Logical foundation of symmetric implicational methods for fuzzy reasoning

Abstract

The symmetric implicational methods for fuzzy reasoning characterizes the solution B^* (A^*) of the formula (A → ₁B) → ₂ (A^* → ₁B^*) for the fuzzy modus ponens (fuzzy modus tollens), where →₁ and →₂ are two different implications. In this study, we provide a predicate formal representation of the solution for the symmetric implicational methods based on the LΠ formal logic system, including detailed logic proofs. We bring the symmetric implicational methods within a logical framework and provide a sound logic foundation for the symmetric implicational methods of fuzzy reasoning.

Keywords

Fuzzy reasoning symmetric implicational method

1 Introduction

It is well known that the most fundamental inference forms of fuzzy reasoning are fuzzy modus ponens (FMP) and fuzzy modus tollens (FMT), which can be respectively expressed as follows (see [18, 19]).

FMP(A, B, A^*). For a given rule A → B and an input A^*, calculate the output B^*.

FMT(A, B, B^*). For a given rule A → B and an input B^*, calculate the output A^*.

In order to deal with the above inference forms, Zadeh [18] proposed the compositional rule of inference (CRI) method. Since the CRI method was proposed, many other methods of fuzzy reasoning have been proposed introduced (see [14 , 20]). As a revision or a supplement for the Zadeh’s CRI method, Wang [16] proposed the full implication triple I (TI) method, whose solution was the smallest B^* ∈ F (V) (or the largest A^* ∈ F (U)) such that, for any u ∈ U, v ∈ V,

$(A (u) \to B (v)) \to (A^{*} (u) \to B^{*} (v))$ (1) takes its maximum, where → is also a fuzzy implication.

Tang et al. [14] generalized the TI method to the symmetric implicational (SI) method, which changed (1) into

$(A (u) \to_{1} B (v)) \to_{2} (A^{*} (u) \to_{1} B^{*} (v)) .$ (2)

In the research area of fuzzy reasoning, one of the interesting problems is the analysis of reliability and reasonableness of fuzzy reasoning. To solve this problem, one approach is to formalize the fuzzy reasoning algorithms and their consistency. In this way, fuzzy reasoning is put rigorously into the framework of pure fuzzy logic and thus have a rigorous logic foundation. Novák [9] discussed CRI algorithms based on the fuzzy predicate logic BL (see [7]). Pei [10 –12] considered the formalization problem of the TI algorithms based on the monoidal t-norm based fuzzy logic MTL (see Esteva and Godo [4]) and revised Kleene logic K^* (see [4, 13]). Luo et al. [8] studied the formalization problem for quintuple implication (QI) algorithms [20] based on MTL. However, to the best of our knowledge, the logical foundation of the SI algorithms has not been studied previously. Our main aim is to formalize the solutions for SI algorithms to the FMP and FMT problems and provide a sound logic foundation for the SI algorithms of fuzzy reasoning. But one problem is: which logic framework of formal systems? They are many formal systems such as BL, MTL and K^*. Since the SI method allows different fuzzy implications to be employed, these formal systems dealing with a fixed (class of) fuzzy implication are not suitable logical framework for SI method. The formal system LΠ [1 , 5] contains three common implications, Lukasiewicz, product and Gödel implications. Many researchers have considered various methods of fuzzy reasoning including SI methods for these three implications [14 , 20]. Thus in this paper, we consider the problem of formalizing the solutions for SI method based on LΠ.

The remainder of this paper is organized as follows. Section 2 provides some basic definitions and LΠ logic. Section 3 gives the predicate formal representation of the solutions for the SI algorithms based on LΠ logic and the strict logic proofs. We give our conclusions in Section 4.

