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Note on witnessed Gödel logics with Delta
Annals of Pure and Applied Logic 161 (2009) 121–127
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Note on witnessed Gödel logics with Delta Matthias Baaz ∗ , Oliver Fasching Vienna University of Technology, Wiedner Hauptstraße 8–10/E104, A-1040 Vienna, Austria
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Article history: Available online 1 July 2009 MSC: 03B52 03B52 03C80
abstract Witnessed Gödel logics are based on the interpretation of ∀ (∃) by minimum (maximum) instead of supremum (infimum). Witnessed Gödel logics appear for many practical purposes more suited than usual Gödel logics as the occurrence of proper infima/suprema is practically irrelevant. In this note we characterize witnessed Gödel logics with absoluteness operator 4 w.r.t. witnessed Gödel logics using a uniform translation. © 2009 Elsevier B.V. All rights reserved.
Keywords: Gödel logics Disentangled fragment Relative completeness
1. Introduction First-order Gödel logics are among the most prominent intermediate logics. They are defined using closed subsets of the [0, 1]-interval containing 0 and 1, where 1 represents truth. Alternatively, they are logics of countable linearly ordered Kripke semantics with constant domain; cf. [6]. The dominant connective of Gödel logics is ⊃, which is projective. All other connectives and quantifiers are evaluated in a natural way according to the linear structure of the truth values. There is an asymmetry in Gödel logics concerning the status of true and false. It is possible to separate false from not false while it is impossible to separate true from not true. For this reason an absoluteness operator 4 is introduced, which is prominent in many non-classical logics. The 4-operator goes back to [11] and was studied in [1,12]. In a proof theoretic sense, quantified formulas are partial descriptions of infimum/supremum expressions associated with uniform provability. Universal/existential quantifiers are induced by iterated conjunction/disjunction. It is therefore natural to include the property of realization by a constituent in the description of quantifiers. These quantifiers are called witnessed quantifiers. In this paper we characterize the relation of Gödel logics with/without absoluteness operator to witnessed Gödel logics. Logics with linearly ordered truth values are especially suited for an investigation of witnessed quantifiers because of the obvious infimum/minimum (supremum/maximum) relation. This note can be therefore considered as a first step towards a general theory of witnessed quantifiers. 2. Language and semantics We will consider first-order languages with free and bound variables, predicate and function symbols, logical connectives
∨, ∧, ⊃, ⊥, and quantifiers ∀, ∃. By L4 we denote the extension of a language L by a unary operator 4. We understand ¬A as an abbreviation for A ⊃ ⊥, A ↔ B for (A ⊃ B) ∧ (B ⊃ A), and A ≺ B for (B ⊃ A) ⊃ B. Terms and formulas are defined in the usual way.
∗
Corresponding author. E-mail address: [email protected] (M. Baaz).
0168-0072/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.apal.2009.05.011
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M. Baaz, O. Fasching / Annals of Pure and Applied Logic 161 (2009) 121–127
Definition 1. Suppose we are given a truth value set V , i.e. a closed set V with {0, 1} ⊆ V ⊆ [0, 1]. A V -interpretation I (of L and L4 ) consists of (1) a nonempty set |I|, the domain of I, (2) a function P I : |I|n → V for each n-ary predicate symbol P, (3) a function f I : |I|n → |I| for each n-ary function symbol f , and (4) a value aI ∈ V for each free variable a. Definition 2. Let I be a V -interpretation of L. We denote by LI the extension of L by constant symbols for all elements of |I| and we define c I := c for all c ∈ |I|. We extend the interpretation to terms and atoms by putting I(⊥) := 0, I(f (u1 , . . . , un )) := f I (I(u1 ), . . . , I(un )) and I(P (t1 , . . . , tn )) := P I (I(t1 ), . . . , I(tn )). For composite formulas we define:
I(A ∧ B) := min(I(A), I(B)), I(A ∨ B) := max(I(A), I(B)),
I(A ⊃ B) :=
if I(A) ≤ I(B) I(B) if I(A) > I(B), 1
I(∀x A(x)) := inf{I(A(u)); u ∈ |I|}, I(∃x A(x)) := sup{I(A(u)); u ∈ |I|}. For L4 we additionally define:
I(4A) :=
1
if I(A) = 1
0
if I(A) < 1.
Note that
I(¬A) =
1
if I(A) = 0
0
if I(A) > 0,
I(A ↔ B) = I(A ≺ B) =
1
if I(A) = I(B)
min{I(A), I(B)}
if I(A) 6= I(B),
if I(A) < I(B) or I(A) = I(B) = 1 I(B) else.
1
Definition 3. If I(A) = 1, we write I |= A and say I satisfies A. For any truth value set V we define the (non4 witnessed) Gödel logic GV (resp. GV ) based on V as the set of all formulas A in L (resp. L4 ) such that I |= A for all V -interpretations I. Witnessed interpretations are interpretations fulfilling I(∀xA(x)) = min{I(A(u)); u ∈ |I|} and 4 I(∃xA(x)) = max{I(A(u)); u ∈ |I|}. For any truth value set V we define the witnessed Gödel logic GWV (resp. GWV ) based on V as the set of all formulas A in L (resp. L4 ) such that I |= A for all witnessed V -interpretations I. For any of the above logics G, we write G A instead of A ∈ G. Definition 4. For a set Γ of formulas, we define an entailment I:Γ A by inf{I(B); B ∈ Γ } ≤ I(A). If I:Γ A holds for all V -interpretations I, we write Γ V A and say that Γ entails A in GV . 3. Witnessed Gödel logics Lemma 1. Let I be a V -interpretation, and let 0 ≤ d ≤ 1. We put h(x) := x for x ≤ d and h(x) := 1 for x > d. Let I0 be the 0 interpretation that arises from I by changing P I to P I (A) := h(P I (A)) for atomic A; in particular, I0 = |I|. (1) Then I0 is a V 0 -interpretation where V 0 := V r (d, 1) ⊆ V , and I0 (B) = h(I(B)) for all ∀-free formulas B. (2) If, moreover, for each F (a) with I(∀xF (x)) = d there exists some u∗ ∈ |I| such that d = I(F (u∗ )), then I0 (B) = h(I(B)) holds for all formulas B. (3) If I is witnessed, so is I0 and I0 (B) = h(I(B)) holds for all formulas B. Proof. Part (1) is Lemma 3.4 in [4]. Part (2). We may assume d < 1, for otherwise I = I0 . The claim is proved by induction on the complexity of the formula B. Let infu abbreviate infu∈|I| in what follows. We need to prove infu I0 (A(u)) = h(I(∀xA(x))) under the induction hypothesis I0 (A(a)) = h(I(A(a))). Since h(I(A(v))) ≥ h(infu I(A(u))) for all v ∈ |I|, we have infu h(I(A(u))) ≥ h(infu I(A(u))), and therefore it suffices to prove infu h(I(A(u))) ≤ h(e), where e := infu I(A(u)). In the case d > e, there is a series (um )m with d > I (A(um )) and I (A(um )) & e. From h(I (A(um ))) = I (A(um )) we get h(I (A(um ))) & e; thus infu h(I (A(u))) ≤ e = h(e), as required. In the case d < e, we have d < I(A(u)) for all u ∈ |I|. Thus h(I(A(u))) = 1 for all u ∈ |I|, and we have 1 = h(e), as required. In the case d = e, we find u∗ ∈ |I| with d = I(A(u∗ )) due to the extra assumption on I. We have infu h(I(A(u))) ≤ h(I(A(u∗ ))) = h(d) = h(e), as required. Part (3). Since I is witnessed, we find u∗ , u∗ ∈ |I| with I(∀xA(x)) = I(A(u∗ )) and I(∃xA(x)) = I(A(u∗ )). Hence, by (2), I0 (B) = h(I(B)) holds for all formulas B. In particular, I0 (∀xA(x)) = h(I(∀xA(x))) = h(I(A(u∗ ))) = I0 (A(u∗ )) and I0 (∃xA(x)) = h(I(∃xA(x))) = h(I(A(u∗ ))) = I0 (A(u∗ )). Therefore I0 is witnessed.
