Herbrand s Theorem, Skolemization, and Proof Systems for First-Order Łukasiewicz Logic

Transcription

1 Herbrand s Theorem, Skolemization, and Proof Systems for First-Order ukasiewicz Logic Matthias Baaz 1 and George Metcalfe 2 1 Institute of Discrete Mathematics and Geometry, Technical University Vienna, Wiedner Hauptstrasse 8-10, A-1040 Wien, Austria baaz@logic.at 2 Department of Mathematics, Vanderbilt University 1326 Stevenson Center, Nashville TN 37240, USA george.metcalfe@vanderbilt.edu Abstract. An approximate Herbrand theorem is established for first-order infinitevalued ukasiewicz logic and used to obtain a proof-theoretic proof of Skolemization for this logic. These results are used to define proof systems for first-order ukasiewicz logic in the framework of hypersequents; in particular, a calculus lacking cut-elimination for the fragment characterized by MV-algebras, a cutfree calculus with an infinitary rule for the full logic, and a cut-free calculus with finitary rules for the one-variable fragment. 1 Introduction Infinite-valued ukasiewicz logic was introduced for philosophical reasons by Jan ukasiewicz in the 1930s [16], and is among the most important and widely studied of all non-classical logics. It is considered, along with Gödel logic and Product logic, one of the fundamental t-norm based fuzzy logics most suitable for formalizing reasoning in the context of vagueness (see [12] for details). Moreover, the algebraic semantics of ukasiewicz logic, MV-algebras, are the subject of intensive research in Algebra, having deep connections with abelian l-groups and, via McNaughton s theorem [17], piecewise continuous linear functions on [0, 1] (for an in-depth treatment, see [11]). The first-order counterpart of ukasiewicz logic is obtained by generalizing the classical interpretations of the quantifiers and to infima and suprema in [0, 1]. Unlike the situation in Classical logic, however, the valid formulas of are not recursively enumerable, a result obtained by Scarpellini [26] and later sharpened by Ragaz to Π 2 - completeness [24]. Axiomatizations of with infinitary rules have been obtained nevertheless; by Hay [14], Belluce and Chang [7, 6] (see also Mostowski [20]), and Hájek [12]. Also, various fragments of have been investigated. Validity and satisfiability in the one-variable fragment were proved decidable by Rutledge [25], while satisfiability for the monadic fragment was proved Π 1 -complete by Ragaz [24]; the complexity of validity for this fragment being an open problem. Validity for the fragment characterized by validity in so-called safe MV-algebras is Σ 1 -complete [7]. Also of interest are (decidable) fragments suitable for fuzzy description logics, investigated by Straccia [27] and Hájek [13, 27], or fuzzy logic programming, as studied e.g. by Vojtás [28].

2 The aim of this paper is to provide a proof-theoretic basis for first-order ukasiewicz logic as a starting point for both applications and deeper algorithmic investigation. We begin with a simple topological proof of the fact that while the usual Herbrand theorem does not hold for, an approximate Herbrand theorem (proved in a more complicated fashion by Novák in [23]) can be obtained instead. Essentially, for any valid existential formula, there exist Herbrand disjunctions for successive approximations to validity: for any r < 1 a disjunction exists that always takes a value greater than r. Following a similar proof for first-order Gödel logic [2], we then use this result to give a proof-theoretic proof that first-order ukasiewicz logic admits Skolemization. The proof-theoretic treatment, unlike semantic proofs (obtained e.g. in the wider setting of Continuous Model Theory in [8]), implies that any system of functions can be considered as Skolem functions obeying some constraints, such as commutativity. This in turn allows greater flexibility for replacing functions in formulas with quantified terms. We also make use of the approximate Herbrand theorem and Skolemization to define proof systems for (fragments of) in the framework of hypersequents, a generalization of Gentzen sequents consisting of a multiset intuitively, a disjunction of sequents. Hypersequents were introduced by Avron in [1] and have been used to provide proof systems for a wide range of (first-order) fuzzy logics [5, 3, 18, 10]. In particular, a cut-free calculus G for propositional ukasiewicz logic was developed in [19] using a non-standard interpretation of hypersequents. Here we add hypersequent versions of the usual quantifier rules for Gentzen s LK and LJ (as used e.g. for first-order Gödel logic in [5]) to G, obtaining a calculus that, when extended with a cut rule is complete with respect to safe MV-algebras. However, as we show with a suitable counter-example, cut-elimination does not hold for this extended calculus. On the other hand, adding an infinitary rule to G gives a cut-free system that is complete for the full logic. Finally, a cut-free calculus with finitary rules is obtained for the one-variable fragment of by relaxing the eigenvariable condition in the quantifier rules of G. 3 2 First-Order ukasiewicz Logic We will make use here of a usual first-order language with countable sets of predicate symbols, (object) constants, function symbols (with positive arity), and variables; quantifiers and ; a binary connective ; and a (logical) constant, where: A = def A = def A B = def A B A B = def ( A B) A B = def (A B) B A B = def ( A B) n.a = def A... A }{{} n A n = def A... A }{{} n In fact, for first-order ukasiewicz logic, as in the classical case, it is possible to define ( x)a(x) as ( x) A(x), but we will find it more convenient here to treat both quantifiers as primitive. 3 An earlier version of this paper has appeared as [4].

