Express following sentence with predicates, quantifiers, and logical operators.

discrete mathematicslogicquantifiers

first, here is following sentence and my solving process.

● sentence : The negation of a contradiction is a tautology.

● my solving process

if

  1. $x$ : proposition
  2. $P(x)$ : $x$ is tautology.
  3. $C(x)$ : $x$ is contradiction.
  4. $-C(x)$ : $x$ is not contradiction.

then,
it can express like this:

$$\forall x(\neg C(x)\implies P(x))$$

The answer at answer sheet was: $$\forall x(C(x)\implies P(\neg x))$$

I'm not sure if my answer is a correct answer, because If the negative in $C(x)$ is $\neg C(x)$, then:
I'm confused if $C(\neg x)$ can mean the same thing.

Best Answer

Your answer says: $$\text{if proposition }x\text{ is not a contradiction then it is a tautology}$$ This is not true because a proposition $x$ that is not a contradiction and is (also) not a tautology certainly exists.

Observe that $\neg C(x)$ says that $x$ is not a contradiction and that $C(\neg x)$ says that $\neg x$ is a contradiction.

These statements are not the same.

It might be that $x$ is not a contradiction and is not a tautology.

In that case $\neg C(x)$ is a true statement but $C(\neg x)$ is not a true statement.

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