G"odel homomorphisms - Semantic Scholar
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Fundamenta Informaticae 2012 1–15
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IOS Press
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G¨odel homomorphisms as G¨odel modal operators
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Oliver Fasching∗
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Institute of Discrete Mathematics and Geometry Vienna University of Technology Wiedner Hauptstrasse 8/E104.2, 1040 Vienna, Austria [email protected]
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Abstract. We extend propositional G¨odel logic by a unary modal operator, which we interpret as G¨odel homomorphisms, i.e. functions [0, 1] → [0, 1] that distribute over the interpretations of the binary connectives of G¨odel logic. We show that validity in the propositional fragment has a simple superintuitionistic Hilbert-type proof system, which is not structurally complete, and that validity does not change if we use the function class of continuous, strictly increasing functions. We also give proof systems for restrictions to sub- and superdiagonal functions.
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Keywords: G¨odel logic; superintuitionistic logic; modal logic; truth stressers.
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Introduction
Modal logic extensions are widely studied for BL and other fuzzy core logics, see e.g. [3]. In this context, unary connectives with a range of different features are often called operators or hedges, see, e.g., [6] and also their list of references. Baaz’ ∆-operator can be understood as an early extremal example, already treated in [8]. As [6] remark, it is often difficult to find a semantics that enjoys standard completeness, with G¨odel logics being an exception. Therefore we confine ourselves in this technical paper to propositional G¨odel logics and to interpretations of operators as functions in [0, 1]. As G¨odel logic is the t-norm based logic of relative comparison on [0, 1], it is natural to study the extensions by unary operators that are interpreted by [0, 1]-functions that are (1) order isomorphisms, preserving 0 and 1, or (2) G¨odel homomorphisms, i.e. homomorphisms of the binary functions on [0, 1] induced by the binary connectives of G¨odel logic. ∗
supported by the Austrian Science Fund (FWF): P22416
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O. Fasching / G¨odel homomorphisms as G¨odel modal operators
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We present proof systems for them in Corollary 4.2 and Theorem 3.1, resp. The said functions do not act as truth stressers (subdiagonal) or truth depressers (superdiagonal) but Corollaries 4.1 and 4.2 show that these restrictions can be easily rendered by proof systems. In Proposition 3.1, we give an exposition of Dummett’s method [5] of proving completeness of IPL + LIN for G¨odel logic; this yields a much more comprehensible proof of Theorem 3.1 than reproving all necessary steps as done in [2] for a similar but simpler result. It might appear that the restriction to a [0, 1]-valued semantics instead of a Kripke semantics oversimplifies the task of finding a sound and complete proof system but Theorem 3.1 and Corollary 4.3 show that a straightforward translation of the desired properties into axioms or rules is not enough; axiom (G1) in Definition 2.6 must be taken into account. As [2] demonstrates, a seemingly harmless [0, 1]-valued operator in the first-order fragment of (unwitnessed) G¨odel logic may even transfer so much semantical expressiveness from Łukasiewicz logic to make the logic fail to have a r.e. proof system. One of the disadvantages of our approach is, however, that the semantics we consider always makes o(A ⊃ B) ↔ (oA ⊃ oB) true, which is stronger than the normality axiom o(A ⊃ B) ⊃ (oA ⊃ oB) and thus might be unsuitable for applications. The structure of the paper is as follows: In Section 2, we give basic definitions, the definition of the semantics and we prove soundness of the proof system Go w.r.t. G¨odel homomorphisms. In Section 3, we give definitions only needed for the proof of completeness, which is then shown in Theorem 3.1. We also prove that Go is not structurally complete. In Section 4, we show the independence of the axiom scheme (G1) from the other axioms by using a very simple Kripke semantics with crisp accessibility relations; further key results are Corollaries 4.1 and 4.2 for order isomorphisms.
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Preliminaries
Definition 2.1. The following function definitions allow us to succinctly express G¨odel interpretations of formulas: For all x, y ∈ R, let x E y := 1 for x ≤ y; x E y := y for y < x; x C y := 1 for x < y; x C y := y for y ≤ x; x ./ y := 1 for x = y; x ./ y := min{x, y} for x 6= y. These binary functions clearly induce functions [0, 1] × [0, 1] → [0, 1]. Definition 2.2. The set F of G¨odel homomorphisms consists of all functions f : [0, 1] → [0, 1] such that f (min{x, y}) = min{f (x), f (y)}, f (max{x, y}) = max{f (x), f (y)}, and f (x E y) = f (x) E f (y) for all x, y ∈ [0, 1]. For each d ∈ [0, 1], define the lifting function hd : [0, 1] → [0, 1] by hd (x) := x for x ≤ d and hd (x) := 1 for d < x. Clearly, h1 is the identity on [0, 1]. We immediately obtain: Proposition 2.1. (a) A function f : [0, 1] → [0, 1] is in F is and only if f (1) = 1 and x < y ⇒ (f (x) < f (y) ∨ f (y) = 1) for all x, y ∈ [0, 1]. (b) F contains the constant function 1 and all lifting functions. (c) F is closed under composition and multiplication, i.e. (x 7→ f (g(x))) ∈ F and (x 7→ f (x) · g(x)) ∈ F for all f, g ∈ F. Definition 2.3. Let F0 denote the set of strictly increasing, continuous, piecewise linear functions f : [0, 1] → [0, 1] such that f (1) = 1. By the above, F0 ( F.
O. Fasching / G¨odel homomorphisms as G¨odel modal operators
(MP) (IPL1) (IPL2) (IPL3) (IPL4)
A A⊃B B ⊥⊃A (A ∧ B) ⊃ A (A ∧ B) ⊃ B A ⊃ B ⊃ (A ∧ B)
(IPL5)
A ⊃ (A ∨ B)
(IPL6) (IPL7) (IPL8) (IPL9)
B ⊃ (A ∨ B) (A ⊃ C) ⊃ (B ⊃ C) ⊃ ((A ∨ B) ⊃ C) A⊃B⊃A (A ⊃ (B ⊃ C)) ⊃ (A ⊃ B) ⊃ (A ⊃ C)
Figure 1.
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Proof system IPL
Definition 2.4. The propositional language Lo of all propositional formulas Frmo contains the binary connectives ∧, ∨, ⊃, the nullary ⊥, the unary o, and the propositional variables Var. Its o-free fragment is denoted by L. We abbreviate ⊥ ⊃ ⊥ by >, (A ⊃ B) ∧ (B ⊃ A) by A ↔ B, and (B ⊃ A) ⊃ B by A ≺ B. Let K ⊆ [0,1] [0, 1]. Given f ∈ K, we say that a function I : Frmo → [0, 1] is an f -G¨odelinterpretation, or for short f -interpretation, if I(oA) = f (I(A)), I(⊥) = 0, I(A ∧ B) = min{I(A), I(B)}, I(A ∨ B) = max{I(A), I(B)} and I(A ⊃ B) = I(A) E I(B) holds for all A, B ∈ Frmo . A function I : Frmo → [0, 1] is an K-G¨odel-interpretation, or for short K-interpretation, if it is an f -G¨odelinterpretation for some f ∈ K. It clearly suffices to specify f ∈ K and I : Var → [0, 1] to define an f -G¨odel-interpretation. We say that A ∈ Frmo is K-valid if I(A) = 1 for all K-interpretations. G¨odel interpretations and validity in L are defined as reducts so that they match the usual definition. We tacitly use the fact that I(A ≺ B) = I(A) C I(B) and I(A ↔ B) = I(A) ./ I(B) hold for any (f -)G¨odel interpretation I. Definition 2.5. Figure 1 presents IPL, which is some standard Hilbert-style proof system of intuitionistic logic. Let (LIN) denote the axiom scheme (A ⊃ B) ∨ (B ⊃ A) of prelinearity. We put G := IPL + LIN. We will make use of the following well-known facts [5], proved by Dummett.
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Theorem 2.1. (a) G is sound and complete for L w.r.t. G¨odel semantics: An L-formula A is valid if and only if G ` A; indeed, the G-derivation of A is effective. (b) G ` (A ↔ B) ⊃ (E[A] ↔ E[B]) for all L-formulas A, B and any L-context E[·].
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Definition 2.6. Let Go be the proof system G extended by the necessitation rule
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A , oA
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o(A ⊃ B) ↔ (oA ⊃ oB)
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the axiom scheme and the following axiom schemes for all natural ` ≥ 1: (((A ⊃ B) ∧ (o` B ≺ o` A)) ∨ ((A ≺ B) ∧ (o` B ⊃ o` A))) ⊃ (A ∨ o` B).
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(G1)
To emphasise that we do not use (G1) in certain proofs, we define the proof system G0o by G+(GN)+(G0).
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Lemma 2.1. Go is sound, i.e. if Go proves an Lo -formula H, then H is F-valid and, in particular, F0 -valid. Proof: We hint only how to prove the soundness of (G1) because the rest is routine: We will derive a contradiction from the assumption that I is an f -interpretation for some f ∈ F such that I((((A ⊃ B) ∧ (o` B ≺ o` A)) ∨ ((A ≺ B) ∧ (o` B ⊃ o` A))) ⊃ (A ∨ o` B)) < 1 for some ` ≥ 1. Observe g := f ` ∈ F. Putting a := I(A), b := I(B) we have (∗1 ): 1 ≥ max{min{aEb, g(b)Cg(a)}, min{aCb, g(b) E g(a)}} > max{a, g(b)} ≥ g(b). We have b ≤ a because a < b, together with g(b) < 1, implies g(a) < g(b) and then (∗1 ) becomes absurd. As a = b implies g(a) = g(b) and thus refutes (∗1 ), we have b < a. It follows g(b) ≤ g(a) and then max{min{a E b, g(b) C g(a)}, min{a C b, g(b) E g(a)}} = b, which also refutes (∗1 ). t u
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Completeness
The proof of completeness of Go needs several lengthy preparations and will take up almost all of the rest of the paper. The basic idea, which goes back to Dummett, will be described just before Proposition 3.1. Definition 3.1. For the following notation, fix two fresh binary formal symbols and ⇔. Let A0 , . . . , AN be formulas in Lo such that ⊥, > ∈ {A0 , . . . , AN } so that N ≥ 1, in particular. If we have symbols i ∈ { , ⇔} for all i ∈ {0, . . . , N − 1} and if B0 , . . . , BN is a permutation of the list A0 , . . . , AN such that B0 = ⊥ and BN = >, the formal notation ⊥ 0 B1 1 . . . N −2 BN −1 N −1 > is called an (A0 , . . . , AN )-chain and abbreviates the formula (⊥ ♦0 B1 ) ∧ (B1 ♦1 B2 ) ∧ . . . ∧ (BN −1 ♦N −1 >); here ♦i :=≺ if i = , and ♦i := ↔ if i =⇔. Let C be an (A0 , . . . , AN )-chain of the form ⊥ 0 B1 1 . . . N −2 BN −1 N −1 >. (The following definitions are made w.r.t. this notation and not to the formula the chain abbreviates.) We define eqc(C) as the number of the n such that n = ⇔ so that 0 ≤ eqc(C) ≤ N . The binary relations C and ⇔C on {0, . . . , N } are given by i C j :⇔ i < j ∧ ∃k(i ≤ k < j ∧ k = ) and i ⇔C j :⇔ ∀k((i ≤ k < j ∨ j ≤ k < i) ⊃ k = ⇔). We write i C j for (i C j) ∨ (i ⇔C j). (Observe that C and ⇔C depend only on the i but not on the order of the formulas Ai in C.) We say C is void if 0 ⇔C N , or equivalently, if eqc(C) = N . For example, the (⊥, >, X, Y, Z)-chain C := ⊥ Z ⇔ Y X ⇔ > abbreviates (⊥ ≺ Z) ∧ (Z ↔ Y ) ∧ (Y ≺ X) ∧ (X ↔ >), and for p⊥ := 0, pZ := 1, pY := 2, pX := 3, p> := 4, we have, e.g., p⊥ C pZ , pZ C pX , pZ C p> , pY C p> and pX ⇔C p> . The following proposition renders the main steps in Dummett’s completeness proof [5] for G w.r.t. o-free G¨odel semantics in a way that admits an easy generalisation to Lo . As all the steps are (implicitly) contained in his paper, we will not re-prove it here but just make some remarks: A formula X is valid w.r.t. G¨odel semantics if and only if X evaluates to 1 under all (the finitely many) ordering types of orderings of the value 1 and of the (finitely many) variables of X. This semantical procedure to test validity can be simulated step-by-step by G, where chains take over the task of ordering types: The relations < and = correspond to ≺ and ↔, however, ≺ does not precisely render
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< since I(X ≺ >) = 1 holds even if I(X) = 1. This does not cause problems as there is no formula in G¨odel logic that can crisply distinguish the value 1. The method relies among other things on the following facts: (1) Deep substitution as in Theorem 2.1(b) is possible, which makes the method fail for other t-norm based logics. This is needed because the evaluation procedure must be able to simplify a proper subformula (by Proposition 3.1(d)). (2) The proof system is strong enough to perform case distinctions, i.e. G ` (A ≺ B) ∨ (A ↔ B) ∨ (B ≺ A). Proposition 3.1(a) will generalise this, and C plays the rˆole of one particular case among all distinguished cases. Proposition 3.1. Let A be a list of L-formulas A0 , . . . , AN such that ⊥, > ∈ {A0 , . . . , AN }. We have then: V (a) For every L-formula X, we can construct a derivation G ` X ↔ C (C ⊃ X) where C runs through all A-chains. If C is a void A-chain, we can construct a derivation G ` C ↔ ⊥. For the remaining items suppose also that E[·] is an L-context, that i, j, k ∈ {0, . . . , N } and that C is an A-chain of the form B0 0 B1 1 . . . N −2 BN −1 N −1 BN with all n ∈ { , ⇔}; observe B0 = ⊥ and BN = >. Then (b)–(e) hold and all derivations therein are effective. (b) We have either i C j, or i ⇔C j, or j C i. The relation ⇔C is an equivalence relation; C and C are transitive; i = implies i C i + 1; and i =⇔ implies i ⇔C i + 1. We have i C k if i C j ⇔C k or i ⇔C j C k. (c) If i C j then G ` C ⊃ (Bi ≺ Bj ) and G ` C ⊃ (Bi ⊃ Bj ). If i ⇔C j then G ` C ⊃ (Bi ↔ Bj ), G ` C ⊃ (Bi ⊃ Bj ), and G ` C ⊃ (Bj ⊃ Bi ). (d) If i C j then G ` C ⊃ (E[Bj ⊃ Bi ] ↔ E[Bi ]). If i C j then G ` C ⊃ (E[Bi ⊃ Bj ] ↔ E[>]). G ` C ⊃(E[Bi ∧Bj ]↔E[Bi ]), G ` C ⊃(E[Bj ∧Bi ]↔E[Bi ]), G ` C ⊃(E[Bi ∨Bj ]↔E[Bj ]), G ` C ⊃ (E[Bj ∨ Bi ] ↔ E[Bj ]). (e) (Projection property) For every variable-free L-context F [ · n ]0≤n≤N , i.e. with N + 1 gaps, there is an effective P ∈ {0, . . . , N } such that G ` C ⊃ (F [An ]n ↔ AP ); we will say G can evaluate F [An ]n to AP under C. If P ⇔C N , we have G ` C ⊃ F [An ]n . If contexts are handled with care (see also Proposition 3.4(a)), Proposition 3.1 immediately generalises from L to Lo : Proposition 3.2. Proposition 3.1 holds also for Lo under the following provisions: (1) The language of G must be extended to Lo , i.e. axiom schemes can be instantiated with Lo -formulas, likewise (MP); in particular, Go or G0o can be used instead of G to derive the formulas shown. (2) The gaps in the contexts must not be in the scope of any ring.—By (a)–(e) of this proposition, we shall refer to the corresponding parts of Proposition 3.1. Theorem 2.1(a) and Proposition 3.1(d) yield some results we will use later: Proposition 3.3. (a) Let H0 , . . . , Hn be L-formulas. If G ` H0 ↔ H1 , G ` H1 ↔ H2 , . . . , G ` Hn−1 ↔ Hn , then G ` H0 ↔ Hn .
