On EQ-Fuzzy Logics with Delta ConnectiveMartin Dyba, Vilém Novák

University of Ostrava, IRAFM, 30. dubna 22, 701 03 Ostrava 1, Czech Republic∗

Abstract

In this paper, extension of the EQ-logic by the ∆-connective is introduced. The former is a new kindof many-valued logic which based on EQ-algebra oftruth values, i.e. the algebra in which fuzzy equal-ity is the fundamental operation and implication isderived from it. First, we extend the EQ-algebraby the ∆ operation and then introduce axioms andinference rules of EQ∆-logic. We also prove the de-duction theorem formulated using fuzzy equalities.

Keywords: EQ-algebra, EQ-logic, equational logic,delta connective

1. Introduction

In this paper, we continue development of the the-ory of a special class of fuzzy logics which arebased on EQ-algebras. The latter are special al-gebras in which the fundamental operation is thatof fuzzy equality. The concept of EQ-algebra wasintroduced in [1] and in more detail elaborated in[2] and [3, 4]. It was motivated by the paper ofL. Henkin [5] who introduced type theory (higher-order logic) in which equality is the sole connective.The fuzzy version of type theory introduced in [6]is based on residuated lattice (more specifically, onthe IMTL-algebra) in which equivalence (i.e. fuzzyequality on truth values) is defined as biresiduationa ↔ b = (a → b) ∧ (b → a). Since this is a derivedoperation, a question arises whether it is possibleto introduce a special algebra of truth values forfuzzy type theory (FTT) with fuzzy equality beingthe fundamental operation (it can be demonstratedthat it cannot be the sole operation as is the caseof classical type theory). A possible answer to thisquestion is the concept of EQ-algebra. The implica-tion operation is derived from the fuzzy equality andis relaxed from the multiplication. Therefore, it isno more the residuation. The role of multiplicationin EQ-algebras thus becomes only auxiliary and themultiplication serves as a specific connector. Con-sequently, EQ-algebras are more general than resid-uated lattices in the sense that every residuated lat-tice is an EQ-algebra but not vice-versa. Moreover,the multiplication in EQ-algebra is, in general, non-commutative and can be even non-associative with-out any influence on the implication which remainsunique because of not being tied with the former.∗This work was supported by the Institutional Research

Plan MSM 6198898701.

The relation between both kinds of algebras is quitesubtle and still needs to be studied in more detail.

After establishing the algebra of truth values,EQ-fuzzy type theory has been introduced in [7].First, the basic EQ-FTT was established and itsgeneralized completeness property was proved. Fur-ther, several extensions were introduced, amongthem also the above mentioned IMTL-FTT.

All this finally raised the question how EQ-logicbased purely on EQ-algebra can be introduced. Thefirst papers on this topic are [8, 9]. EQ-logic can betaken as a step towards realization of Leibniz ideathat “a fully satisfactory logical calculus must be anequational one” (cf. [10]). It turned out, however,that the situation is not as straightforward as inthe case of residuated (fuzzy) logics (cf. [11]). Theequality seems to be more fundamental than impli-cation but it is in a certain sense more restricted.First of all, though the original EQ-algebras allowedalso distinct elements to be equal in the degree 1,for logic this is unacceptable and only the, so called,good EQ-algebras must be considered, i.e. algebrasin which the property a ∼ 1 = a holds true. Butstill, properties of the obtained basic EQ-logic arelimited. Namely, despite its completeness, the gen-eralized deduction theorem does not hold. This de-ficiency has fatal consequences, e.g. for possible de-velopment of the predicate EQ-logic since its com-pleteness seems hardly to be proved. This gives riseto introducing the ∆-connective well known alreadyfrom the residuated fuzzy logics. Unlike the latter,where introducing ∆ is an interesting but dispens-able option, the role of it in the fuzzy equality-basedlogics seems to be much deeper. For, it enables usto grasp firmly the classical equality inside fuzzylogics and the resulting system has nicer proper-ties. Philosophically, it demonstrates that classicalequality is fundamental principle which can neverbe fully relaxed and that the mentioned goodnessaxiom is not sufficient.