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Remark 1. Since
I(∃x(A(x) ⊃ ∀yA(y))) =
{I(A(u)); u ∈ |I|} attains its minimum I(∀xA(x)) otherwise
1
and
I(∃x(∃yA(y) ⊃ A(x))) =
{I(A(u)); u ∈ |I|} attains its maximum I(∃xA(x)) otherwise,
1
it is easy to see that I(∃x(A(x) ⊃ ∀yA(y))) = 1 holds if and only if {I(A(x)); x ∈ |I|} attains its minimum. In contrast to that, I(∃x(∃yA(y) ⊃ A(x))) = 1 holds if and only if I(∃xA(x)) = 1 or {I(A(x)); x ∈ |I|} attains its maximum. Definition 5. Let W be the set of the formulas ∃x(A(x) ⊃ ∀yA(y)) and ∃x(∃yA(y) ⊃ A(x)) for all formulas A(a) in L, and let HV := {B; W V B}. Theorem 1. We have GWV = HV . Proof. If A ∈ HV , then infB∈W I(B) ≤ I(A) for all V -interpretations I. For all witnessed V -interpretations I we have I(B) = 1 for all B ∈ W ; thus 1 ≤ I(A). Therefore A ∈ GWV . Conversely, if A ∈ / HV , there is a V -interpretation I such that inf{I(B); B ∈ W } > d := I(A). Note that 0 ≤ d < 1. We prove by contradiction that, for any F (a) with I(∀xF (x)) = d, there is x∗ ∈ |I| such that d = I(F (x∗ )): If {I(F (u)); u ∈ |I|} does not attain its infimum, we have I(∃x(F (x) ⊃ ∀yF (y))) = I(∀xF (x)) by Remark 1; but this contradicts I(∀xF (x)) = d < inf{I(B); B ∈ W } ≤ I(∃x(F (x) ⊃ ∀yF (y))). From Lemma 1(2) it follows that there is a V 0 -interpretation I0 , where V 0 := V r (I0 (A), 1) ( V , such that 1 = inf{I0 (B); B ∈ W } > I0 (A). For any formula D(a), define U := {I0 (D(x)); x ∈ I0 }. From 1 = I0 (∃x(D(x) ⊃ ∀yD(y))) it follows that U attains its minimum. From 1 = I0 (∃x(∃yD(y) ⊃ D(x))) it follows that U attains its maximum or sup U = 1; but even in the latter case, U attains its maximum since U ⊆ V 0 and 1 is isolated in V 0 . This means that I0 is witnessed. Since I0 (A) < 1, we have A ∈ / GWV . Corollary 1. GW[0,1] is axiomatized by the axioms and rules of G[0,1] with W as additional axioms. We have GW[0,1] = GWV for all uncountable V . Proof. Clearly G[0,1] ∪ W ⊆ GW[0,1] . Conversely, suppose GW[0,1] |= A so that W |=[0,1] A by Theorem 1. Let H be the proof system of intuitionistic predicate calculus + (A ⊃ B) ∨ (B ⊃ A) + ∀x(A(x) ∨ C ) ⊃ (∀xA(x) ∨ C ), where x does not occur in C . By [9] (also cf. [10,8]), the entailment for G[0,1] is axiomatized by H ; in particular, there is a conjunction W 0 of finitely many formulas in W such that H ` W 0 ⊃ A. Thus H + W ` A. We have GW[0,1] ⊆ GWV for any truth value set V . Let V be uncountable so that there is a perfect kernel V 0 of V . Any witnessed interpretation in [0, 1] can be transformed to a witnessed interpretation in {0} ∪ V 0 ∪ {1}.1 Therefore GWV ⊆ GW[0,1] and thus GWV = GW[0,1] for all uncountable V . Remark 2. GW[0,1] is the only recursively enumerable (r.e.), infinitely valued GWV since the argument for GV also shows that GWV , V countable, is not r.e. (cf. [3]). As all GV are core fuzzy logics in the sense of [7], extensions of ∃x(A(x) ⊃ ∀yA(y)), ∃x(∃yA(y) ⊃ A(x)) provide completeness w.r.t. witnessed models based on L-algebras (cf. Theorem 8 in [7]). Corollary 2. All classically valid quantifier rules are valid in GWV . Proof. The only classically valid quantifier shift rules not always valid in Gödel logics are (∀xA(x) ⊃ B) ⊃ ∃x(A(x) ⊃ B) and (A ⊃ ∃xB(x)) ⊃ ∃x(A ⊃ B(x)) (see [4], Section 3). They can be easily obtained from the valid formulas (A(a) ⊃ ∀xA(x)) ⊃ (∀xA(x) ⊃ B) ⊃ (A(a) ⊃ B) and (∃xA(x) ⊃ A(a)) ⊃ (B ⊃ ∃xA(x)) ⊃ (B ⊃ A(a)) together with the formulas in W . Definition 6. We put V↑ := {1 − 1k ; k ≥ 1} ∪ {1}, H↑ := HV↑ and G↑ := GV↑ .
T
T
Proposition 1. (1) We have V finite GV = V finite GWV = G↑ = GW↑ . (2) GWV = GV holds if and only if V is finite or V is order isomorphic to V↑ . Proof. (1) By Remark 1, we have I(A) = 1 for all V↑ -interpretations I and all A ∈ W . Thus H↑ = G↑ , and now GW↑ = G↑ by Theorem 1. NoteT that GV = GU if V and U are finite and have same cardinality. Clearly GV = GWV for finite V . Thus we T have V finite GV = V finite GWV = G↑ by Proposition 3.1 and Theorem 3.2 in [4]. (2) One direction follows from (1). Conversely, assume GWV = GV . By Corollary 2.26 in [3] we only need to prove that all quantifier shift rules are valid in GV , but this is true due to Corollary 2. 4. Eliminating ∆ in witnessed Gödel logics 4
In this section we characterize GWV w.r.t. GWV .
1 Personal communication with N. Preining; cf. [8].
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Proposition 2. The equivalence theorem holds for GV , GWV in the form
∀x(A(x) ↔ B(x)) ⊃ (C (A(t )) ↔ C (B(t ))) 4
4
and for GV , GWV in the form
4∀x(A(x) ↔ B(x)) ⊃ C (A(t ) ↔ C (B(t ))) where t may contain bound variables. Proof. By induction on the complexity of C . For 4, the proof uses 4A ⊃ A, 4A ⊃ 44A, and A ⇒ 4A.
4
Definition 7. Structural normal form for formulas A in L . Replace stepwise from the inside to the outside all subsemiformulas G(E x), with bound variables E x, by fresh defining predicates FG (E x). Now, struc(A) is
^
4∀Ex(FG (Ex) ↔ G(Ex)) ⊃ FA
G
where the conjunction ranges over all subsemiformulas G of A. 4∀E x(FG (E x) ↔ G(E x)) is the defining equivalence for G. 4
4
4
4
Proposition 3. GV A if and only if GV struc(A). GWV A if and only if GWV struc(A). Proof. Replacing FG (E x) by G(E x) by means of the equivalence theorem, and canceling 4∀E x(G(E x) ↔ G(E x)), we obtain
^
4∀Ex(FG (Ex) ↔ G(Ex)) ⊃ FA
G
⇔
^
4∀Ex(FG (Ex) ↔ G(Ex)) ⊃ A
G
⇔ A. The proof uses 4-axioms and rules besides the argumentation of Proposition 2.