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O. Fasching / G¨odel homomorphisms as G¨odel modal operators
(b) Let C be an A-chain B0 0 B1 1 . . . N −2 BN −1 N −1 BN for a list A of L-formulas. (Notice B0 = ⊥ and BN = >.) If i ∈ {0, . . . , N }, we have G ` (C ⊃ Bi ) ⊃ (C ↔ D), where D is the A-chain B0 0 . . . i−2 Bi−1 i−1 Bi ⇔ Bi+1 ⇔ . . . ⇔ BN . If i, j ∈ {0, . . . , N }, we have G ` (C ⊃ (Bi ∨ Bj )) ⊃ (C ↔ D), where D is the A-chain B0 0 . . . k−1 Bk ⇔ Bk+1 ⇔ . . . ⇔ BN for k := max{i, j}. The formal derivations in our systems have to be handled with more care than in [2], [1], [7] because the semantics here neither admits the deduction theorem nor the full equivalence axiom scheme: We have Go + A ` oA but any (x 7→ x2 )-interpretation I with I(A) := 12 yields I(A ⊃ oA) = 41 < 1. Extending I by I(B) := 1, we find I((A↔B)⊃(oA↔oB)) = 14 < 1. Neither the formula oA⊃A H´ajek considered √ in [9] can be valid since the (x 7→ x)-interpretation J with J(C) := 14 yields J(oC ⊃ C) = 14 < 1. However, Theorem 2.1(b) immediately generalises to part (a) of the following proposition. Proposition 3.4. We have for all formulas A, B in Lo : (a) G0o ` (A ↔ B) ⊃ (E[A] ↔ E[B]) if the gap in the Lo -context E is not in the scope of any ring. (b) G0o ` ok (A ⊃ B) ↔ (ok A ⊃ ok B) for all k ∈ N. (c) If G0o ` ok (A ⊃ B), then G0o ` ok+m A ⊃ ok+m B for all m ∈ N. (d) We have G0o ` ok (A B) ↔ (ok A ok B) for ∈ {∧, ∨, ⊃, ≺, ↔} and every k ∈ N. Proof: We will often just state the G-provability of an L-formula and leave it to the reader to check its validity and to apply Theorem 2.1(a) to obtain its formal derivation. (a) If the gap of E is not in the scope of any ring, (A ↔ B) ⊃ (E[A] ↔ E[B]) is an instance of a o-free, G-provable formula (A0 ↔ B 0 ) ⊃ (E 0 [A0 ] ↔ E 0 [B 0 ]), where A0 and B 0 are fresh variables and E 0 is an appropriately chosen o-free context. (b) We skip the trivial case k = 0. Suppose we have already shown G0o ` ok (A ⊃ B) ↔ (ok A ⊃ ok B) so that (∗1 ): G0o ` ok (A ⊃ B) ⊃ (ok A ⊃ ok B) and (∗2 ): G0o ` (ok A ⊃ ok B) ⊃ ok (A ⊃ B) follow from (IPL2) and (IPL3). Applying (GN) and (G0) to (∗1 ) yields G0o ` ook (A ⊃ B) ⊃ o(ok A ⊃ ok B). By (a) and (G0), we find G0o ` ook (A ⊃ B) ⊃ (ook A ⊃ ook B). From (∗2 ), we obtain G0o ` (ook A ⊃ ook B) ⊃ ook (A ⊃ B) in a similar way. Now (IPL4) yields the claimed G0o ` (ok+1 A ⊃ ok+1 B) ⊃ ok+1 (A ⊃ B). (c) Apply (GN) m-times to a given G0o -proof of ok (A ⊃ B) to obtain ok+m (A ⊃ B). The claim now follows from (b). (d, case ⊃) See (b). (d, case ≺) We have G0o ` ok ((B⊃A)⊃B)↔(ok (B⊃A)⊃ok B) and G0o ` ok (B⊃A)↔(ok B⊃ok A) by (b). The claim now follows from (a). (d, case ∧) Apply (c) to (IPL2), to (IPL3), and to (IPL4) to obtain ok (A∧B)⊃ok A, ok (A∧B)⊃ok B, and ok A ⊃ ok (B ⊃ (A ∧ B)); the latter yields ok A ⊃ (ok B ⊃ ok (A ∧ B)) by (a) and (b). Since (C 0 ⊃A0 )⊃(C 0 ⊃B 0 )⊃(A0 ⊃B 0 ⊃C 0 )⊃(C 0 ↔(A0 ∧B 0 )) is G-provable, we find ok (A∧B)↔(ok A∧ok B). (d, case ↔) The claim follows from the cases for ∧ and ⊃. (d, case ∨) The G-provable formula (B ⊃ A) ⊃ ((A ∨ B) ⊃ A) yields ok (B ⊃ A) ⊃ ok ((A ∨ B) ⊃ A) by (c). Applying both (a) and (b) twice, we find (ok B ⊃ ok A) ⊃ ok (A ∨ B) ⊃ ok A. We obtain (ok B ⊃ ok A) ⊃ ok (A ∨ B) ⊃ (ok A ∨ ok B) from the G-provable (R ⊃ S ⊃ T ) ⊃ (R ⊃ S ⊃ (T ∨ U )). Likewise, we find a G0o -proof of (ok A ⊃ ok B) ⊃ ok (A ∨ B) ⊃ (ok A ∨ ok B). By (IPL7) and (LIN), we have ok (A ∨ B) ⊃ (ok A ∨ ok B). For the converse direction, apply (c) to (IPL5) and (IPL6) to obtain ok A ⊃ ok (A ∨ B) and ok B ⊃ ok (A ∨ B). Now, ok (A ∨ B) ↔ (ok A ∨ ok B) follows from the G-provable (C 0 ⊃ (A0 ∨ B 0 )) ⊃ (A0 ⊃ C 0 ) ⊃ (B 0 ⊃ C 0 ) ⊃ (C 0 ↔ (A0 ∨ B 0 )).
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Definition 3.2. The ring depth of an Lo -formula is defined by rdp(V ) := 0 for all variables V , rdp(⊥) := 0, rdp(oA) := rdp(A) + 1, and rdp(A B) := max{rdp(A), rdp(B)} for ∈ {∧, ∨, ⊃}. A formula is in ring normal form if no binary connective is in the scope of any ring, i.e. ring powers ok can occur only directly in front of ⊥ or of a variable. The following proposition reduces the problem of finding a Go -proof for a valid Lo -formula A to valid formulas in ring normal form. Proposition 3.5. For every Lo -formula H, there exists an Lo -formula H 0 in ring normal form of same ring depth such that G0o ` H ↔ H 0 . Proof: Using Proposition 3.3(a), we transform H step-by-step into the required form. As long as there is a subformula ok (A B) of H that is not in the scope of a ring, use Proposition 3.4(a,d) to replace it by ok A ok B of equal ring depth; here, k ≥ 1, and A and B are arbitrary formulas in Lo . This algorithm clearly terminates and when it terminates, H 0 is in ring normal form. t u Definition 3.3. We will always consider a fixed R ∈ N r {0} and a fixed list of finitely many distinct variables V1 , . . . , VM so that we suppress the dependence of the following definitions on them. Let Z denote the list ⊥, o⊥, . . . , oR ⊥, V1 , oV1 , . . . , oR V1 , . . . , VM , oVM , . . . , oR VM , >. We write A ∈ Z to indicate that A occurs in Z. As all elements in Z are distinct, we can define w.r.t. a Z-chain C := B0 0 B1 1 . . . N −2 BN −1 N −1 BN that a formula A ∈ Z has position i in C if A = Bi . Suppressing also the dependence on C, we use P0r , Pnr , P> for the position of or ⊥, or Vm , > in C; here r ∈ {0, . . . , R} and m ∈ {1, . . . , M }, i.e., we treat ⊥ like a variable in this respect and consequently put V0 := ⊥ to unify the notation. Definition 3.4. A non-void Z-chain C is reduced, using the notation of Definition 3.3, if for all m, n ∈ r P s , we have {0, . . . , M }, r, s ∈ {0, . . . , R}, ` ∈ Z such that 0 ≤ r + ` ≤ R, 0 ≤ s + ` ≤ R and Pm n r+` s+` r+` s+` Pm Pn or Pm ⇔ Pn ⇔ P> . Lemma 3.1. Let C be a Z-chain and use the notation Z from Definition 3.3. Then we can find a Z-chain D such that Go ` C ↔ D and such that D is reduced or void. Proof: We use all notations from Definition 3.3 so that, e.g., V0 is now defined. We will often tacitly apply Proposition 3.2(b). To prove the claim, it suffices by Proposition 3.3(a) to reduce C stepwise. The following steps (1) and (2) of this reduction loop keep C in Z-chain form. As they will be seen to also properly increase the bounded eqc(C), they are applicable only for a finite number of times and hence the loop terminates. Since we may assume that the loop immediately terminates if C becomes void, we will establish the claim by eventually proving that a non-void chain C is reduced if none of the reduction steps is applicable.—We write , ⇔, instead of C , ⇔C , C . r P s , P s+` P r+` , P s+` P for some (1) We claim that C can be further reduced if Pm > n n m n m, n ≤ M , r, s ≤ R, ` ≥ 1 with max{r, s} + ` ≤ R. Proof: Proposition 3.2(c) shows Go ` C ⊃ (or Vm ≺ os Vn ) and Go ` C ⊃ (os+` Vn ⊃ or+` Vm ), which we apply together with (G1) to an appropriate instance of G ` (U ⊃ X) ⊃ (U ⊃ Y ) ⊃ (((X ∧ Y ) ∨
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Q) ⊃ W ) ⊃ (U ⊃ W ) in order to obtain Go ` C ⊃ (or Vm ∨ os+` Vn ). Proposition 3.3(c) now yields a Z-chain C 0 such that Go ` C ↔ C 0 , such that C 0 results from C just by replacing certain -symbols by r ⇔ 0 P or P s+` ⇔ 0 P . We have eqc(C) < eqc(C 0 ) since P r P ⇔-symbols, and such that Pm > > > C C n m and Pns+` P> . (2) We claim that C can be further reduced whenever there are m, n ≤ M , r, s ≤ R, ` ≥ 1 with r+` P s+` , P s P r , P s P . The proof is analogous to the one max{r, s} + ` ≤ R such that Pm > n n m n of (1). Using (1) for ` > 0 and (2) for ` < 0 in Definition 3.4, it easily follows that C is reduced. t u Proposition 3.6. Let C be a reduced Z-chain and use the notation of Definition 3.3. Then: r ⇔ P s P and ` ∈ Z such that 0 ≤ r + ` ≤ R and 0 ≤ s + ` ≤ R, then P r+` ⇔ P s+` . (a) If Pm > n m n r ⇔ P r+1 P for some r < R, then P 0 ⇔ . . . ⇔ P r ⇔ P r+1 ⇔ . . . ⇔ P R . (b) If Pm > m m m m m r P r+1 for some r < R then P 0 P 1 . . . P r P r+1 . . . P R−1 P R . (c) If Pm m m m m m m m r+1 P r for some r < R then P R P R−1 . . . P r+1 P r . . . P 1 P 0 and (d) If Pm m m m m m m m R P0 . Pm m s P r , P s P , P r P s , P r P , then either P s ⇔ P r ⇔ P s ⇔ P r , or P r P r , (e) If Pm > > n m n n m n m m n m n s. or Pns Pm (f) For each m ≤ M , exactly one of the following cases holds: 0 ⇔ P1 ⇔ ... ⇔ PR ⇔ P , (A1) Pm > m m 0 ⇔ P1 ⇔ ... ⇔ PR P , (A2) Pm > m m 0 P1 ... PR P , (A3) Pm > m m R P R−1 . . . P 0 P , (A4) Pm > m m 0 P 1 . . . P k ⇔ P k+1 ⇔ . . . ⇔ P R ⇔ P for some k with 0 < k ≤ R, (A5) Pm > m m m m R P R−1 . . . P k ⇔ P k−1 ⇔ . . . ⇔ P 0 ⇔ P for some k with 0 ≤ k < R. (A6) Pm > m m m m r+` P s+` since C is reduced, which would imply P r P s or P r ⇔ Proof: (a) We cannot have Pm n m n m s r r+` cannot hold. The claim Pn ⇔ P> , contradicting Pm ⇔ Pns P> . We similarly find that Pns+` Pm now follows from Proposition 3.2(b). (b) follows immediately from (a). (c) Since C is reduced, we have P r+` P r+`+1 or P r+` ⇔ P r+`+1 ⇔ P> , i.e. P r+` P r+`+1 k P r P for all k ≤ m, we have, by (b), that for all ` ∈ Z such that 0 ≤ r + ` + 1 ≤ R. Since Pm > m k k+1 there is no k < r with Pm ⇔ Pm , which proves the claim. (d) has a proof similar to the one of (c). r P r and or P s P s since this (e) The cases are clearly disjoint, e.g. we cannot have both Pm n n m r r s s r P r , nor s would yield Pm Pm Pn Pn Pm , which is absurd. Suppose we neither have Pm n s . Since P r P r , P r P and C is reduced, we see that we cannot have P s P s ; We Pns Pm > n m n m n r is impossible. Thus P s P s P r P r P s , as claimed. similarly obtain that Pnr Pm n m m n n (f) By Proposition 3.2(b), the cases are clearly distinct. In order to prove their disjunction, we disr P r+1 for some r ≤ R − 1, we can choose r maximal with tinguish several cases: In the case of Pm m R P , and we see (A5) if r + 1 = R that property and apply (d). We find (A3) if r + 1 = R and Pm > R ⇔ P . Thus we may assume r + 2 ≤ R. Maximality of r yields P r+2 P r+1 so that and Pm > m m r+2 ⇔ P r+1 ⇔ P since C is reduced. The claim is now proved for this case because (A5) is estabPm > m r+1 P r for some r ≤ R − 1, we find in a similar way that (A4) or (A6) holds. lished. In the case of Pm m In the remaining case, we see (A1) or (A2), which completes the proof.
