Definition of absolute value

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  • #26
Erland
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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?
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.

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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  • #27
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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.

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.
Let us take your proof step by step.

How do you know that ##a^2>0##
is false if a =0
 
  • #28
Erland
Science Advisor
738
136
Let us take your proof step by step.

How do you know that ##a^2>0##
is false if a =0
I repeat: If you want a PROOF, make a formal derivation from the axioms! The intention of the "semantical" method was NOT to do that, but to give arguments that, hopefully, would convince you. This seems to have failed. If you are not already convinced that ##0^2>0## is false, nothing else I say will convince you of that, except possibly a derivation from the axioms.

But what would be the point of that, since you anyway question simple consequences of the axioms and inference rules of logic (which differ in different systems but have the same power to prove theorems), such that one can go from ##\forall x\forall y P(x,y)## to ##\forall x P(x,x)##?
This latter can be derived in few steps with any logical system in use. As an exercise, derive with the system in your logic textbook (and I don't know which book you have, so I cannot do it until you tell me which system is used in it).
 

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