4
Theorem 2. There is a translation T from formulas with 4 to formulas without 4 such that GWV A if and only GWV AT . Proof. We construct AT as follows: Write the defining equivalence in struc(A) as
^
4∀Ex(F4H (Ex) ↔ 4H (Ex)) ∧
^
4∀Ex(FG (Ex) ↔ G(Ex)) ⊃ FA
G
H
where the first big conjunction contains all defining equivalences for 4-subsemiformulas and the second one the remaining ones. We define AT as
^ _ ^ ∃Ex(¬F4H (Ex) ∧ H (Ex)) . ∀Ex(FG (Ex) ↔ G(Ex)) ⊃ FA ∨ (∀Ex(¬F4H (Ex) ∨ F4H (Ex)) ∧ ∀Ex(F4H (Ex) ⊃ H (Ex))) ∧ H
G
H
4
Suppose GWV AT holds. It suffices to prove GWV struc(A). We add 4∀Ex(F4H (Ex) ↔ 4H (Ex)) to the antecedent in AT . Use the equivalence theorem, 4B ⊃ B and ¬4B ∨ 4B to delete ∀E x(¬F4H (E x) ∨ F4H (E x)) and ∀E x(F4H (E x) ⊃ H (E x)). Convert ∃E x(¬F4H (E x) ∧ H (E x)) to ∃E x(¬4H (E x) ∧ H (E x)). Apply A ` 4A, 4(A ⊃ B) ⊃ (4A ⊃ 4B), 4A ⊃ 44A, 4A ⊃ A, 4(A ∨ B) ⊃ (4A ∨ 4B) to obtain 4
GWV
^
4∀Ex(F4H (Ex) ↔ 4H (Ex)) ∧
H
^
V
_ 4∀Ex(FG (Ex) ↔ G(Ex)) ⊃ FA ∨ 4∃Ex(¬4H (Ex) ∧ H (Ex)) .
G
H
4
We refute 4∃E x(¬4H (E x) ∧ H (E x)) by 4∃E xA(E x) ⊃ ∃E x4A(E x), which is valid for all GWV , and by
4(¬4H (Ea) ∧ H (Ea)) ⊃ (4¬4H (Ea) ∧ 4H (Ea)) (4¬4H (Ea) ∧ 4H (Ea)) ⊃ (¬4H (Ea) ∧ 4H (Ea)). 4
This proves GWV struc(A). For the converse direction, suppose GWV 2 AT holds. By Lemma 1(3), there is 0 ≤ w < 1 and a witnessed interpretation I such that for all G and H we have
I(∀Ex(¬F4H (Ex) ∨ F4H (Ex))) = 1, I(∀Ex(F4H (Ex) ⊃ H (Ex))) = 1, I(∀Ex(FG (Ex) ↔ G(Ex))) = 1 for all G not of the form 4H 0 , I(FA ) ≤ w, I(∀Ex(¬F4H (Ex) ∧ H (Ex))) ≤ w.
E: If I(F4H (Ea)) = 1 then I(H (Ea)) = 1 and now I(F4H (Ea)) = I(4H (Ea)) = 1. If We show that I(F4H (E a)) = I(4H (E a)) for all a I(F4H (E a)) 6= 1 then I(F4H (E a)) = 0, I(¬F4H (E a)) = 1, and therefore I(H (E a)) < 1, thus I(F4H (E a)) = I(4H (E a)) = 0. 4 Therefore I(struc(A)) ≤ w , and thus GWV 2 A.
M. Baaz, O. Fasching / Annals of Pure and Applied Logic 161 (2009) 121–127
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4
Remark 3. If 1 is isolated in V and there is no infinite descending chain in V , then GV A if and only GV AT : The proof 4
runs on the same lines as the theorem. Note that GV 4∃E xA(E x) ⊃ ∃E x4A(E x) since 1 is isolated in V , and the condition in Lemma 1(2) is fulfilled since V has no infinite descending chain. Corollary 3. Let 4 denote the following 4-rule and 4-axioms: A ` 4A, 4A ⊃ A, 4A ⊃ 44A, 4A ∨¬4A, 4(A ⊃ B) ⊃ (4A ⊃ 4B), 4(A ∨ B) ⊃ (4A ∨ 4B), ∃x4A(x) ↔ 4∃xA(x). 4 Then GW[0,1] is axiomatized by GW[0,1] + 4. 4
Proof. Clearly the 4-rule and the 4-axioms hold in GW[0,1] . 4
Conversely, suppose GW[0,1] |= A. By Theorem 2, GW[0,1] |= AT . It is then easy to see that GW[0,1] + 4 ` A.
Example 1. Let A be the Barcan formula ∀x4B(x) ⊃ 4∀xB(x). Then struc(A) is
4∀x(F1 (x) ↔ 4B(x)) ∧ 4(F2 ↔ ∀xF1 (x)) ∧ ∧ 4(F3 ↔ ∀xB(x)) ∧ 4(F4 ↔ 4F3 ) ∧ ∧ 4(FA ↔ (F2 ⊃ F4 )) ⊃ FA , and AT is u ⊃ v , where u := ∀x(¬F1 (x) ∨ F1 (x)) ∧ (¬F4 ∨ F4 ) ∧
∧ ∀x(F1 (x) ⊃ B(x)) ∧ (F4 ⊃ F3 ) ∧ ∧ (F2 ↔ ∀xF1 (x)) ∧ (F3 ↔ ∀xB(x)) ∧ ∧ (FA ↔ (F2 ⊃ F4 )) and
v := FA ∨ ∃x(¬F1 (x) ∧ B(x)) ∨ (¬F4 ∧ F3 ). Assume a counterexample to AT ; then we can find I such that I(u) = 1 and I(v) < 1. We obtain I(F2 ⊃ F4 ) = I(FA ) ≤ I(v) < 1, and hence I(F4 ) < 1. But then I(F4 ) = 0 and we have I(F3 ) < 1 and it follows that I(∀xB(x)) < 1. Since there is a with I(B(a)) < 1, we have I(F1 (a)) < 1; thus I(F1 (a)) = 0. We conclude I(∀xF1 (x)) = 0 and I(F2 ) = 0; hence I(F2 ⊃ F4 ) = 1, which is impossible. Remark 4. The Barcan formula ∀x4B(x) ⊃ 4∀xB(x) can be derived in any extension of G[0,1] using only propositional axioms and rules for 4. Use transitivity to obtain it from the following derivations:
(1) `4B(a) ∨ ¬4B(a) (2) `∀x(4B(x) ∨ ¬4B(x)) from 1 (3) `∀x4B(x) ∨ ∃x¬4B(x) from 2 (4) `4(∀x4B(x) ∨ ∃x¬4B(x)) from 3 (5) `4∀x4B(x) ∨ 4∃x¬4B(x) from 4 (6) `4∀x4B(x) ∨ ∃x¬4B(x) from 5 (7) `¬(4B(a) ∧ ¬4B(a)) (8) `¬(∃x(4B(x) ∧ ¬4B(x))) from 7 (9) `¬(∀x4B(x) ∧ ∃x¬4B(x)) from 8 (10) `∀x4B(x) ∧ ∃x¬4B(x) ⊃ 4∀x4B(x) from 9 (11) `∃x¬4B(x) ⊃ ∀x4B(x) ⊃ 4∀x4B(x) from 10 (12) `4∀x4B(x) ⊃ ∀x4B(x) ⊃ 4∀x4B(x) (13) `∀x4B(x) ⊃ 4∀x4B(x) from 6, 12, 11. In the following we summarize the possibilities of formula translations between the different types of Gödel logics. By G1 → G2 we express that there is a translation of formulas from G1 to G2 , preserving their validity. 4 GV → GV : by identity. 4
GV 6→ GV : since G[0, 1 ]∪[ 2 ,1] = G[0, 1 ]∪{1} but G 3
3
3
4 [0, 13 ]∪[ 23 ,1]
2 4∃xA(x) ⊃ ∃x4A(x) and G
4
GWV → GWV : by identity. 4
GWV → GWV : by Theorem 2. GV 6→ GWV : since GW{0}∪[ 1 ,1] = GW[0,1] but G{0}∪[ 1 ,1] 6= G[0,1] . 2 2
4 [0, 31 ]∪{1}
4∃xA(x) ⊃ ∃x4A(x).