O. Fasching / G¨odel homomorphisms as G¨odel modal operators
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Definition 3.5. Let C be a reduced Z-chain and use the notation of Definition 3.3. Let mSn for any a P b ∧ ∃c, d ≤ R. P c P d so that S is a reflexive and m, n ≤ M abbreviate ∃a, b ≤ R. Pm n n m symmetric relation. By ∼, we denote the transitive closure of S, i.e. ∼ is an equivalence relation on {0, . . . , M }. Proposition 3.7. Let C be a reduced Z-chain and use the notation of Definition 3.3. Then: 0 P R , then P 0 P R or P 0 ⇔ P R ⇔ P . (a) If m ∼ n, Pm > m n n n n R P 0 , then P R P 0 or P 0 ⇔ P R ⇔ P . (b) If m ∼ n, Pm > m n n n n 0 ⇔ P R P , then P 0 ⇔ P R ⇔ P 0 ⇔ P R . (c) If m ∼ n, Pm > m m m n n (d) The equivalence classes of ∼ can be linearly ordered by , i.e. for all n1 6∼ n2 , we have either a P b for all a, b ≤ R and m ∼ n and m ∼ n , or P a P b for all a, b ≤ R and m ∼ n Pm 1 1 2 2 1 1 m2 m2 m1 1 and m2 ∼ n2 . Proof: We begin with a few observations: (1) We claim mSk if mSn, nSk, Pn0 ⇔ PnR : By inspection of (A), we have Pn0 ⇔ Pni for all i ≤ R. 0 ⇔ P R and P 0 ⇔ P 0 ⇔ P R . In case Pn0 P> , we apply (B3) to mSn and to kSn so that Pn0 ⇔ Pm m n k k R 0 R 0 a b for some a, b ≤ R by mSn Thus Pm ⇔ Pm ⇔ Pk so that mSk. In case Pn ⇔ P> , we find Pn Pm b ⇔ P d , thus and Pnc Pkd for some c, d ≤ R by nSk. It follows from P> ⇔ Pna ⇔ Pna that P> ⇔ Pm k mSk. This establishes the claim. 0 P R : Suppose the contrary so (2) We claim Pn0 PnR or Pn0 ⇔ PnR ⇔ P> if mSn and Pm m 0 P R , P R P 0 . It follows that P R P R or P 0 P 0 or by Proposition 3.6(e), that mSn, Pm m n n m n n m 0 ⇔ P R ⇔ P 0 ⇔ P R would be absurd. We cannot have P R P R since this would imply since Pm m n n m n 0 P R P R P 0 so that, applying Proposition 3.6(f) twice, would yield P 0 P i P R Pm m n n m m m j PnR Pm Pn0 for all i, j ≤ R, contradicting mSn. We similarly obtain that we cannot have 0 , which establishes the claim.—Likewise, we see (3) and (4): Pn0 Pm R P0 . (3) We have PnR Pn0 or Pn0 ⇔ PnR ⇔ P> if mSn and Pm m 0 R 0 R 0 R P . (4) We have Pm ⇔ Pm ⇔ Pn ⇔ Pn if mSn and Pm ⇔ Pm > Since m ∼ n, there are I ∈ N and ki ≤ R for all i < I such that mSk0 Sk1 S . . . SkI−1 Sn. We may assume I ≥ 1 by (2). We may also assume that Pk0i ⇔ PkRi for no i < I for otherwise (1) would allow us to pass to mSk0 S . . . Ski−1 Ski+1 S . . . Sn. Now, (a) follows inductively by (2). Using (3) and (4), we obtain similar proofs for (b) and (c). a P b , then P c P d for all c, d ≤ R: By m 6∼ k, we cannot have (5) Whenever mSn 6∼ k and Pm n k k a P b implies P e P f for all e, f ≤ R. Suppose we had, to the mSk and thus, by definition of S, Pm m k k contrary, Pkd Pnc for some c, d ≤ R; since nSk cannot hold, we see by definition of S that Pkf Png e P g for all e, g ≤ R, contradicting mSn.—Similarly: for all f, g ≤ R and hence Pm n b , then P c P d for all c, d ≤ R. (6) Whenever mSn 6∼ k and Pka Pm n k Let n1 6∼ n2 so that, in particular, not n1 Sn2 and thus we have either Pn01 Pn02 or Pn02 Pn01 . Now, (d) follows inductively (like in (a)) from (5) and (6). t u The proof idea of Lemma 3.2 below is to construct, for each chain C, an f ∈ F0 and an f -G¨odelinterpretation I : Var → [0, 1] that tries to embed the total preorder given by C and ⇔C in a suitable
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O. Fasching / G¨odel homomorphisms as G¨odel modal operators
way on [0, 1]. As the equivalence classes in the sense of Proposition 3.7(d) are linearly ordered, we will first give in Proposition 3.8 such a realisation for each equivalence class and then compose them in Lemma 3.2. In contrast to the situations in [1] and [2], observe that C and ⇔C are not exactly r ⇔ P can be realised as values in [0, 1); then I(C) < 1 follows but realised as < and = but points Pm C > the realisation is still good enough to prove I(C) > I(os Vk ) for all Pks C P> . Dropping this feature of exact realisation makes it possible to find an f ∈ F0 r F.—The following proposition has a rather technical proof; we leave several easy details to the reader. Proposition 3.8. Let C be a reduced Z-chain and use the notation from Definition 3.3. Let w ≤ M and denote by W := {n ≤ M ; n ∼ w} its ∼-equivalence class. Denote the condition 0 ∈ W by (∗0 ), and let (∗1 ) denote the condition that Pnr ⇔ P> for some r ≤ R, n ∈ W . Suppose 0 ≤ α < β ≤ 1, but, however, we require α = 0 if (∗0 ), and β = 1 if (∗1 ). Moreover, let (∗↑ ) and (∗↓ ) denote the conditions that Pn0 PnR for some n ∈ W , and PnR Pn0 for some n ∈ W , resp. Then there are Jn ∈ [α, β] for every n ∈ W and a strictly increasing, continuous, piecewise linear function f : [α, β] → [α, β] with the following properties: If (∗0 ), then J0 = 0. If (∗0 ) fails, then f (α) = α. We have f (β) = β. Moreover, for all r, s ≤ R and n, m ≤ M and x ∈ [α, β]: We have α ≤ f r (Jn ) ≤ β and, in particular, f r (Jn ) is well-defined. If (∗0 ) fails, then α < f r (Jn ). If (∗1 ) fails, then f r (Jn ) < β. If (∗↑ ), then x ≤ f (x). If (∗↓ ), then f (x) ≤ x. If neither (∗↑ ) nor (∗↓ ), s P implies f r (J ) = f s (J ) < 1; in particular, P r P then x = f (x). We have that Pnr ⇔ Pm n m > > n r r s implies f (Jn ) < 1; that Pn Pm implies f r (Jn ) < f s (Jm ); and that Pn0 ⇔ PnR ⇔ P> implies Jn = 1.—Finally, observe that (∗0 ) fails if (∗↓ ). Proof: If we neither have (∗↑ ) nor (∗↓ ), we have Pn0 ⇔ PnR for all n ∈ W . Take f (x) := x and define Jn for all n ∈ W as follows: Put Jn := 0 if w = 0; put Jn := 1 if Pw0 ⇔ P> ; and Jn := (α + β)/2 otherwise. It can be easily verified by Proposition 3.7(c) that exactly one of this cases must occur and that the claimed properties follow then. Let us now suppose (∗↓ ). By Proposition 3.7(b), we see that for each m ∈ W only the cases (A1), (A4), (A6) in Proposition 3.6(f) are possible; in particular, we cannot have (∗0 ) since C is not void and since P00 is the least position in C. Let W0 := W ∪ {0}, H := {(r, n); r ≤ R, n ∈ W0 } ∪ {>} and define relations ≺ and ↔ on H by the following: For all r, s ≤ R and n, k ∈ W , we put (r, n) ≺ (s, k) :⇔ PnR−r PkR−s ; (r, n) ↔ (s, k) :⇔ PnR−r ⇔ PkR−s ; (r, n) ≺ > :⇔ PnR−r P> ; (r, n) ↔ > :⇔ PnR−r ⇔ P> ; (r, 0) ≺ (s, 0) :⇔ r < s; (r, 0) ↔ (s, 0) :⇔ r = s; and put (r, 0) ≺ (s, k) always. We use Proposition 3.2(b), 0 6∼ n, Proposition 3.6(f) and (a), and the fact that C is reduced to verify the conditions of Lemma 5 in [2]. The conditions involving the dummy pairs (r, 0) can be easily checked separately. Thus there are q ∈ [0, 1] ∩ Q and gnr ∈ [0, 1] ∩ Q for all r ≤ R and n ∈ W0 such that for all r, s ≤ R and n, k ∈ W we have: PnR−r PkR−s if and only if gnr < gks ; PnR−r ⇔ PkR−s if and only if gnr = gks ; PnR−r P> if and only if gnr < 1; PnR−r ⇔ P> if and only if gnr = 1; moreover, 0 = g00 < gnr always; and gnr+1 = min{1, q + gnr } if r < R. Since there is n ∈ W with PnR Pn0 by assumption, it follows gn0 < gnR = min{1, Rq + gn0 } and hence x+(R+2)q 0 < q and gn0 < 1. Let d(x) := α + (β − α)( 41 + 2(2+(R+2)q) ). Put Jn := 1 for all n ∈ W R with Pn ⇔ P> , which is possible since then β = 1; and put Jn := d(gn0 ) for all n ∈ W with q(β−α) PnR P> . Put c := 2(2+(R+2)q) . Let f be the function that linearly interpolates the coordinates (α, α), (d(−(R + 1)q), d(−(R + 1)q) − c), (d(1 + q), d(1 + q) − c), (β, β). It is easy to verify that 0 r 0 0 α < α + β−α 4 ≤ d(−(R + 1)q) < d(−(R + 1)q) + c = d(−Rq) < d(gn − rq) = f (gn ) = d(gn ) − rc ≤
O. Fasching / G¨odel homomorphisms as G¨odel modal operators
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d(1) = d(1 + q) − c < d(1 + q) ≤ α + 3(β−α) < β for all n ∈ W with Pn0 P> . Observe that d is 4 strictly increasing. The claimed implications can now be easily verified. For (∗↑ ), we only sketch the proof: Apply Proposition 3.7(a) and then Proposition 3.6(f). The basic idea is to put (r, n) ≺ (s, k) :⇔ Pnr Pks and (r, n) ↔ (s, k) :⇔ Pnr ⇔ Pks . Depending on the absence of (∗0 ) or (∗1 ), dummy bottom or top elements have to be added; for that purpose, we extend W by 0 if (∗0 ) does not hold. If (∗0 ), Lemma 5 in [2] guarantees g00 = 0. Now, (∗↑ ) guarantees q > 0 and hence c > 0, where c is defined as before. In case of (∗0 ), we have α = 0 by assumption so that we put xβ d(x) := 2(2+(R+2)q) and let f linearly interpolate (0, c), (d(1 + Rq), d(1 + Rq) + c), (β, β); in case that x ) and let f linearly interpolate (α, α), (∗0 ) does not hold, we put d(x) := α + (β − α)( 14 + 2(2+(R+2)q) (d(0), d(0) + c), (d(1 + Rq), d(1 + Rq) + c), (β, β) so that α < α + β−α 4 ≤ d(0). We put Jn := 1 if Pn0 ⇔ P> and Jn := d(gn0 ) otherwise. No matter if (∗0 ) or not, we have d(0) ≤ d(gn0 + rq) = f r (gn0 ) = d(gn0 ) + rc ≤ d(1 + Rq) < d(1 + Rq) + c ≤ α + 3(β−α) < β for all r ≤ R and n ∈ W such that 4 Pn0 P> . The claimed implications can now be easily verified. t u Lemma 3.2. Let C be a reduced Z-chain and use the notation from Definition 3.3. Then there is an F0 -G¨odel interpretation I such that I(C ⊃ ot Vk ) < 1 for all t ≤ R and k ≤ M with Pkt P> . (Remark: The case k = 0 is permitted.) Proof: By Proposition 3.7(d), induces a linear order on the equivalence classes of ∼. By Proposition 3.7(a)–(c), the same is true even if ∼ is extended from {0, . . . , M } to {0, . . . , M, >} by defining m ∼ > r ⇔ P and by completing it to an equivalence relation. Let W , . . . , W ⊆ {0, . . . , :⇔ ∃r ≤ R. Pm 0 E > M, >} denote all equivalence classes with W0 being -minimal and WE -maximal. Observe that r for all r ≤ R and m ≤ M , and that W contains > by construction, W0 contains 0 since P00 Pm E including the possibility that WE is just {>}. e+1 e For each e ∈ {0, . . . , E} construct fe on [ E+1 , E+1 ] and Je,n , for all n ∈ We , according to Proposition 3.8; however, observe the following: For WE , we have to put JE,> := 1 since the said Proposition does not assign a value to JE,> . If Pn0 ⇔ PnR ⇔ P> for all n ∈ WE , the conditions of the 1 said Proposition are not met and we have to put JE,n := 1 then. Since W0 is used for [0, E+1 ] and WE E is used for [ E+1 , 1], the boundary conditions implied by (∗0 ) and (∗1 ) in Proposition 3.8 are fulfilled. Since the (We )e are a partition of {0, . . . , M, >}, it is well-defined S to put I(Vn ) := Je,n for all e e n ∈ We . Since fe ( E+1 ) = E+1 for all e ∈ {1, . . . , E + 1}, also f := e fe is a well-defined function in F0 . Observe for later use that Proposition 3.8 yields the following implications: I(or Vn ) = I(os Vm ) < 1 s P since n ∼ m then. I(or V ) < 1 holds if P r P . We also prove holds if Pnr ⇔ Pm n > > n s : If n ∼ m, this follows from Proposition 3.8 so that we assume n 6∼ m. I(or Vn ) < I(os Vm ) if Pnr Pm By Proposition 3.7(d), we have n ∈ Wi and m ∈ Wj for some i, j ≤ E with i < j. Again by Proposition j i+1 ≤ E+1 < I(os Vm ) by applying Proposition 3.7(d), we find n 6∼ > and m 6∼ 0 and thus I(or Vn ) < E+1 3.8 twice. Recall the notation from Definition 3.3; we claim I(C) = min{ min
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By Proposition 3.2(c) and soundness of Go , we have 1 = I(C ⊃(Bi ≺ Bj )) and thus I(C) ≤ I(Bi ≺ Bj ) for all i, j with i C j and, similarly, 1 = I(C ⊃ (Bi ↔ Bj )) and thus I(C) ≤ I(Bi ↔ Bj ) for
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O. Fasching / G¨odel homomorphisms as G¨odel modal operators
all i, j with i ⇔C j. The claim now follows since the converse inequality holds because of I(C) = min{I(⊥ ♦0 B1 ), I(B1 ♦1 B2 ), . . . , I(BN −1 ♦N −1 >)}, where ♦i :=≺ if i = , and ♦i := ↔ if i =⇔. Hence I(C) = min{minPnr Pms I(or Vn ≺ os Vm ), minPnr ⇔Pms I(or Vn ↔os Vm ), minPnr P> I(or Vn ≺ >), minPnr ⇔P> I(or Vn ↔ >)}; here the running indices r, s and m, n are suppressed. As proved earlier, we have I(ot Vk ) < 1. It follows for all n, m ≤ M and r, s ≤ R: We have I(ot Vk ) < 1 = I(or Vn ≺ >) by definition of C. If Pnr ⇔ P> then Pkt Pnr and thus, s then I(or V ) < I(os V ) and as proved earlier, I(ot Vk ) < I(or Vn ) = I(or Vn ↔ >). If Pnr Pm n m s and P s P t , then P r ⇔ P s P so that thus I(ot Vk ) < 1 = I(or Vn ≺ os Vm ). If Pnr ⇔ Pm > m n m k s and P t P s , then P t P r so I(or Vn ) = I(os Vm ) and thus I(or Vn ↔ os Vm ) = 1. If Pnr ⇔ Pm m n k k that I(ot Vk ) < I(or Vn ) and I(ot Vk ) < I(os Vm ), thus I(ot Vk ) < min{I(or Vn ), I(os Vm )} ≤ I(or Vn ) ./ I(os Vm ) = I(or Vn ↔ os Vm ). Hence I(ot Vk ) < I(C) and thus I(C ⊃ ot Vk ) = I(ot Vk ) < 1. t u Theorem 3.1. For any Lo -formula H, the following are equivalent:
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(1) Go ` H.
