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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.


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


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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


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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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