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4
4
4 [0, 12 ]∪{1} 4
GV 6→ GWV : GW
4
4
4
= GW[0,1] but G[0, 1 ]∪{1} 4∃xA(x) ⊃ ∃x4A(x) and G[0,1] 2 4∃xA(x) ⊃ ∃x4A(x). 2
4
4
4
GWV → GV and GWV → GV : For any formula A, we have A ∈ GWV (A ∈ GWV ) if and only if C ⊃ A ∈ GV (4C ⊃ A ∈ GV ), where C is the conjunction of the universal closure of all formulas ∃x(B(x) ⊃ ∀yB(y)), where ∀yB(y) is any subsemiformula of A and of all formulas ∃x(∃yB(y) ⊃ B(x)) where ∃yB(y) is any subsemiformula of A. 4
4
Theorem 3. Let G, GW, G4 , GW4 be the sets of all Gödel logics GV , GWV , GV , GWV . (1) Then G, GW, G4 , GW4 are countable. (2) Let L contain infinitely many predicate symbols. Then there are infinitely many infinitely valued logics in G, in GW, in G4 and in GW4 . Proof. (1) G is countable by Corollary 40 in [5]. G4 is countable by analogous arguments. By the above remark it follows that GW is countable since G is countable. Now GW4 is countable by Theorem 2. (2) In the proof of Corollary 40 in [5] it is shown that there are countable many countable valued logics in G. It is sufficient to show that GW4 contains infinitely many different Gödel logics with infinite V since then, by Theorem 2, the claim also holds for GW. Define A C B as ¬4(B ⊃ A) so that I(A C B) = 1 if and only if I(A) < I(B) and, moreover, I(A C B) ∈ {0, 1}. Let Fn := ∀x((Xn C Pn (x)) ∧ (Pn (x) C Xn+1 )) ∧ ∃x∃y¬4(Pn (x) ↔ Pn (y)), Gn := ∀x((∀yPn (y) C Pn (x)) ⊃ ∃z (∀yPn (y) C Vn 4 Pn (z ) ∧ (Pn (z ) C Pn (x)))) and An := k=1 Fk ∧ Gk . It is easy to check that GWV ¬An if and only if there are ≤ n − 1 decreasing cumulation points in V r {0, 1}. Now choose countable sets V1 ( V2 ( . . . such that each Vn has exactly n decreasing cumulation points. We then obtain 4 4 GWV1 ) GWV2 ) . . .. 5. The disentangled/S5 fragment The disentangled fragment consists of formulas without function symbols where no quantifier is in the scope of another quantifier (cf. [2]). This fragment can be understood as representing the Leibniz notation of necessity (validity in all worlds). Therefore this can be called an S5-fragment because the universal (existential) quantifier may be interpreted as (♦). We write Dis(L) for the disentangled fragment of a logic L. Proposition 4. Let A be ∃E xA0 (E x), A0 q. f. with constants C = {c1 , . . . , ck } as the only function symbols. (Add one constant if there 4 is none in A0 .) Let m be the number of predicate symbols in A and let a be their maximal arity. Then GWV A if and only if, for all 4
I with |I| = C , we have GWU A, where |U | = n := 2 + m · ka . An analogous statement holds for GWV . Proof. The restriction of |I| for I(A) < 1 does not change the interpretation of A to 1. The problem is reduced to deciding W the validity of dj ∈C A(d1 , . . . , dr ). Any propositional formula with v variables is valid in any Gödel logic if and only if it is valid in the Gödel logic with v + 2 truth values. 4
4
Theorem 4. Dis(GWV ) is decidable. Dis(GWV ) = Dis(GWDV 0 ) for all infinitely-valued V , V 0 . 4
4
Proof. We have A ∈ Dis(GWV ) if and only if struc(A) ∈ Dis(GWV ): no quantifier is in the scope of another one. Therefore, 4
4
E). This is equivalent to GWV ∃EyB(c1 , . . . , ck , yE), and in for some quantifier free B, we have GWV struc(A) ↔ ∀E x∃E yB(E x, y 4
E), where |U | = n with n as in the above proposition. turn to GWU ∃E yB(c1 , . . . , ck , y
Remark 5. Since Dis(G↑ ) = Dis(GW[0,1] ) by Proposition 1 and by Corollary 1, the decidability has been shown independently by Baaz, Ciabattoni and Fermüller [2] and by Hájek. Contrary to the witnessed case, there are countably many different infinitely-valued Dis(GV ) and consequently countably 4 many different infinitely-valued Dis(GV ) (see proof of Prop. 4.2 in [1]). 6. Final remark A different approach to witnessed quantifier interpretations is to combine the interpretation with a default value. min {I(A(u)); u ∈ |I|} if the infimum is a minimum I(∀Wx A(x)) := 1 otherwise
I(∃Wx A(x)) :=
I(∀¬Wx A(x)) := I(∃¬Wx A(x)) :=
max{I(A(u)); u ∈ |I|} if the supremum is a maximum 1
otherwise inf{I(A(u)); u ∈ |I|} if the infimum is no minimum
1
sup{I(A(u)); u ∈ |I|}
1 These quantifiers can be defined by
otherwise if the supremum is no maximum otherwise.
• ∀W xA(x) :⇔ ∃x(A(x) ⊃ ∀yA(y)) ⊃ ∀yA(y),
M. Baaz, O. Fasching / Annals of Pure and Applied Logic 161 (2009) 121–127
127
• ∃W xA(x) :⇔ ∃x(∃yA(y) ⊃ A(x)) ⊃ ∃yA(y), • ∀¬W xA(x) :⇔ (∃x(A(x) ⊃ ∀yA(y)) ⊃ ∀yA(y)) ⊃ ∀yA(y), • ∃¬W xA(x) :⇔ (∃x(∃yA(y) ⊃ A(x)) ⊃ ∃yA(y)) ⊃ ∃yA(y). Note that (∃x(A(x) ⊃ ∀yA(y))) ∨ (∃x(A(x) ⊃ ∀yA(y)) ⊃ ∀xA(x)) and (∃x(∃yA(y) ⊃ A(x)))) ∨ (∃x(∃yA(y) ⊃ A(x)) ⊃ ∃xA(x)) are valid in all Gödel logics. It is easy to show that ∀xA(x) ↔ ∀W xA(x)∧∀¬W xA(x) and ∃xA(x) ↔ ∃W xA(x)∧∃¬W xA(x) are valid in all Gödel logics. Acknowledgement The second author was partially supported by FWF project P19872. References [1] M. Baaz, Infinite-valued Gödel logics with 0–1-projections and relativizations, in: Gödel ’96, in: Lecture Notes Logic, vol. 6, 1996, pp. 23–33. [2] M. Baaz, A. Ciabattoni, C. Fermüller, Monadic fragments of Gödel logics: Decidability and undecidability results, in: Logic for Programming, Artificial Intelligence and Reasoning 2007, 14th Internat. Conf., LPAR 2007, in: Lecture Notes in Computer Science, vol. 4790, Springer, Berlin, 2007, pp. 77–91. [3] M. Baaz, N. Preining, R. Zach, First-order Gödel logics, Annals of Pure and Applied Logic 147 (1–2) (2007) 23–47. [4] M. Baaz, N. Preining, R. Zach, Characterization of the axiomatizable prenex fragments of first-order Gödel logics, in: 33rd IEEE Internat. Symposium on Multiple-Valued Logic, ISMVL 2003, IEEE Computer Society, 2003, pp. 175–180. [5] A. Beckmann, M. Goldstern, N. Preining, Continuous Fraïssé Conjecture, Order, Springer, 2008 (in press). [6] A. Beckmann, N. Preining, Linear Kripke frames and Gödel logics, Journal of Symbolic Logic 72 (1) (2007) 26–44. [7] P. Cintula, P. Hájek, On theories and models in fuzzy predicate logics, Journal of Symbolic Logic 71 (3) (2006) 863–880. [8] N. Preining, Complete recursive axiomatizability of Gödel logics, Ph.D. Thesis, Vienna University of Technology, 2003. [9] M. Takano, Another proof of the strong completeness of the intuitionistic fuzzy logic, Tsukuba Journal of Mathematics 11 (1) (1987) 101–105. [10] G. Takeuti, S. Titani, Intuitionistic fuzzy logic and intuitionistic fuzzy set theory, Journal of Symbolic Logic 94 (1984) 851–866. [11] G. Takeuti, S. Titani, Global intuitionistic fuzzy set theory, in: In the Mathematics of Fuzzy Systems, ISR Interdiscip. Systems, 1986, pp. 291–301. [12] S. Titani, Completeness of global intuitionistic set theory, Journal of Symbolic Logic 62 (2) (1997) 506–526.
