# Logic axiom of simplification.

1. Jun 19, 2009

### matheinste

Hello all

I cannot find a simple explanation of the meaning of this axiom, probably because it is considered so obvioius that it needs no explanation. Can anyone explain in words.

$${a}\rightarrow{({b}\rightarrow{a})}$$

Thanks. Matheinste.

2. Jun 19, 2009

### CRGreathouse

I'm not sure what you're looking for.

If a is false, then the statement reduces to "false implies stuff" which is by definition true. If a is true the statement reduces to "b implies true" which is also by definition true.

3. Jun 19, 2009

### honestrosewater

a -> (b -> a) is also equivalent to (a & b) -> a:

a -> (b -> a)
~a v (~b v a) [p -> q <=> ~p v q]
(~a v ~b) v a [(p v q) v r <=> p v (q v r)]
~(a & b) v a [~(p & q) <=> ~p v ~q]
(a & b) -> a

This formula also follows from the assumptions that (i) a formula always implies itself (p -> p) and (ii) lengthening a formula doesn't remove any of the formulas that the original implied ((p -> q) -> ((p & r) -> q)).

4. Jun 19, 2009

### matheinste

Thanks also to GRGreathouse. I see it now.

To Compuchip. Yes, I mistakenly repeated the thread but did not know how to remove the second posting.

Thanks. Matheinste.

5. Jun 20, 2009

### Preno

It's a formula expressing the fact that a is deducible from a,b. It's a particular instance of the structural rule of weakening (which says that if A |- B, then A,phi |- B).