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Moreover, the derivation in (1) is effective, and Go is a conservative extension of G. Proof: By Lemma 2.1, (1) implies (2). Since F0 ⊂ F, (2) implies (3). Let now H be an F0 -valid Lo -formula. We have to show Go ` H. Let V1 , . . . , VM denote all the variables of H. By Proposition 3.5, there is a Lo -formula H 0 with Go ` H ↔ H 0 and with same ring depth R. By soundness, we have 1 = I(H ↔ H 0 ) = I(H) ./ I(H 0 ) = 1 ./ I(H 0 ) = I(H 0 ) for any F0 - or F-interpretation I. Now that we have fixed R and M , we can use the notation from Definition 3.3: Let now C be an arbitrary Z-chain; we have to prove Go ` C ⊃H 0 . If C is void, Proposition 3.2(a) yields Go ` C ↔⊥ and thus Go ` C ⊃ H 0 , so that we may suppose that C is non-void. Thus, by Lemma 3.1, there is a reduced Z-chain C 0 with Go ` C ↔ C 0 . By Proposition 3.2(e), there is W ∈ Z such that Go ` C 0 ⊃ (H 0 ↔ W ). If W = >, we clearly have Go ` C 0 ⊃ H 0 and thus also Go ` C ⊃ H 0 , as required; hence we may assume W = ot Vk for some t ≤ R and k ≤ M . If we had Pkt C 0 P> , then there is an F0 interpretation I such that I(C 0 ⊃ot Vk ) < 1 by Lemma 3.2, but soundness and Go ` C 0 ⊃(H 0 ↔W ) imply 1 = I(C 0 ⊃ (H 0 ↔ ot Vk )) = I(C 0 ) E I(H 0 ↔ ot Vk ) = I(C 0 ) E (I(H 0 ) ./ I(ot Vk )) = I(C 0 ) E I(ot Vk ) = I(C 0 ⊃ ot Vk ), which is absurd. Therefore, we have Pkt ⇔C 0 P> . This and the fact that H 0 is in ring normal form yields Go ` C 0 ⊃ H 0 from the last sentence in Proposition 3.2(e). Since Go ` C ↔ C 0 , we have Go ` C ⊃ H 0 . V The previous paragraph yields Go ` C (C ⊃ H 0 ), where C runs through all Z-chains. From Proposition 3.2(a), we obtain Go ` H 0 , and thus Go ` H from Go ` H ↔ H 0 . Tracing the construction of the derivation of H, we can readily see that all steps are effective. t u Since f ∈ F0 , f (x) = 1, x ∈ [0, 1] implies x = 1, we immediately obtain:
O. Fasching / G¨odel homomorphisms as G¨odel modal operators
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Corollary 3.1. If Go ` oH, then Go ` H.
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Let 1 denote the constant function 1; observe 1 ∈ F. The proof system Go + oA is clearly sound for all {1}-valid formulas. Due to the 1-G¨odel-interpretation I defined by I(X) := 0 for all X ∈ Var, we find Go + oA 6` A. Hence we see:
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Corollary 3.2. Go is not structurally complete.
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Extra axioms and axiom independence
In this section, we present further extensions of Go and prove the necessity to include (G1) in Go . The method of proof of Theorem 3.1 allows an easy generalisation when additional axioms oA ⊃ A, A ⊃ oA or o⊥ ⊃ ⊥ are present. For the axiom oA ⊃ A, see [9], in particular. Define F↓ := {f ∈ F : f (x) ≤ x for all x ∈ [0, 1]}, F↑ := {f ∈ F : x ≤ f (x) for all x ∈ [0, 1]}, and similarly F0↓ and F0↑ . Notice that all functions in F↓ and F0↓ are strictly increasing. Corollary 4.1. (a) For any Lo -formula H, the following are equivalent: (1) G0o + oA ⊃ A ` H. (2) H is F↓ -valid. (3) H is F0↓ -valid. (b) For any Lo -formula H, the following are equivalent: (1) G0o + A ⊃ oA ` H. (2) H is F↑ -valid. (3) H is F0↑ -valid. Proof: We only prove (a) since (b) is completely analogous. For soundness, use Lemma 2.1 and the fact that I(oA ⊃ A) = f (I(A)) E I(A) = 1 whenever I is an f -interpretation and f ∈ F↓ . We now prove completeness: Let G↓ denote the proof system G0o + oA ⊃ A. We denote by (L) the G-provable formula ((X ≺ Y ) ∧ (Y ⊃ X)) ⊃ (X ∧ Y ). G↓ proves (o` B ≺ o` A) ⊃ (B ≺ A) and (o` B ⊃ o` A) ⊃ (B ⊃ A) by Proposition 3.4(a,d). It is easy to show by (L) that G↓ proves every instance of (G1). We have to inspect the proof of Theorem 3.1 and refine it slightly: We call a reduced chain C ↓-reduced if Pnk+1 C Pnk holds for all positions. In Lemma 3.1, any chain C can be further reduced by (L) and by oA ⊃ A if Pnk C Pnk+1 holds, thus we find G↓ ` C ↔ D for some ↓-reduced chain D. Hence the case (∗↑ ) in Proposition 3.8 can never appear and now Lemma 3.2 yields an F0↓ -interpretation. The proof of Theorem 3.1 does not require any modification. t u By the same method as before, we also obtain: Corollary 4.2. (a) For any Lo -formula H, the following are equivalent: (1) Go + o⊥ ⊃ ⊥ ` H. (2) H is {f ∈ F : f (0) = 0}-valid. (3) H is {f ∈ F0 : f (0) = 0}-valid. (b) For any Lo -formula H, the following are equivalent: (1) G0o + A ⊃ oA + o⊥ ⊃ ⊥ ` H. (2) H is {f ∈ F↑ : f (0) = 0}-valid. (3) H is {f ∈ F0↑ : f (0) = 0}-valid. To show that (G1) cannot be omitted in the proof system Go , we give another semantics of Lo in Kripke style. For general Kripke semantics for G¨odel logics, see [4]. Definition 4.1. We can give the Lo -formulas a Kripke semantics as follows: Let E := N×Var [0, 1]. Extend any e ∈ E to e : N × Frmo → [0, 1] by e(w, ⊥) := 0, e(w, A ∧ B) := min{e(w, A), e(w, B)},
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e(w, A ∨ B) := max{e(w, A), e(w, B)}, e(w, A ⊃ B) := e(w, A) E e(w, B), and e(w, oA) := e(w + 1, A). We say that A ∈ Frmo is K-valid if e(w, A) = 1 for all e ∈ E and all w ∈ N.
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Definition 4.2. Let GK denote the proof system G0o + o⊥ ⊃ ⊥.
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Proposition 4.1. GK is sound and complete in Lo w.r.t. the semantics from Definition 4.1; in fact, for every K-valid Lo -formula, one can effectively find a GK -derivation. Proof: We have e(w, o(A ⊃ B)) = e(w + 1, A ⊃ B) = e(w + 1, A) E e(w + 1, B) = e(w, oA ⊃ oB) for all e ∈ E and w ∈ N, thus e(w, o(A ⊃ B) ↔ (oA ⊃ oB)) = 1. We also have e(w, ⊥ ↔ o⊥) = e(w, ⊥) ./ e(w + 1, ⊥) = 0 ./ 0 = 1 for all e ∈ E and w ∈ N. Clearly, the validity of A implies the validity of oA. It is also easy to check that all Lo -instances of G are valid. Thus GK is sound w.r.t. this semantics. Observe that GK ` ok ⊥ ↔ ⊥ for any k ≥ 0: We clearly have GK ` ⊥ ⊃ ok ⊥ for any k ≥ 0. For the converse direction, we may suppose k ≥ 1 and proceed by induction on k Apply (GN) k times to o⊥ ⊃ ⊥ and use Proposition 3.4(c) to find GK ` ok+1 ⊥ ⊃ ok ⊥. By the induction hypothesis GK ` ok ⊥ ⊃ ⊥, we obtain GK ` ok+1 ⊥ ⊃ ⊥, as claimed. We now prove completeness. Let H be K-valid. By Proposition 3.5 and soundness of GK , we may assume that H is in ring normal form so that H = F [ok Vm ]k≤R, 0≤m≤M such that R is the ring depth of H, V0 := ⊥, V1 , . . . , VM ∈ Var, F = F [·k,m ]k,m is a context neither containing o, neither ⊥, nor any variable. We apply GK ` ou ⊥ ↔ ⊥ for all u ≤ R and Proposition 3.4(a) to eliminate all rings in front of ⊥. By soundness of GK , we may assume that H = F 0 [ok Vm ]k≤R, 1≤m≤M where F 0 is a context that contains neither o, neither any variable, but may contain ⊥. Let Wk,m for k ≤ R and 1 ≤ m ≤ M be fresh variables. Then F 0 [Wk,m ]k≤R, 1≤m≤M is valid in the usual G¨odel semantics in L because to every G¨odel interpretation I in L we can define a Kripke interpretation e(k, Vm ) := I(Wk,m ) for k ≤ R, and 1 ≤ m ≤ M (and e(k, A) := 0 for k > R or A ∈ Var r {V1 , . . . , VM }) so that I(F 0 [Wk,m ]k,m ) = e(0, F 0 [ok Vm ]k,m ) = 1. By Theorem 2.1(a), we have G ` F 0 [Wk,m ]k,m and we can also build the Lo -instance G ` F 0 [ok Vm ]k,m if we allow Lo -instances in the axioms of G. In particular, GK ` H, as claimed. t u In fact, we will only use soundness of GK to prove that the scheme (G1) cannot be omitted from the proof systems in Theorem 3.1 and Corollary 4.2. Corollary 4.3. For every fixed ` ≥ 1, the axiom scheme (G1) does not follow from GK and, in particular, not from G0o . Proof: Define e(0, A) := 51 , e(0, B) := 25 , e(w, A) := 45 , e(w, B) := 35 for all w ≥ 1. For any ` ≥ 1, we find e(0, (G1)) = 45 < 1 for the semantics from Definition 4.1. Thus GK cannot prove this (G1)-instance. t u
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References [1] Baaz M., Fasching O.: G¨odel logics with monotone operators, Fuzzy Sets and Systems, in press, http://dx. doi.org/10.1016/j.fss.2011.04.012, 2012. [2] Baaz M., Fasching O.: Monotone operators on G¨odel logic, submitted, http://www.logic.at/people/ fasching/files/baaz-fasching-preprint-monotone-operators-on-godel-logic.pdf. [3] Bou F., Esteva F., Godo L., Rodr´ıguez R. O.: On the Minimum Many-Valued Modal Logic over a Finite Residuated Lattice, J. Log. Comput., 21, 2011, 739-790. http://dx.doi.org/10.1093/logcom/exp062. [4] Caicedo X., Rodr´ıguez R. O.: Bi-modal G¨odel logic over [0,1]-valued Kripke frames. CoRR, abs/1110.2407, 2011. http://arxiv.org/abs/1110.2407. [5] Dummett M.: A propositional calculus with denumerable matrix. J. Symbolic Logic, 24, 1959, 97–106. http: //www.jstor.org/stable/2964753. [6] Esteva F.; Godo L.; Noguera C.: Fuzzy logics with truth hedges revisited. 7th Conference of the European Society of Fuzzy Logic and Technology, EUSFLAT - LFA 2011, Atlantis Press, Aix-Les-Bains, 2011, 146–152.