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Note on witnessed Gödel logics with Delta Matthias Baaz ∗ , Oliver Fasching Vienna University of Technology, Wiedner Hauptstraße 8–10/E104, A-1040 Vienna, Austria
article
info
Article history: Available online 1 July 2009 MSC: 03B52 03B52 03C80
abstract Witnessed Gödel logics are based on the interpretation of ∀ (∃) by minimum (maximum) instead of supremum (infimum). Witnessed Gödel logics appear for many practical purposes more suited than usual Gödel logics as the occurrence of proper infima/suprema is practically irrelevant. In this note we characterize witnessed Gödel logics with absoluteness operator 4 w.r.t. witnessed Gödel logics using a uniform translation. © 2009 Elsevier B.V. All rights reserved.
Keywords: Gödel logics Disentangled fragment Relative completeness
1. Introduction First-order Gödel logics are among the most prominent intermediate logics. They are defined using closed subsets of the [0, 1]-interval containing 0 and 1, where 1 represents truth. Alternatively, they are logics of countable linearly ordered Kripke semantics with constant domain; cf. [6]. The dominant connective of Gödel logics is ⊃, which is projective. All other connectives and quantifiers are evaluated in a natural way according to the linear structure of the truth values. There is an asymmetry in Gödel logics concerning the status of true and false. It is possible to separate false from not false while it is impossible to separate true from not true. For this reason an absoluteness operator 4 is introduced, which is prominent in many non-classical logics. The 4-operator goes back to [11] and was studied in [1,12]. In a proof theoretic sense, quantified formulas are partial descriptions of infimum/supremum expressions associated with uniform provability. Universal/existential quantifiers are induced by iterated conjunction/disjunction. It is therefore natural to include the property of realization by a constituent in the description of quantifiers. These quantifiers are called witnessed quantifiers. In this paper we characterize the relation of Gödel logics with/without absoluteness operator to witnessed Gödel logics. Logics with linearly ordered truth values are especially suited for an investigation of witnessed quantifiers because of the obvious infimum/minimum (supremum/maximum) relation. This note can be therefore considered as a first step towards a general theory of witnessed quantifiers. 2. Language and semantics We will consider first-order languages with free and bound variables, predicate and function symbols, logical connectives
∨, ∧, ⊃, ⊥, and quantifiers ∀, ∃. By L4 we denote the extension of a language L by a unary operator 4. We understand ¬A as an abbreviation for A ⊃ ⊥, A ↔ B for (A ⊃ B) ∧ (B ⊃ A), and A ≺ B for (B ⊃ A) ⊃ B. Terms and formulas are defined in the usual way.
∗
Corresponding author. E-mail address: [email protected] (M. Baaz).
0168-0072/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.apal.2009.05.011
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Definition 1. Suppose we are given a truth value set V , i.e. a closed set V with {0, 1} ⊆ V ⊆ [0, 1]. A V -interpretation I (of L and L4 ) consists of (1) a nonempty set |I|, the domain of I, (2) a function P I : |I|n → V for each n-ary predicate symbol P, (3) a function f I : |I|n → |I| for each n-ary function symbol f , and (4) a value aI ∈ V for each free variable a. Definition 2. Let I be a V -interpretation of L. We denote by LI the extension of L by constant symbols for all elements of |I| and we define c I := c for all c ∈ |I|. We extend the interpretation to terms and atoms by putting I(⊥) := 0, I(f (u1 , . . . , un )) := f I (I(u1 ), . . . , I(un )) and I(P (t1 , . . . , tn )) := P I (I(t1 ), . . . , I(tn )). For composite formulas we define:
I(A ∧ B) := min(I(A), I(B)), I(A ∨ B) := max(I(A), I(B)),
I(A ⊃ B) :=
if I(A) ≤ I(B) I(B) if I(A) > I(B), 1
I(∀x A(x)) := inf{I(A(u)); u ∈ |I|}, I(∃x A(x)) := sup{I(A(u)); u ∈ |I|}. For L4 we additionally define:
I(4A) :=
1
if I(A) = 1
0
if I(A) < 1.
Note that
I(¬A) =
1
if I(A) = 0
0
if I(A) > 0,
I(A ↔ B) = I(A ≺ B) =
1
if I(A) = I(B)
min{I(A), I(B)}
if I(A) 6= I(B),
if I(A) < I(B) or I(A) = I(B) = 1 I(B) else.
1
Definition 3. If I(A) = 1, we write I |= A and say I satisfies A. For any truth value set V we define the (non4 witnessed) Gödel logic GV (resp. GV ) based on V as the set of all formulas A in L (resp. L4 ) such that I |= A for all V -interpretations I. Witnessed interpretations are interpretations fulfilling I(∀xA(x)) = min{I(A(u)); u ∈ |I|} and 4 I(∃xA(x)) = max{I(A(u)); u ∈ |I|}. For any truth value set V we define the witnessed Gödel logic GWV (resp. GWV ) based on V as the set of all formulas A in L (resp. L4 ) such that I |= A for all witnessed V -interpretations I. For any of the above logics G, we write G A instead of A ∈ G. Definition 4. For a set Γ of formulas, we define an entailment I:Γ A by inf{I(B); B ∈ Γ } ≤ I(A). If I:Γ A holds for all V -interpretations I, we write Γ V A and say that Γ entails A in GV . 3. Witnessed Gödel logics Lemma 1. Let I be a V -interpretation, and let 0 ≤ d ≤ 1. We put h(x) := x for x ≤ d and h(x) := 1 for x > d. Let I0 be the 0 interpretation that arises from I by changing P I to P I (A) := h(P I (A)) for atomic A; in particular, I0 = |I|. (1) Then I0 is a V 0 -interpretation where V 0 := V r (d, 1) ⊆ V , and I0 (B) = h(I(B)) for all ∀-free formulas B. (2) If, moreover, for each F (a) with I(∀xF (x)) = d there exists some u∗ ∈ |I| such that d = I(F (u∗ )), then I0 (B) = h(I(B)) holds for all formulas B. (3) If I is witnessed, so is I0 and I0 (B) = h(I(B)) holds for all formulas B. Proof. Part (1) is Lemma 3.4 in [4]. Part (2). We may assume d < 1, for otherwise I = I0 . The claim is proved by induction on the complexity of the formula B. Let infu abbreviate infu∈|I| in what follows. We need to prove infu I0 (A(u)) = h(I(∀xA(x))) under the induction hypothesis I0 (A(a)) = h(I(A(a))). Since h(I(A(v))) ≥ h(infu I(A(u))) for all v ∈ |I|, we have infu h(I(A(u))) ≥ h(infu I(A(u))), and therefore it suffices to prove infu h(I(A(u))) ≤ h(e), where e := infu I(A(u)). In the case d > e, there is a series (um )m with d > I (A(um )) and I (A(um )) & e. From h(I (A(um ))) = I (A(um )) we get h(I (A(um ))) & e; thus infu h(I (A(u))) ≤ e = h(e), as required. In the case d < e, we have d < I(A(u)) for all u ∈ |I|. Thus h(I(A(u))) = 1 for all u ∈ |I|, and we have 1 = h(e), as required. In the case d = e, we find u∗ ∈ |I| with d = I(A(u∗ )) due to the extra assumption on I. We have infu h(I(A(u))) ≤ h(I(A(u∗ ))) = h(d) = h(e), as required. Part (3). Since I is witnessed, we find u∗ , u∗ ∈ |I| with I(∀xA(x)) = I(A(u∗ )) and I(∃xA(x)) = I(A(u∗ )). Hence, by (2), I0 (B) = h(I(B)) holds for all formulas B. In particular, I0 (∀xA(x)) = h(I(∀xA(x))) = h(I(A(u∗ ))) = I0 (A(u∗ )) and I0 (∃xA(x)) = h(I(∃xA(x))) = h(I(A(u∗ ))) = I0 (A(u∗ )). Therefore I0 is witnessed.