2 LΠ logic

LΠ logic [1 , 5] is a combination of the Lukasiewicz and the product logics, which are two extensions of the basic logic BL introduced by Hájek in [7]. BL is based on two basic connectives & and → and one truth constant $\bar{0}$ . The derived connectives ¬, ∨ , ∧ , ↔ are defined as follows:

$\neg A = A \to \bar{0}$

A ∧ B = A & (A → B)

A ∨ B = ((A → B) → B) ∧ ((B → A) → A)

A ↔ B = (A → B) & (B → A)

BL consists of the deduction rule MP (modus ponens) and the following axioms:

(A → B) → ((B → C) → (A → C))

(A & B) → A

(A & B) → (B & A)

A & (A → B) → B & (B → A)

(A → (B → C)) → ((A & B) → C)

((A & B) → C) → (A → (B → C))

((A → B) → C) → (((B → A) → C) → C)

$\bar{0} \to A$

The predicate version of the BL is built in [7], here are the logical axioms on quantifiers:

(∀ x) A (x) → A (t), where t is substitutable for x in A;

A (t) → (∃ x) A (x), where t is substitutable for x in A;

(∀ x) (A → B) → (A → (∀ x) B), where x is not free in A;

(∀ x) (A → B) → ((∃ x) A → B), where x is not free in B;

(∀ x) (A ∨ B) → ((∀ x) A ∨ B), where x is not free in B.

Theorem 2.1. [7]. The following formulas are theorems of the system BL∀:

A → A

(A → (B → C)) → (B → (A → C))

(∀ x) A (x) → A (x)

(∀ x) A (x) → (∃ x) A (x)

A → (B → A & B)

A → (B → A)

(A → B) → (A & C → B & C)

(B → C) → ((A → B) → (A → C))

A (x) → (∃ x) A (x)

{A → B, B → C} ⊢ A → C.

BL can be extended to the Lukasiewicz and the product logics:

Łukasiewicz logic:

BL + (Ł) ¬¬ A → A

Product logic Π:

BL + (Π1) ¬¬ A → ((A → A & B) → (¬¬ B & B))

+ (Π2) $A \land \neg A \to \bar{0}$

The LΠ logic [5] has one truth constant $\bar{0}$ and three basic connectives: Łukasiewicz implication →_L, product implication →_P and product conjunction ⊙. The derived connectives are defined as follows:

Łukasiewicz negation:

$\neg_{L} A \overset{def}{=} A \to_{L} \bar{0}$

Product negation:

$\neg_{P} A \overset{def}{=} A \to_{P} \bar{0}$

0-1 projector:

$Δ A \overset{def}{=} \neg_{P} \neg_{L} A$

Łukasiewicz conjunction:

$A \otimes B \overset{def}{=} \neg_{L} (A \to_{L} \neg_{L} B)$

Gödel conjunction:

$A \land B \overset{def}{=} A \otimes (A \to_{L} B)$

Max disjunction:

$A \lor B \overset{def}{=} (A \to_{L} B) \to_{L} B$

Gödel implication:

$A \to_{G} B \overset{def}{=} (A \to_{L} B) \lor B$

Łukasiewicz equivalence:

$A \equiv_{L} B \overset{def}{=} (A \to_{L} B) \otimes (B \to_{L} A)$

Product equivalence:

$A \equiv_{P} B \overset{def}{=} (A \to_{P} B) ⊙ (B \to_{P} A)$

Truncated Addition:

$A \oplus B \overset{def}{=} \neg_{L} A \Rightarrow_{L} B$

Truncated Subtraction:

$A ⊖ B \overset{def}{=} A \otimes \neg_{L} B$

The LΠ logic consists of the following axioms

Axioms of the Łukasiewicz logic

Axioms of the product logic Π without (BL7)

¬_PA → _L ¬ _LA

Δ (A → _LB) ≡ _L (A → _PB)

A ⊙ (B ⊖ C) ≡ _L (A ⊙ B) ⊖ (A ⊙ C)

and the deduction rules are the modus ponens MP for Łukasiewicz implication and the necessitation of Δ (from A infer ΔA).

Cintula [2] defined a new deduction rule, denoted ModusT: if one of the formulas A → _LB, A → _PB, A → _GB is a theorem, then all of them are theorems.