3 Terms t, s and formulas A, B, C are built inductively from the elements of this language in the usual manner, adopting standard notions of subformulas, scope, and distinguished bound variables x, y, z and free variables a, b. Function-free, quantifier-free, and one-variable formulas are those built using no function symbols, no quantifiers, or just one bound variable, respectively. 4 Also, we will call quantifier-free formulas containing no variables, propositional. A sequence of terms t 1,..., t n is often written t, denoting a formula A with variables among x by A( x), and A with each x i replaced by t i for i = 1... n by A( t). Finite multisets of formulas, written directly as [A 1,..., A n ] where [] is the empty multiset, are denoted Γ,. We write Γ 1 Γ 2 for the multiset sum of Γ 1 and Γ 2 and use Γ to indicate that is a sub-multiset of Γ, and multiset-builder notation [ : ] for constructing multisets satisfying a particular property. We also let Γ n stand for the multiset sum of n copies of Γ, and [A 1,..., A n ] for the formula A 1... A n. -interpretations I = (D, v I ) consist of a non-empty set D and a valuation v I that maps constants and object variables to elements of D; n-ary function symbols to functions from D n into D; and n-ary predicate symbols to functions from D n into [0, 1]. As usual, v I is extended to all terms inductively by the condition v I (f(t 1,..., t n )) = v I (f)(v I (t 1 ),..., v I (t n )) for any n-ary function symbol f and terms t 1,..., t n. For an n-ary predicate symbol p and terms t 1,..., t n : v I (p(t 1,..., t n )) = v I (p)(v I (t 1 ),..., v I (t n )) For a variable x and element d D, let v I [x d] be the valuation obtained from v I by changing v I (x) to d. Then v I is extended to all formulas by: v I ( ) = 0 v I (A B) = min(1, 1 v I (A) + v I (B)) v I (( x)a(x)) = inf{v I [x a](a(x)) : a D} v I (( x)a(x)) = sup{v I [x a](a(x)) : a D} I satisfies a formula A if v I (A) = 1, and A is -valid, written = A, if A is satisfied by all -interpretations. Two formulas A and B are -equivalent, written A B, if v I (A) = v I (B) for all -interpretations v I. It will also be crucial in this paper to consider the following notion of approximate validity for {>, } and r [0, 1]: = r A iff v I (A) r for all -interpretations I. While the problem of checking -validity of formulas, or approximate validity when is, is Π 2 -complete [24], checking approximate validity when is > and r is rational is Σ 1 -complete [6]. For one-variable formulas, i.e. the one-variable fragment, checking validity is decidable [25]. For the monadic fragment, having predicate symbols of arity at most one, checking satisfiability is known to be Π 1 -complete [24], but the complexity of checking -validity is an open problem. Finally, checking the -validity of propositional formulas is known to be decidable, indeed co-np complete [22]. 4 One-variable formulas usually allow the given variable to have both free and bound occurrences: this is not permitted here due to the distinction between free and bound variables, but has no impact on decidability or complexity.

4 Interpretations and validity can also be generalized to a wider class of algebraic structures; an MV-algebra is an algebra A = L,,, 0 such that: 1. L,, 0 is a commutative monoid. 2. x = x. 3. x 0 = ( x y) y = ( y x) x. where x y iff x y = 0, with defined operations 1 = def 0 and: x y = def ( x y) x y = def ( x y) y x y = def x y x y = def ( (x y) y) An MV-chain is a linearly ordered MV-algebra; i.e. satisfying x y or y x for all x, y L. The most important example of an MV-chain is [0, 1],,, 0 where x y = min(1, x + y) and x = 1 x. In this case x y = min(1, 1 x + y), x y = max(0, x + y 1), x y = min(x, y), and x y = max(x, y). Let A = L,,, 0 be an MV-chain. Then A-interpretations are defined as for -interpretations except that n-ary predicate symbols are mapped by v I to functions from D n into L, and v I is extended to all formulas by: v I ( ) = 0 { vi (A) v v I (A B) = I (B) if v I (A) and v I (B) are defined undefined otherwise { inf{vi [x d](a(x)) : d D} if the infimum exists v I (( x)a(x)) = undefined otherwise { sup{vi [x d](a(x)) : d D} if the supremum exists v I (( x)a(x)) = undefined otherwise As in the previous definition, I satisfies a formula A if v I (A) = 1, and A is A-valid if A is satisfied by all A-interpretations. Notice that if the required infimum or supremum does not exist in L, then the values of the corresponding universal or existential formulas, and all those in which they occur as subformulas, are undefined. The standard practice for dealing with such situations is simply to stipulate that they do not occur. That is, we treat only safe MV-chains A where v I (A) is defined for all formulas A and A-interpretations I. Clearly, the standard MVchain [0, 1],,, 0 mentioned above is safe; and indeed, as is trivial to check, validity in this algebra coincides with -validity. MV-algebras arise as the algebraic semantics of propositional ukasiewicz logic; that is, for each propositional formula A: = A iff A is A-valid for every MV-algebra A See [11] for details. Note however that at the first-order level, not only do we have to restrict our attention to safe MV-chains, but also just the right-to-left direction holds in the above equivalence.