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Remark 1. Since
I(∃x(A(x) ⊃ ∀yA(y))) =
{I(A(u)); u ∈ |I|} attains its minimum I(∀xA(x)) otherwise
1
and
I(∃x(∃yA(y) ⊃ A(x))) =
{I(A(u)); u ∈ |I|} attains its maximum I(∃xA(x)) otherwise,
1
it is easy to see that I(∃x(A(x) ⊃ ∀yA(y))) = 1 holds if and only if {I(A(x)); x ∈ |I|} attains its minimum. In contrast to that, I(∃x(∃yA(y) ⊃ A(x))) = 1 holds if and only if I(∃xA(x)) = 1 or {I(A(x)); x ∈ |I|} attains its maximum. Definition 5. Let W be the set of the formulas ∃x(A(x) ⊃ ∀yA(y)) and ∃x(∃yA(y) ⊃ A(x)) for all formulas A(a) in L, and let HV := {B; W V B}. Theorem 1. We have GWV = HV . Proof. If A ∈ HV , then infB∈W I(B) ≤ I(A) for all V -interpretations I. For all witnessed V -interpretations I we have I(B) = 1 for all B ∈ W ; thus 1 ≤ I(A). Therefore A ∈ GWV . Conversely, if A ∈ / HV , there is a V -interpretation I such that inf{I(B); B ∈ W } > d := I(A). Note that 0 ≤ d < 1. We prove by contradiction that, for any F (a) with I(∀xF (x)) = d, there is x∗ ∈ |I| such that d = I(F (x∗ )): If {I(F (u)); u ∈ |I|} does not attain its infimum, we have I(∃x(F (x) ⊃ ∀yF (y))) = I(∀xF (x)) by Remark 1; but this contradicts I(∀xF (x)) = d < inf{I(B); B ∈ W } ≤ I(∃x(F (x) ⊃ ∀yF (y))). From Lemma 1(2) it follows that there is a V 0 -interpretation I0 , where V 0 := V r (I0 (A), 1) ( V , such that 1 = inf{I0 (B); B ∈ W } > I0 (A). For any formula D(a), define U := {I0 (D(x)); x ∈ I0 }. From 1 = I0 (∃x(D(x) ⊃ ∀yD(y))) it follows that U attains its minimum. From 1 = I0 (∃x(∃yD(y) ⊃ D(x))) it follows that U attains its maximum or sup U = 1; but even in the latter case, U attains its maximum since U ⊆ V 0 and 1 is isolated in V 0 . This means that I0 is witnessed. Since I0 (A) < 1, we have A ∈ / GWV . Corollary 1. GW[0,1] is axiomatized by the axioms and rules of G[0,1] with W as additional axioms. We have GW[0,1] = GWV for all uncountable V . Proof. Clearly G[0,1] ∪ W ⊆ GW[0,1] . Conversely, suppose GW[0,1] |= A so that W |=[0,1] A by Theorem 1. Let H be the proof system of intuitionistic predicate calculus + (A ⊃ B) ∨ (B ⊃ A) + ∀x(A(x) ∨ C ) ⊃ (∀xA(x) ∨ C ), where x does not occur in C . By [9] (also cf. [10,8]), the entailment for G[0,1] is axiomatized by H ; in particular, there is a conjunction W 0 of finitely many formulas in W such that H ` W 0 ⊃ A. Thus H + W ` A. We have GW[0,1] ⊆ GWV for any truth value set V . Let V be uncountable so that there is a perfect kernel V 0 of V . Any witnessed interpretation in [0, 1] can be transformed to a witnessed interpretation in {0} ∪ V 0 ∪ {1}.1 Therefore GWV ⊆ GW[0,1] and thus GWV = GW[0,1] for all uncountable V . Remark 2. GW[0,1] is the only recursively enumerable (r.e.), infinitely valued GWV since the argument for GV also shows that GWV , V countable, is not r.e. (cf. [3]). As all GV are core fuzzy logics in the sense of [7], extensions of ∃x(A(x) ⊃ ∀yA(y)), ∃x(∃yA(y) ⊃ A(x)) provide completeness w.r.t. witnessed models based on L-algebras (cf. Theorem 8 in [7]). Corollary 2. All classically valid quantifier rules are valid in GWV . Proof. The only classically valid quantifier shift rules not always valid in Gödel logics are (∀xA(x) ⊃ B) ⊃ ∃x(A(x) ⊃ B) and (A ⊃ ∃xB(x)) ⊃ ∃x(A ⊃ B(x)) (see [4], Section 3). They can be easily obtained from the valid formulas (A(a) ⊃ ∀xA(x)) ⊃ (∀xA(x) ⊃ B) ⊃ (A(a) ⊃ B) and (∃xA(x) ⊃ A(a)) ⊃ (B ⊃ ∃xA(x)) ⊃ (B ⊃ A(a)) together with the formulas in W . Definition 6. We put V↑ := {1 − 1k ; k ≥ 1} ∪ {1}, H↑ := HV↑ and G↑ := GV↑ .
T
T
Proposition 1. (1) We have V finite GV = V finite GWV = G↑ = GW↑ . (2) GWV = GV holds if and only if V is finite or V is order isomorphic to V↑ . Proof. (1) By Remark 1, we have I(A) = 1 for all V↑ -interpretations I and all A ∈ W . Thus H↑ = G↑ , and now GW↑ = G↑ by Theorem 1. NoteT that GV = GU if V and U are finite and have same cardinality. Clearly GV = GWV for finite V . Thus we T have V finite GV = V finite GWV = G↑ by Proposition 3.1 and Theorem 3.2 in [4]. (2) One direction follows from (1). Conversely, assume GWV = GV . By Corollary 2.26 in [3] we only need to prove that all quantifier shift rules are valid in GV , but this is true due to Corollary 2. 4. Eliminating ∆ in witnessed Gödel logics 4
In this section we characterize GWV w.r.t. GWV .
1 Personal communication with N. Preining; cf. [8].
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Proposition 2. The equivalence theorem holds for GV , GWV in the form
∀x(A(x) ↔ B(x)) ⊃ (C (A(t )) ↔ C (B(t ))) 4
4
and for GV , GWV in the form
4∀x(A(x) ↔ B(x)) ⊃ C (A(t ) ↔ C (B(t ))) where t may contain bound variables. Proof. By induction on the complexity of C . For 4, the proof uses 4A ⊃ A, 4A ⊃ 44A, and A ⇒ 4A.
4
Definition 7. Structural normal form for formulas A in L . Replace stepwise from the inside to the outside all subsemiformulas G(E x), with bound variables E x, by fresh defining predicates FG (E x). Now, struc(A) is
^
4∀Ex(FG (Ex) ↔ G(Ex)) ⊃ FA
G
where the conjunction ranges over all subsemiformulas G of A. 4∀E x(FG (E x) ↔ G(E x)) is the defining equivalence for G. 4
4
4
4
Proposition 3. GV A if and only if GV struc(A). GWV A if and only if GWV struc(A). Proof. Replacing FG (E x) by G(E x) by means of the equivalence theorem, and canceling 4∀E x(G(E x) ↔ G(E x)), we obtain
^
4∀Ex(FG (Ex) ↔ G(Ex)) ⊃ FA
G
⇔
^
4∀Ex(FG (Ex) ↔ G(Ex)) ⊃ A
G
⇔ A. The proof uses 4-axioms and rules besides the argumentation of Proposition 2.