Theorem 2.2 [2]. ModusT is a derived deduction rule in the LΠ logic

The predicate version of the LΠ logic, denoted LΠ∀, is given by adding the following axiomatic schema:

(∀ x) A (x) → _LA (t), where t is substitutable for x in A;

(∀ x) (A → _LB) → _L (A → _L (∀ x) B), where x is not free in A;

and the inference rule generalization, denoted GEN, i.e., from A infer (∀ x) A.

Theorem 2.3 [2]. The LΠ∀ logic extend Lukasiewicz and product predicate logics.

3 Formalization of symmetric implicational algorithms

In this section, we formalize the SI algorithms of fuzzy reasoning. First, we introduce the following concepts.

Definition 3.1. Suppose that A and A^* are two unary predicates with the type (s₁), and B is a unary predicate with the type (s₂). If there exists some unary predicate B^* with the type (s₂) such that the following two conditions hold, then we call B^* the formal symmetric implicational solution of FMP, briefly, FSTI solution of FMP:

⊢ (∀ x) (∀ y) ((A (x) → ₁B (y)) → ₂ (A^* (x) → ₁B^* (y)));

If C is any unary predicate with the type (s₂) satisfying ⊢ (∀ x) (∀ y) ((A (x) → ₁B (y)) → ₂ (A^* (x) → ₁C (y))) , then ⊢ (∀ y) (B^* (y) → ₂C (y)).

Where →₁, → ₂ ∈ {→ _L, → _P, → _G}.

Definition 3.2. Suppose that A is a unary predicate with the type (s₁), and B and B^* are two unary predicates with the type (s₂). If there exists some unary predicate A^* with the type (s₁) such that the condition (SI1) and the following condition hold, then we call A^* the formal symmetric implicational solution of FMT, briefly, FSTI solution of FMT:

If D is any unary predicate with the type (s₂) satisfying ⊢ (∀ x) (∀ y) ((A (x) → ₁B (y)) → ₂ (D (x) → ₁B^* (y))) , then ⊢ (∀ x) (D (x) → ₂A^* (x)).

Where →₁, → ₂ ∈ {→ _L, → _P, → _G}.

Remark 3.1. In this paper, we only consider the case →₁ ≠ → ₂. Formalization problem of the TI algorithms for the case →₁ = → ₂ are considered in [11, 12].

Definition 3.3. Suppose that A and A^* are two unary predicates with type (s₁), B is a unary predicate with type (s₂), and A is normal, i.e., ⊢ (∃ x) A (x). We say that a FSI algorithm of the FMP problem is L-reductive if B^* ∼ _LB (i.e., both ⊢B^* → _LB and ⊢B → _LB^* hold) whenever A^* = A.

Definition 3.4. Suppose that A is a unary predicate with the type (s₁), and B and B^* are two unary predicates with the type (s₂) and B is co-normal, i.e., ⊢ (∃ y) ¬ B (y). We say that a FSI algorithm of the FMT problem is L-reductive, if we have A^* ∼ _LA whenever B^* = B.

3.1 Formalization of SI algorithms for FMP

For the SI algorithm of the FMP problem, we can obtain the following results.

Theorem 3.1. If →₁ = → _P, → ₂ = → _L, then the FSTI solution B^* of FMP is given by the following formulas: $B^{*} = (\exists x) ((A (x) \to_{P} B) ⊙ A^{*} (x)),$ (3) for variable y with the sort s₂, $B^{*} (y) = (\exists x) ((A (x) \to_{P} B (y)) ⊙ A^{*} (x)) .$ (4)

Proof. The following formula sequence is a proof of (SI1):

((A (x) → _PB) ⊙ A^* (x)) → _P (∃ x) ((A (x) → _PB) ⊙ A^* (x)) (∃1)

(((A (x) → _PB) ⊙ A^* (x)) → _P (∃ x) ((A (x) → _PB) ⊙ A^* (x))) →_P ((A (x) → _PB) → _P (A^* (x) → _P (∃ x) ((A (x) → _PB) ⊙ A^* (x)))) (BL5b)