5 (1) A (B A) (2) (A B) ((B C) (A C)) (3) ((A B) B) ((B A) A) (4) ((A ) (B )) (B A) ( 1) ( x)a(x) A(t) ( 2) ( x)(a B) (A ( x)b) where x is not free in A A A B B (mp) A ( x)a (gen) Fig. 1: The Hilbert System H In this paper we will make brief use of a fundamental result for MV-algebras: the categorical equivalence of the category of MV-algebras with the category of abelian l- groups with strong unit. Recall that an abelian l-group is an algebra G = L,,, +,, 0 such that L,, is a lattice, L, +,, 0 is an abelian group, and + is order preserving; i.e. x y implies z + x z + y for all x, y, z L. Making use of the additive notation, we write x y for y + x, nx for x x (n times), and Γ for the sum x x n of a multiset of elements Γ = [x 1,..., x n ] in G where [] = 0. A strong unit for G is a member u of L such that for any x G, nu x for some n N. Then more precisely, what we need is the following lemma: Lemma 1 ([21]). For every safe MV-chain A, there is a linearly ordered abelian l- group G = L,,, +,, 0 with strong unit u, Ξ(A) = (G, u), such that A is isomorphic to [0, u],,, 0 where [0, u] = {x L : 0 x u}, x y = def u (x + y), and x = def u x, and the sups and infs of A coincide with the corresponding sups and infs of [0, u],,, 0. For convenience, we can assume that A just is the algebra [0, u],,, 0 of Ξ(A). Finally, we note that the axiomatization in Figure 2, introduced by Hájek in [12] (simplifying previous axiomatizations of Hay [14] and Belluce and Chang [7, 6]), corresponds to validity in safe MV-chains, where formulas are defined without distinguishing between free and bound variables, and A = def A, Theorem 1 ([12]). H A iff A is A-valid for all safe MV-chains A. 3 An Approximate Herbrand Theorem Our aim in this section will be to show the failure of the Herbrand theorem for first-order ukasiewicz logic, compensated for somewhat by the success instead of an approximate Herbrand theorem. Before moving on to the deficiencies of, however, let us first consider one of its more attractive features: like Classical logic, has the full quota of quantifier shifts. That is, we have the following -equivalences, where by definition x does not occur free in A: A ( x)b ( x)(a B) ( x)b A ( x)(b A) A ( x)b ( x)(a B) ( x)b A ( x)(b A))

6 Let us write (Q x)a( x) for a formula (Q 1 x 1 )... (Q n x n )A(x 1,..., x n ) where Q i {, } for i = 1... n. A prenex formula is a formula (Q x)p ( x) where P is quantifierfree. Then in as in Classical logic, given any formula A, we can rewrite all bound variables to new variables and use the above equivalences (left-to-right) as rewrite rules to push all quantifiers to the outside. Theorem 2. Any formula is -equivalent to a prenex formula. Let us now recall some basic notions relating to Herbrand s theorem. Let A be a formula, and let C and F be the constants and function symbols occurring in A, respectively, adding a constant if the former is empty. The Herbrand universe U(A) of A is the set of ground (i.e. containing no variables) terms built using C and F; i.e. U(A) = n=0 U n(a) where: U 0 (A) = C U n+1 (A) = U n (A) {f(t 1,..., t k ) : t 1,..., t k U n (A) and f F with arity k} Letting P be the predicate symbols of A, the Herbrand base B(A) of A is defined as: B(A) = {p(t 1,..., t k ) : t 1,..., t k U(A) and p P with arity k} For a logic L, the usual Herbrand Theorem for existential formulas states that a formula ( x)p ( x) where P is quantifier-free, is L-valid iff some disjunction n P ( t i ) is L- valid, where t 1,..., t n U(P ). However, as we will now show, this formulation does not hold for. First, observe that = ( x)p(x) ( y)p(y) and hence, using the quantifier-shifting equivalences above: = ( y)( x)(p(x) p(y)) It then follows by a simple semantic argument that: = ( y)(p(f(y)) p(y)) So if Herbrand s theorem holds for, then for some constant c and n 1: n = (p(f i (c)) p(f i 1 (c))) where f 0 (c) = c and f i+1 (c) = f(f i (c)). But now define v I (p(f i (c))) = i/n, giving: v I (p(f i (c))) > v I (p(f i 1 (c))) for i = 1... n It follows that v I ( n (p(f i (c)) p(f i 1 (c))) < 1, which is a contradiction, so the Herbrand theorem fails. Take another look at the formula n (p(f i (c)) p(f i 1 (c))), however. Although this is not -valid, it comes within one nth of being so. For any r < 1 1/n: = >r n (p(f i (c)) p(f i 1 (c)))

