th International Symposium on Multiple-Valued Logic. Proceedings. (Fukuoka, Japan, May 1998), pp. 108–113. IEEE Computer Society Press, Los Alamitos, 1998. Compact Propositional Gödel Logics Matthias Baaz Technische Universität Wien Institut für Algebra und Diskrete Mathematik E118.2 A–1040 Vienna, Austria baaz@logic.at Richard Zach∗ Technische Universität Wien Institut für Computersprachen E185.2 A–1040 Vienna, Austria zach@logic.at Abstract Entailment in propositional Gödel logics can be defined in a natural way. While all infinite sets of truth values yield the same sets of tautologies, the entailment relations differ. It is shown that there is a rich structure of infinite-valued Gödel logics, only one of which is compact. It is also shown that the compact infinite-valued Gödel logic is the only one which interpolates, and the only one with an r.e. entailment relation. 1 Introduction Traditionally, the study of many-valued logics has identified logics with their sets of tautologies and centered on their study as such. With the notable exception of Kleene logic, which lacks tautologies altogether, the study of the entailment relations of many-valued logics has taken the back seat. It has, however, become increasingly obvious that such a study is called for, especially in cases where many-valued logics are applied in computer science to reasoning about various domains. For instance, Avron [1] has argued that Gödel logics are suited to formalize properties of concurrency and advocated a view of logics primarily as entailment relations. The present paper aims to take a first step in the investigation of the structure and properties of entailment in propositional Gödel logics. Unless otherwise noted, we will consider a logic as given by an entailment relation, not as a set of tautologies. Gödel logics were introduced by Kurt Gödel [5] and also extensively studied, as sets of tautologies, by Dummett [4]. The set of truth values can always be taken to be a subset of the real interval [0,1], containing 0 and 1 and closed under greatest lower bound. 1 is the designated truth value. The language we consider consist of a denumerably infinite set ∗Research supported by FWF Grant P–10282 MAT of variables var, the logical constants> and⊥, and connectives ∧, ∨,→. A valuation v on a set of truth values V is a function mapping var to V . The valuation is extended in the standard way to formulas by: v(>) = 1 v(⊥) = 0 v(φ∧ψ) = min(v(φ),v(ψ)) v(φ∨ψ) = max(v(φ),v(ψ)) v(φ→ ψ) = { 1 if v(φ)≤ v(ψ) v(ψ) otherwise. Negation is defined by ¬φ = φ→ ⊥; equivalence by φ ≡ ψ = (φ→ ψ)∧ (ψ→ φ). An entailment pair is a pair Π φ, where Π is a set of formulas. We extend the definition of valuations to entailment pairs by v(Π  φ) = { 1 if inf{v(ψ) : ψ ∈Π} ≤ v(φ) v(φ) otherwise for Π any set of formulas. It is easily seen that v(φ) = v( /0  φ) if we take the inf over the empty set to be 1. We may thus define: V φ and ΠV φ iff v(φ) = 1 and v(Π φ) = 1 for all valuations v over a given set of truth values V , respectively. Furthermore, we define: Taut(V ) = {φ : V φ} Ent(V ) = {〈Π,φ〉 : Π V φ} It is well known that Taut(V ) only depends on the cardinality of V , and in particular that Taut(V ) = Taut(V ′) whenever V and V ′ are both infinite. However, in general Ent(V ) and Ent(V ′) will differ even if V and V ′ have the same infinite cardinality but differ in order type. 1.1. DEFINITION