Why can I make the claim: $\forall x \left[ x = c \lor x \neq c \right ]$ for some object $c$?

Question

I find that I frequently make use of the following claim in different arguments (when I am trying to exhaust all possibilities):

$\forall x \left[ x = c \lor x \neq c \right ]$ for some object $c$.

Why is it that I am permitted to make this claim? In particular, I guess I am asking what definition/axiom of equality allows me to deduce this universal statement. For real numbers, I am familiar with trichotomy, which states that $\forall x \in \mathbb R \left [ x=c \lor x \lt c \lor x \gt c \right ]$ for some object $c$...but the aforementioned universal claim is certainly even more fundamental.

Answer 1

Suppose otherwise. Then there exists an $a$ in the domain such that $\lnot (a=c\lor a\neq c)$, which means, by de Morgan's laws, $\lnot (a=c)\land\lnot(a\neq c)$. But this means $a\neq c$ and $a=c$, a contradiction.

Answer 2

Keep in mind that $x \neq y$ is just shorthand for $\neg(x = y)$. So what you're really proving is $\forall x (x = c \lor \neg (x = c))$.

Given an arbitrary $x$, the fact that $x = c$ or $\neg (x = c)$ follows from the law of excluded middle.

Note that in settings where the law of excluded middle is absent, one cannot always assert $\forall x (x = c \lor x \neq c)$. However, there are many cases in which one can assert this.

For example, one can prove using induction and the Peano axioms that $\forall y \forall x (x = y \lor x \neq y)$, where the domain of discourse is the natural numbers.

However, when one is dealing with sets, one cannot assert that $\forall x (x = \{\emptyset\} \lor x \neq \{\emptyset\})$.

This is because if one could assert this, then consider an arbitrary proposition $P$. Let $w = \{x \in \{\emptyset\} \mid P\}$. Then $w = \{\emptyset\}$ if and only if $P$. So $P \lor \neg P$ is equivalent to $w = \{\emptyset\} \lor w \neq \{\emptyset\}$. So asserting $\forall x (x = \{\emptyset\} \lor x \neq \{\emptyset\})$ is equivalent to asserting the law of excluded middle.

Answer 3

From your comment for your accepted answer, it's worthwhile to point out that you're still conflating law of excluded middle (LEM) with another law of noncontradiction (LNC).

For LEM:

It is one of the so called three laws of thought, along with the law of noncontradiction, and the law of identity... (p $\lor$ ¬p)

For LNC:

Formally this is expressed as the tautology ¬(p ∧ ¬p)... The law is not to be confused with the law of excluded middle which states that at least one, "p is the case" or "p is not the case" holds.

So under classic propositional logic you learned in school, obviously they're equivalent per DeMorgan, however, in intuitionistic logic LNC still holds while LEM together with double negation have been removed. On the other hand, in paraconsistent logic such as relevance logic inconsistency is tolerated and thus rejects the LNC also called principle of explosion sometimes.