7 That is, we can characterize successive Herbrand approximations to ( y)(p(f(y)) p(y)) that come arbitrarily close to 1. This is an example of a more general phenomenon; captured by the following approximate Herbrand theorem: Theorem 3. = ( x)p ( x) where P is quantifier-free iff for all r < 1: = >r n P ( t i ) for some t 1,..., t n U(P ) Proof. We refer to [29] for all topological terminology. Suppose that = >r n P ( t i ) for all r < 1. Then = >r ( x)p ( x) for all r < 1, so clearly = ( x)p ( x). For the other direction, fix r < 1. Notice that any mapping from B(P ) into [0, 1] a member either of [0, 1] k for some k if B(P ) is finite, or of the Hilbert cube [0, 1] ω if B(P ) is countably infinite may be viewed as a valuation v I for an interpretation I. In either case ([0, 1] ω using the Tychonoff Theorem), [0, 1] B(P ) is a compact space with respect to the product topology. Now for each t U(P ) define: S( t) = {v I [0, 1] B(P ) : v I (P ( t)) r} Since P is quantifier-free and the propositional connectives and are interpreted by continuous functions on [0, 1], each S( t) is a closed subset of [0, 1] B(P ). Consider: We have two possibilities: S = {S( t) : t U(P )} 1. Suppose that for some {S( t 1 ),..., S( t n )} S: S( t i ) = Then for every interpretation I, v I (P ( t i )) > r for some i {1,..., n}. I.e. n P ( t i ) as required. = >r 2. Otherwise, every finite subset of S has a non-empty intersection and since S is a collection of closed subsets of [0, 1] B(P ), by the finite intersection property for compact spaces, S has a non-empty intersection. I.e., there exists v I such that v I (P ( t)) r for all t U(P ). So v I (( x)p ( x)) r, a contradiction. This approximate Herbrand theorem has a nice corollary. Let F = ( x)( ȳ)p ( x, ȳ) where P is both quantifier-free and function-free. Then = F iff = ( ȳ)p ( c, ȳ) for new constants c. Let C be the (finite) set of constants occurring in ( ȳ)p ( c, ȳ), adding one if the set is empty. Using the previous theorem: = F iff for all r < 1, = >r n P ( t i ) for some t i C iff = P ( c) But checking validity in propositional ukasiewicz logic is decidable, so we have established the following: c C

8 Proposition 1. The function-free -fragment of is decidable. Notice also that any formula in the function-free one-variable fragment of is - equivalent to a formula ( x)( ȳ)p ( x, ȳ) where P is quantifier-free and function-free. Just use the -equivalences to push the universal quantifiers to the front of the formula first; permitted by the restriction of the original formula to one variable. Corollary 1. The function-free one-variable fragment of is decidable. 4 Skolemization We will now use the approximate Herbrand theorem in a rather neat way: to provide a proof-theoretic proof of Skolemization for. Note that while for Classical logic, the usual process involves removing existential quantiifers and preserving satisfiability, here we follow common terminology for fuzzy logics (see e.g. [2]) and remove universal quantifiers, hoping rather to preserve validity. Let A be a prenex formula and assume harmlessly that the ith occurrence of is labelled i and that no function symbol f i of any arity occurs in A. Then the Skolem form A S of A is defined by induction as follows: (1) If A is of the form ( x)p ( x) where P is quantifier-free, then A S is ( x)p ( x). (2) If A is of the form ( x)( i y)b( x, y), then A S is (( x)b( x, f i ( x)) S. Our aim is to prove that = A iff = A S for any prenex formula A. The first step towards establishing the difficult right-to-left direction of this equivalence, will be to show that if a Herbrand disjunction for the Skolem form of a prenex formula is approximately c valid, then the formula is itself valid to the same degree. Lemma 2. Let ( x)p F ( x) be the Skolem form of (Qȳ)P (ȳ). Then [(Qȳ)P (ȳ)] is derivable from any finite non-empty sub-multiset of [P F ( t) : t U(P F )] using the rules: Γ [A(t)] Γ [( x)a(x)] Γ [A(s)] Γ [( x)a(x)] Γ [A, A] Γ [A] Γ Γ [A] where in the leftmost rule, t is any ground term not occurring in Γ or A. Proof. We assume that the occurrences of in (Qȳ)P (ȳ) are labelled with the corresponding functions f( z) of ( x)p F ( x) introduced by Skolemization and that substituting for a variable in such formulas extends to substituting also in the labels. Let Γ 0 = [(Qȳ)P (ȳ)], and given Γ j, let Γ j+1 be the smallest multiset satisfying: (1) Γ j Γ j+1. (2) If ( x)b(x) Γ j+1 and f( t) labels, then B(f( t)) Γ j+1. (3) If ( x)b(x) Γ j+1, then B(s) Γ j+1 for all s U j (P F ).