Let Vk = {1−1/n : 1≤ k−1}∪{0} 108 V↑ = {1−1/n : n ∈ ω−{0}}∪{1} V↓ = {1/n : n ∈ ω−{0}}∪{0} V∞ = [0,1] If V is a truth value set, GV is the Gödel logic corresponding to that set, i.e., GV = Ent(V ). In particular, Gn, G↓, G↑, and G∞ stand for Ent(Vn), Ent(V↓), Ent(V↑), and Ent(V∞), respectively. Note that v((X → Z)→ Z) = { 1 if v(X)> v(Z) v(Z) otherwise. It is then easily seen that Π1 V↓ Y , Π2 V↑ Y but Π1 2V↑ Y , 1 Π2 2V↓ Y and also Π1 2V∞ Y and Π2 2V∞ Y , where Let Π1 = {(X2→ X1)→ X1,(X3→ X2)→ X2, . . . X1→ Y,X2→ Y, . . .} Π2 = {(X1→ X2)→ X2,(X2→ X3)→ X3, . . . X1→ Y,X2→ Y, . . .} Thus G↓ 6= G↑ 6= G∞ 6= G↓, even though all three truth value sets are infinite and ∣∣V↓∣∣= ∣∣V↑∣∣. Our aim shall thus be to establish the inclusion relationships of the various Gödel logics and in particular to characterize those which have a compact entailment relation. Entailment is compact iff, as usual, Π φ implies that for some finite Π′ ⊆ Π, Π′  φ. As corollaries, we will obtain characterizations of the logics with r.e. entailment relations and those with interpolating entailment. 2 Structure of entailment relations The following proposition will be used frequently in what follows. 2.1. PROPOSITION Let w ∈ V for V a set of truth values, and v a valuation on V . Define vw(X) = { v(X) if v(X)≤ w 1 otherwise. Then vw(ψ) = { v(ψ) if v(ψ)≤ w 1 otherwise for all ψ. Proof. By induction on the complexity of ψ. If ψ a variable, the claim is true by definition of vw. If ψ = ψ′ ∧ψ′′, and v(ψ′)≤ v(ψ′′), then vw(ψ) = v(ψ′) = v(ψ) if v(ψ′)≤w or vw(ψ) = 1 = v(ψ) if v(ψ′) > w. Similarly if ψ is a disjunction. If ψ = ψ′ → ψ′′, we distinguish three cases. If v(ψ) = w = 1, then v(ψ′) ≤ v(ψ′′) ≤ w. By induction hypothesis, vw(ψ′) = v(ψ′) and vw(ψ′′) = v(ψ′′), so vw(ψ) = 1 = v(ψ). Now suppose that v(ψ)≤w< 1. Then v(ψ′′)≤w and by induction hypothesis, vw(ψ′′) = v(ψ′′). Since v′(ψ′)≥ v(ψ′), we have vw(ψ) = v(ψ). Finally, suppose that v(ψ) > w. Then either v(ψ′) > v(ψ′′) = v(ψ) > w, or v(ψ′) ≤ v(ψ′′). In the former case, vw(ψ′) = vw(ψ′′) = 1, and so vw(ψ) = 1. In the latter case, again, we have two cases: (1) v(ψ′) ≤ w. Then vw(ψ′) = v(ψ′) ≤ vw(ψ′′). (2) v(ψ′) > w, and so vw(ψ′) = vw(ψ′′) = 1. In either case, vw(ψ) = 1.2  One might wonder whether a different definition of the entailment relation in Gödel logic might give different results. Another standard way of defining entailment in manyvalued logics is: Π φ iff for all v, (∀ψ ∈Π)(v(ψ) = 1)⇒ v(φ) = 1 This definition yields the same results, as the following proposition shows: 2.2. PROPOSITION Π V φ iff Π V φ Proof. Only if: Immediate. If: Suppose v witnesses Π 2 φ, i.e., v(Π  φ) = w < 1. Then, by Lemma 2.1, vw(ψ) = 1 for all ψ ∈Π and vw(φ) = w< 1.  This allows us to use the characterization of  or as proves convenient. It is an easy but fundamental result that Taut(V ) and Ent(V ) depend only on the order type of V . This is made precise in the following 2.3. PROPOSITION Let v and v′ be valuations, not necessarily on the same sets of truth values, such that v(X) = 1 iff v′(X) = 1, v(X)< v(Y ) iff v′(X)< v′(Y ), and v(X) = v(Y ) iff v′(X) = v′(Y ) (for all X , Y ). Then v(φ) = 1 iff v′(φ) = 1 and v(φ) = v(X) iff v′(φ) = v′(X).3 2.4. PROPOSITION (1) If |var(φ)| ≤ n and Vn+2 φ, then V φ for all V with |V | ≥ n + 2. (2) Taut(V )⊆ Taut(V ′) for |V | ≤ |V ′|.4 Proof. This follows immediately from Proposition 2.3. The bound in (1) is tight, as the example of X ∨¬X shows for n = 1.  