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Logic axiom of simplification.

  1. Jun 19, 2009 #1
    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.

    [tex]{a}\rightarrow{({b}\rightarrow{a})}[/tex]

    Thanks. Matheinste.
     
  2. jcsd
  3. Jun 19, 2009 #2

    CRGreathouse

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    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.
     
  4. Jun 19, 2009 #3

    honestrosewater

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    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)).
     
  5. Jun 19, 2009 #4
    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.
     
  6. Jun 20, 2009 #5
    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).
     
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