In this paper, we extend the basic EQ-logic by the∆ connective and the appropriate axioms and proveseveral properties of it including the completeness.It is notable that we also prove a slightly modifiedform of the deduction theorem whose formulationcopes well with the character of EQ-logic for whichformulations using equivalences and also equationalstyle of proofs is more natural.

EUSFLAT-LFA 2011 July 2011 Aix-les-Bains, France

© 2011. The authors - Published by Atlantis Press 156

2. EQ-logic: An overview

In this section we recall basic definitions and prop-erties of EQ-algebras and basic EQ-logic.

2.1. EQ-algebrasDefinition 1A non-commutative EQ-algebra E is an algebra oftype (2, 2, 2, 0), i.e.

E = 〈E,∧,⊗,∼,1〉 ,

where for all a, b, c, d ∈ E:

(E1) 〈E,∧,1〉 is a commutative idempotentmonoid (i.e. ∧-semilattice with top element1). We put a ≤ b iff a ∧ b = a, as usual.

(E2) 〈E,⊗,1〉 is a monoid and ⊗ is isotone w.r.t.≤ .

(E3) a ∼ a = 1

(E4) ((a ∧ b) ∼ c)⊗ (d ∼ a) ≤ c ∼ (d ∧ b)(E5) (a ∼ b)⊗ (c ∼ d) ≤ (a ∼ c) ∼ (b ∼ d)

(E6) (a ∧ b ∧ c) ∼ a ≤ (a ∧ b) ∼ a(E7) a⊗ b ≤ a ∼ bThe operation ∧ is called meet, ⊗ is called mul-

tiplication and ∼ is the fuzzy equality.For all a, b ∈ E we put

a→ b = (a ∧ b) ∼ a (1)

and call this operation an implication.The following theorem demonstrates the funda-

mental properties of the fuzzy equality and impli-cation.

Theorem 1Let E be an EQ-algebra. The following holds for alla, b, c ∈ E:

(a) Symmetry: a ∼ b = b ∼ a ,(b) Transitivity: (a ∼ b)⊗ (b ∼ c) ≤ a ∼ c ,(c) Transitivity of implication:

(a→ b)⊗ (b→ c) ≤ a→ c .

We distinguish several special classes of EQ-algebras (see [2]). Among them, very important forEQ-logics, are good EQ-algebras which satisfy theproperty

a ∼ 1 = a, a ∈ E.Note that in good EQ-algebras, only one way of theadjunction condition is satisfied:

a ≤ b→ c implies a⊗ b ≤ c.

The opposite implication does not hold in general.

2.2. Basic EQ-logic

Basic EQ-logic was introduced in [9]. It has threebasic binary connectives ∧∧∧,&&&,≡ and the truth con-stant >. Implication is a derived operation definedby

A⇒⇒⇒ B := (A∧∧∧B) ≡ A.The set of all formulas of the given language J is de-noted by FJ . The algebra of truth values is formedby a good non-commutative EQ-algebra.The following formulas are logical axioms of the

basic EQ-logic:

(EQ1) (A ≡ >) ≡ A(EQ2) A∧∧∧B ≡ B ∧∧∧A(EQ3) (A©B)©C ≡ A©(B©C), © ∈ {∧∧∧,&&&}(EQ4) A∧∧∧A ≡ A(EQ5) A∧∧∧ > ≡ A(EQ6) A&&&> ≡ A(EQ7) >&&&A ≡ A(EQ8a) ((A∧∧∧B)&&&C)⇒⇒⇒ (B&&&C)

(EQ8b) (C&&&(A∧∧∧B))⇒⇒⇒ (C&&&B)

(EQ9) ((A∧∧∧B) ≡ C)&&&(D ≡ A)⇒⇒⇒ (C ≡ (D ∧∧∧B))