2.5. COROLLARY Taut(V∞) = ⋂ n∈ω Taut(Vn) 2.6. PROPOSITION For every φ there is a normal form N(φ) such that V∞ φ≡ N(φ). Furthermore, if for every n, the set {N(φ) : var(φ) = {X1, . . . ,Xn}} is finite and depends only on n. 109 Proof. The idea of the proof is easy: By Theorem 2.3, only the order of the variables induced by a given valuation is relevant to determine the truth value of a formula. Such an ordering can be expressed by a formula of the language, using (X → Y )∧ ((Y → X)→ X) for: v(X) < v(Y ) or v(X) = v(Y ) = 1, and X ≡Y for v(X) = v(Y ). Every such formula implies that φ is equivalent to a variable in var(φ), 1, or 0. Form the disjunction over all such implications. For a detailed proof, see [3], Theorem 5.  Gödel logics were invented as a tool to study propositional intuitionistic logic. Consequently it is not surprising that there is a tight connection between Kripke semantics and Gödel logics. It is well known that Taut(V∞) equals the set of formulas valid in all linearly ordered Kripke structures. The connection extends to entailment; the truth value set corresponding to such structures with respect to entailment is V↓. 2.7. DEFINITION A linear Kripke structure k is a function from var to {0}ω ∪ {0}<ω _ {1}ω (i.e., 0–1 sequences which, once 1, remain 1). We extend k to formulas by: k(>)i = 1 k(⊥)i = 0 k(φ∧ψ)i = { 1 if k(φ)i = k(ψ)i = 1 0 otherwise k(φ∨ψ)i = { 0 if k(φ)i = k(ψ)i = 0 1 otherwise k(φ→ ψ)i = { 1 if, for all j ≥ i, k(φ) j = 0 or k(ψ) j = 1 0 otherwise. The logic of linear intuitionistic Kripke structures LI is given by Π φ iff for all k, (∀ψ ∈Π)(k(ψ)0 = 1)⇒ k(φ)0 = 1 2.8. PROPOSITION G↓ = LI Proof. Let k be a Kripke structure and v a V↓-valuation. Then k(X) = { 0 if k(X) = {0}ω 1/(n + 1) if n is least s.t. k(X)n = 1 is a V↓-valuation, and v(X)i = { 0 if v(X)< 1/(i + 1) 1 otherwise defines a Kripke structure. It is easily shown, by induction on formulas, that  v = v and k = k. Consequently, k(φ)0 = 1 iff k(φ) = 1 and v(φ) = 1 iff v(φ)0 = 1. Using the characterization of  in terms of of Proposition 2.2, the result follows.  An analogous result holds for first-order Gödel and linear intuitionistic logic [2]. 2.9. LEMMA Suppose v(Π  φ) = w. Then vw(Π  φ) = w. Proof. Suppose, first, that v(Π)> w. Then vw(ψ) = 1 for all ψ ∈ Π, and vw(φ) = v(φ) = w. Otherwise, w = 1 and v(ψ) ≤ w for some ψ ∈ Π and so vw(Π) ≤ w. In this case, v(Π  φ) = vw(Π  φ) = 1.  2.10. PROPOSITION G↑ = ⋂ n∈ω Gn. Proof. Every Vn valuation is also a V↑ valuation, so Ent(Vn) ⊆ Ent(V↑). On the other hand, if v(Π  φ) = 1− 1/n = w, then vw as defined in Lemma 2.1 is a Vn+1valuation with vw(Π  φ) = w, and so Π 2Vn+1 φ. 5  2.11. PROPOSITION Let V be any truth value set. Then Π 2V∞ φ if Π 2V φ. 2.12. COROLLARY Ent(V∞) = ⋂ V ⊆ [0,1] V countable Ent(V ) 2.13. DEFINITION We define the following families of truth value sets: Uk = {i/(k + 1)+ 1/ j(k + 1) : 0≤ i≤ k−1; j > 0}6 Dk = {i/(k + 1)−1/ j(k + 1) : i = 1, . . . ,k; j > 0}∪ ∪{i/(k + 1); i = 0, . . . ,k + 1} Correspondingly, we define sets of formulas Γk = {(Ci+1→Ci)→Ci, Ci→Ck : i = 0, . . . ,k−1} Υ` = {(X `i+1→ X `i )→ X `i ,C`→ X `i , X `i →C`+1 : i ∈ ω} ∆` = {(X `i → X `i+1)→ X `i+1,C`→ X `i , X `i →C`+1 : i ∈ ω} Υk = Γk ∪ k−1⋃ `=0 Υ` ∆k = Γk ∪ k−1⋃ `=0 ∆` Uk has the order type 1 + kω∗, while Dk has order type kω + 2. In particular, U1 = V↓.7 Uk 0 ** * * * *} {{ } 1k+1 ** * * * *} {{ } 2k+1 * * * kk+1 1 C0 X0i C1 X 1 i C2 . . . Ck Dk 0 { }} { * * * * ** 1k+1 { }} { * * * * ** 2k+1 * * * k k+1 1 2.14. PROPOSITION (1) ∆r Dk Cr for all r. 110 (2) ∆r 2Uk Cr for r ≤ k. (3) ∆r Uk Cr for all r > k. (4) Υr Uk Cr for all r. (5) Υr 2Dk Cr for r ≤ k. (6) Υr Dk Cr for all r > k. (7) If Γ 2Dk φ then Γ 2Dr φ for all r ≥ k. (8) If Γ 2Uk φ then Γ 2Ur φ for all r ≥ k.8 We thus have the following picture: G2 )G3 ) . . .) ⋂ Gi = )Ent(U1) = G↓)Ent(U2)) . . . = G↑ )( )( )G∞ ) Ent(D1) )Ent(D2)) . . . The hierarchy result just described shows that there are at least א0 many different Gödel logics between G↑ and G∞. This suggests the question: 2.15. PROBLEM Are there even 2א0 or 22 א0 logics below G∞? We will show in the next section that an infinite-valued Gödel logic is compact iff its set of truth values contains a densely ordered subset. It should be pointed out right here that almost all infinite-valued logics are not compact. In fact, there is only one compact infinite-valued Gödel logic, namely G∞, as Proposition 3.7 will show. 3 Classification of compact Gödel logics We now turn to the characterization of those Gödel logics whose entailment relations are compact, as defined by the following 3.1. DEFINITION GV is compact if, whenever Π V φ there is a finite Π′ ⊆Π so that Π′ V φ. 3.2. PROPOSITION GV is compact if V is finite. Proof. Let Π = {ψ1,ψ2, . . .}, and let X = {X0, X1, . . .}, be an enumeration of variables occuring in Π, ψ0 such that all variables in ψi occur before the variables in ψi+1. We show that either {ψ1, . . . ,ψk}  ψ0 or Ψ 2 ψ0. Let T be the complete semantic tree on X , i.e., T = V<ω. An element of T of length k is a valuation of X0, . . . , Xk−1. Since V is finite, T is finitary. Let T ′ be the subtree of T defined by: v∈ T ′ if for every initial segment v′ of v and every k such that all the variables in ψ0, . . . ,ψk are among X0, . . . , X`(v′), v ′({ψ1, . . . ,ψk}) > v′(ψ0). In other words, branches in T ′ terminate at nodes v′ where v′({ψ1, . . . ,ψk}) ≤ ψ0. Now if T ′ is finite, there is a k such that ψ1, . . . ,ψk V ψ0. Otherwise, since T ′ is finitary, it contains an infinite branch. Let v be the limit of the partial valuations in that branch. Obviously, since V is finite, v(Π)> v(ψ0) and so Π 2V ψ0.  3.3. PROPOSITION Suppose Π contains only finitely many variables. Then Π′ V φ, for some finite Π′ ⊆ Π, provided Π V φ. Proof. By Proposition 2.6, there are only finitely many non-equivalent formulas on k variables. Choose a representative from each equivalence class to obtain Π′.  3.4. THEOREM Suppose V ⊇ W with W densely ordered and |W | ≥ 2. Then GV is compact. Proof. Let X be a set of variables. A chain on X is an arrangement of X in a linear order. Formally, a chain C on X is a sequence of pairs 〈Xi,oi〉where oi ∈ {<,=,>}where Xi appears exactly once. A valuation v respects C if v(Xi) = v(Xi+1) if oi is =, v(Xi) > v(Xi+1) if oi is >, and v(Xi) < v(Xi+1) if oi is <. If X is finite, there are only finitely many chains on X . We construct a tree in stages as follows: The initial node is labeled by 0 < 1 and an empty valuation. Stage n + 1: A node N constructed in stage n is labeled by a chain on the variables X1, . . . , Xn and a valuation vN of X1, . . . , Xn respecting the chain. N receives successor nodes, one for each possibility of extending the chain by inserting Xn+1. The labels of each successor node N′ are