4
Theorem 2. There is a translation T from formulas with 4 to formulas without 4 such that GWV A if and only GWV AT . Proof. We construct AT as follows: Write the defining equivalence in struc(A) as
^
4∀Ex(F4H (Ex) ↔ 4H (Ex)) ∧
^
4∀Ex(FG (Ex) ↔ G(Ex)) ⊃ FA
G
H
where the first big conjunction contains all defining equivalences for 4-subsemiformulas and the second one the remaining ones. We define AT as
^ _ ^ ∃Ex(¬F4H (Ex) ∧ H (Ex)) . ∀Ex(FG (Ex) ↔ G(Ex)) ⊃ FA ∨ (∀Ex(¬F4H (Ex) ∨ F4H (Ex)) ∧ ∀Ex(F4H (Ex) ⊃ H (Ex))) ∧ H
G
H
4
Suppose GWV AT holds. It suffices to prove GWV struc(A). We add 4∀Ex(F4H (Ex) ↔ 4H (Ex)) to the antecedent in AT . Use the equivalence theorem, 4B ⊃ B and ¬4B ∨ 4B to delete ∀E x(¬F4H (E x) ∨ F4H (E x)) and ∀E x(F4H (E x) ⊃ H (E x)). Convert ∃E x(¬F4H (E x) ∧ H (E x)) to ∃E x(¬4H (E x) ∧ H (E x)). Apply A ` 4A, 4(A ⊃ B) ⊃ (4A ⊃ 4B), 4A ⊃ 44A, 4A ⊃ A, 4(A ∨ B) ⊃ (4A ∨ 4B) to obtain 4
GWV
^
4∀Ex(F4H (Ex) ↔ 4H (Ex)) ∧
H
^
V
_ 4∀Ex(FG (Ex) ↔ G(Ex)) ⊃ FA ∨ 4∃Ex(¬4H (Ex) ∧ H (Ex)) .
G
H
4
We refute 4∃E x(¬4H (E x) ∧ H (E x)) by 4∃E xA(E x) ⊃ ∃E x4A(E x), which is valid for all GWV , and by
4(¬4H (Ea) ∧ H (Ea)) ⊃ (4¬4H (Ea) ∧ 4H (Ea)) (4¬4H (Ea) ∧ 4H (Ea)) ⊃ (¬4H (Ea) ∧ 4H (Ea)). 4
This proves GWV struc(A). For the converse direction, suppose GWV 2 AT holds. By Lemma 1(3), there is 0 ≤ w < 1 and a witnessed interpretation I such that for all G and H we have
I(∀Ex(¬F4H (Ex) ∨ F4H (Ex))) = 1, I(∀Ex(F4H (Ex) ⊃ H (Ex))) = 1, I(∀Ex(FG (Ex) ↔ G(Ex))) = 1 for all G not of the form 4H 0 , I(FA ) ≤ w, I(∀Ex(¬F4H (Ex) ∧ H (Ex))) ≤ w.
E: If I(F4H (Ea)) = 1 then I(H (Ea)) = 1 and now I(F4H (Ea)) = I(4H (Ea)) = 1. If We show that I(F4H (E a)) = I(4H (E a)) for all a I(F4H (E a)) 6= 1 then I(F4H (E a)) = 0, I(¬F4H (E a)) = 1, and therefore I(H (E a)) < 1, thus I(F4H (E a)) = I(4H (E a)) = 0. 4 Therefore I(struc(A)) ≤ w , and thus GWV 2 A.
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125
4
Remark 3. If 1 is isolated in V and there is no infinite descending chain in V , then GV A if and only GV AT : The proof 4
runs on the same lines as the theorem. Note that GV 4∃E xA(E x) ⊃ ∃E x4A(E x) since 1 is isolated in V , and the condition in Lemma 1(2) is fulfilled since V has no infinite descending chain. Corollary 3. Let 4 denote the following 4-rule and 4-axioms: A ` 4A, 4A ⊃ A, 4A ⊃ 44A, 4A ∨¬4A, 4(A ⊃ B) ⊃ (4A ⊃ 4B), 4(A ∨ B) ⊃ (4A ∨ 4B), ∃x4A(x) ↔ 4∃xA(x). 4 Then GW[0,1] is axiomatized by GW[0,1] + 4. 4
Proof. Clearly the 4-rule and the 4-axioms hold in GW[0,1] . 4
Conversely, suppose GW[0,1] |= A. By Theorem 2, GW[0,1] |= AT . It is then easy to see that GW[0,1] + 4 ` A.
Example 1. Let A be the Barcan formula ∀x4B(x) ⊃ 4∀xB(x). Then struc(A) is
4∀x(F1 (x) ↔ 4B(x)) ∧ 4(F2 ↔ ∀xF1 (x)) ∧ ∧ 4(F3 ↔ ∀xB(x)) ∧ 4(F4 ↔ 4F3 ) ∧ ∧ 4(FA ↔ (F2 ⊃ F4 )) ⊃ FA , and AT is u ⊃ v , where u := ∀x(¬F1 (x) ∨ F1 (x)) ∧ (¬F4 ∨ F4 ) ∧
∧ ∀x(F1 (x) ⊃ B(x)) ∧ (F4 ⊃ F3 ) ∧ ∧ (F2 ↔ ∀xF1 (x)) ∧ (F3 ↔ ∀xB(x)) ∧ ∧ (FA ↔ (F2 ⊃ F4 )) and
v := FA ∨ ∃x(¬F1 (x) ∧ B(x)) ∨ (¬F4 ∧ F3 ). Assume a counterexample to AT ; then we can find I such that I(u) = 1 and I(v) < 1. We obtain I(F2 ⊃ F4 ) = I(FA ) ≤ I(v) < 1, and hence I(F4 ) < 1. But then I(F4 ) = 0 and we have I(F3 ) < 1 and it follows that I(∀xB(x)) < 1. Since there is a with I(B(a)) < 1, we have I(F1 (a)) < 1; thus I(F1 (a)) = 0. We conclude I(∀xF1 (x)) = 0 and I(F2 ) = 0; hence I(F2 ⊃ F4 ) = 1, which is impossible. Remark 4. The Barcan formula ∀x4B(x) ⊃ 4∀xB(x) can be derived in any extension of G[0,1] using only propositional axioms and rules for 4. Use transitivity to obtain it from the following derivations:
(1) `4B(a) ∨ ¬4B(a) (2) `∀x(4B(x) ∨ ¬4B(x)) from 1 (3) `∀x4B(x) ∨ ∃x¬4B(x) from 2 (4) `4(∀x4B(x) ∨ ∃x¬4B(x)) from 3 (5) `4∀x4B(x) ∨ 4∃x¬4B(x) from 4 (6) `4∀x4B(x) ∨ ∃x¬4B(x) from 5 (7) `¬(4B(a) ∧ ¬4B(a)) (8) `¬(∃x(4B(x) ∧ ¬4B(x))) from 7 (9) `¬(∀x4B(x) ∧ ∃x¬4B(x)) from 8 (10) `∀x4B(x) ∧ ∃x¬4B(x) ⊃ 4∀x4B(x) from 9 (11) `∃x¬4B(x) ⊃ ∀x4B(x) ⊃ 4∀x4B(x) from 10 (12) `4∀x4B(x) ⊃ ∀x4B(x) ⊃ 4∀x4B(x) (13) `∀x4B(x) ⊃ 4∀x4B(x) from 6, 12, 11. In the following we summarize the possibilities of formula translations between the different types of Gödel logics. By G1 → G2 we express that there is a translation of formulas from G1 to G2 , preserving their validity. 4 GV → GV : by identity. 4
GV 6→ GV : since G[0, 1 ]∪[ 2 ,1] = G[0, 1 ]∪{1} but G 3
3
3
4 [0, 13 ]∪[ 23 ,1]
2 4∃xA(x) ⊃ ∃x4A(x) and G
4
GWV → GWV : by identity. 4
GWV → GWV : by Theorem 2. GV 6→ GWV : since GW{0}∪[ 1 ,1] = GW[0,1] but G{0}∪[ 1 ,1] 6= G[0,1] . 2 2
4 [0, 31 ]∪{1}
4∃xA(x) ⊃ ∃x4A(x).
