23rd International Symposium on Multiple Valued Logic. Sacramento, CA, May 1993 Proceedings. (IEEE Press, Los Alamitos, 1993) pp. 208{213 Systematic Construction of Natural Deduction Systems for Many-valued Logics Matthias Baaz∗ Christian G. Ferm¨ullery Richard Zachy Technische Universit¨atWien, Austria Abstract sion.) Each position i corresponds to one of the truth values, vm is the distinguished truth value. The in- A construction principle for natural deduction sys- tended meaning is as follows: Derive that at least tems for arbitrary finitely-many-valued first order log- one formula of Γm takes the value vm under the as- ics is exhibited. These systems are systematically ob- sumption that no formula in Γi takes the value vi tained from sequent calculi, which in turn can be au- (1 i m 1). ≤ ≤ − tomatically extracted from the truth tables of the log- Our starting point for the construction of natural ics under consideration. Soundness and cut-free com- deduction systems are sequent calculi. (A sequent is pleteness of these sequent calculi translate into sound- a tuple Γ1 ::: Γm, defined to be satisfied by an ness, completeness and normal form theorems for the interpretationj iffj for some i 1; : : : ; m at least one 2 f g natural deduction systems. formula in Γi takes the truth value vi.) For each pair of an operator 2 or quantifier Q and a truth value vi 1 Introduction we construct a rule introducing a formula of the form 2(A ;:::;A ) or (Qx)A(x), respectively, at position i The study of natural deduction systems for many- 1 n of a sequent. The resulting calculi are shown to be valued logics can be motivated by the following two is- sound and cut-free complete by Sch¨utte'sreduction sues: (1) Many-valued logics provide a general frame- tree method. work for the investigation of properties of classical (two-valued) sytems. (2) A general construction of Every sequent rule introducing a formula at a non- sound and complete natural deduction calculi leads to distinguished position is converted into an elimina- an adequate syntactical (proof-theoretic) characteri- tion rule; the sequent rule introducing a formula at zation of many-valued logics for which one wants to the distinguished position is transformed into an in- emphasize the r^oleof a particular truth value. (For troduction rule (in the sense of natural deduction). standard logics, such as the families of G¨odelund Any natural deduction derivation can be translated Lukasiewicz logics, one usually considers such distin- into a derivation of the corresponding sequent calcu- guished truth values.) lus. On the other hand, any cut-free sequent calcu- We consider finitely-many-valued first order logics lus proof translates into a normal natural deduction with arbitrary truth-functional connectives and dis- derivation. (Here normal means that for no branch tribution quantifiers (see Definition 2.2). A natural of the proof tree an elimination follows an introduc- deduction derivation for a logic with the truth values tion; this excludes maximal segments in the sense of Prawitz [1971].) Consequently, the natural deduc- v1; : : : ; vm is defined as a derivation f g tion sytems are sound and complete and every deriva- Γ ::: Γ tion can be transformed into a normal derivation. 1 . m 1 j . j − Such derivations consist of \analytical" paths. Γm 2 Preliminaries where Γi (1 i m) are sets of formulas (Γ1 ::: ≤ ≤ j j Γm 1 represents the assumptions, Γm is the conclu- 2.1. Definition A language for a logic L consists − L ∗Technische Universit¨at Wien, Institut f¨ur Algebra und of: (1) free variables, (2) bound variables, (3) predi- Diskrete Mathematik E118.2, Wiedner Hauptstraße 8{10, A- cate symbols, (4) propositional connectives, (5) quan- 1040 Wien, Austria, [email protected] tifiers, and (6) auxiliary symbols: \(", \)", \," yTechnische Universit¨at Wien, Institut f¨ur Computer- sprachen E185.2, Resselgasse 3/1, A-1040 Wien, Austria, We use a, b, c, . to denote free variables; x, y, z, chrisf, zach @logic.tuwien.ac.at . to denote bound variables; P , Q, R, . to denote f g predicate symbols; 2 to denote connectives; and Q to where a is a new free variable, and M(d=a) is de- denote quantifiers, all possibly indexed. fined as the interpretation equal to M, except that M(d=a)a = d. 2.2. Definition A matrix L for a language is given by: L 3 Sequent calculi (1) a nonempty set of truth values V = v1; : : : ; vm of size m, f g Sequent calculi for classical logic were intro- (2) an abstract algebra V with domain V of appro- duced by Gentzen [1934] and were later general- priate type: For every n-place connective 2 of ized to the many-valued case by Schroter¨ [1955], there is an associated truth function 2: V n VL, Rousseau [1967], and others. More recently, equiv- ! (3) for every quantifier Q, an associatede truth func- alent formulations for tableaux calculi were given tion Q: }(V ) V (see, e.g., Carnielli [1987] or Hahnle¨ [1991]). The n f;g ! A languagee and a matrix for it together fully deter- method used here can also be used to obtain calculi mine a logic L. L is said to be m-valued. for transformation into clause form for many-valued resolution (see Baaz and Fermuller¨ [1992]). The intended meaning of a truth function for a propo- sitional connective is obvious and perfectly analogous 3.1. Definition An (m-valued) sequent is an m- to the two-valued case. A truth function for a quanti- tuple of finite sets Γi of formulas, denoted thus: fier is a mapping from nonempty sets of truth values to truth values: given a quantified formula (Qx)F (x), Γ1 Γ2 ::: Γm j j j such a set of truth values describes the situation where For convenience, we will abbreviate Γ ∆ by Γ; ∆; the instances of F take exactly the truth values in this [ set as values under a given interpretation. Γ F by Γ; F ; and sometimes Γ1 ::: Γm by [m f g j j Γl l=1. We say that F stands (or occurs) at place i, 2.3. Example The matrix for the three-valued G¨odel j j if F Γi; vi then is the truth value corresponding to logic G3 consists of: place2i. (1) The set of truth values V = f; ; t (2) The truth functions for the connectives,f ∗ g e.g.: 3.2. Definition An interpretation M is said to sat- isfy a sequent Γ1 ::: Γm, if there is an i (1 i m) t f f t j j ≤ ≤ : _ ∗ ⊃ ∗ and a formula F Γi, s.t. valM(F ) = vi. A sequent is t f t t t t f t t t called valid, if it is2 satisfied under every interpretation. t f t t ∗ ∗ ∗ ∗ ∗ ∗ f t f t f t f t 3.3. Definition An introduction rule for a connec- ∗ ∗ tive 2 at place i in the logic L is a schema of the (3) The truth functions for the quantifiers and form: (generalized and ): 8 9 ^ _ j j j j t = t t = t DΓ1 ; ∆1 ::: Γm; ∆mE 8f g 9f g j j j I et; = et; = t 2 2:i 8f ∗} ∗ 9f ∗} Γ1 ::: Γi; 2(A1;:::;An) ::: Γm e t; f = f e t; f = t j j j j 8f g 9f g j et; ; f = f et; ; f = t where the arity of 2 is n, I is a finite set, Γl = j I Γl , 8f ∗ g 9f ∗ g S 2 e = e = ∆j A ;:::;A and the following condition holds: 8{∗} ∗ 9{∗} ∗ l 1 n e ; f = f e ; f = Let⊆ fM be an interpretation.g Then the following are 8{∗ g 9{∗ g ∗ e f = f e f = f equivalent: 8f g 9f g e e 2.4. Definition An interpretation M is a mapping of (1) 2(A1;:::;An) takes the truth value vi under M. j j all free variables to elements of a given domain D, and (2) For j I, M satisfies the sequents ∆1 ::: ∆m. of predicate symbols to functions of type Dn V . 2 j j ! A valuation valM is a mapping that extends the in- It should be stressed that the introduction rules for a terpretation to formulas via the truth functions given connective at a given place are far from being unique: in the matrix. We only give the precise definition of Let the expression Avl denote the statement \A takes the valuation function for a quantified formula: the truth value vl". Then every introduction rule for 2(A1;:::;An) at place i corresponds to a con- junction of disjunctions of some Avl which is true valM (Qx)G(x) = Q valM(d=a)G(a); [ e d D iff 2(A1;:::;An) takes the truth value vi (namely, 2 m vl j j I l=1 A ∆j A ). Any such conjunctive normal j; w = vl A(t ) vl V j . Again, it should be V 2 W W 2 l 6 g [ f j 2 n g vi form for 2(A1;:::;An) will do. stressed that in general this is not the only possible In particular, the truth table for 2 immediately rule. yields a complete conjunctive normal form, the cor- 3.6. Example Consider the universal quantifier in responding rule is as in Definition 3.3, with: I V n 8 ⊆ three-valued G¨odellogic G3 given in Example 2.3. In- is the set of all n-tuples j = (w1; : : : ; wn) of truth val- j tuitively, ( x)A(x) takes the value f, if A(t) is false for ues such that 2(w1; : : : ; wn) = vi; and ∆ = Ak 8 6 l f j some t; t, if A(a) is true for all a; and , if A(t) takes 1 k n; vl =ewk .