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JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 1437
Δ( φ →ψ) →Δ( −ψ →− φ)
(Order Reversing)
Δφ
∨¬Δ φ
Δ( φ ∨ψ) →( Δφ ∨Δ ψ)
Δφ
→ φ
Δφ
→ΔΔ φ
Δ( φ →ψ) →( Δφ →Δ ψ)
where
¬ φ is φ → 0 . Deduction rules of UL − h∈ (0, 1] are those
of UL Δ h∈ (0,, 1] that is, modus ponens and generalization: from
ϕ derive Δ ϕ .
Definition 4 [17] A ŁΠG Δ
-algebra is a structure
L =< L,∗,⇒,∩,∪, 01 , ,Δ > which is a ŁΠG
algebra
expanded by an unary operation Δ in which the
following formulas are true:
Δx∪( Δx⇒ 0) = 1
Δ( x ∪ y)
≤ Δx∪Δ
y
Δx
≤ x
Δx
≤ ΔΔ x
( Δx) ∗( Δ( x⇒ y))
≤ Δ y
Δ 1=
1
Definition 5 [17] A ŁΠG − -algebra is a structure L
=< L,∗,⇒,∩,∪, 01 , ,Δ,− > which is a ŁΠG Δ
-algebra
expanded by an unary operation -, and satisfying the
following conditions:
(1) −− x = x
(2) Δ( x ⇒ y) =Δ( −y⇒ − x)
(3) Δx
∨¬Δ x = 1
(4) Δ( x ∨ y) ≤( Δx∨Δ
y)
(5) Δx ≤ x
(6) Δx ≤ ΔΔ x
(7) ( Δx) ∗( Δ( x⇒ y))
≤ Δ y
(8) Δ 1=
1
Theorem 3 [17] (Soundness) Each formula provable in
UL − h∈ (0, 1] is a L-tautology for each ŁΠG − -algebra.
Theorem 4 [17] (Completeness) The system UL − h∈ (0, 1] is
complete, i.e. If φ , then φ . In more detail, for each
formula φ , the following are equivalent:
(i) φ is provable in UL − h∈ (0,, 1] i.e. ϕ ,
(ii) φ is an L-tautology for each ŁΠG − -algebra L,
(iii)φ is an L-tautology for each linearly ordered ŁΠG − -
algebra L.
III. PREDICATE FORMAL SYSTEM ∀ h (0 1]
UL − ∈ ,
In order to build first-order predicate formal deductive
system based on 1-level universal AND operator, we give
the first-order predicate language as following:
First-order language J consists of symbols set and
generation rules:
The symbols set of J consist of as following:
(1) Object variables: x, yzx , ,
1, y1, z1, x2, y2, z2,;
(2) Object constants: abca , , ,
1, b1, c1,, Truth constants:
01 , ;
(3) Predicate symbols: PQRP , , ,
1, Q1, R1,;
(4) Connectives: &,→, Δ,− ;
(5) Quantifiers: ∀ (universal quantifier), ∃
(existential quantifier);
(6) Auxiliary symbols: (, ), , .
The symbols in (1)- (3) are called non-logical symbols
of language J. The object variables and object constants
of J are called terms. The set of all object constants is
denoted by Var (J), The set of all object variables is
denoted by Const (J), The set of all terms is denoted by
Term (J). If P is n-ary predicate symbol, t 1
, t 2
,, t n
are
terms, then Pt (
1, t2,, t n
) is called atomic formula.
The formula set of J is generated by the following
three rules in finite times:
(i) If P is atomic formula, then P∈ J ;
(ii) If PQ , ∈ J, then P&Q, P→ Q,ΔP∈ J,−P∈ J ;
(iii) If P∈ J , and x ∈ Var( J ) , then
( ∀ x) P,∃ ( x)
P∈ J .
The formulas of J can be denoted by φ, ϕψ , , φ1, ϕ1, ψ1,.
Further connectives are defined as following:
φ ∧ ψ is φ & ( φ → ψ)
,
φ ∨ ψ is (( φ →ψ) →ψ) ∧( ψ →φ) → φ)
,
¬ φ is φ → 0 ,
φ ≡ ψ is ( φ →ψ) & ( ψ → φ)
.
Definition 6The axioms and deduction rules of predicate
formal system ∀ UL − h∈ (0,
1] as following:
(i)The following formulas are axioms of ∀ UL − h∈ (0,
1] :
(U1) ( φ →ψ) →(( ψ → χ) →( φ → χ))
(U2) ( φ & ψ)
→ φ
(U3) ( φ & ψ) → ( ψ & φ)
(U4) φ &( φ →ψ) →( ψ &( ψ → φ))
(U5) ( φ →( ψ → χ)) →(( φ&
ψ) → χ)
(U6) (( φ & ψ) → χ) →( φ →( ψ → χ))
(U7) (( φ →ψ) → χ) →((( ψ →φ) → χ) → χ)
(U8) 0 → φ
(U9) ( φ →φ&
ψ) →(( φ →0) ∨ψ ∨(( φ →φ&
φ)
∧
( ψ → ψ & ψ)))
(U10) ( − −ϕ)
≡ ϕ
(U11) Δ( ϕ →ψ) →Δ( −ψ →− ϕ)
(U12) Δφ ∨¬Δ φ
(U13) Δ( φ ∨ψ) →( Δφ∨Δ
ψ)
(U14) Δφ → φ
(U15) Δφ →ΔΔ φ
(U16) Δ( φ →ψ) →( Δφ →Δ ψ)
(U17) ( ∀x) φ( x) → φ( t)
(t substitutable for x in φ ( x)
)
(U18) φ() t →( ∃ x) φ( x)
(t substitutable for x in φ ( x)
)
(U19) ( ∀x)( χ →φ) →( χ →( ∀ x) φ)
(x is not free in
χ )
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