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[7] Fasching O.: Operator extensions of G¨odel logics, PhD thesis, Vienna University of Technology, http:// permalink.obvsg.at/AC07810559, 2011.
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[8] H´ajek P.: Metamathematics of Fuzzy Logic, Kluwer, 1998.
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[9] H´ajek P.: On very true. Fuzzy sets and systems, 124, 2001, 329–333. http://www.sciencedirect.com/ science/article/pii/S0165011401001038.
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Fundamenta Informaticae 2012 1–15
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IOS Press
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G¨odel homomorphisms as G¨odel modal operators
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Oliver Fasching∗
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Institute of Discrete Mathematics and Geometry Vienna University of Technology Wiedner Hauptstrasse 8/E104.2, 1040 Vienna, Austria [email protected]
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Abstract. We extend propositional G¨odel logic by a unary modal operator, which we interpret as G¨odel homomorphisms, i.e. functions [0, 1] → [0, 1] that distribute over the interpretations of the binary connectives of G¨odel logic. We show that validity in the propositional fragment has a simple superintuitionistic Hilbert-type proof system, which is not structurally complete, and that validity does not change if we use the function class of continuous, strictly increasing functions. We also give proof systems for restrictions to sub- and superdiagonal functions.
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Keywords: G¨odel logic; superintuitionistic logic; modal logic; truth stressers.
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Introduction
Modal logic extensions are widely studied for BL and other fuzzy core logics, see e.g. [3]. In this context, unary connectives with a range of different features are often called operators or hedges, see, e.g., [6] and also their list of references. Baaz’ ∆-operator can be understood as an early extremal example, already treated in [8]. As [6] remark, it is often difficult to find a semantics that enjoys standard completeness, with G¨odel logics being an exception. Therefore we confine ourselves in this technical paper to propositional G¨odel logics and to interpretations of operators as functions in [0, 1]. As G¨odel logic is the t-norm based logic of relative comparison on [0, 1], it is natural to study the extensions by unary operators that are interpreted by [0, 1]-functions that are (1) order isomorphisms, preserving 0 and 1, or (2) G¨odel homomorphisms, i.e. homomorphisms of the binary functions on [0, 1] induced by the binary connectives of G¨odel logic. ∗
supported by the Austrian Science Fund (FWF): P22416
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We present proof systems for them in Corollary 4.2 and Theorem 3.1, resp. The said functions do not act as truth stressers (subdiagonal) or truth depressers (superdiagonal) but Corollaries 4.1 and 4.2 show that these restrictions can be easily rendered by proof systems. In Proposition 3.1, we give an exposition of Dummett’s method [5] of proving completeness of IPL + LIN for G¨odel logic; this yields a much more comprehensible proof of Theorem 3.1 than reproving all necessary steps as done in [2] for a similar but simpler result. It might appear that the restriction to a [0, 1]-valued semantics instead of a Kripke semantics oversimplifies the task of finding a sound and complete proof system but Theorem 3.1 and Corollary 4.3 show that a straightforward translation of the desired properties into axioms or rules is not enough; axiom (G1) in Definition 2.6 must be taken into account. As [2] demonstrates, a seemingly harmless [0, 1]-valued operator in the first-order fragment of (unwitnessed) G¨odel logic may even transfer so much semantical expressiveness from Łukasiewicz logic to make the logic fail to have a r.e. proof system. One of the disadvantages of our approach is, however, that the semantics we consider always makes o(A ⊃ B) ↔ (oA ⊃ oB) true, which is stronger than the normality axiom o(A ⊃ B) ⊃ (oA ⊃ oB) and thus might be unsuitable for applications. The structure of the paper is as follows: In Section 2, we give basic definitions, the definition of the semantics and we prove soundness of the proof system Go w.r.t. G¨odel homomorphisms. In Section 3, we give definitions only needed for the proof of completeness, which is then shown in Theorem 3.1. We also prove that Go is not structurally complete. In Section 4, we show the independence of the axiom scheme (G1) from the other axioms by using a very simple Kripke semantics with crisp accessibility relations; further key results are Corollaries 4.1 and 4.2 for order isomorphisms.
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Preliminaries
Definition 2.1. The following function definitions allow us to succinctly express G¨odel interpretations of formulas: For all x, y ∈ R, let x E y := 1 for x ≤ y; x E y := y for y < x; x C y := 1 for x < y; x C y := y for y ≤ x; x ./ y := 1 for x = y; x ./ y := min{x, y} for x 6= y. These binary functions clearly induce functions [0, 1] × [0, 1] → [0, 1]. Definition 2.2. The set F of G¨odel homomorphisms consists of all functions f : [0, 1] → [0, 1] such that f (min{x, y}) = min{f (x), f (y)}, f (max{x, y}) = max{f (x), f (y)}, and f (x E y) = f (x) E f (y) for all x, y ∈ [0, 1]. For each d ∈ [0, 1], define the lifting function hd : [0, 1] → [0, 1] by hd (x) := x for x ≤ d and hd (x) := 1 for d < x. Clearly, h1 is the identity on [0, 1]. We immediately obtain: Proposition 2.1. (a) A function f : [0, 1] → [0, 1] is in F is and only if f (1) = 1 and x < y ⇒ (f (x) < f (y) ∨ f (y) = 1) for all x, y ∈ [0, 1]. (b) F contains the constant function 1 and all lifting functions. (c) F is closed under composition and multiplication, i.e. (x 7→ f (g(x))) ∈ F and (x 7→ f (x) · g(x)) ∈ F for all f, g ∈ F. Definition 2.3. Let F0 denote the set of strictly increasing, continuous, piecewise linear functions f : [0, 1] → [0, 1] such that f (1) = 1. By the above, F0 ( F.
O. Fasching / G¨odel homomorphisms as G¨odel modal operators
(MP) (IPL1) (IPL2) (IPL3) (IPL4)
A A⊃B B ⊥⊃A (A ∧ B) ⊃ A (A ∧ B) ⊃ B A ⊃ B ⊃ (A ∧ B)
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Figure 1.
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Proof system IPL
Definition 2.4. The propositional language Lo of all propositional formulas Frmo contains the binary connectives ∧, ∨, ⊃, the nullary ⊥, the unary o, and the propositional variables Var. Its o-free fragment is denoted by L. We abbreviate ⊥ ⊃ ⊥ by >, (A ⊃ B) ∧ (B ⊃ A) by A ↔ B, and (B ⊃ A) ⊃ B by A ≺ B. Let K ⊆ [0,1] [0, 1]. Given f ∈ K, we say that a function I : Frmo → [0, 1] is an f -G¨odelinterpretation, or for short f -interpretation, if I(oA) = f (I(A)), I(⊥) = 0, I(A ∧ B) = min{I(A), I(B)}, I(A ∨ B) = max{I(A), I(B)} and I(A ⊃ B) = I(A) E I(B) holds for all A, B ∈ Frmo . A function I : Frmo → [0, 1] is an K-G¨odel-interpretation, or for short K-interpretation, if it is an f -G¨odelinterpretation for some f ∈ K. It clearly suffices to specify f ∈ K and I : Var → [0, 1] to define an f -G¨odel-interpretation. We say that A ∈ Frmo is K-valid if I(A) = 1 for all K-interpretations. G¨odel interpretations and validity in L are defined as reducts so that they match the usual definition. We tacitly use the fact that I(A ≺ B) = I(A) C I(B) and I(A ↔ B) = I(A) ./ I(B) hold for any (f -)G¨odel interpretation I. Definition 2.5. Figure 1 presents IPL, which is some standard Hilbert-style proof system of intuitionistic logic. Let (LIN) denote the axiom scheme (A ⊃ B) ∨ (B ⊃ A) of prelinearity. We put G := IPL + LIN. We will make use of the following well-known facts [5], proved by Dummett.
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Theorem 2.1. (a) G is sound and complete for L w.r.t. G¨odel semantics: An L-formula A is valid if and only if G ` A; indeed, the G-derivation of A is effective. (b) G ` (A ↔ B) ⊃ (E[A] ↔ E[B]) for all L-formulas A, B and any L-context E[·].
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Definition 2.6. Let Go be the proof system G extended by the necessitation rule
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(G1)
To emphasise that we do not use (G1) in certain proofs, we define the proof system G0o by G+(GN)+(G0).
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Lemma 2.1. Go is sound, i.e. if Go proves an Lo -formula H, then H is F-valid and, in particular, F0 -valid. Proof: We hint only how to prove the soundness of (G1) because the rest is routine: We will derive a contradiction from the assumption that I is an f -interpretation for some f ∈ F such that I((((A ⊃ B) ∧ (o` B ≺ o` A)) ∨ ((A ≺ B) ∧ (o` B ⊃ o` A))) ⊃ (A ∨ o` B)) < 1 for some ` ≥ 1. Observe g := f ` ∈ F. Putting a := I(A), b := I(B) we have (∗1 ): 1 ≥ max{min{aEb, g(b)Cg(a)}, min{aCb, g(b) E g(a)}} > max{a, g(b)} ≥ g(b). We have b ≤ a because a < b, together with g(b) < 1, implies g(a) < g(b) and then (∗1 ) becomes absurd. As a = b implies g(a) = g(b) and thus refutes (∗1 ), we have b < a. It follows g(b) ≤ g(a) and then max{min{a E b, g(b) C g(a)}, min{a C b, g(b) E g(a)}} = b, which also refutes (∗1 ). t u
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Completeness
The proof of completeness of Go needs several lengthy preparations and will take up almost all of the rest of the paper. The basic idea, which goes back to Dummett, will be described just before Proposition 3.1. Definition 3.1. For the following notation, fix two fresh binary formal symbols and ⇔. Let A0 , . . . , AN be formulas in Lo such that ⊥, > ∈ {A0 , . . . , AN } so that N ≥ 1, in particular. If we have symbols i ∈ { , ⇔} for all i ∈ {0, . . . , N − 1} and if B0 , . . . , BN is a permutation of the list A0 , . . . , AN such that B0 = ⊥ and BN = >, the formal notation ⊥ 0 B1 1 . . . N −2 BN −1 N −1 > is called an (A0 , . . . , AN )-chain and abbreviates the formula (⊥ ♦0 B1 ) ∧ (B1 ♦1 B2 ) ∧ . . . ∧ (BN −1 ♦N −1 >); here ♦i :=≺ if i = , and ♦i := ↔ if i =⇔. Let C be an (A0 , . . . , AN )-chain of the form ⊥ 0 B1 1 . . . N −2 BN −1 N −1 >. (The following definitions are made w.r.t. this notation and not to the formula the chain abbreviates.) We define eqc(C) as the number of the n such that n = ⇔ so that 0 ≤ eqc(C) ≤ N . The binary relations C and ⇔C on {0, . . . , N } are given by i C j :⇔ i < j ∧ ∃k(i ≤ k < j ∧ k = ) and i ⇔C j :⇔ ∀k((i ≤ k < j ∨ j ≤ k < i) ⊃ k = ⇔). We write i C j for (i C j) ∨ (i ⇔C j). (Observe that C and ⇔C depend only on the i but not on the order of the formulas Ai in C.) We say C is void if 0 ⇔C N , or equivalently, if eqc(C) = N . For example, the (⊥, >, X, Y, Z)-chain C := ⊥ Z ⇔ Y X ⇔ > abbreviates (⊥ ≺ Z) ∧ (Z ↔ Y ) ∧ (Y ≺ X) ∧ (X ↔ >), and for p⊥ := 0, pZ := 1, pY := 2, pX := 3, p> := 4, we have, e.g., p⊥ C pZ , pZ C pX , pZ C p> , pY C p> and pX ⇔C p> . The following proposition renders the main steps in Dummett’s completeness proof [5] for G w.r.t. o-free G¨odel semantics in a way that admits an easy generalisation to Lo . As all the steps are (implicitly) contained in his paper, we will not re-prove it here but just make some remarks: A formula X is valid w.r.t. G¨odel semantics if and only if X evaluates to 1 under all (the finitely many) ordering types of orderings of the value 1 and of the (finitely many) variables of X. This semantical procedure to test validity can be simulated step-by-step by G, where chains take over the task of ordering types: The relations < and = correspond to ≺ and ↔, however, ≺ does not precisely render
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< since I(X ≺ >) = 1 holds even if I(X) = 1. This does not cause problems as there is no formula in G¨odel logic that can crisply distinguish the value 1. The method relies among other things on the following facts: (1) Deep substitution as in Theorem 2.1(b) is possible, which makes the method fail for other t-norm based logics. This is needed because the evaluation procedure must be able to simplify a proper subformula (by Proposition 3.1(d)). (2) The proof system is strong enough to perform case distinctions, i.e. G ` (A ≺ B) ∨ (A ↔ B) ∨ (B ≺ A). Proposition 3.1(a) will generalise this, and C plays the rˆole of one particular case among all distinguished cases. Proposition 3.1. Let A be a list of L-formulas A0 , . . . , AN such that ⊥, > ∈ {A0 , . . . , AN }. We have then: V (a) For every L-formula X, we can construct a derivation G ` X ↔ C (C ⊃ X) where C runs through all A-chains. If C is a void A-chain, we can construct a derivation G ` C ↔ ⊥. For the remaining items suppose also that E[·] is an L-context, that i, j, k ∈ {0, . . . , N } and that C is an A-chain of the form B0 0 B1 1 . . . N −2 BN −1 N −1 BN with all n ∈ { , ⇔}; observe B0 = ⊥ and BN = >. Then (b)–(e) hold and all derivations therein are effective. (b) We have either i C j, or i ⇔C j, or j C i. The relation ⇔C is an equivalence relation; C and C are transitive; i = implies i C i + 1; and i =⇔ implies i ⇔C i + 1. We have i C k if i C j ⇔C k or i ⇔C j C k. (c) If i C j then G ` C ⊃ (Bi ≺ Bj ) and G ` C ⊃ (Bi ⊃ Bj ). If i ⇔C j then G ` C ⊃ (Bi ↔ Bj ), G ` C ⊃ (Bi ⊃ Bj ), and G ` C ⊃ (Bj ⊃ Bi ). (d) If i C j then G ` C ⊃ (E[Bj ⊃ Bi ] ↔ E[Bi ]). If i C j then G ` C ⊃ (E[Bi ⊃ Bj ] ↔ E[>]). G ` C ⊃(E[Bi ∧Bj ]↔E[Bi ]), G ` C ⊃(E[Bj ∧Bi ]↔E[Bi ]), G ` C ⊃(E[Bi ∨Bj ]↔E[Bj ]), G ` C ⊃ (E[Bj ∨ Bi ] ↔ E[Bj ]). (e) (Projection property) For every variable-free L-context F [ · n ]0≤n≤N , i.e. with N + 1 gaps, there is an effective P ∈ {0, . . . , N } such that G ` C ⊃ (F [An ]n ↔ AP ); we will say G can evaluate F [An ]n to AP under C. If P ⇔C N , we have G ` C ⊃ F [An ]n . If contexts are handled with care (see also Proposition 3.4(a)), Proposition 3.1 immediately generalises from L to Lo : Proposition 3.2. Proposition 3.1 holds also for Lo under the following provisions: (1) The language of G must be extended to Lo , i.e. axiom schemes can be instantiated with Lo -formulas, likewise (MP); in particular, Go or G0o can be used instead of G to derive the formulas shown. (2) The gaps in the contexts must not be in the scope of any ring.—By (a)–(e) of this proposition, we shall refer to the corresponding parts of Proposition 3.1. Theorem 2.1(a) and Proposition 3.1(d) yield some results we will use later: Proposition 3.3. (a) Let H0 , . . . , Hn be L-formulas. If G ` H0 ↔ H1 , G ` H1 ↔ H2 , . . . , G ` Hn−1 ↔ Hn , then G ` H0 ↔ Hn .