9 Notice first that each Γ j can be derived from Γ j+1 using the given rules. The only difficulty could be that for (2) the term f( t) occurs already in a formula in the multiset. However, since each occurrence of is labelled with a different function symbol f and the arguments of this function are determined uniquely by the terms chosen for the preceding occurrences of, the formula to be added in such case must already be a member of the multiset. The desired result is then a consequence of the following: Claim: if t U(P F ), then P F ( t) Γ j for some j N. This is proved by induction on the number of quantifiers in (Qȳ)P (ȳ), and implies that any finite non-empty sub-multiset of [P F ( t) : t U(P F )] is a sub-multiset of some Γ j. But then since Γ 0 can be derived from Γ j using the rules, the result follows using the rightmost rule backwards to remove the superfluous formulas from Γ j. In fact the rightmost rule in the preceding lemma is not really needed for the required derivations: we can remove the superfluous formulas and those derived directly from them, and still obtain a derivation. Also, the remaining rules applied downwards are terminating. So given a Herbrand disjunction, the above lemma provides an algorithm for finding a quantified form. Moreover, we can establish the following approximate Skolemization result: Proposition 2. Let ( x)p F ( x) be the Skolem form of (Qȳ)P (ȳ). For any r [0, 1]: n if = r P F ( t i ) for some t 1,..., t n U(P F ), then = r (Qȳ)P (ȳ) Proof. For each rule in Lemma 2, with premise Γ and conclusion, it is easy to see that = r Γ implies = r. So the result follows by a simple induction on the height of a derivation using these rules of [(Qȳ)P (ȳ)] from [P F ( t 1 ),..., P F ( t n )]. We now establish Skolemization for the prenex formulas of by combining this last result with the approximate Herbrand theorem. Theorem 4. Let ( x)p F ( x) be the Skolem form of (Qȳ)P (ȳ). Then: = ( x)p F ( x) iff = (Qȳ)P (ȳ) Proof. The right-to-left direction follows easily using standard quantifier properties of. For the other direction, suppose that = ( x)p F ( x). By Theorem 3, for all r < 1, there exist tuples of terms, t 1,..., t n, in U(P F ) such that = >r n P F ( t i ). But then by Proposition 2, for all r < 1, = r (Qȳ)P (ȳ). Hence = (Qȳ)P (ȳ) as required. Skolemization allows us to extend the approximate Herbrand theorem to the whole of : we just put the formula into prenex form, use Theorem 4 to find an appropriate existential formula, and apply Theorem 3. Corollary 2. Let A be a formula and let ( x)p F ( x) be the Skolem form of an equivalent prenex formula for A. Then = A iff = ( x)p F ( x) iff for all r < 1: = >r n P F ( t i ) for some t 1,..., t n U(P F )

10 Finally, we can use this approximate Herbrand theorem to sketch an alternative proof of a completeness result for the Hilbert system H provided in [12] (following [14, 7, 6]). First notice that for any formula B and k N + : 1 1/k v I (B) iff v I ( B) 1/k iff kv I ( B) 1 iff (k 1)v I ( B) v I (B) iff 1 v I (((k 1). B) B) iff 1 v I (B B k 1 ) Now for any formula A, let ( x)p F ( x) be the Skolem form of a prenex formula equivalent to A. We note without proof that A is H -derivable from ( x)p F ( x). Moreover, if = A, then by the approximate Herbrand theorem, for all k N + : = 1 1/k n P F ( t i ) for some t 1,..., t n U(P F ) So by the above reasoning and the propositional completeness of H, for all k N + : H B B k 1 where B = n P F ( t i ) for some t 1,..., t n U(P F ) Since ( x)p F ( x) is H -derivable from any such B, so for all k N +, A A k 1 is H -derivable from B B k 1, and we have: Theorem 5 ([12]). = A iff H A A k for all k N +. 5 The Hypersequent Calculus G We will define proof systems for ukasiewicz logic in the framework of hypersequents; finite multisets of sequents, written: Γ Γ n n where Γ i and i are finite multisets of formulas for i = 1... n. Validity is extended to hypersequents as follows: Definition 1. Let G = Γ Γ n n be a hypersequent. Then G is -valid, written = G, if for all -interpretations I: [vi (A) 1 : A Γ i ] [v I (B) 1 : B i ] for some i {1,..., n}. More generally, for a safe MV-algebra A, G is A-valid if for all A-interpretations I: [vi (A) u : A Γ i ] [v I (B) u : B i ] for some i {1,..., n}. in Ξ(A) = (G, u), the abelian l-group G = L,,, +,, 0 with strong unit u.