(EQ10) (A ≡ B)&&&(C ≡ D)⇒⇒⇒ (A ≡ C) ≡ (D ≡ B)

(EQ11) (A⇒⇒⇒ (B ∧∧∧ C))⇒⇒⇒ (A⇒⇒⇒ B)

The deduction rules of basic EQ-logic are equa-nimity rule (EA) and Leibniz rule (Leib):

(EA) A,A ≡ BB

, (Leib) A ≡ BC[p := A] ≡ C[p := B]

where by C[p := A] we denote a formula (in theproofs also called as a “C-part”) resulting from C byreplacing all occurrences of a propositional variablep in C by the formula A. The basic notions of truthevaluation, tautology, theory, etc. are defined asusual.The main properties of the basic EQ-logic needed

in the sequel are summarized in the following lem-mas.

Lemma 1(a) A ` A ≡ >, (rule (T2))

(b) ` A ≡ A,(c) A ≡ B ` B ≡ A,(d) A,A⇒⇒⇒ B ` B, (Modus Ponens)

(e) A⇒⇒⇒ B,B⇒⇒⇒ C ` A⇒⇒⇒ C,

(f) ` (A ≡ B) ≡ (B ≡ A),

(g) A⇒⇒⇒ (B ≡ C), B ≡ D ` A⇒⇒⇒ (D ≡ C),

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(h) ` (A ≡ B)⇒⇒⇒ (A⇒⇒⇒ B),

(i) ` (A ≡ B)&&&(C ≡ D)⇒⇒⇒ (A ≡ C) ≡ (B ≡ D),

(j) A⇒⇒⇒ B,C ⇒⇒⇒ D ` (A&&&C)⇒⇒⇒ (B&&&D),

(k) ` B⇒⇒⇒ (A⇒⇒⇒ B),

(l) ` (A ≡ B)&&&(B ≡ C)⇒⇒⇒ (A ≡ C),

(m) ` (A&&&(A ≡ B))⇒⇒⇒ B.

For the proof, see [9]. Let us emphasize that theformal proofs proceed mainly in an equational stylewhich is more natural for EQ-logics. This style isalso used in the proof below.

Lemma 2(a) ` (A ≡ B)⇒⇒⇒ ((A∧∧∧ C) ≡ (B ∧∧∧ C)),

(b) ` ((A ≡ B)&&&(C ≡ D))⇒⇒⇒ ((A∧∧∧C) ≡ (B∧∧∧D)).

proof: (a)

((A ∧∧∧ C) ≡ (A ∧∧∧ C))&&&(B ≡ A) ⇒⇒⇒ ((A ∧∧∧ C) ≡(B ∧∧∧ C)) (EQ9)

⇔ 〈(Leib) + Lemma 1(b) + rule (T2); “C-part”:p&&&(B ≡ A)⇒⇒⇒ ((A∧∧∧ C) ≡ (B ∧∧∧ C))

>&&&(B ≡ A)⇒⇒⇒ ((A∧∧∧ C) ≡ (B ∧∧∧ C))

⇔ 〈(Leib) + (EQ7); “C-part”:p⇒⇒⇒ ((A∧∧∧ C) ≡ (B ∧∧∧ C))

(B ≡ A)⇒⇒⇒ ((A∧∧∧ C) ≡ (B ∧∧∧ C))

⇔ 〈(Leib) + Lemma 1(f); “C-part”:p⇒⇒⇒ ((A∧∧∧ C) ≡ (B ∧∧∧ C))

(A ≡ B)⇒⇒⇒ ((A∧∧∧ C) ≡ (B ∧∧∧ C))

(b) First we use Lemma 2(a) ` (A ≡ B)⇒⇒⇒ ((A∧∧∧C) ≡ (B ∧∧∧ C)). Then we use Lemma 2(a) againand also (EQ2) and the Leibniz rule to obtain `(C ≡ D) ⇒⇒⇒ ((B ∧∧∧ C) ≡ (B ∧∧∧ D)). We use bothformulas as assumptions in Lemma 1(j) to get `((A ≡ B)&&&(C ≡ D))⇒⇒⇒ ((A∧∧∧C) ≡ (B∧∧∧C))&&&((B∧∧∧C) ≡ (B ∧∧∧ D)). Finally using Lemma 1(l) and (e)we obtain Lemma 2(b).