(A (x) → _PB) → _P (A^* (x) → _P (∃ x) ((A (x) → _PB) ⊙ A^* (x))) (1°, 2°, MP)

(A (x) → _PB) → _L (A^* (x) → _P (∃ x) ((A (x) → _PB) ⊙ A^* (x))) (3°, ModusT)

(∀ x) (∀ y) ((A (x) → _PB) → _L (A^* (x) → _P (∃ x) ((A (x) → _PB) ⊙ A^* (x)))) (4°, GEN)

The following formula sequence is a proof of (SI2):

(∀ x) (∀ y) ((A (x) → _PB (y)) → _L (A^* (x) → _PC (y))) (Hypot)

(A (x) → _PB (y)) → _L (A^* (x) → _PC (y)) (∀1)

(A (x) → _PB (y)) → _P (A^* (x) → _PC (y)) (2°, ModusT)

((A (x) → _PB (y)) → _P (A^* (x) → _PC (y))) →_P ((A (x) → _PB (y)) ⊙ A^* (x) → _PC (y)) (BL5a)

(A (x) → _PB (y)) ⊙ A^* (x) → _PC (y) (3°, 4°, MP)

((A (x) → _PB (y)) ⊙ A^* (x) → _PC (y)) →_P (∀ x) ((A (x) → _PB (y)) ⊙ A^* (x) → _PC (y))

(∀1)

(∀ x) ((A (x) → _PB (y)) ⊙ A^* (x) → _PC (y))

(5°, 6°, MP)

(∀ x) ((A (x) → _PB (y)) ⊙ A^* (x) → _PC (y)) →_P (∃ x) ((A (x) → _PB (y)) ⊙ A^* (x)) → _PC (y)

(∃2)

(∃ x) ((A (x) → _PB (y)) ⊙ A^* (x)) → _PC (y)

(7°, 8°, MP)

B^* (y) → _PC (y) (9°, Abbr.)

B^* (y) → _LC (y) (10°, ModusT)

(∀ y) (B^* (y) → _LC (y)) (11°, GEN)

Theorem 3.2 If →₁ = → _G, → ₂ = → _L, then the FSTI solution B^* of FMP is given by the following formulas: $B^{*} = (\exists x) ((A (x) \to_{G} B) ⊙ A^{*} (x)),$ (5) for variable y with the sort s₂, $B^{*} (y) = (\exists x) ((A (x) \to_{G} B (y)) ⊙ A^{*} (x)) .$ (6)

Theorem 3.3 If →₁ = → _G, → ₂ = → _P, then the FSTI solution B^* of FMP is given by the following formulas: $B^{*} = (\exists x) ((A (x) \to_{G} B) ⊙ A^{*} (x)),$ (7) for variable y with the sort s₂, $B^{*} (y) = (\exists x) ((A (x) \to_{G} B (y)) ⊙ A^{*} (x)) .$ (8)

3.2 Formalization of SI algorithms for FMT

For the SI algorithm of the FMT problem, we can obtain the following results.

Theorem 3.4. If →₁ = → _P, → ₂ = → _L, then the FSTI solution A^* of FMT is given by the following formulas: $A^{*} = (\forall y) ((A \to_{P} B (y)) \to_{P} B^{*} (y)),$ (9) for variable x with the sort s₁, $A^{*} (x) = (\forall y) ((A (x) \to_{P} B (y)) \to_{P} B^{*} (y)) .$ (10)

Proof. The following formula sequence is a proof of (SI1):

((A (x) → _PB (y)) → _PB^* (y)) → _P ((A (x) → _PB (y)) → _PB^* (y)) (T1)

((A (x) → _PB (y)) → _P (((A (x) → _PB (y)) → _PB^* (y)) → _PB^* (y)) ) → _P ((A (x) → _PB (y)) → _P (((A (x) → _PB (y)) → _PB^* (y)) → _PB^* (y)) ) (T2)

(A (x) → _PB (y)) → _P (((A (x) → _PB (y)) → _PB^* (y)) → _PB^* (y)) (1°, 2°, MP)

