of modal systems, Abstracts of VIIIth Soviet Symposium: “Logic and Methodology of Science”, Vilnius 1982, pp. 15–20 (in Russ.) [3] S. A. Kripke, Semantical analysis of modal logic I, Normal modal propositional calculi, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 9 (1963), pp. 67–96. [4] K. Schütte, Vollständige Systeme modaler und intuitionisticher Logik, Ergebnisse der Mathematik und ihrer Grenzgebiete 42 (1968).