¤The proof of the following theorems can be found

in [9].

Theorem 2 (Soundness)The basic EQ-fuzzy logic is sound, i.e. ` A implies|= A.

Theorem 3 (Completeness)The following is equivalent for every formula A:

(a) ` A,(b) e(A) = 1 for every good non-commutative EQ-

algebra E and a truth evaluation e : FJ −→ E.

3. EQ∆∆∆-logic

In this section we introduce propositional EQ-logicwith ∆∆∆ connective∗) which we will call basic EQ∆∆∆-logic. Its semantics is based on a good non-commutative EQ∆-algebra.

3.1. EQ∆-algebra

Definition 2An EQ∆-algebra is an algebra

E∆ = 〈E,∧,⊗,∼,∆,1〉 (2)

which is a good non-commutative EQ-algebra E ex-tended by a unary additional operation ∆ : E → Efulfilling the following axioms:

(E∆∆∆1) ∆a ≤ ∆∆a

(E∆∆∆2) ∆(a ∼ b) ≤ ∆a ∼ ∆b

(E∆∆∆3) ∆(a ∧ b) = ∆a ∧∆b

(E∆∆∆4) ∆a = ∆a⊗∆a.

(E∆∆∆5) ∆(a ∼ b) ≤ (a⊗ c) ∼ (b⊗ c)

(E∆∆∆6) ∆(a ∼ b) ≤ (c⊗ a) ∼ (c⊗ b)

Axioms (E∆∆∆1)–(E∆∆∆4) have already been intro-duced in [7]. Axioms (E∆∆∆5) and E∆∆∆6) characterizebehavior of the multiplication with respect to thecrisp equality. They turn out to be necessary forfurther development of EQ-logic, especially of itspredicate version (which is not yet existing).

b a

0

c

d

e f

1

Figure 1: Eight element EQ∆-algebra.

Example 1An example of a finite non-trivial EQ∆-algebra isthe following: we take the lattice structure in Figure1. The multiplication and fuzzy equality are defined

∗)This is also called Baaz delta.

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as follows:

⊗ 0 a b c d e f 10abcdef1

0 0 0 0 0 0 0 00 0 0 0 0 0 0 a0 0 0 0 0 0 0 b0 0 0 0 0 0 a c0 0 0 0 d d d d0 0 0 0 d d d e0 0 0 0 d d d f0 a b c d e f 1

∼ 0 a b c d e f 10abcdef1

1 e f d c a b 0e 1 d f c a c af d 1 e c c b bd f e 1 c c c cc c c c 1 f e da a c c f 1 d eb c b c e d 1 f0 a b c d e f 1

Using (1) we obtain the implication:

→ 0 a b c d e f 10abcdef1

1 1 1 1 1 1 1 1e 1 e 1 1 1 1 1f f 1 1 1 1 1 1d f e 1 1 1 1 1c c c c 1 1 1 1a a c c f 1 f 1b c b c e e 1 10 a b c d e f 1

The ∆ operation is defined by

∆(x) =

0 if x = 0, a, b, cd if x = d, e, f1 if x = 1

Note that the multiplication ⊗ is not commuta-tive. Indeed, for example c ⊗ f = a but f ⊗ c = 0.Moreover, this algebra is also non-residuated since,e.g., 0 = c⊗ e ≤ 0, but c � e→ 0.

It is easy to verify the following properties ofEQ∆-algebra:

Lemma 3Let E∆ be an EQ∆-algebra. For all a, b ∈ E it holdsthat

(a) ∆1 = 1 ,

(b) ∆a ≤ a ,

(c) If a ≤ b then ∆a ≤ ∆b ,

(d) ∆(a→ b) ≤ ∆a→ ∆b .