(∀ v) ((A (x) → _PB (v)) → _PB^* (v)) → _P ((A (x) B (y)) → _PB^* (y)) (T3)

((∀ v) ((A (x) → _PB (v)) → _PB^* (v)) → _P ((A (x) B (y)) → _PB^* (y)) ) →_P ((((A (x) → _PB (y)) → _PB^* (y)) → _PB^* (y)) → _P ((∀ v) ((A (x) → _PB (v)) → _PB^* (v)) → _PB^* (y)) )

(BL1)

(((A (x) → _PB (y)) → _PB^* (y)) → _PB^* (y)) →_P ((∀ v) ((A (x) → _PB (v)) → _PB^* (v)) → _PB^* (y)) (4°, 5°, MP)

(A (x) → _PB (y)) → _P (A^* (x) → _PB^* (y)) (3°, 4°, 6°, HS)

(A (x) → _PB (y)) → _L (A^* (x) → _PB^* (y)) (7°, ModusT)

(∀ x) (∀ y) ((A (x) → _PB (y)) → _L (A^* (x) → _PB^* (y))) (8°, GEN)

The following formula sequence is a proof of (SI3):

(∀ x) (∀ y) ((A (x) → _PB (y)) → _L (D (x) → _PB^* (y))) (Hypot.)

(A (x) → _PB (y)) → _L (D (x) → _PB^* (y)) (1°, T3, MP)

(A (x) → _PB (y)) → _P (D (x) → _PB^* (y)) (2°, ModusT)

((A (x) → _PB (y)) → _P (D (x) → _PB^* (y)) ) →_P (D (x) → _P ((A (x) → _PB (y)) → _PB^* (y)) ) (T2)

D (x) → _P ((A (x) → _PB (y)) → _PB^* (y)) (3°, 4°, MP)

(∀ y) (D (x) → _P ((A (x) → _PB (y)) → _PB^* (y))) (5°, GEN)

D (x) → _P (∀ y) ((A (x) → _PB (y)) → _PB^* (y)) (6°, ∀2, MP)

D (x) → _PA^* (x) (7°, Abbr.)

D (x) → _LA^* (x) (8°, ModusT)

(∀ x) (D (x) → _LA^* (x)) (9°, GEN)

Similarly, we have the following results.

Theorem 3.5. If →₁ = → _G, → ₂ = → _L, then the FSTI solution A^* of FMT is given by the following formulas: $A^{*} = (\forall y) ((A \to_{G} B (y)) \to_{G} B^{*} (y)),$ (11) for variable x with the sort s₁, $A^{*} (x) = (\forall y) ((A (x) \to_{G} B (y)) \to_{G} B^{*} (y)) .$ (12)

Theorem 3.6. If →₁ = → _G, → ₂ = → _P, then the FSTI solution A^* of FMT is given by the following formulas: $A^{*} = (\forall y) ((A \to_{G} B (y)) \to_{G} B^{*} (y)),$ (13) for variable x with the sort s₁, $A^{*} (x) = (\forall y) ((A (x) \to_{G} B (y)) \to_{G} B^{*} (y)) .$ (14)

Theorem 3.7.

FSTI solution B^* of FMP problem given by Theorem 3.1 is L-reductive.

FSTI solution B^* of FMP problem given by Theorem 3.2 is L-reductive.

FSTI solution B^* of FMP problem given by Theorem 3.3 is L-reductive.

FSTI solution A^* of FMT problem given by Theorem 3.4 is L-reductive.

FSTI solution A^* of FMT problem given by Theorem 3.5 is L-reductive.

FSTI solution A^* of FMT problem given by Theorem 3.6 is L-reductive.

Proof. Here, we only prove (i). Suppose that A^* = A is normal, then $B^{*} (y) = (\exists x) ((A (x) \to_{P} B (y)) ⊙ A (x)) .$ First, we prove ⊢ (∀ y) (B^* (y) → _LB (y)).