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4
4
4 [0, 12 ]∪{1} 4
GV 6→ GWV : GW
4
4
4
= GW[0,1] but G[0, 1 ]∪{1} 4∃xA(x) ⊃ ∃x4A(x) and G[0,1] 2 4∃xA(x) ⊃ ∃x4A(x). 2
4
4
4
GWV → GV and GWV → GV : For any formula A, we have A ∈ GWV (A ∈ GWV ) if and only if C ⊃ A ∈ GV (4C ⊃ A ∈ GV ), where C is the conjunction of the universal closure of all formulas ∃x(B(x) ⊃ ∀yB(y)), where ∀yB(y) is any subsemiformula of A and of all formulas ∃x(∃yB(y) ⊃ B(x)) where ∃yB(y) is any subsemiformula of A. 4
4
Theorem 3. Let G, GW, G4 , GW4 be the sets of all Gödel logics GV , GWV , GV , GWV . (1) Then G, GW, G4 , GW4 are countable. (2) Let L contain infinitely many predicate symbols. Then there are infinitely many infinitely valued logics in G, in GW, in G4 and in GW4 . Proof. (1) G is countable by Corollary 40 in [5]. G4 is countable by analogous arguments. By the above remark it follows that GW is countable since G is countable. Now GW4 is countable by Theorem 2. (2) In the proof of Corollary 40 in [5] it is shown that there are countable many countable valued logics in G. It is sufficient to show that GW4 contains infinitely many different Gödel logics with infinite V since then, by Theorem 2, the claim also holds for GW. Define A C B as ¬4(B ⊃ A) so that I(A C B) = 1 if and only if I(A) < I(B) and, moreover, I(A C B) ∈ {0, 1}. Let Fn := ∀x((Xn C Pn (x)) ∧ (Pn (x) C Xn+1 )) ∧ ∃x∃y¬4(Pn (x) ↔ Pn (y)), Gn := ∀x((∀yPn (y) C Pn (x)) ⊃ ∃z (∀yPn (y) C Vn 4 Pn (z ) ∧ (Pn (z ) C Pn (x)))) and An := k=1 Fk ∧ Gk . It is easy to check that GWV ¬An if and only if there are ≤ n − 1 decreasing cumulation points in V r {0, 1}. Now choose countable sets V1 ( V2 ( . . . such that each Vn has exactly n decreasing cumulation points. We then obtain 4 4 GWV1 ) GWV2 ) . . .. 5. The disentangled/S5 fragment The disentangled fragment consists of formulas without function symbols where no quantifier is in the scope of another quantifier (cf. [2]). This fragment can be understood as representing the Leibniz notation of necessity (validity in all worlds). Therefore this can be called an S5-fragment because the universal (existential) quantifier may be interpreted as (♦). We write Dis(L) for the disentangled fragment of a logic L. Proposition 4. Let A be ∃E xA0 (E x), A0 q. f. with constants C = {c1 , . . . , ck } as the only function symbols. (Add one constant if there 4 is none in A0 .) Let m be the number of predicate symbols in A and let a be their maximal arity. Then GWV A if and only if, for all 4
I with |I| = C , we have GWU A, where |U | = n := 2 + m · ka . An analogous statement holds for GWV . Proof. The restriction of |I| for I(A) < 1 does not change the interpretation of A to 1. The problem is reduced to deciding W the validity of dj ∈C A(d1 , . . . , dr ). Any propositional formula with v variables is valid in any Gödel logic if and only if it is valid in the Gödel logic with v + 2 truth values. 4
4
Theorem 4. Dis(GWV ) is decidable. Dis(GWV ) = Dis(GWDV 0 ) for all infinitely-valued V , V 0 . 4
4
Proof. We have A ∈ Dis(GWV ) if and only if struc(A) ∈ Dis(GWV ): no quantifier is in the scope of another one. Therefore, 4
4
E). This is equivalent to GWV ∃EyB(c1 , . . . , ck , yE), and in for some quantifier free B, we have GWV struc(A) ↔ ∀E x∃E yB(E x, y 4
E), where |U | = n with n as in the above proposition. turn to GWU ∃E yB(c1 , . . . , ck , y
Remark 5. Since Dis(G↑ ) = Dis(GW[0,1] ) by Proposition 1 and by Corollary 1, the decidability has been shown independently by Baaz, Ciabattoni and Fermüller [2] and by Hájek. Contrary to the witnessed case, there are countably many different infinitely-valued Dis(GV ) and consequently countably 4 many different infinitely-valued Dis(GV ) (see proof of Prop. 4.2 in [1]). 6. Final remark A different approach to witnessed quantifier interpretations is to combine the interpretation with a default value. min {I(A(u)); u ∈ |I|} if the infimum is a minimum I(∀Wx A(x)) := 1 otherwise
I(∃Wx A(x)) :=
I(∀¬Wx A(x)) := I(∃¬Wx A(x)) :=
max{I(A(u)); u ∈ |I|} if the supremum is a maximum 1
otherwise inf{I(A(u)); u ∈ |I|} if the infimum is no minimum
1
sup{I(A(u)); u ∈ |I|}
1 These quantifiers can be defined by
otherwise if the supremum is no maximum otherwise.
• ∀W xA(x) :⇔ ∃x(A(x) ⊃ ∀yA(y)) ⊃ ∀yA(y),
M. Baaz, O. Fasching / Annals of Pure and Applied Logic 161 (2009) 121–127
127
• ∃W xA(x) :⇔ ∃x(∃yA(y) ⊃ A(x)) ⊃ ∃yA(y), • ∀¬W xA(x) :⇔ (∃x(A(x) ⊃ ∀yA(y)) ⊃ ∀yA(y)) ⊃ ∀yA(y), • ∃¬W xA(x) :⇔ (∃x(∃yA(y) ⊃ A(x)) ⊃ ∃yA(y)) ⊃ ∃yA(y). Note that (∃x(A(x) ⊃ ∀yA(y))) ∨ (∃x(A(x) ⊃ ∀yA(y)) ⊃ ∀xA(x)) and (∃x(∃yA(y) ⊃ A(x)))) ∨ (∃x(∃yA(y) ⊃ A(x)) ⊃ ∃xA(x)) are valid in all Gödel logics. It is easy to show that ∀xA(x) ↔ ∀W xA(x)∧∀¬W xA(x) and ∃xA(x) ↔ ∃W xA(x)∧∃¬W xA(x) are valid in all Gödel logics. Acknowledgement The second author was partially supported by FWF project P19872. References [1] M. Baaz, Infinite-valued Gödel logics with 0–1-projections and relativizations, in: Gödel ’96, in: Lecture Notes Logic, vol. 6, 1996, pp. 23–33. [2] M. Baaz, A. Ciabattoni, C. Fermüller, Monadic fragments of Gödel logics: Decidability and undecidability results, in: Logic for Programming, Artificial Intelligence and Reasoning 2007, 14th Internat. Conf., LPAR 2007, in: Lecture Notes in Computer Science, vol. 4790, Springer, Berlin, 2007, pp. 77–91. [3] M. Baaz, N. Preining, R. Zach, First-order Gödel logics, Annals of Pure and Applied Logic 147 (1–2) (2007) 23–47. [4] M. Baaz, N. Preining, R. Zach, Characterization of the axiomatizable prenex fragments of first-order Gödel logics, in: 33rd IEEE Internat. Symposium on Multiple-Valued Logic, ISMVL 2003, IEEE Computer Society, 2003, pp. 175–180. [5] A. Beckmann, M. Goldstern, N. Preining, Continuous Fraïssé Conjecture, Order, Springer, 2008 (in press). [6] A. Beckmann, N. Preining, Linear Kripke frames and Gödel logics, Journal of Symbolic Logic 72 (1) (2007) 26–44. [7] P. Cintula, P. Hájek, On theories and models in fuzzy predicate logics, Journal of Symbolic Logic 71 (3) (2006) 863–880. [8] N. Preining, Complete recursive axiomatizability of Gödel logics, Ph.D. Thesis, Vienna University of Technology, 2003. [9] M. Takano, Another proof of the strong completeness of the intuitionistic fuzzy logic, Tsukuba Journal of Mathematics 11 (1) (1987) 101–105. [10] G. Takeuti, S. Titani, Intuitionistic fuzzy logic and intuitionistic fuzzy set theory, Journal of Symbolic Logic 94 (1984) 851–866. [11] G. Takeuti, S. Titani, Global intuitionistic fuzzy set theory, in: In the Mathematics of Fuzzy Systems, ISR Interdiscip. Systems, 1986, pp. 291–301. [12] S. Titani, Completeness of global intuitionistic set theory, Journal of Symbolic Logic 62 (2) (1997) 506–526.