- #26

Erland

Science Advisor

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- 136

If you want to prove that something is a theorem, you should make a derivation from the axioms. My intention with the "semantic" method was not to prove that the sentence in question is a theorem, but to convince you that it is true, so that the proposed definition of absolute value does not lead to something false. The sentence in the previous post can be derived in a few steps from the given definition of absolute value, by using logical axioms and rules of inference (exactly how depends on which logical axioms and rules we have; these differ in different systems, for example natural deduction or a Hilbert style axiom system, but it is easy in all cases) based on the idea that if something holds for all x and y, it also holds for all x and y such that y=|x|, or y=x, as pointed out in an earlier post.Before we examine the correctness of your proof,

Can you use your semantical method of true and false values to find out whether the formula:

##\forall a[a\neq 0\Longrightarrow a^2>0]## is a theorem or not?

But of course, I can try to convince you with the semantic method that ##\forall a[a\neq 0\Longrightarrow a^2>0]## is true.

Case 1: ##a=0##. Then ##a\neq 0## is false and ##a^2> 0## is false. Hence, the implication ##a\neq 0\Longrightarrow a^2> 0## is true.

Case 2: ##a>0##. Then ##a\neq 0## is true and ##a^2> 0## is true. Hence, the implication ##a\neq 0\Longrightarrow a^2> 0## is true.

Case 3: ##a<0##. Then ##a\neq 0## is true and ##a^2> 0## is true. Hence, the implication ##a\neq 0\Longrightarrow a^2> 0## is true.

Thus, ##a\neq 0\Longrightarrow a^2> 0## is true for all (real) values of ##a##. In other words, ##\forall a[a\neq 0\Longrightarrow a^2>0]## is true.

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