the corresponding extended chain and an extension of vN which respects the extended chain. The value vN′(Xn+1) is chosen inside W , i.e., the endpoints of W may not be chosen as values. Since W is densely ordered, this ensures that such a choice can be made at every stage. We call a branch of T closed at node N (constructed at stage n) if for some finite Π′ ⊆ Π such that var(Π′)∪ var(φ) ⊆ {X1, . . . ,Xn} it holds that vN(Π′) ≤ vN(φ). T is closed if it is closed on every branch. In that case, for some finite Π′ ⊆Π, we have Π′  φ. If T is not closed, it contains an infinite branch. Let v be the limit of the vN of nodes N on the infinite branch. It holds that v(ψ) > v(φ) for all ψ ∈ Π, for otherwise the branch would be closed at the first stage where all the variables in ψ were assigned values. Let w = v(φ). By Proposition 2.1, vw(φ) = v(φ) and vw(Π) = 1, and so Π 2 φ, a contradiction.  3.5. THEOREM Suppose V does not contain a densely ordered subset. Then GV is not compact. 111 Proof. We define a sequence of sets of variables Xk and formulas Γk as follows: Xk = {Xr : r = i/2k,0≤ i≤ 2k} Γ′k = {(Xr→ Xs)→ Xs,(Xs→ Xt)→ Xt : t = (i−1)/2k,s = i/2k,r = (i + 1)/2k, 1≤ i≤ 2k−1,2(k−1)|i} Γ′′k = {Xr→ Z : r = i/2k,0≤ i≤ 2k} Γk = Γ′k ∪Γ′′k Γ = ⋃ k∈ω Γk Intuitively, Γ′k expresses that the Xi/2k for 0 ≤ i ≤ 2k are linearly ordered so that Xi/2k < X(i+1)/2k . ⋃ k∈ω Γ′k expresses that the variables Xr are densely ordered: v(Xi/2k )< v(X(2i+1)/2k+1) < v(X(i+1)/2k ). Now if V has at least 2 k + 2 truth values v0 < * * *< v2k+1 < 1, the valuation v v(Xi/2k ) = vi v(Z) = v2k+1 will have v(Γk) = 1 and v(Z)< 1, so Γk 2V Z. On the other hand, if V does not contain a densely ordered subset, then Γ V Z. In fact the only v such that v(Γ) = 1 is v(Xr) = 1 for all r, and v(Z) = 1.  We have thus succeeded in characterizing the compact propositional Gödel logics. They are all those where the set of truth values V is either finite or contains a nontrivial densely ordered subset. 3.6. THEOREM The compact Gödel logics are exactly those given by the truth-value sets Gn (n≥ 2) and G∞. This follows from the next proposition, together with the fact that all infinite-valued Gödel logics have the same tautologies. 3.7. PROPOSITION Let 1 and 2 be the entailment relations of two compact logics, each satisfying modus ponens, the deduction theorem, and having the same tautologies. Then 1 = 2. Proof. Suppose Π 1 φ. Then for some {ψ1, . . . ,ψn} 1 φ. By the deduction theorem for 1, we have 1 ψ1→***→ ψn → φ. Since 1 φ iff 2 φ, 2 ψ1 → *** → ψn → φ, and by modus ponens, Π 2 φ.  3.8. DEFINITION A logic GV is model compact if, for any Π, Π has a model provided all finite subsets Π′ ⊆ Π have models. The notion of model compactness coincides with compactness in the case of classical logic, but not in the general case of infinite-valued Gödel logics. The following theorem is due to Petr Hájek: 3.9. THEOREM All Gödel logics are model compact. Proof. Observe that for any valuation v and any formula φ, if v(φ) = 1 then also v0(φ) = 1 (see Proposition 2.1). Hence, if v is a model for Π in any logic, v0 is a model for Π in G2. Hence, Π has a model in GV iff it has a model in G2. Since G2 is model compact, so is GV for any V .  