3.2. Basic EQ∆∆∆-logic

Let J be a language of EQ∆∆∆-logic, FJ a set of allformulas in the language J and

E∆ = 〈E,∧,⊗,∼,∆,1〉

be a good non-commutative EQ∆-algebra. The lan-guage of EQ∆∆∆-logic is the language of basic EQ-logicexpanded by the unary connective ∆∆∆.A truth evaluation of formulas is a mapping e :

FJ −→ E defined as follows: if A is a propositionalvariable p then e(p) ∈ E. Otherwise,

e(>) = 1,e(A∧∧∧B) = e(A) ∧ e(B),e(A&&&B) = (e(A)⊗ e(B)),e(A ≡ B) = e(A) ∼ e(B),

e(∆∆∆A) = ∆e(A)

for all formulas A,B ∈ FJ .

3.2.1. Logical axioms and inference rulesDefinition 3Axioms of EQ∆∆∆-logic are those of basic EQ-logicplus the following ones:

(EQ∆∆∆1) ∆∆∆A⇒⇒⇒∆∆∆∆∆∆A

(EQ∆∆∆2) ∆∆∆(A ≡ B)⇒⇒⇒ (∆∆∆A ≡∆∆∆B)

(EQ∆∆∆3) ∆∆∆(A∧∧∧B) ≡ (∆∆∆A∧∧∧∆∆∆B)

(EQ∆∆∆4) ∆∆∆A ≡ (∆∆∆A&&&∆∆∆A)

(EQ∆∆∆5) ∆∆∆(A ≡ B)⇒⇒⇒ ((A&&&C) ≡ (B&&&C))

(EQ∆∆∆6) ∆∆∆(A ≡ B)⇒⇒⇒ ((C&&&A) ≡ (C&&&B))

Deduction rules of EQ∆∆∆-logic are equanimity rule,Leibniz rule and Necessitation rule:

(N) A

∆∆∆A.

3.2.2. Main propertiesLemma 4All axioms of EQ∆∆∆-logic are tautologies.

proof: This is straightforward using the axiomsand properties of EQ∆-algebra. ¤

Lemma 5The deductive rules of EQ∆∆∆-logic are sound in thefollowing sense. Let e : FJ −→ E be a truth evalu-ation:

(a) If e(A) = 1 and e(A ≡ B) = 1 then e(B) = 1.

(b) If e(B ≡ C) = 1 then e(A[p := B] ≡ A[p :=C]) = 1 for any formula A.

(c) If e(A) = 1 then e(∆∆∆A) = 1.

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proof: By straightforward verification. ¤The following formulas are provable in EQ∆∆∆-logic.

Lemma 6(a) ∆∆∆A⇒⇒⇒ A,

(b) ∆∆∆> ≡ >,(c) ∆∆∆(A⇒⇒⇒ B)⇒⇒⇒ (∆∆∆A⇒⇒⇒∆∆∆B),

(d) ∆∆∆(A&&&B)⇒⇒⇒ (∆∆∆A&&&∆∆∆B),

(e) ∆∆∆((A ≡ B)&&&(C ≡ D))⇒⇒⇒((A&&&C) ≡ (B&&&D)).

proof: (a)

∆∆∆(A ≡ >)⇒⇒⇒ ((A&&&>) ≡ (>&&&>)) (EQ∆∆∆5)

⇔ 〈(Leib) + (EQ6); “C-part”: ∆∆∆(A ≡ >)⇒⇒⇒(p ≡ (>&&&>))

∆∆∆(A ≡ >)⇒⇒⇒ (A ≡ (>&&&>))

⇔ 〈(Leib) + (EQ6); “C-part”: ∆∆∆(A ≡ >)⇒⇒⇒(A ≡ p)

∆∆∆(A ≡ >)⇒⇒⇒ (A ≡ >)

