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

Question

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:

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

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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.

Answer 1

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.

Answer 2

I don't like either answer.

First, your

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

is definitely not correct. $\neg C(x)$ means that $x$ is not a contradiction, and thus this whole statements ends up saying "If $x$ is not a contradiction, then $x$ is a tautology", which is not at all the same as saying that "If $x$ is a contradiciton, then its negation is a tautology".

Indeed, note that for classical logic, the statement "If $x$ is a contradiction, then its negation is a tautology" is true: Yes, the negation of any contradiction is a tautology. However, the translation of your sentence, "If $x$ is not a contradiction, then $x$ is a tautology", is false: just because $x$ is not a contradiction, doesn't mean ti is a tautology; it could be neither (a contingency). So, this immediately shows that your symbolization does not capture the intended English statement.

However, I don't like

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

either. It commits a terrible use/mention mistake! I understand it tries to puts a $\neg$ in front of the $x$ in order for $P(\neg x)$ to make a claim about the sentence that is the negation of $x$ ... but it doesn't work that way at all. Inside a logical formula, the use of a negation is to negate some other formula, not to extend some formula with some other symbol. Indeed, the use of $\neg x$ is simply ungrammatical: you can only negate formulas, not objects!

So, what to do? Well, with the given key, you really can't do it.

Here are two ways that could work. First, you could use a 2-place relation like $N(x,y)$ that means "$x$ is the negation of $y$". With that, you can translate the sentence as:

$$\forall x \forall y ((C(x) \land N(y,x)) \to P(y))$$

Another way is to use a function symbol $n$, where $n(x)$ denotes the negation of $x$. With that, we get:

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

which is close to what the answer sheet has, but crucially different as well.