4 Interpolation in Gödel logics Interpolation is a property usually defined for logics as considered as a set of sentences. A logic L weakly interpolates if, whenever L  φ→ ψ, there is a sentence σ so that var(σ) ⊆ var(φ)∩ var(ψ) and L  φ→ σ and L  σ→ ψ. The sentence σ is called sn interpolant of φ and ψ. We may extend this defnition to entailment relations as follows: 4.1. DEFINITION Gödel logic GV interpolates if, for all Π1, Π2 and φ, such that Π1∪Π2 V φ, there is σ with var(σ)⊆ var(Π1)∩var(Π2∪{φ}) and Π1 V σ and Π2∪{σ} V φ. Since ψ1, . . . ,ψn V φ iff V ψ1 → ** * → ψn → φ, GV interpolates weakly iff it interpolates with respect to finite sets. The following observation is immediate: 4.2. PROPOSITION If GV is compact, then GV interpolates iff it weakly interpolates. Our aim in this section is to show that weak interpolation and interpolation fall apart when V is infinite: GV weakly interpolates if V is infinite, but only interpolates when GV is compact as well. 4.3. THEOREM GV weakly interpolates iff |V |= 2 or |V |= 3 or V is infinite. Proof. See [6, 3]  4.4. PROPOSITION Let V be infinite but not contain a nontrivial densely ordered subset. Then GV does not interpolate. Proof. Consider the set Γ from the proof of Theorem 3.5. We know that Γ V Z. Suppose there was an interpolant σ so that Γ′ V σ and Γ′′∪{σ} V Z. Let Γ′ = {ψ ∈ Γ′ : var(ψ)⊆ var(σ)} Γ′′ = {ψ ∈ Γ′′ : var(ψ)⊆ var(σ)∪{Z}} 112 Then Γ′ V σ and Γ′′ ∪ {σ} V Z. To see this, suppose v(Γ′) = 1 and v(σ)< 1. Then v′ defined by v′(X) = { v(X) if X ∈ var(σ) 1 otherwise would evaluate all formulas in Γ′− Γ′ to 1, contradicting Γ′ V σ. A similar contradiction follows for the second part of the claim by considering v′′(X) = { v(X) if X ∈ var(σ)∪{Z} 0 otherwise. Of course, Γ′ and Γ′′ are both finite. By the cut rule for V it follows that Γ′ ∪ Γ′′ V Z. This contradicts the proof of Theorem 3.5.  Note that the proof actually establishes a stronger result: The definition of an interpolant requires the variables in σ to be contained in the intersection of var(Γ′) and var(Γ′ ∪ {Z}). We have shown that no formula whatsoever, i.e., not even one that does not satisfy that condition, can serve as an interpolating formula. 4.5. THEOREM GV interpolates iff |V | = 2 or |V | = 3 or V contains a densely ordered subset. That is, the only interpolating Gödel logics are G2, G3, and G∞. Proof. By Theorem 4.3 and Propositions 4.2 and 4.4.  It is not necessary for a logic to be compact in order to interpolate. For instance, consider the many-valued logic on V↓ with constants >, ⊥ for 1 and 0, respectively, and the operator ♣ given by: ♣(1/n) = 1/(n + 1) and ♣(0) = 0. Π  φ iff one of the following hold: (1) ♣k(X) ∈Π with k unbounded, (2) ⊥ ∈Π, (3) φ =♣`(X) and ♣k(X) ∈Π for some k ≥ `, (4) φ = ♣`(>) and ♣k(X) ∈ Π or ♣k(>) ∈ Π for some k ≥ `. In each case, an interpolant can easily be found for any partition of Π. However, the logic is not compact, as the example {♣k(>) : k ∈ ω} ⊥ illustrates. 5 R.e. entailment relations We conclude with a somewhat curious result. Propositional logics are considered "easy" in the sense that the validity problem of most is decidable, usually within reasonable bounds, in contrast to first-order logic. This situation changes drastically if entailment is considered instead. The following proposition shows that the question of whether a recursive set of formulas entails another formula is highly undecidable for non-compact Gödel logics. This potentially has serious consequences for the implementation and application of Gödel logics in inference mechanisms. 