Xjyuk

We have the law of non-contradiction and the law of excluded middle, but looking at it, it seems that both of them are the same thing, or at least one of them logically implies the other.

Is there a reason why we chose to say that there are two laws, when one of them is redundant?

NOT(A and NOT A) = NOT A or A = A or NOT A

Both laws basically mean the same thing. One is a restatement of the other. Am I wrong in any way?

They're not equivalent, but they do seem very close together in most contexts when you assume a bivalent (two truth valued) logic.

But they pull apart when it comes to several controversial decisions we have to make in formal semantics and the philosophy of language. Lets consider two prominent examples.

Supervaluationism: This is one solution to the problem of vagueness, which the is problem that many, perhaps even most, predicates don't have determine meanings in terms of their truth. For example, a 7ft man is tall, but precisely where is the line between tallness and non-tallness? Supervaluationism demarcates the LEM from a related rule, the law of bivalence, which says that either P is true or ¬P is true (and thus P is false). On classical logic, this is how the LEM is rendered, because these are the only options possible. But strictly speaking, LEM states that either P is true or is not true. Supervaluationism, and I won't go into detail, says that LEM is true, but that there are "untrue" but not false statements. So, on supervaluationism, the law of bivalence is denied, but excluded middle accepted. If this seems weird, or unwarrented, note I haven't explained the benefits or motivations of supervaluationism at all here, I'm just telling you what it says as an example of where excluded middle pulls apart from how we'd interpret it when learning about propositional logic in itself.

The point of that example is to show that how the Law of Excluded middle manifests in one system is not the full characterisation of it; it's a metalogical property, and has different implications in different systems. So while it could be equivalent to non-contradiction in one system, it might not in another.

Now consider modal conditionals such as "if it were the case that x, then it would be the case that y". Let ">" be the symbol for these statements. Conditional law of excluded middle is then rendered:

"(A > B) V (A > B)".

But consider the following two conditionals (note that Bizet is a french singer and Verdi is an Italian one; i didn't know this when i came across this so was confused:

(BV1) If Bizet and Verdi were compatriots, Verdi would be French.

(BV2) If Bizet and Verdi were compatriots, Bizet would be Italian.

Note that BV2 entails the negation of BV1 and vice versa. It's a very defensible claim that neither one of these are true. And this is the line taken in perhaps the most widely accepted semantics for counterfactuals; Lewis'

Therefore conditional excluded middle fails on Lewis' system. But does the law of non contradiction? No. So they aren't the same thing on some very mainstream systems of conditional logic. For a system that accepts excluded middle for conditionals, see Stalnaker's theory. To learn more about LEwis' system, this is the best guide I've seen: https://math.berkeley.edu/~buehler/Counterfactuals%20Notes.pdf

Warning: this is a technical textbook. But it's rextremely clear.

I hope this clarifies things. The takeaway lesson: don't consider the properties of some logical laws as being identical to their consequences in some system, no matter how ubiquitous that system is. LEM is a metalogical property, and it manifests differently in different logical contexts.

This answer is offered as a supplement to Daniel Prendergast's answer. I hope to address the following question:

Both laws basically mean the same thing. One is a restatement of the other. Am I wrong in any way?

Wikipedia offers a way to look at the law of the excluded middle differently from the law of non-contradiction if one thinks of A and not A creating a dichotomy of the logical space of two truth-values true and false:

The law of non contradiction and the law of excluded middle create a dichotomy in "logical space", wherein the two parts are "mutually exclusive" and "jointly exhaustive". The law of non-contradiction is merely an expression of the mutually exclusive aspect of that dichotomy, and the law of excluded middle, an expression of its jointly exhaustive aspect.

From the perspective of a dichotomy, NOT(A and NOT A), the law of non-contradiction, means that A and NOT A are mutually exclusive. They do not overlap. If they are a dichotomy of the set of two truth-values true and false, neither A nor NOT A can be both true and false.

Furthermore, A or NOT A, the law of the excluded middle, means that they are jointly exhaustive of those two truth-values. Together they cover both truth-values true and false.

Wikipedia contributors. (2019, September 11). Law of noncontradiction. In Wikipedia, The Free Encyclopedia. Retrieved 23:46, September 19, 2019, from https://en.wikipedia.org/w/index.php?title=Law_of_noncontradiction&oldid=915201284

It's actually an artifact of how we are taught. If you look at original books on logic they don't actually say that a certain law is fundamental or not. That not how you go about thinking. But it does help beginners in a subject to guide where to begin, to help distinguish the important from the non-important. Its probably also an artifact from fetishing the axiomatic system which do use such a system axiomatically. But of course, Euclid didn't invent geometry - somebody else did.

~(A & ~A) The Law of Non-contradiction is that a statement can not be both true and false. It does not prohibit the assignment of some other value, but restricts how many values may be assigned.

(A v ~A) The Law of Excluded Middle is that a statement must true or false. It does not prohibit statements from having multiple values, but restricts what those values may be.

Now, in Classical Logic these are equivalent statements. However, not all logics are classical.