logically equivalent (original) (raw)

For example, for any integer z, the statement “z is positive” is equivalent to “z is not negative and z≠0”.

More generally, one says that a formula A is a logical consequence of a set Γ of formulas, written

if whenever every formula in Γ is true, so is A. If Γ is a singleton consisting of formula B, we also write

Using this, one sees that

To see this: if ⊧A↔B, then A→B and B→A are both true, which means that if A is true so is B and that if B is true so is A, or A⊧B and B⊧A. The argument can be reversed.