5.1. PROPOSITION The set E = {〈e,pφq〉 : {ψ : {e}(pψq) = 0} V φ} is r.e. iff GV is compact. Hence, the only r.e. Gödel logics are Gn and G∞. Proof. If: We can enumerate E by enumerating finite subsets of Π = {ψ : {e}(pψq) = 0} and testing for implication. If Π V φ, this search terminates eventually since GV is compact. Only if: Suppose GV is not compact. Let f (e) be the index of the predicate defined by { f (e)}(pψq) = 0 iff ψ ∈ Γi and (∀ j ≤ i)({e}( j) = 0) where Γi is as in the proof of Theorem 3.5. Then 〈 f (e),pZq〉 ∈ E iff {e} is total and constant equal to 0. That problem, however, is Π02-complete [7, Theorem IV.3.2].  Acknowledgement. The authors would like to thank Petr Hájek and Jonathan Lebowitsch for helpful comments. References [1] A. Avron. Hypersequents, logical consequence and intermediate logics for concurrency. Ann. Math. Artificial Intelligence, 4:225–248, 1991. [2] M. Baaz, A. Leitsch, and R. Zach. Incompleteness of an infinite-valued first-order Gödel logic and of some temporal logics of programs. In E. Börger, et al., eds., Computer Science Logic. Selected Papers from CSL'95, pages 1–15, Berlin, 1996. Springer. [3] M. Baaz and H. Veith. Interpolation in fuzzy logic. Technical Report DBAI-TR-97-9, Technische Universität Wien, Vienna, Austria, 1997. [4] M. Dummett. A propositional calculus with denumerable matrix. J. Symbolic Logic, 24:97–106, 1959. [5] K. Gödel. Zum intuitionistischen Aussagenkalkül. Anz. Akad. Wiss. Wien, 69:65–66, 1932. [6] L. Maksimova. Craig's interpolation theorem and amalgamable varieties. Dokl. Akad. Nauk SSSR, 237(6):1281–1284, 1977. [7] R. I. Soare. Recursively enumerable sets and degrees. Springer, Berlin, 1987. 113 Errata We would like to thank Petr Cintula for pointing out a number of misprints and possible improvements in the text. These reached us only after the proceedings volume went to print; they are added in this version as notes. 1. This should read Π1 V↑ Y , not Π1 2V↑ Y . 2. Petr Cintula has communicated the following simpler argument for the induction step: If w = 1 the claim is trivial. So suppose w < 1. (1) v(φ) ≤ v(ψ). Then vw(φ) ≤ vw(ψ) and so vw(φ → ψ) = 1, while v(φ → ψ) = 1 > w. (2) w < v(ψ) < v(φ). Then vw(φ→ ψ) = 1, whereas v(φ→ ψ) = v(ψ) > w. (3) v(ψ) ≤ w and v(ψ) < v(φ). In this case, vw(ψ) = v(ψ)< v(φ)≤ vw(φ), so vw(φ→ψ) = vw(ψ), while v(φ→ ψ) = v(ψ)≤ w. 3. Petr Cintula points out that Ent(V ) ⊆ Ent(V ′) if there is an injection f :V ′ → V which preserves strict order, i.e., f (v) < f (v′) iff v < v′. This follows quite easily from Proposition 2.3: If v is a valuation on V ′ and v(Π  φ)< 1, then f v is a valuation on V (in fact, on f (V ′) ⊆ V ), and f v(Π  φ) < 1. As an immediate corollary we obtain the fact stated just before Proposition 2.3, i.e., that Ent(V ) = Ent(V ′) if V and V ′ are order-isomorphic. 4. This should read |V | ≥ |V ′|, not |V | ≤ |V ′|. 5. Note that we defined Vn = {1−1/k : k ≤ n−1}∪{1}. However, using the consideration of note 3, the result holds for any sequence V ∗n with |V ∗n | = n. In particular, if v(Π  φ) = 1− 1/n = w, then there is an order isomorphism f between {v ∈ V↑ : v ≤ w} ∪ {1} and V ∗n+1 and so f v(Π  φ) = f (w)< 1. 6. The definition of Uk should read Uk = {i/(k + 1)+ 1/ j(k + 1) : 0≤ i≤ k−1; j > 0}∪ ∪{i/(k + 1);0≤ i≤ k + 1} 7. This is not strictly true, however, U1 and V↓ are order isomorphic and thus Ent(U1) = Ent(V↓) by note 3. 8. (7) and (8) follow from note 3.