Bulletin of the Section of Logic Volume 17:3/4 (1988), pp. 92 98 reedition 2005 [original edition, pp. 92 103] P. Bystrov TBLEUX VRINTS OF SOME MODL ND RELEVNT SYSTEMS The tableaux-constructions have a number of properties hich advantageously distinguish them from equivalent axiomatic systems (see [1]). The proofs in the form of tableaux-constructions have a full accordance ith 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 ell 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 ith serious difficulties. Tableaux variants (in Beth-Kripke style) are constructed only for a fe normal modal systems. s 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 (R ) and E (E ) using Beth s tableaux construction ith indexed formulas. Let axiomatic propositional system S4 be given in usual ay. xiom schemes: 0. Schemes of axioms of classical propositional logic 1. 2. ( B) ( B) 3. Rules: modus ponens;. If e have as additional axiom schemes: 4. ( ( ) ) ( )
Tableaux Variants of some Modal and Relevant Systems 93 5. 6. ( B) ( B ) 7. ( ) systems S4.1, S4.2, S4.3 and S4.4 can be obtained by means of adding to S4 one axiom scheme 4, 5, 6 and 7 accordingly. To construct the tableau variants T4.1, T4.2, T4.3 and T4.4 of axiomatic systems in question e define the folloing notions: Index is the sequence of natural numbers beginning ith 0 in hich no number is repeated. Let u,,,,... be idexes and i, k, l, m, n,... - natural numbers. R is a binary relation such that ur iff = u, k or = u or for some l 1 there exist such indexes 1, 2,..., l that 1 = u, l = and i R i+1 for any i (1 i < l). Index is called subindex of index iff R. If,, &B, B, B are ell formed formulae (in usual sense), then,, & B, B, B ill be indexed formulae. No e introduce tableaux systems T4.1 - T4.4 by means of the folloing rules. Rules of construction: (BR) Usual rules for construction of Beth s tableaux ith the folloing addition: index of formula subjected to non-modal rule application is transferred ithout 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 folloing ay. If B occurs on the right of (sub)tableau, rite on the left of it and B on its right. If B occurs on the left of (sub)tableau, split it into to subtableau riting on the right of first subtableau and B on the left of the second. ( ) T If formula u occurs on the right of (sub)tableau, rite u,k on the right of it, k being a number hich does not occur in indexes described to formulae occurring in this (sub)tableau. ( ) T 4.1 If formula occurs on the left of (sub)tableau rite on the left of it, here
94 P. Bystrov ( ) T 4.2 ( ) T 4.3 ( ) T 4.4 a) R or b) = u, k and = u, m, n if formula u,m has already occurred on the left of this (sub)tableau. rule is obtained from ( ) T.4.1 by replacement of point b) by c) = u, n and = u, m, n index = u, n being subindex of at least one of indexes described to formulae hich occur in this (sub)tableau. rule is obtained from ( ) T 4.1 by replacement of point b) ith d) = (here 0) if some formula of the form B occurs on the left and formula on the right of this (sub)tableau. rule is obtained from ( ) T 4.1 by replacement of point b) ith e) if = u, k and formula u occurs on the left of this (sub)tableau, then is an arbitrary index such that 0. Rule of closure: (CR) If some formula ith 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), ( ) T, ( ) T 4.1. Other systems T4.2, T4.3 and T4.4 are obtained from T4.1 by means of replacement of the rule ( ) T 4.1 ith rule ( ) T 4.2, ( ) T 4.3, ( ) T 4.4, respectively. Proof of equivalency beteen SM and TM can be given semantically, in Kripke style, using the notion of equivalency of tableaux and corresponding models for SM 1). But more constructive syntactical proof can be given also. Let indexed sequent be an expression of the form Γ Θ here 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 G J in the folloing ay:
Tableaux Variants of some Modal and Relevant Systems 95 Basic sequent (axiom): Γ 1,, Γ 2 Θ 1,, Θ 2. Rules: 2), B, Γ Θ B, B, Γ Θ ; B, Γ Θ Γ Θ, B, Γ Θ, ; B, Γ Θ, Γ Θ, ;, Γ Θ,, Γ Θ ; Calculus G J 4.1 is obtained from G J by addition of the rules: Γ Θ, u, u,k Γ Θ,, here k does not occur in indexes described to any formula occurrence in conclusion sequent;,, Γ Θ, Γ Θ, here R or ( ) = u, m, n and = u, k if formula u,m occurs in Γ. We can obtain calculi G J 4.2, G J 4.3, G J 4.4 from G J 4.1 by replacing condition ( ) by one of the folloing conditions respectively: ( ) 4.2 ( ) 4.3 ( ) 4.4 = u, m, n and = u, n index = u, m being subindex of at least one of indexes described to formulae in conclusion sequent; ( 0) if some formula of the form B occurs in Γ and formula occurs in Θ; is an arbitrary index such that 0 if = u, k and formula u occurs in Γ. Cut is eliminable in each of those systems. So, if e define in appropriate ay the notion of representing formula of the indexed sequent Γ Θ, it can be proved that representing formula of the axiom of G J M is theorem