⇔ 〈(Leib) + (EQ1); “C-part”: ∆∆∆p⇒⇒⇒ p

∆∆∆A⇒⇒⇒ A

(b) Follows from (EQ1) in the form ` (∆∆∆> ≡>) ≡∆∆∆> and from `∆∆∆> using (EA).(c)

∆∆∆((A∧∧∧B) ≡ A)⇒⇒⇒ (∆∆∆(A∧∧∧B) ≡∆∆∆A) (EQ∆∆∆2)

⇔ 〈(Leib) + EQ∆∆∆3; “C-part”: ∆∆∆((A∧∧∧B) ≡ A)⇒⇒⇒(p ≡∆∆∆A)〉

∆∆∆((A∧∧∧B) ≡ A)⇒⇒⇒ ((∆∆∆A∧∧∧∆∆∆B) ≡∆∆∆A) (i.e. (c))

(e) First we use Lemma 6(d)`∆∆∆((A ≡ B)&&&(C ≡ D))

⇒⇒⇒ (∆∆∆(A ≡ B)&&&∆∆∆(C ≡ D)).Using assumptions (EQ∆∆∆5) and ` ∆∆∆(C ≡ D) ⇒⇒⇒∆∆∆(C ≡ D) in Lemma 1(j) we obtain` (∆∆∆(A ≡ B)&&&∆∆∆(C ≡ D))

⇒⇒⇒ (((A&&&C) ≡ (B&&&C))&&&∆∆∆(C ≡ D)) .In the same way (but using (EQ∆∆∆6)) we get` (((A&&&C) ≡ (B&&&C))&&&∆∆∆(C ≡ D))⇒⇒⇒ (((A&&&C) ≡ (B&&&C))&&&((B&&&C) ≡ (B&&&D))) .

Finally from the previous formulas and Lemma 1(l)in the form` (((A&&&C) ≡ (B&&&C))&&&((B&&&C) ≡ (B&&&D)))

⇒⇒⇒ ((A&&&C) ≡ (B&&&D))using Lemma 1(e) we find desired formula. ¤The following is the deduction theorem formu-

lated in the style natural for EQ-logic.

Theorem 4 (Deduction theorem)For each theory T and formulas A,B,C it holdsthat

T ∪ {A ≡ B} ` C iff T `∆∆∆(A ≡ B)⇒⇒⇒ C.

proof: [Hilbert style] Let T ∪ {A ≡ B} ` C.The proof proceeds by induction on the length ofthe proof of C.

(a) If C := (A ≡ B) then T ` ∆∆∆(A ≡ B)⇒⇒⇒ (A ≡B) by Lemma 6(a).

(b) C is an axiom of T then T ` ∆∆∆(A ≡ B) ⇒⇒⇒ Cusing Lemma 1(k) and Lemma 1(d).

(c) Let C have been obtained using rule (EA) bythe proof

. . . , D,D ≡ C,C.Then

(L.1) T ` ∆∆∆(A ≡ B) ⇒⇒⇒ (∆∆∆(A ≡ B)&&&∆∆∆(A ≡B)) ((EQ∆∆∆4), Lemma 1(h), (d))

(L.2) T ` (∆∆∆(A ≡ B)&&&∆∆∆(A ≡ B)) ⇒⇒⇒(D&&&(D ≡ C))(2x inductive assumption, Lemma 1(j))

(L.3) T ` (D&&&(D ≡ C))⇒⇒⇒ C (Lemma 1(m))(L.4) T `∆∆∆(A ≡ B)⇒⇒⇒ C (L.1, L.2, L.3 and

Lemma 1(e))

(d) Let C := D[p := E] ≡ D[p := F ] have beenobtained using rule (Leib) by the proof

. . . , E ≡ F,D[p := E] ≡ D[p := F ].

Then the proof proceeds by induction on thecomplexity of the formula C.