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(b) Let C be an A-chain B0 0 B1 1 . . . N −2 BN −1 N −1 BN for a list A of L-formulas. (Notice B0 = ⊥ and BN = >.) If i ∈ {0, . . . , N }, we have G ` (C ⊃ Bi ) ⊃ (C ↔ D), where D is the A-chain B0 0 . . . i−2 Bi−1 i−1 Bi ⇔ Bi+1 ⇔ . . . ⇔ BN . If i, j ∈ {0, . . . , N }, we have G ` (C ⊃ (Bi ∨ Bj )) ⊃ (C ↔ D), where D is the A-chain B0 0 . . . k−1 Bk ⇔ Bk+1 ⇔ . . . ⇔ BN for k := max{i, j}. The formal derivations in our systems have to be handled with more care than in [2], [1], [7] because the semantics here neither admits the deduction theorem nor the full equivalence axiom scheme: We have Go + A ` oA but any (x 7→ x2 )-interpretation I with I(A) := 12 yields I(A ⊃ oA) = 41 < 1. Extending I by I(B) := 1, we find I((A↔B)⊃(oA↔oB)) = 14 < 1. Neither the formula oA⊃A H´ajek considered √ in [9] can be valid since the (x 7→ x)-interpretation J with J(C) := 14 yields J(oC ⊃ C) = 14 < 1. However, Theorem 2.1(b) immediately generalises to part (a) of the following proposition. Proposition 3.4. We have for all formulas A, B in Lo : (a) G0o ` (A ↔ B) ⊃ (E[A] ↔ E[B]) if the gap in the Lo -context E is not in the scope of any ring. (b) G0o ` ok (A ⊃ B) ↔ (ok A ⊃ ok B) for all k ∈ N. (c) If G0o ` ok (A ⊃ B), then G0o ` ok+m A ⊃ ok+m B for all m ∈ N. (d) We have G0o ` ok (A B) ↔ (ok A ok B) for ∈ {∧, ∨, ⊃, ≺, ↔} and every k ∈ N. Proof: We will often just state the G-provability of an L-formula and leave it to the reader to check its validity and to apply Theorem 2.1(a) to obtain its formal derivation. (a) If the gap of E is not in the scope of any ring, (A ↔ B) ⊃ (E[A] ↔ E[B]) is an instance of a o-free, G-provable formula (A0 ↔ B 0 ) ⊃ (E 0 [A0 ] ↔ E 0 [B 0 ]), where A0 and B 0 are fresh variables and E 0 is an appropriately chosen o-free context. (b) We skip the trivial case k = 0. Suppose we have already shown G0o ` ok (A ⊃ B) ↔ (ok A ⊃ ok B) so that (∗1 ): G0o ` ok (A ⊃ B) ⊃ (ok A ⊃ ok B) and (∗2 ): G0o ` (ok A ⊃ ok B) ⊃ ok (A ⊃ B) follow from (IPL2) and (IPL3). Applying (GN) and (G0) to (∗1 ) yields G0o ` ook (A ⊃ B) ⊃ o(ok A ⊃ ok B). By (a) and (G0), we find G0o ` ook (A ⊃ B) ⊃ (ook A ⊃ ook B). From (∗2 ), we obtain G0o ` (ook A ⊃ ook B) ⊃ ook (A ⊃ B) in a similar way. Now (IPL4) yields the claimed G0o ` (ok+1 A ⊃ ok+1 B) ⊃ ok+1 (A ⊃ B). (c) Apply (GN) m-times to a given G0o -proof of ok (A ⊃ B) to obtain ok+m (A ⊃ B). The claim now follows from (b). (d, case ⊃) See (b). (d, case ≺) We have G0o ` ok ((B⊃A)⊃B)↔(ok (B⊃A)⊃ok B) and G0o ` ok (B⊃A)↔(ok B⊃ok A) by (b). The claim now follows from (a). (d, case ∧) Apply (c) to (IPL2), to (IPL3), and to (IPL4) to obtain ok (A∧B)⊃ok A, ok (A∧B)⊃ok B, and ok A ⊃ ok (B ⊃ (A ∧ B)); the latter yields ok A ⊃ (ok B ⊃ ok (A ∧ B)) by (a) and (b). Since (C 0 ⊃A0 )⊃(C 0 ⊃B 0 )⊃(A0 ⊃B 0 ⊃C 0 )⊃(C 0 ↔(A0 ∧B 0 )) is G-provable, we find ok (A∧B)↔(ok A∧ok B). (d, case ↔) The claim follows from the cases for ∧ and ⊃. (d, case ∨) The G-provable formula (B ⊃ A) ⊃ ((A ∨ B) ⊃ A) yields ok (B ⊃ A) ⊃ ok ((A ∨ B) ⊃ A) by (c). Applying both (a) and (b) twice, we find (ok B ⊃ ok A) ⊃ ok (A ∨ B) ⊃ ok A. We obtain (ok B ⊃ ok A) ⊃ ok (A ∨ B) ⊃ (ok A ∨ ok B) from the G-provable (R ⊃ S ⊃ T ) ⊃ (R ⊃ S ⊃ (T ∨ U )). Likewise, we find a G0o -proof of (ok A ⊃ ok B) ⊃ ok (A ∨ B) ⊃ (ok A ∨ ok B). By (IPL7) and (LIN), we have ok (A ∨ B) ⊃ (ok A ∨ ok B). For the converse direction, apply (c) to (IPL5) and (IPL6) to obtain ok A ⊃ ok (A ∨ B) and ok B ⊃ ok (A ∨ B). Now, ok (A ∨ B) ↔ (ok A ∨ ok B) follows from the G-provable (C 0 ⊃ (A0 ∨ B 0 )) ⊃ (A0 ⊃ C 0 ) ⊃ (B 0 ⊃ C 0 ) ⊃ (C 0 ↔ (A0 ∨ B 0 )).
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Definition 3.2. The ring depth of an Lo -formula is defined by rdp(V ) := 0 for all variables V , rdp(⊥) := 0, rdp(oA) := rdp(A) + 1, and rdp(A B) := max{rdp(A), rdp(B)} for ∈ {∧, ∨, ⊃}. A formula is in ring normal form if no binary connective is in the scope of any ring, i.e. ring powers ok can occur only directly in front of ⊥ or of a variable. The following proposition reduces the problem of finding a Go -proof for a valid Lo -formula A to valid formulas in ring normal form. Proposition 3.5. For every Lo -formula H, there exists an Lo -formula H 0 in ring normal form of same ring depth such that G0o ` H ↔ H 0 . Proof: Using Proposition 3.3(a), we transform H step-by-step into the required form. As long as there is a subformula ok (A B) of H that is not in the scope of a ring, use Proposition 3.4(a,d) to replace it by ok A ok B of equal ring depth; here, k ≥ 1, and A and B are arbitrary formulas in Lo . This algorithm clearly terminates and when it terminates, H 0 is in ring normal form. t u Definition 3.3. We will always consider a fixed R ∈ N r {0} and a fixed list of finitely many distinct variables V1 , . . . , VM so that we suppress the dependence of the following definitions on them. Let Z denote the list ⊥, o⊥, . . . , oR ⊥, V1 , oV1 , . . . , oR V1 , . . . , VM , oVM , . . . , oR VM , >. We write A ∈ Z to indicate that A occurs in Z. As all elements in Z are distinct, we can define w.r.t. a Z-chain C := B0 0 B1 1 . . . N −2 BN −1 N −1 BN that a formula A ∈ Z has position i in C if A = Bi . Suppressing also the dependence on C, we use P0r , Pnr , P> for the position of or ⊥, or Vm , > in C; here r ∈ {0, . . . , R} and m ∈ {1, . . . , M }, i.e., we treat ⊥ like a variable in this respect and consequently put V0 := ⊥ to unify the notation. Definition 3.4. A non-void Z-chain C is reduced, using the notation of Definition 3.3, if for all m, n ∈ r P s , we have {0, . . . , M }, r, s ∈ {0, . . . , R}, ` ∈ Z such that 0 ≤ r + ` ≤ R, 0 ≤ s + ` ≤ R and Pm n r+` s+` r+` s+` Pm Pn or Pm ⇔ Pn ⇔ P> . Lemma 3.1. Let C be a Z-chain and use the notation Z from Definition 3.3. Then we can find a Z-chain D such that Go ` C ↔ D and such that D is reduced or void. Proof: We use all notations from Definition 3.3 so that, e.g., V0 is now defined. We will often tacitly apply Proposition 3.2(b). To prove the claim, it suffices by Proposition 3.3(a) to reduce C stepwise. The following steps (1) and (2) of this reduction loop keep C in Z-chain form. As they will be seen to also properly increase the bounded eqc(C), they are applicable only for a finite number of times and hence the loop terminates. Since we may assume that the loop immediately terminates if C becomes void, we will establish the claim by eventually proving that a non-void chain C is reduced if none of the reduction steps is applicable.—We write , ⇔, instead of C , ⇔C , C . r P s , P s+` P r+` , P s+` P for some (1) We claim that C can be further reduced if Pm > n n m n m, n ≤ M , r, s ≤ R, ` ≥ 1 with max{r, s} + ` ≤ R. Proof: Proposition 3.2(c) shows Go ` C ⊃ (or Vm ≺ os Vn ) and Go ` C ⊃ (os+` Vn ⊃ or+` Vm ), which we apply together with (G1) to an appropriate instance of G ` (U ⊃ X) ⊃ (U ⊃ Y ) ⊃ (((X ∧ Y ) ∨
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Q) ⊃ W ) ⊃ (U ⊃ W ) in order to obtain Go ` C ⊃ (or Vm ∨ os+` Vn ). Proposition 3.3(c) now yields a Z-chain C 0 such that Go ` C ↔ C 0 , such that C 0 results from C just by replacing certain -symbols by r ⇔ 0 P or P s+` ⇔ 0 P . We have eqc(C) < eqc(C 0 ) since P r P ⇔-symbols, and such that Pm > > > C C n m and Pns+` P> . (2) We claim that C can be further reduced whenever there are m, n ≤ M , r, s ≤ R, ` ≥ 1 with r+` P s+` , P s P r , P s P . The proof is analogous to the one max{r, s} + ` ≤ R such that Pm > n n m n of (1). Using (1) for ` > 0 and (2) for ` < 0 in Definition 3.4, it easily follows that C is reduced. t u Proposition 3.6. Let C be a reduced Z-chain and use the notation of Definition 3.3. Then: r ⇔ P s P and ` ∈ Z such that 0 ≤ r + ` ≤ R and 0 ≤ s + ` ≤ R, then P r+` ⇔ P s+` . (a) If Pm > n m n r ⇔ P r+1 P for some r < R, then P 0 ⇔ . . . ⇔ P r ⇔ P r+1 ⇔ . . . ⇔ P R . (b) If Pm > m m m m m r P r+1 for some r < R then P 0 P 1 . . . P r P r+1 . . . P R−1 P R . (c) If Pm m m m m m m m r+1 P r for some r < R then P R P R−1 . . . P r+1 P r . . . P 1 P 0 and (d) If Pm m m m m m m m R P0 . Pm m s P r , P s P , P r P s , P r P , then either P s ⇔ P r ⇔ P s ⇔ P r , or P r P r , (e) If Pm > > n m n n m n m m n m n s. or Pns Pm (f) For each m ≤ M , exactly one of the following cases holds: 0 ⇔ P1 ⇔ ... ⇔ PR ⇔ P , (A1) Pm > m m 0 ⇔ P1 ⇔ ... ⇔ PR P , (A2) Pm > m m 0 P1 ... PR P , (A3) Pm > m m R P R−1 . . . P 0 P , (A4) Pm > m m 0 P 1 . . . P k ⇔ P k+1 ⇔ . . . ⇔ P R ⇔ P for some k with 0 < k ≤ R, (A5) Pm > m m m m R P R−1 . . . P k ⇔ P k−1 ⇔ . . . ⇔ P 0 ⇔ P for some k with 0 ≤ k < R. (A6) Pm > m m m m r+` P s+` since C is reduced, which would imply P r P s or P r ⇔ Proof: (a) We cannot have Pm n m n m s r r+` cannot hold. The claim Pn ⇔ P> , contradicting Pm ⇔ Pns P> . We similarly find that Pns+` Pm now follows from Proposition 3.2(b). (b) follows immediately from (a). (c) Since C is reduced, we have P r+` P r+`+1 or P r+` ⇔ P r+`+1 ⇔ P> , i.e. P r+` P r+`+1 k P r P for all k ≤ m, we have, by (b), that for all ` ∈ Z such that 0 ≤ r + ` + 1 ≤ R. Since Pm > m k k+1 there is no k < r with Pm ⇔ Pm , which proves the claim. (d) has a proof similar to the one of (c). r P r and or P s P s since this (e) The cases are clearly disjoint, e.g. we cannot have both Pm n n m r r s s r P r , nor s would yield Pm Pm Pn Pn Pm , which is absurd. Suppose we neither have Pm n s . Since P r P r , P r P and C is reduced, we see that we cannot have P s P s ; We Pns Pm > n m n m n r is impossible. Thus P s P s P r P r P s , as claimed. similarly obtain that Pnr Pm n m m n n (f) By Proposition 3.2(b), the cases are clearly distinct. In order to prove their disjunction, we disr P r+1 for some r ≤ R − 1, we can choose r maximal with tinguish several cases: In the case of Pm m R P , and we see (A5) if r + 1 = R that property and apply (d). We find (A3) if r + 1 = R and Pm > R ⇔ P . Thus we may assume r + 2 ≤ R. Maximality of r yields P r+2 P r+1 so that and Pm > m m r+2 ⇔ P r+1 ⇔ P since C is reduced. The claim is now proved for this case because (A5) is estabPm > m r+1 P r for some r ≤ R − 1, we find in a similar way that (A4) or (A6) holds. lished. In the case of Pm m In the remaining case, we see (A1) or (A2), which completes the proof.