96 P. Bystrov of SM and all rules of G J M are rules derived from SM in the folloing 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 G J M because e can prove α in G J M if α is an axiom of SM and rules of SM are derivable in G J M. Then the folloing proposition is valid for G J M and SM. Proposition 1.. formula α can be proved in SM iff pure sequent α can be proved in G J M. No let us consider the set of indexed sequents the expression of the form S 1 ; S 2 ;... ; S m, here for each i (i = 1, 2,..., m) S i is an indexed sequent. Then calculus of the sets of indexed sequents B J M e obtain by adding to the axiom-schemata S 1 ; S 2 ;... ; S n, here n > 0 and for each i (i = 1, 2,..., n) S i is a basic sequent of G J M to folloing rules: U; S ; W U; S; W ; U; S ; W ; U; S ; W U; S; W here U, W are sets of indexed sequents (may be empty), S (S, S ) - premis(es) and S is a conclusion of some rule of G J M. It is easy to prove that calculi B J M and G J M are deductively equivalent. On the other hand, any inference of B J M can be transformed into correct TM-inference. So, the G J M and TM are deductively equivalent and on the ground of Proposition 1 e have the folloing: Proposition 2. SM. ; The system TM is deductively equivalent to the system 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) ith respect to elementary 3) 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) hen the last sentence in formulation of (CR) is replaced by the folloing: (Sub)tableau is closed iff
Tableaux Variants of some Modal and Relevant Systems 97 all its subtableaux are and there is a closure ith respect to each elementary subformula of the formula occurring on the right of initial tableau. Moreover, e replace the rules for conditional from the set of rules (BR) by the folloing to rules. ( r ) If formula B occurs on the right of (sub)tableau, rite,k on the left and,k B on the right of this (sub)tableau (k being a number not occurring in indexes described to formulae in this (sub)tableau). ( l ) If formula B occurs on the left of (sub)tableau, split this (sub)tableau into to alternative subtableaux and rite on the right of the first and B on the left of the second subtableau, being index such that R and graphically coincides ith or. The resulting set of rules (BR) together ith (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) ( B) and (2) (&B) are not theorems of R. Let the occurrence of the formula in the formulae of the form (1), (2) be called essential occurrence. Then the rule of closure (CR) results from (CR) if e modify the last sentence of its formulation in the folloing ay: (Sub)tableau is closed iff all its subtableaux are and there is a closure ith 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 ith (CR) gives the relevant system R. If e add to the formulation of (CR) the condition: ll elementary subformulae ith respect to hich closure takes place have one and the same index ; the rule (CE) results. ccordingly, if the condition: ll essential occurrences of each elementary subformula ith respect to hich
98 P. Bystrov closure takes place have one and the same index is added to (CR), e have the rule (CE). (BR) +(CE) gives strongly relevant system E. (BR) + (CE) gives relevant (in usual sense) system E. s for interrelation beteen R, E and ell knon relevant systems R, E it can be proved that a class of theorems of the system R (E ) 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 ritten just in the same ay preserving indexes in premis(es) ithout change. 3) formula is elementary iff it has the form, B, & B or B, here and B are propositional variables. References [1] E. W. Beth, The foundations of mathematics, msterdam 1959. [2] P. I. Bystrov, Calculi of indexed sequents and tableaux constructions of modal systems, bstracts of VIIIth Soviet Symposium: Logic and Methodology of Science, Vilnius 1982, pp. 15 20 (in Russ.) [3] S.. 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).