(i) If D is either > or q (other than p) thenD[p := E] ≡ D[p := F ] = D ≡ Dand T ` ∆∆∆(A ≡ B) ⇒⇒⇒ (D ≡ D) fol-lows from Lemma 1(b), Lemma 1(k) andLemma 1(d).

(ii) If D is p then it follows immediately fromthe inductive assumption.

(iii) Let D be G© H, where © ∈ {∧∧∧,&&&,≡}.Then we need to proveT `∆∆∆(A ≡ B)⇒⇒⇒ ((G©H)[p := E]

≡ (G©H)[p := F ])thusT `∆∆∆(A ≡ B)⇒⇒⇒ ((G[p := E]©H[p := E]) ≡ (G[p := F ]©H[p := F ]))and thus shortlyT `∆∆∆(A ≡ B)

⇒⇒⇒ ((G′©H ′) ≡ (G′′©H ′′),where G′ := G[p := E], H ′ := H[p :=E], G′′ := G[p := F ] and H ′′ := H[p :=F ].Let first D be G∧∧∧H. Then(L.1) T `∆∆∆(A ≡ B)

⇒⇒⇒ (∆∆∆(A ≡ B)&&&∆∆∆(A ≡ B))((EQ∆∆∆4), Lemma 1(h), (d))

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(L.2) T ` (∆∆∆(A ≡ B)&&&∆∆∆(A ≡ B)) ⇒⇒⇒((G′ ≡ G′′)&&&(H ′ ≡ H ′′))

(2x inductive assumption,Lemma 1(j))

(L.3) T ` ((G′ ≡ G′′)&&&(H ′ ≡ H ′′))⇒⇒⇒ ((G′ ∧∧∧H ′) ≡ (G′′ ∧∧∧H ′′))

(Lemma 2(b))(L.4) T `∆∆∆(A ≡ B)⇒⇒⇒ ((G′ ∧∧∧H ′)

≡ (G′′ ∧∧∧H ′′))(L.1, L.2, L.3 and Lemma 1(e))

(iv) Let D be G ≡ H. Then(L.1) T `∆∆∆(A ≡ B)

⇒⇒⇒ (∆∆∆(A ≡ B)&&&∆∆∆(A ≡ B))((EQ∆∆∆4), Lemma 1(h), (d))

(L.2) T ` (∆∆∆(A ≡ B)&&&∆∆∆(A ≡ B)) ⇒⇒⇒((G′ ≡ G′′)&&&(H ′ ≡ H ′′))

(2x inductive assumption,Lemma 1(j))

(L.3) T ` ((G′ ≡ G′′)&&&(H ′ ≡ H ′′))⇒⇒⇒ ((G′ ≡ H ′) ≡ (G′′ ≡ H ′′))

(Lemma 1(i))(L.4) T `∆∆∆(A ≡ B)⇒⇒⇒ ((G′ ≡ H ′)

≡ (G′′ ≡ H ′′))(L.1, L.2, L.3 and Lemma 1(e))

(v) Let D be G&&&H.(L.1) T ` (∆∆∆(A ≡ B) ⇒⇒⇒ ((G′ ≡

G′′)&&&(H ′ ≡ H ′′)) (see above)(L.2) T ` ∆∆∆(A ≡ B) ⇒⇒⇒ ∆∆∆∆∆∆(A ≡ B)

((EQ∆∆∆1))(L.3) T ` ∆∆∆∆∆∆(A ≡ B) ⇒⇒⇒ ∆∆∆((G′ ≡

G′′)&&&(H ′ ≡ H ′′)) (L.1, rule (N),Lemma 6(c), Lemma 1(d))

(L.4) T `∆∆∆((G′ ≡ G′′)&&&(H ′ ≡ H ′′))⇒⇒⇒ ((G′&&&H ′) ≡ (G′′&&&H ′′))

(Lemma 6(e))(L.5) T `∆∆∆(A ≡ B)⇒⇒⇒ ((G′&&&H ′)

≡ (G′′&&&H ′′))(L.2, L.3, L.4 and Lemma 1(e))

(e) Let C := ∆∆∆D have been obtained using rule (N)by the proof

. . . , D,∆∆∆D.