11 Initial Sequents A A (id) (Λ) A ( ) Structural Rules: G G Γ (ew) G Γ Γ G Γ (ec) G Γ G Γ, A (wl) G Γ 1, Γ 2 1, 2 G Γ 1 1 Γ 2 2 (split) G Γ 1 1 G Γ 2 2 G Γ 1, Γ 2 1, 2 (mix) Logical Rules G Γ, B A, G Γ, A B ( ) G Γ G Γ, A B, G Γ A B, ( ) Fig. 2: The Hypersequent Calculus G These definitions where hypersequents are interpreted using sums of elements in abelian l-groups rather than formulas of are certainly non-standard. In particular, multisets of formulas on the left and right of sequents are not interpreted using the operations of an MV-algebra, but outside the logic using the addition + of the corresponding abelian l-group. 5 Nevertheless, note that for formulas we still retain the usual notion of validity, i.e. = A iff = A The hypersequent calculus G for propositional ukasiewicz logic introduced in [19] is displayed in Figure 5, noting that Γ 1, Γ 2 stands for the multiset union Γ 1 Γ 2, and Γ, A for Γ [A], and that G denotes an arbitrary side-hypersequent occurring in both the premises and conclusion of a rule. G is cut-free by definition and all the rules have the subformula property. Moreover, natural rules for the defined connectives can be obtained, e.g. for and : G Γ, A G Γ, A B ( )1 G Γ, B G Γ, A B ( )2 G Γ, A G Γ, B G Γ, A B ( ) G Γ A, G Γ B, G Γ A B, ( ) G Γ A, G Γ A B, ( )1 G Γ B, G Γ A B, ( )2 Note that the standard version of the implication right rule is derivable when only one formula appears on the right (since the left premise G Γ in this case of ( ) is 5 We remark that in [19], this interpretation is represented as an embedding of ukasiewicz logic into Abelian logic, the logic of abelian l-groups.

12 derivable using (ew), (wl), and (Λ)): G Γ, A B G Γ A B ( ) 1 Example 1. We illustrate this calculus with a derivation of the key axioms of (3): B B (id) A A (id) B B (id) A A (id) (mix) (mix) B, A A, B B, B A A ( ) B, A A, B B, B A, A A, B (wl) ( ) B, B A A, A B (A B) B, B A A ( ) (A B) B (B A) A ( )1 ((A B) B) ((B A) A) ( )1 Hypersequents are not needed to prove this or indeed any of the other propositional axioms (1)-(4) for ; nevertheless, they are essential to prove other -valid formulas such as A (B ((A (A C)) ((B (B C)) C))). A semantic completeness proof for G, making use of the invertibility of ( ) and a derived version of ( ), was presented in [19]. Theorem 6 ([19]). Let G be a propositional hypersequent. Then G G iff = G. It follows easily from this theorem that the following cut rule and (inter-derivable) cancellation rule are admissible for G: G Γ 1, A 1 G Γ 2 A, 2 G Γ 1, Γ 2 1, 2 (cut) Syntactic eliminations of these rules were provided in [9]. G Γ, A A, G Γ (can) Theorem 7 ([9]). Cut/cancellation elimination holds for G + (cut) and G + (can). 6 Adding Quantifiers One of the most attractive features of sequent calculi such as Gentzen s LJ and LK for Intuitionistic logic and Classical logic, respectively, is that the first-order calculus is obtained by extending the propositional part with natural rules for the quantifiers. This feature is shared by hypersequent calculi for many first-order fuzzy logics such as first-order Gödel logic [5] and Monoidal t-norm logic [3] where the rules added are just hypersequent versions of those for LJ and LK. In this section we consider the effect of adding these usual quantifier rules to G, warning the reader in advance that progress to the first-order level will not be quite so smooth for this logic. Definition 2. Let G be G with the added rules: G Γ, A(t) G Γ, ( x)a(x) ( ) G Γ, A(a) G Γ, ( x)a(x) ( ) G Γ A(a), G Γ ( x)a(x), ( ) G Γ A(t), G Γ ( x)a(x), ( )

13 where the eigenvariable a does not occur in the conclusion of ( ) or ( ). Example 2. Consider the following G -derivation of the axioms for ( 2): B(a) B(a) (id) A A (id) (mix) B(a), A A, B(a) A B(a), A B(a) ( ) ( x)(a B), A B(a) ( ) ( x)(a B), A ( x)b ( ) ( x)(a B) A ( x)b ( )1 ( x)(a B) (A ( x)b) ( )1 Notice that it is crucial here that x cannot occur free in A, since in that case A(a) would appear on the left in the second application of (id), with A(x) on the right. Establishing soundness for G with respect to the interpretation = is straightforward. Theorem 8. If G H, then = H. Proof. By induction on the height of a derivation of H in G. Since the soundness of the other rules is checked in [19], we can just consider the quantifier rules. Also note that we can ignore the side-hypersequent G occurring in both the premises and conclusion of the rules, since clearly if an inequality holds for a sequent of G in a premise, then it holds also for the same sequent in the conclusion. For ( ), suppose that for every interpretation I: [vi (C) 1 : C Γ ] (v I (A(a)) 1) + [v I (D) 1 : D ] where a does not occur in Γ,, or A. But now notice that for any interpretation I and K R, if K v I [a d](a(a)) for all d D, then also K inf{v I [x d](a(x)) : d D} = v I (( x)a(x)). Hence: [vi (C) 1 : C Γ ] (v I (( x)a(x)) 1) + [v I (D) 1 : D ] as required. For ( ), suppose that for some term t: [vi (C) 1 : C Γ ] + (v I (A(t)) 1) [v I (D) 1 : D ] Then since v I (( x)a(x)) v I (A(t)): [vi (C) 1 : C Γ ] + (v I (( x)a(x)) 1) [v I (D) 1 : D ] Cases for the existential quantifier rules are very similar. Moreover, if we extend G with (cut) or (can), then we obtain soundness and completeness with respect to safe MV-chains and the axiomatization H. In fact, this is about as much as we could hope for, since is not recursively enumerable.