O. Fasching / G¨odel homomorphisms as G¨odel modal operators
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Definition 3.5. Let C be a reduced Z-chain and use the notation of Definition 3.3. Let mSn for any a P b ∧ ∃c, d ≤ R. P c P d so that S is a reflexive and m, n ≤ M abbreviate ∃a, b ≤ R. Pm n n m symmetric relation. By ∼, we denote the transitive closure of S, i.e. ∼ is an equivalence relation on {0, . . . , M }. Proposition 3.7. Let C be a reduced Z-chain and use the notation of Definition 3.3. Then: 0 P R , then P 0 P R or P 0 ⇔ P R ⇔ P . (a) If m ∼ n, Pm > m n n n n R P 0 , then P R P 0 or P 0 ⇔ P R ⇔ P . (b) If m ∼ n, Pm > m n n n n 0 ⇔ P R P , then P 0 ⇔ P R ⇔ P 0 ⇔ P R . (c) If m ∼ n, Pm > m m m n n (d) The equivalence classes of ∼ can be linearly ordered by , i.e. for all n1 6∼ n2 , we have either a P b for all a, b ≤ R and m ∼ n and m ∼ n , or P a P b for all a, b ≤ R and m ∼ n Pm 1 1 2 2 1 1 m2 m2 m1 1 and m2 ∼ n2 . Proof: We begin with a few observations: (1) We claim mSk if mSn, nSk, Pn0 ⇔ PnR : By inspection of (A), we have Pn0 ⇔ Pni for all i ≤ R. 0 ⇔ P R and P 0 ⇔ P 0 ⇔ P R . In case Pn0 P> , we apply (B3) to mSn and to kSn so that Pn0 ⇔ Pm m n k k R 0 R 0 a b for some a, b ≤ R by mSn Thus Pm ⇔ Pm ⇔ Pk so that mSk. In case Pn ⇔ P> , we find Pn Pm b ⇔ P d , thus and Pnc Pkd for some c, d ≤ R by nSk. It follows from P> ⇔ Pna ⇔ Pna that P> ⇔ Pm k mSk. This establishes the claim. 0 P R : Suppose the contrary so (2) We claim Pn0 PnR or Pn0 ⇔ PnR ⇔ P> if mSn and Pm m 0 P R , P R P 0 . It follows that P R P R or P 0 P 0 or by Proposition 3.6(e), that mSn, Pm m n n m n n m 0 ⇔ P R ⇔ P 0 ⇔ P R would be absurd. We cannot have P R P R since this would imply since Pm m n n m n 0 P R P R P 0 so that, applying Proposition 3.6(f) twice, would yield P 0 P i P R Pm m n n m m m j PnR Pm Pn0 for all i, j ≤ R, contradicting mSn. We similarly obtain that we cannot have 0 , which establishes the claim.—Likewise, we see (3) and (4): Pn0 Pm R P0 . (3) We have PnR Pn0 or Pn0 ⇔ PnR ⇔ P> if mSn and Pm m 0 R 0 R 0 R P . (4) We have Pm ⇔ Pm ⇔ Pn ⇔ Pn if mSn and Pm ⇔ Pm > Since m ∼ n, there are I ∈ N and ki ≤ R for all i < I such that mSk0 Sk1 S . . . SkI−1 Sn. We may assume I ≥ 1 by (2). We may also assume that Pk0i ⇔ PkRi for no i < I for otherwise (1) would allow us to pass to mSk0 S . . . Ski−1 Ski+1 S . . . Sn. Now, (a) follows inductively by (2). Using (3) and (4), we obtain similar proofs for (b) and (c). a P b , then P c P d for all c, d ≤ R: By m 6∼ k, we cannot have (5) Whenever mSn 6∼ k and Pm n k k a P b implies P e P f for all e, f ≤ R. Suppose we had, to the mSk and thus, by definition of S, Pm m k k contrary, Pkd Pnc for some c, d ≤ R; since nSk cannot hold, we see by definition of S that Pkf Png e P g for all e, g ≤ R, contradicting mSn.—Similarly: for all f, g ≤ R and hence Pm n b , then P c P d for all c, d ≤ R. (6) Whenever mSn 6∼ k and Pka Pm n k Let n1 6∼ n2 so that, in particular, not n1 Sn2 and thus we have either Pn01 Pn02 or Pn02 Pn01 . Now, (d) follows inductively (like in (a)) from (5) and (6). t u The proof idea of Lemma 3.2 below is to construct, for each chain C, an f ∈ F0 and an f -G¨odelinterpretation I : Var → [0, 1] that tries to embed the total preorder given by C and ⇔C in a suitable
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way on [0, 1]. As the equivalence classes in the sense of Proposition 3.7(d) are linearly ordered, we will first give in Proposition 3.8 such a realisation for each equivalence class and then compose them in Lemma 3.2. In contrast to the situations in [1] and [2], observe that C and ⇔C are not exactly r ⇔ P can be realised as values in [0, 1); then I(C) < 1 follows but realised as < and = but points Pm C > the realisation is still good enough to prove I(C) > I(os Vk ) for all Pks C P> . Dropping this feature of exact realisation makes it possible to find an f ∈ F0 r F.—The following proposition has a rather technical proof; we leave several easy details to the reader. Proposition 3.8. Let C be a reduced Z-chain and use the notation from Definition 3.3. Let w ≤ M and denote by W := {n ≤ M ; n ∼ w} its ∼-equivalence class. Denote the condition 0 ∈ W by (∗0 ), and let (∗1 ) denote the condition that Pnr ⇔ P> for some r ≤ R, n ∈ W . Suppose 0 ≤ α < β ≤ 1, but, however, we require α = 0 if (∗0 ), and β = 1 if (∗1 ). Moreover, let (∗↑ ) and (∗↓ ) denote the conditions that Pn0 PnR for some n ∈ W , and PnR Pn0 for some n ∈ W , resp. Then there are Jn ∈ [α, β] for every n ∈ W and a strictly increasing, continuous, piecewise linear function f : [α, β] → [α, β] with the following properties: If (∗0 ), then J0 = 0. If (∗0 ) fails, then f (α) = α. We have f (β) = β. Moreover, for all r, s ≤ R and n, m ≤ M and x ∈ [α, β]: We have α ≤ f r (Jn ) ≤ β and, in particular, f r (Jn ) is well-defined. If (∗0 ) fails, then α < f r (Jn ). If (∗1 ) fails, then f r (Jn ) < β. If (∗↑ ), then x ≤ f (x). If (∗↓ ), then f (x) ≤ x. If neither (∗↑ ) nor (∗↓ ), s P implies f r (J ) = f s (J ) < 1; in particular, P r P then x = f (x). We have that Pnr ⇔ Pm n m > > n r r s implies f (Jn ) < 1; that Pn Pm implies f r (Jn ) < f s (Jm ); and that Pn0 ⇔ PnR ⇔ P> implies Jn = 1.—Finally, observe that (∗0 ) fails if (∗↓ ). Proof: If we neither have (∗↑ ) nor (∗↓ ), we have Pn0 ⇔ PnR for all n ∈ W . Take f (x) := x and define Jn for all n ∈ W as follows: Put Jn := 0 if w = 0; put Jn := 1 if Pw0 ⇔ P> ; and Jn := (α + β)/2 otherwise. It can be easily verified by Proposition 3.7(c) that exactly one of this cases must occur and that the claimed properties follow then. Let us now suppose (∗↓ ). By Proposition 3.7(b), we see that for each m ∈ W only the cases (A1), (A4), (A6) in Proposition 3.6(f) are possible; in particular, we cannot have (∗0 ) since C is not void and since P00 is the least position in C. Let W0 := W ∪ {0}, H := {(r, n); r ≤ R, n ∈ W0 } ∪ {>} and define relations ≺ and ↔ on H by the following: For all r, s ≤ R and n, k ∈ W , we put (r, n) ≺ (s, k) :⇔ PnR−r PkR−s ; (r, n) ↔ (s, k) :⇔ PnR−r ⇔ PkR−s ; (r, n) ≺ > :⇔ PnR−r P> ; (r, n) ↔ > :⇔ PnR−r ⇔ P> ; (r, 0) ≺ (s, 0) :⇔ r < s; (r, 0) ↔ (s, 0) :⇔ r = s; and put (r, 0) ≺ (s, k) always. We use Proposition 3.2(b), 0 6∼ n, Proposition 3.6(f) and (a), and the fact that C is reduced to verify the conditions of Lemma 5 in [2]. The conditions involving the dummy pairs (r, 0) can be easily checked separately. Thus there are q ∈ [0, 1] ∩ Q and gnr ∈ [0, 1] ∩ Q for all r ≤ R and n ∈ W0 such that for all r, s ≤ R and n, k ∈ W we have: PnR−r PkR−s if and only if gnr < gks ; PnR−r ⇔ PkR−s if and only if gnr = gks ; PnR−r P> if and only if gnr < 1; PnR−r ⇔ P> if and only if gnr = 1; moreover, 0 = g00 < gnr always; and gnr+1 = min{1, q + gnr } if r < R. Since there is n ∈ W with PnR Pn0 by assumption, it follows gn0 < gnR = min{1, Rq + gn0 } and hence x+(R+2)q 0 < q and gn0 < 1. Let d(x) := α + (β − α)( 41 + 2(2+(R+2)q) ). Put Jn := 1 for all n ∈ W R with Pn ⇔ P> , which is possible since then β = 1; and put Jn := d(gn0 ) for all n ∈ W with q(β−α) PnR P> . Put c := 2(2+(R+2)q) . Let f be the function that linearly interpolates the coordinates (α, α), (d(−(R + 1)q), d(−(R + 1)q) − c), (d(1 + q), d(1 + q) − c), (β, β). It is easy to verify that 0 r 0 0 α < α + β−α 4 ≤ d(−(R + 1)q) < d(−(R + 1)q) + c = d(−Rq) < d(gn − rq) = f (gn ) = d(gn ) − rc ≤
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d(1) = d(1 + q) − c < d(1 + q) ≤ α + 3(β−α) < β for all n ∈ W with Pn0 P> . Observe that d is 4 strictly increasing. The claimed implications can now be easily verified. For (∗↑ ), we only sketch the proof: Apply Proposition 3.7(a) and then Proposition 3.6(f). The basic idea is to put (r, n) ≺ (s, k) :⇔ Pnr Pks and (r, n) ↔ (s, k) :⇔ Pnr ⇔ Pks . Depending on the absence of (∗0 ) or (∗1 ), dummy bottom or top elements have to be added; for that purpose, we extend W by 0 if (∗0 ) does not hold. If (∗0 ), Lemma 5 in [2] guarantees g00 = 0. Now, (∗↑ ) guarantees q > 0 and hence c > 0, where c is defined as before. In case of (∗0 ), we have α = 0 by assumption so that we put xβ d(x) := 2(2+(R+2)q) and let f linearly interpolate (0, c), (d(1 + Rq), d(1 + Rq) + c), (β, β); in case that x ) and let f linearly interpolate (α, α), (∗0 ) does not hold, we put d(x) := α + (β − α)( 14 + 2(2+(R+2)q) (d(0), d(0) + c), (d(1 + Rq), d(1 + Rq) + c), (β, β) so that α < α + β−α 4 ≤ d(0). We put Jn := 1 if Pn0 ⇔ P> and Jn := d(gn0 ) otherwise. No matter if (∗0 ) or not, we have d(0) ≤ d(gn0 + rq) = f r (gn0 ) = d(gn0 ) + rc ≤ d(1 + Rq) < d(1 + Rq) + c ≤ α + 3(β−α) < β for all r ≤ R and n ∈ W such that 4 Pn0 P> . The claimed implications can now be easily verified. t u Lemma 3.2. Let C be a reduced Z-chain and use the notation from Definition 3.3. Then there is an F0 -G¨odel interpretation I such that I(C ⊃ ot Vk ) < 1 for all t ≤ R and k ≤ M with Pkt P> . (Remark: The case k = 0 is permitted.) Proof: By Proposition 3.7(d), induces a linear order on the equivalence classes of ∼. By Proposition 3.7(a)–(c), the same is true even if ∼ is extended from {0, . . . , M } to {0, . . . , M, >} by defining m ∼ > r ⇔ P and by completing it to an equivalence relation. Let W , . . . , W ⊆ {0, . . . , :⇔ ∃r ≤ R. Pm 0 E > M, >} denote all equivalence classes with W0 being -minimal and WE -maximal. Observe that r for all r ≤ R and m ≤ M , and that W contains > by construction, W0 contains 0 since P00 Pm E including the possibility that WE is just {>}. e+1 e For each e ∈ {0, . . . , E} construct fe on [ E+1 , E+1 ] and Je,n , for all n ∈ We , according to Proposition 3.8; however, observe the following: For WE , we have to put JE,> := 1 since the said Proposition does not assign a value to JE,> . If Pn0 ⇔ PnR ⇔ P> for all n ∈ WE , the conditions of the 1 said Proposition are not met and we have to put JE,n := 1 then. Since W0 is used for [0, E+1 ] and WE E is used for [ E+1 , 1], the boundary conditions implied by (∗0 ) and (∗1 ) in Proposition 3.8 are fulfilled. Since the (We )e are a partition of {0, . . . , M, >}, it is well-defined S to put I(Vn ) := Je,n for all e e n ∈ We . Since fe ( E+1 ) = E+1 for all e ∈ {1, . . . , E + 1}, also f := e fe is a well-defined function in F0 . Observe for later use that Proposition 3.8 yields the following implications: I(or Vn ) = I(os Vm ) < 1 s P since n ∼ m then. I(or V ) < 1 holds if P r P . We also prove holds if Pnr ⇔ Pm n > > n s : If n ∼ m, this follows from Proposition 3.8 so that we assume n 6∼ m. I(or Vn ) < I(os Vm ) if Pnr Pm By Proposition 3.7(d), we have n ∈ Wi and m ∈ Wj for some i, j ≤ E with i < j. Again by Proposition j i+1 ≤ E+1 < I(os Vm ) by applying Proposition 3.7(d), we find n 6∼ > and m 6∼ 0 and thus I(or Vn ) < E+1 3.8 twice. Recall the notation from Definition 3.3; we claim I(C) = min{ min
i,j : i C j
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I(Bi ≺ Bj ),
min
i,j : i⇔C j
I(Bi ↔ Bj )}.