(A (x) → _PB (y)) → _P (A (x) → _PB (y)) (T1)

((A (x) → _PB (y)) → _P (A (x) → _PB (y)) ) →_P ((A (x) → _PB (y)) ⊙ A (x) → _PB (y) ) (BL5a)

(A (x) → _PB (y)) ⊙ A (x) → _PB (y)

(1°, 2°, MP)

(∀ x) ((A (x) → _PB (y)) ⊙ A (x) → _PB (y)) (3°, GEN)

(∀ x) ((A (x) → _PB (y)) ⊙ A (x) → _PB (y)) →_P (∃ x) ((A (x) → _PB (y)) ⊙ A (x) → _PB (y)) (T4)

(∃ x) ((A (x) → _PB (y)) ⊙ A (x) → _PB (y))

(4°, 5°, MP)

B^* (y) → _PB (y) (6°, Abbr.)

B^* (y) → _LB (y) (7°, ModusT)

(∀ y) (B^* (y) → _LB (y)) (8°, GEN)

Then we prove ⊢ (∀ y) (B (y) → _LB^* (y)).

A (x) → _P (B (y) → _PA (x) ⊙ B (y)) (T5)

B (y) → _P (A (x) → _PB (y)) (T6)

(B (y) → _P (A (x) → _PB (y)) ) → _P (A (x) ⊙ B (y) → _PA (x) ⊙ (A (x) → _PB (y)) ) (T7)

A (x) ⊙ B (y) → _PA (x) ⊙ (A (x) → _PB (y))

(2°, 3°, MP)

(A (x) ⊙ B (y) → _PA (x) ⊙ (A (x) → _PB (y)) ) →_P ((B (y) → _PA (x) ⊙ B (y)) → _P (B (y) → _PA (x) ⊙ (A (x) → _PB (y))) )

(T8)

(B (y) → _PA (x) ⊙ B (y)) → _P (B (y) → _PA (x) ⊙ (A (x) → _PB (y))) (4°, 5°, MP)

A (x) → _P (B (y) → _PA (x) ⊙ (A (x) → _PB (y))) (1°, 6°, HS)

(A (x) → _PB (y)) ⊙ A (x) → _P (∃ x) ((A (x) → _PB (y)) ⊙ A (x)) (T9)

(B (y) → _PA (x) ⊙ (A (x) → _PB (y))) → _P (B (y) → _P (∃ x) ((A (x) → _PB (y)) ⊙ A^* (x))

(8°, T8, MP)

A (x) → _P (B (y) → _PB^* (y)) (1°, 6°, 9°, HS, Abbr.)

(∀ x) (A (x) → _P (B (y) → _PB^* (y)))

(10°, GEN)

(∀ x) (A (x) → _P (B (y) → _PB^* (y))) →_P (∃ x) A (x) → _P (B (y) → _PB^* (y))

(T4)

(∃ x) A (x) → _P (B (y) → _PB^* (y))

(11°, 12°, MP)

(∃ x) A (x) (Hypot.)

B (y) → _PB^* (y) (13°, 14°, MP)

B (y) → _LB^* (y) (15°, ModusT)

(∀ y) (B (y) → _LB^* (y) (16°, GEN)

4 Conclusion

Based on the variety of TI method of fuzzy reasoning, SI method is an extension of TI method. Obviously, SI method is more flexible because two different implications are used. In this paper, we provided the predicate formal representation and logic proof of the solutions for SI method based on the LΠ logic, which handles three implication connectives, i.e., Łukasiewicz, product and Gödel implications. Moreover, we can formalize the consistency of the SI method in the logical systems LΠ∀. Thus, we have placed SI method for these implications in the framework of pure fuzzy logic and provided a sound logical foundation for SI method of fuzzy reasoning.

As we know, there are other logics that contain two different implication connectives, such as non-commutative fuzzy logics [3, 6]. Thus, it is interesting to put SI method of fuzzy reasoning into the framework of these fuzzy logics.

Footnotes

Acknowledgments

The work is supported by the Opening Foundation of Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing (Grant No.2017CSOBDP0103).

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