14 Theorem 9. For any formula A, the following are equivalent: (1) G +(cut) A. (2) H A. (3) A is A-valid for every safe MV-chain A. Proof. The equivalence of (2) and (3) is Theorem 1. To show that (2) implies (1), we first observe that (gen) and (mp) are admissible for G + (cut) using ( ) and (cut), respectively. It is also easy to see that the axioms (1)-(4) and ( 1)-( 2) are G -derivable, and hence that the derivability of A in H implies the derivability of A in G + (cut). To show that (1) implies (3), we prove the following: Claim: If G +(cut) G, then G is A-valid for every safe MV-chain A. We proceed by induction on the height of a derivation of G in G +(cut). This requires checking the soundness of each rule, recalling that for each safe MV-algebra A and abelian l-group G = L,,, +,, 0 with strong unit u such that Ξ(A) = (G, u), A is (isomorphic to) [0, u],,, 0 where x y = def u (x+y) and x = def u x. In fact, these soundness proofs are almost exactly as for the particular case of the safe MVchain on [0, 1] checked above; in particular the soundness of the quantifier rules follows as in the proof of Theorem 8. As an example, consider ( ), again disregarding the side-hypersequent G. Suppose that for some safe MV-chain A and A-interpretation I: [vi (C) u : C Γ ] + (v I (B) u) [v I (D) u : D ] + (v I (A) u) Then using properties of addition and substraction in Ξ(A): [vi (C) u : C Γ ] + ((u v I (A) + v I (B)) u) [v I (D) u : D ] Hence, since v I (A B) = v I (A) v I (B) = u (u v I (A) + v I (B)): [vi (C) u : C Γ ] + (v I (A B) u) [v I (D) u : D ] as required. Other cases are very similar. This is all very well. However, unfortunately, cut elimination fails for G + (cut) and cancellation elimination fails for G + (can), so we do not have an analytic calculus for this fragment of. For example, ( x)( y)(p(x) p(y)) has the following proof in G + (can): p(a) p(a) (id) p(b) p(b) (id) (id) ( z)p(z) p(a) ( ) p(a) p(a) ( z)p(z) p(b) ( ) (mix) ( z)p(z), p(a) p(b), p(a) ( ) ( z)p(z) p(a) p(b), p(a) ( z)p(z) ( y)(p(a) p(y)), p(a) ( ) ( z)p(z) ( x)( y)(p(x) p(y)), p(a) ( ) ( z)p(z) ( x)( y)(p(x) p(y)), ( z)p(z) ( ) (can) ( x)( y)(p(x) p(y)) But it is easy to see that no G -proof exists for this formula.

15 7 An Infinitary Calculus To obtain a calculus that is complete for the full logic, we clearly need as in the axiomatizations of Hay [14], Belluce and Chang [7, 6], and Hájek [12] an infinitary rule. To establish the completeness of G extended with such a rule, we make essential use of the approximate Herbrand Theorem and Skolemization results of earlier sections. The crucial step is to show that if a Herbrand disjunction of the prenex form of a formula C is -valid, then C is derivable in G. We proceed in similar fashion to the proof of Lemma 2, the complicating factors here being the presence of quantifiers deep within C and the use of rules to decompose the formula into sets of hypersequents. Proposition 3. Let (Qȳ)P (ȳ) be a prenex form of a formula C with Skolem form ( x)p F ( x). If = n P F ( t i ), then G C. Proof. Let the sequence Γ j of multisets of formulas with labelled occurrences of quantifiers be defined for (Qȳ)P (ȳ) exactly as in Lemma 2 and label with the same function symbols the corresponding occurrences of and in C (noting that some occurrences of are transformed to while prenexing and vice versa). We define sets H j of hypersequents as follows. Let H 0 = { C} and given H j, let H j+1 be the result of applying the following operations to H j exhaustively and in order: (i) Replace G Γ, A B with G Γ Γ, B A,. (ii) Replace G Γ A B, with G Γ and G Γ, A B,. (iii) If G Γ ( x)b(x), is in the set and f( t) labels, add G Γ B(f( t)),. (iv) If G Γ, ( x)b(x) is in the set, add G Γ, B(s) for all s U j (A). (v) If G Γ, ( x)b(x) is in the set and f( t) labels, add G Γ, B(f( t)). (vi) If G Γ ( x)b(x), is in the set, add G Γ B(s), for all s U j (A). Using the logical and structural rules of G, for each j: if G G for all G H j+1, then G G for all G H j Hence it is sufficient to show that G G for all G H k for some k N. In fact, by the propositional completeness of G, it is sufficient to show that = prop(g) for all G H j for some j N, where prop(g) consists of the propositional sequents of G; i.e. prop(g) = [S G : S contains only propositional formulas]. Recall now that each Γ j is a multiset of formulas containing only terms from U j (A). Let prop(γ j ) = [A Γ j : A is a propositional formula]. It follows from the construction of the two sequences that for each j N: = prop(γj ) iff = prop(g) for all G H j But by Lemma 2, the left hand side holds for some j N, so the result follows. Theorem 10. = A iff G A A n for all n N +.