By Proposition 3.2(c) and soundness of Go , we have 1 = I(C ⊃(Bi ≺ Bj )) and thus I(C) ≤ I(Bi ≺ Bj ) for all i, j with i C j and, similarly, 1 = I(C ⊃ (Bi ↔ Bj )) and thus I(C) ≤ I(Bi ↔ Bj ) for
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all i, j with i ⇔C j. The claim now follows since the converse inequality holds because of I(C) = min{I(⊥ ♦0 B1 ), I(B1 ♦1 B2 ), . . . , I(BN −1 ♦N −1 >)}, where ♦i :=≺ if i = , and ♦i := ↔ if i =⇔. Hence I(C) = min{minPnr Pms I(or Vn ≺ os Vm ), minPnr ⇔Pms I(or Vn ↔os Vm ), minPnr P> I(or Vn ≺ >), minPnr ⇔P> I(or Vn ↔ >)}; here the running indices r, s and m, n are suppressed. As proved earlier, we have I(ot Vk ) < 1. It follows for all n, m ≤ M and r, s ≤ R: We have I(ot Vk ) < 1 = I(or Vn ≺ >) by definition of C. If Pnr ⇔ P> then Pkt Pnr and thus, s then I(or V ) < I(os V ) and as proved earlier, I(ot Vk ) < I(or Vn ) = I(or Vn ↔ >). If Pnr Pm n m s and P s P t , then P r ⇔ P s P so that thus I(ot Vk ) < 1 = I(or Vn ≺ os Vm ). If Pnr ⇔ Pm > m n m k s and P t P s , then P t P r so I(or Vn ) = I(os Vm ) and thus I(or Vn ↔ os Vm ) = 1. If Pnr ⇔ Pm m n k k that I(ot Vk ) < I(or Vn ) and I(ot Vk ) < I(os Vm ), thus I(ot Vk ) < min{I(or Vn ), I(os Vm )} ≤ I(or Vn ) ./ I(os Vm ) = I(or Vn ↔ os Vm ). Hence I(ot Vk ) < I(C) and thus I(C ⊃ ot Vk ) = I(ot Vk ) < 1. t u Theorem 3.1. For any Lo -formula H, the following are equivalent:
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(1) Go ` H.
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(2) H is F-valid.
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(3) H is F0 -valid.
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Moreover, the derivation in (1) is effective, and Go is a conservative extension of G. Proof: By Lemma 2.1, (1) implies (2). Since F0 ⊂ F, (2) implies (3). Let now H be an F0 -valid Lo -formula. We have to show Go ` H. Let V1 , . . . , VM denote all the variables of H. By Proposition 3.5, there is a Lo -formula H 0 with Go ` H ↔ H 0 and with same ring depth R. By soundness, we have 1 = I(H ↔ H 0 ) = I(H) ./ I(H 0 ) = 1 ./ I(H 0 ) = I(H 0 ) for any F0 - or F-interpretation I. Now that we have fixed R and M , we can use the notation from Definition 3.3: Let now C be an arbitrary Z-chain; we have to prove Go ` C ⊃H 0 . If C is void, Proposition 3.2(a) yields Go ` C ↔⊥ and thus Go ` C ⊃ H 0 , so that we may suppose that C is non-void. Thus, by Lemma 3.1, there is a reduced Z-chain C 0 with Go ` C ↔ C 0 . By Proposition 3.2(e), there is W ∈ Z such that Go ` C 0 ⊃ (H 0 ↔ W ). If W = >, we clearly have Go ` C 0 ⊃ H 0 and thus also Go ` C ⊃ H 0 , as required; hence we may assume W = ot Vk for some t ≤ R and k ≤ M . If we had Pkt C 0 P> , then there is an F0 interpretation I such that I(C 0 ⊃ot Vk ) < 1 by Lemma 3.2, but soundness and Go ` C 0 ⊃(H 0 ↔W ) imply 1 = I(C 0 ⊃ (H 0 ↔ ot Vk )) = I(C 0 ) E I(H 0 ↔ ot Vk ) = I(C 0 ) E (I(H 0 ) ./ I(ot Vk )) = I(C 0 ) E I(ot Vk ) = I(C 0 ⊃ ot Vk ), which is absurd. Therefore, we have Pkt ⇔C 0 P> . This and the fact that H 0 is in ring normal form yields Go ` C 0 ⊃ H 0 from the last sentence in Proposition 3.2(e). Since Go ` C ↔ C 0 , we have Go ` C ⊃ H 0 . V The previous paragraph yields Go ` C (C ⊃ H 0 ), where C runs through all Z-chains. From Proposition 3.2(a), we obtain Go ` H 0 , and thus Go ` H from Go ` H ↔ H 0 . Tracing the construction of the derivation of H, we can readily see that all steps are effective. t u Since f ∈ F0 , f (x) = 1, x ∈ [0, 1] implies x = 1, we immediately obtain:
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Corollary 3.1. If Go ` oH, then Go ` H.
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Let 1 denote the constant function 1; observe 1 ∈ F. The proof system Go + oA is clearly sound for all {1}-valid formulas. Due to the 1-G¨odel-interpretation I defined by I(X) := 0 for all X ∈ Var, we find Go + oA 6` A. Hence we see:
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Corollary 3.2. Go is not structurally complete.
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Extra axioms and axiom independence
In this section, we present further extensions of Go and prove the necessity to include (G1) in Go . The method of proof of Theorem 3.1 allows an easy generalisation when additional axioms oA ⊃ A, A ⊃ oA or o⊥ ⊃ ⊥ are present. For the axiom oA ⊃ A, see [9], in particular. Define F↓ := {f ∈ F : f (x) ≤ x for all x ∈ [0, 1]}, F↑ := {f ∈ F : x ≤ f (x) for all x ∈ [0, 1]}, and similarly F0↓ and F0↑ . Notice that all functions in F↓ and F0↓ are strictly increasing. Corollary 4.1. (a) For any Lo -formula H, the following are equivalent: (1) G0o + oA ⊃ A ` H. (2) H is F↓ -valid. (3) H is F0↓ -valid. (b) For any Lo -formula H, the following are equivalent: (1) G0o + A ⊃ oA ` H. (2) H is F↑ -valid. (3) H is F0↑ -valid. Proof: We only prove (a) since (b) is completely analogous. For soundness, use Lemma 2.1 and the fact that I(oA ⊃ A) = f (I(A)) E I(A) = 1 whenever I is an f -interpretation and f ∈ F↓ . We now prove completeness: Let G↓ denote the proof system G0o + oA ⊃ A. We denote by (L) the G-provable formula ((X ≺ Y ) ∧ (Y ⊃ X)) ⊃ (X ∧ Y ). G↓ proves (o` B ≺ o` A) ⊃ (B ≺ A) and (o` B ⊃ o` A) ⊃ (B ⊃ A) by Proposition 3.4(a,d). It is easy to show by (L) that G↓ proves every instance of (G1). We have to inspect the proof of Theorem 3.1 and refine it slightly: We call a reduced chain C ↓-reduced if Pnk+1 C Pnk holds for all positions. In Lemma 3.1, any chain C can be further reduced by (L) and by oA ⊃ A if Pnk C Pnk+1 holds, thus we find G↓ ` C ↔ D for some ↓-reduced chain D. Hence the case (∗↑ ) in Proposition 3.8 can never appear and now Lemma 3.2 yields an F0↓ -interpretation. The proof of Theorem 3.1 does not require any modification. t u By the same method as before, we also obtain: Corollary 4.2. (a) For any Lo -formula H, the following are equivalent: (1) Go + o⊥ ⊃ ⊥ ` H. (2) H is {f ∈ F : f (0) = 0}-valid. (3) H is {f ∈ F0 : f (0) = 0}-valid. (b) For any Lo -formula H, the following are equivalent: (1) G0o + A ⊃ oA + o⊥ ⊃ ⊥ ` H. (2) H is {f ∈ F↑ : f (0) = 0}-valid. (3) H is {f ∈ F0↑ : f (0) = 0}-valid. To show that (G1) cannot be omitted in the proof system Go , we give another semantics of Lo in Kripke style. For general Kripke semantics for G¨odel logics, see [4]. Definition 4.1. We can give the Lo -formulas a Kripke semantics as follows: Let E := N×Var [0, 1]. Extend any e ∈ E to e : N × Frmo → [0, 1] by e(w, ⊥) := 0, e(w, A ∧ B) := min{e(w, A), e(w, B)},
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e(w, A ∨ B) := max{e(w, A), e(w, B)}, e(w, A ⊃ B) := e(w, A) E e(w, B), and e(w, oA) := e(w + 1, A). We say that A ∈ Frmo is K-valid if e(w, A) = 1 for all e ∈ E and all w ∈ N.
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Definition 4.2. Let GK denote the proof system G0o + o⊥ ⊃ ⊥.
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Proposition 4.1. GK is sound and complete in Lo w.r.t. the semantics from Definition 4.1; in fact, for every K-valid Lo -formula, one can effectively find a GK -derivation. Proof: We have e(w, o(A ⊃ B)) = e(w + 1, A ⊃ B) = e(w + 1, A) E e(w + 1, B) = e(w, oA ⊃ oB) for all e ∈ E and w ∈ N, thus e(w, o(A ⊃ B) ↔ (oA ⊃ oB)) = 1. We also have e(w, ⊥ ↔ o⊥) = e(w, ⊥) ./ e(w + 1, ⊥) = 0 ./ 0 = 1 for all e ∈ E and w ∈ N. Clearly, the validity of A implies the validity of oA. It is also easy to check that all Lo -instances of G are valid. Thus GK is sound w.r.t. this semantics. Observe that GK ` ok ⊥ ↔ ⊥ for any k ≥ 0: We clearly have GK ` ⊥ ⊃ ok ⊥ for any k ≥ 0. For the converse direction, we may suppose k ≥ 1 and proceed by induction on k Apply (GN) k times to o⊥ ⊃ ⊥ and use Proposition 3.4(c) to find GK ` ok+1 ⊥ ⊃ ok ⊥. By the induction hypothesis GK ` ok ⊥ ⊃ ⊥, we obtain GK ` ok+1 ⊥ ⊃ ⊥, as claimed. We now prove completeness. Let H be K-valid. By Proposition 3.5 and soundness of GK , we may assume that H is in ring normal form so that H = F [ok Vm ]k≤R, 0≤m≤M such that R is the ring depth of H, V0 := ⊥, V1 , . . . , VM ∈ Var, F = F [·k,m ]k,m is a context neither containing o, neither ⊥, nor any variable. We apply GK ` ou ⊥ ↔ ⊥ for all u ≤ R and Proposition 3.4(a) to eliminate all rings in front of ⊥. By soundness of GK , we may assume that H = F 0 [ok Vm ]k≤R, 1≤m≤M where F 0 is a context that contains neither o, neither any variable, but may contain ⊥. Let Wk,m for k ≤ R and 1 ≤ m ≤ M be fresh variables. Then F 0 [Wk,m ]k≤R, 1≤m≤M is valid in the usual G¨odel semantics in L because to every G¨odel interpretation I in L we can define a Kripke interpretation e(k, Vm ) := I(Wk,m ) for k ≤ R, and 1 ≤ m ≤ M (and e(k, A) := 0 for k > R or A ∈ Var r {V1 , . . . , VM }) so that I(F 0 [Wk,m ]k,m ) = e(0, F 0 [ok Vm ]k,m ) = 1. By Theorem 2.1(a), we have G ` F 0 [Wk,m ]k,m and we can also build the Lo -instance G ` F 0 [ok Vm ]k,m if we allow Lo -instances in the axioms of G. In particular, GK ` H, as claimed. t u In fact, we will only use soundness of GK to prove that the scheme (G1) cannot be omitted from the proof systems in Theorem 3.1 and Corollary 4.2. Corollary 4.3. For every fixed ` ≥ 1, the axiom scheme (G1) does not follow from GK and, in particular, not from G0o . Proof: Define e(0, A) := 51 , e(0, B) := 25 , e(w, A) := 45 , e(w, B) := 35 for all w ≥ 1. For any ` ≥ 1, we find e(0, (G1)) = 45 < 1 for the semantics from Definition 4.1. Thus GK cannot prove this (G1)-instance. t u
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References [1] Baaz M., Fasching O.: G¨odel logics with monotone operators, Fuzzy Sets and Systems, in press, http://dx. doi.org/10.1016/j.fss.2011.04.012, 2012. [2] Baaz M., Fasching O.: Monotone operators on G¨odel logic, submitted, http://www.logic.at/people/ fasching/files/baaz-fasching-preprint-monotone-operators-on-godel-logic.pdf. [3] Bou F., Esteva F., Godo L., Rodr´ıguez R. O.: On the Minimum Many-Valued Modal Logic over a Finite Residuated Lattice, J. Log. Comput., 21, 2011, 739-790. http://dx.doi.org/10.1093/logcom/exp062. [4] Caicedo X., Rodr´ıguez R. O.: Bi-modal G¨odel logic over [0,1]-valued Kripke frames. CoRR, abs/1110.2407, 2011. http://arxiv.org/abs/1110.2407. [5] Dummett M.: A propositional calculus with denumerable matrix. J. Symbolic Logic, 24, 1959, 97–106. http: //www.jstor.org/stable/2964753. [6] Esteva F.; Godo L.; Noguera C.: Fuzzy logics with truth hedges revisited. 7th Conference of the European Society of Fuzzy Logic and Technology, EUSFLAT - LFA 2011, Atlantis Press, Aix-Les-Bains, 2011, 146–152.
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