Logic as a tool a guide to formal logical reasoning ( PDFDrive ) 207

k 4.3.3 Natural Deduction for first-order logic with equality The system of Natural Deduction presented so far does not involve rules for the equality.. Such rules can be added to produc

k

4.3.3 Natural Deduction for first-order logic with equality

The system of Natural Deduction presented so far does not involve rules for the equality

Such rules can be added to produce a sound and complete system of Natural Deduc-tion for first-order logic with equality, as follows In each of the rules below, t, t i , s i

are terms

(Ref)

t=t

(Sym)

t1=t2

t2=t1

(Tran)

t1=t2, t2 =t3

t1=t3

(ConFunc)

s1=t1, , s n=t n

f(s1, , s n) =f(t1, , t n) for everyn-ary functional symbolf (ConRel)

s1=t1, , s n=t n

p(s1, , s n)→ p(t1, , t n) for everyn-ary predicate symbolp

Example 136 Using Natural Deduction with equality, derive

∀ x ∀ y(f(x) =y → g(y) =x)ND ∀ z(g(f(z)) =z)

where f, g are unary function symbols.

(∀ I) (→ E)f(x) =f(x) (∀E)

(∀E) ∀ x ∀ y(f(x) =y → g(y) =x)

∀ y(f(x) =y → g(y) =x)

f(x) =f(x)→ g(f(x)) =x

g(f(x)) =x

∀ z(g(f(z)) =z) Finally, two important general results are as follows

Theorem 137 [Equivalent replacement] For any formula A(x ) and terms s, t free for x

in A , the following is derivable in ND:

s=t ND A[s/x]↔ A[t/x].

The proof can be done by induction onAand is left as an exercise

The proof of the following fundamental result, for ND with equality, will be outlined

in Section 4.6