16 Proof. For the right-to-left direction, suppose that G A A n for all n N +. Then for all interpretations I, 1 1/n v I (A) for all n N +, so v I (A) = 1. For the left-to-right direction suppose that = A. Let (Qȳ)P (ȳ) be a prenex form of A with Skolem form ( x)p F ( x). By Theorem 3, for all n N + : = >1 1/n m P F ( t i ) for some t 1,..., t m U(P F ) For j = 1... m, let Pj F be P F with each Skolem function f replaced with distinguished new functions f j. Then, by some arithmetical reasoning, for all n N + : m = (P1 F (P2 F... Pn F ))( t i ) for some t 1,..., t m U(P F ) Moreover, P1 F (P2 F... Pn F ) is -equivalent to the Skolem form of a prenex form of A A n 1. Hence by the previous proposition G A A n 1. Corollary 3. = A iff A is derivable in G extended with the rule: [A] n for all n N + A Example 3. Consider our earlier problematic formula ( x)( y)(p(x) p(y)). Proving this in the above system, requires an infinite number of derivations in G. E.g in the case where n = 2, we have: ( ) p(b) ( ) p(c) p(b) p(b) (id) (mix), p(b) p(b), p(c) ( ) p(b) p(c) ( ) p(b), p(b) p(c), p(a) p(b), p(b) p(c) (wl) ( ) p(a) p(b), p(b) p(c) p(a) p(b), ( y)(p(b) p(y)) ( ) p(a) p(b), ( x)( y)(p(x) p(y)) ( ) ( y)(p(a) p(y)), ( x)( y)(p(x) p(y)) ( ) ( x)( y)(p(x) p(y)), ( x)( y)(p(x) p(y)) ( ) 8 The One-Variable Fragment Although infinitary rules are necessary to provide a calculus for the whole of, we may be able to do better in the case of particular fragments. Recall from Section 3 that the one-variable function-free fragment is decidable. We can obtain a cut-free hypersequent calculus for this fragment by liberalising the eigenvariable condition for the rules ( ) and ( ) of G. The idea is to allow for quantifier shifts that could be performed lower down in the proof.

17 Definition 3. Let G 1 be G with the eigenvariable condition in Definition 2 changed to a is either new or removed by ( ) or ( ) at a lower point in the proof. The eigenvariable condition given here is global in the sense that it applies to whole proofs: the rules are not sound in isolation, but only as part of particular derivations. Example 4. Consider the following proof of ( x)(p(x) ( x)p(x)): p(a) p(a) (id) p(a) ( x)p(x) ( ) p(a) ( x)p(x) ( )1 ( x)(p(x) ( x)p(x)) ( ) The introduction of a in the application of ( ) is justified by the fact that a is removed by ( ) two lines further down in the proof. In fact, the subproof ending with p(a) ( x)p(x) is not allowed in isolation: rightly so, since this sequent is not -valid. Theorem 11. Let A be a one-variable function-free formula. Then G 1 A iff = A. Proof. We first define translations A + and A for any one-variable function-free formula A, assuming harmlessly that the ith occurrence of a quantifier Q in A is annotated as Q i and each a i is a free variable not occurring in A: p( x) + = p( x) + = (B C) + = B C + (( i x)b(x)) + = B(a i ) + (( i x)b(x)) + = ( x)b(x) + p( x) = p( x) = (B C) = B + C (( x) i B(x)) = ( x)b(x) (( x) i B(x)) = B(a i ) + It follows easily using admissible quantifier equivalences for that = A iff = A +. Suppose now for the left-to-right direction that G 1 A. Then G A +. We just remove applications of ( ) and ( ) from the G 1 -derivation and replace each occurrence of x bound by a quantifier i or i removed in A + by a i. Hence, by the soundness of G, = A + and therefore also = A. For the right-to-left direction, suppose that = A. Let the prenex form of A + be ( x)p ( x). Then = ( x)p ( x) and by the approximate Herbrand theorem: = P 1 ( t 1 )... P n ( t n ) for some t 1,..., t n U(P ) Hence, by the propositional completeness of G : G P 1 ( t 1 )... P n ( t n ) To prove A we apply (ec) upwards n times to obtain A... A. We then mimic the proof of P 1 ( t 1 )... P n ( t n ) making sure that we choose the matching constants and variables when we encounter occurrences of and. For ( ) and ( ), this is fine since we can choose terms as we like. The only problem that can occur is in the rules ( ) and ( ) but the relaxed eigenvariable condition takes care of this: any variable that we need is either new or removed further down the proof by ( ) or ( ). We remark finally that this approach works also to obtain Gentzen systems decision procedures even for other fragments such as the -function-free formulas of.

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