Then T ` ∆∆∆(A ≡ B) ⇒⇒⇒ ∆∆∆D follows from in-ductive assumption using rule (N), Lemma 6(c),1(d), (EQ∆∆∆1) and Lemma 1(e).

The converse implication follows from rule (N) andLemma 1(d).

¤

Remark 1If we put B := > in this theorem then it is easy todeduce the “standard” form of the delta deductiontheorem:

T ∪ {A} ` C iff T `∆∆∆A⇒⇒⇒ C.

Definition 4Put

A ≈ B iff ` A ≡ B, A,B ∈ FJ .

It follows from Lemmas 1(b), (f) and (l) that ≈is an equivalence on FJ . Let us denote by [A] anequivalence class of A and put E = {[A] | A ∈ FJ}.Finally we define

1 = [>],[A] ∧ [B] = [A∧∧∧B],[A]⊗ [B] = [A&&&B],[A] ∼ [B] = [A ≡ B],

∆[A] = [∆∆∆A].

Lemma 7The algebra E∆ = 〈E,∧,⊗,∼,∆,1〉 is a good non-commutative EQ∆-algebra.

proof: We have to verify axioms of EQ∆-algebra: for axioms (E1)–(E7) and “goodness prop-erty” see the proof of Lemma 10 in [9]. For axioms(E∆1)–(E∆6) see (EQ∆∆∆1)–(EQ∆∆∆6). ¤

Theorem 5 (Soundness)The basic EQ∆∆∆-fuzzy logic is sound.

proof: This follows from Lemma 4 and 5. ¤

Theorem 6 (Completeness)The following is equivalent for every formula A:

(a) ` A,(b) e(A) = 1 for every good non-commutative

EQ∆-algebra E∆ and a truth evaluation e :FJ −→ E.

proof: The implication (a) to (b) is soundness.(b) to (a): By Lemma 7 the algebra E∆ of equiva-

lence classes of formulas is a good non-commutativeEQ∆-algebra. Thus, if (b) holds then it holds alsofor e : FJ −→ E. If e(A) = 1 then it means that[A] = [>], i.e. ` A ≡ > and so, ` A by rule (T1).

¤

4. Conclusion

In this paper, we continue the development of EQ-logics, i.e. logics based on EQ-algebra of truth val-ues. We introduced the ∆-connective whose role infuzzy equality-based logics seems to be more im-portant than in residuated fuzzy logics. Besidesothers, it enables us to prove the deduction theo-rem which in basic EQ-logic without ∆ does nothold. The completeness theorem has been provedin its basic form. With respect to the results from[3] it is clear that it the representability of prelin-ear EQ-algebras can be extended also to prelinear

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EQ∆-algebras. Consequently, extension of the com-pleteness theorem to hold with respect to all linearlyordered EQ∆-algebras will be possible.It is an open question whether there is an ax-

iomatic system with modus ponens and necessita-tion as the only deduction rules in EQ∆-logics. Inthe case of positive answer, however, equanimityand Leibniz rules are more natural than modus po-nens for equality based logic, because the latter in-ference rule is based on implication — derived con-nective in EQ-logics.The future work will focus on further develop-

ment of all introduced kinds of EQ-logics, study oftheir relation to residuated fuzzy logics and also tothe development of predicate EQ-logics. It seems tobe impossible to develop predicate EQ-logics with-out ∆.

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[8] V. Novák and M. Dyba. Non-commutativeEQ-logics and their extensions. In J. P.Carvalho, D. Dubois, U. Kaymak, and J.M.da Costa Sousa, editors, Proc. of the Joint 2009IFSA World Congress and 2009 EUSFLATConf., July 20-24, 2009, pages 1422–1427, Lis-bon, Portugal, 2009. University of Málaga.

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