# Why are there two fundamental laws of logic?

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?

• One implies the other only if you use de Morgan law, double negation removal, and commutativity in the process. And they mean very different things, that something can not be true along with its negation vs something or its negation being true. In intuitionistic logic you have non-contradiction, but not excluded middle (some statements are undecidable, like the continuum hypothesis), in paraconsistent logic vice versa (some statements are both true and false, like "I am false"). – Conifold Sep 19 '19 at 20:39
• I made an edit replacing 3 with two. I think you are only referring to two laws here. Do not hesitate to roll this back or edit further if I misrepresented your position. – Frank Hubeny Sep 19 '19 at 23:53
• They can only imply each other in the context of a certain set of other logical laws/rules/axioms/postulates/assumptions, the ones we are most used to and normally assume and use. But they are far from the only set of logical laws that can be assumed. – RBarryYoung Sep 20 '19 at 14:42

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.

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.

• Good link to Buehler's notes on Lewis's Counterfactuals – Frank Hubeny Sep 19 '19 at 23:13
• Your statement of the conditional law of excluded middle came out as "(A > B) V (A > B)". Surely this can't be what you intended. – Rosie F Sep 20 '19 at 9:22
• Pedantic comment is that neither Verdi nor Bizet were singers. They were composers. – Grzegorz Oledzki Sep 20 '19 at 9:47
• @Grzegorz fair play. It's still a step up from at first not understanding the problem at all becuase I thought they were random names without referents – Daniel Prendergast Sep 20 '19 at 17:47
• Well, if you accept BV1 makes BV2 false, and that BV2 makes BV1 false, I'm not sure where the substantial issue is here – Daniel Prendergast Sep 20 '19 at 20:39

When you say,

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

you are actually doing a few things at once. First, you are using de Morgan's Laws to state

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

Then, you use double negation elimination to state

NOT A or NOT NOT A = NOT A or A

i.e. NOT NOT A = A.

This makes your reasoning circular. Assuming that double negation elimination holds is equivalent to assuming that the Law of Excluded Middle holds (for justification, see e.g. https://math.stackexchange.com/questions/2902033/double-negation-vs-law-of-excluded-middle/2902129).

In addition, assuming de Morgan's Laws is equivalent to assuming the weak Law of Excluded Middle (proof: https://ncatlab.org/nlab/show/weak+excluded+middle).

So essentially, you're assuming that the Law of Excluded Middle holds to prove that you don't need the Law of Excluded Middle.

• Actually, I think you're using circular logic here. If they are indeed equivalent, then the OP could correctly conclude that the Law of Non-Contradiction and Double Negation are the real 2 fundamental laws of logic, no? If the former is `!(P&!P)` and the latter is `!!P>P`, while the Law of the Excluded Middle is `!P|P`, would not these two laws be a better fit as being the fundamentals, allowing the derivation of the LEM? – Andrew Sep 20 '19 at 20:33
• @Andrew The OP claimed that there only needed to be one fundamental law of logic, because they thought that LEM could be derived from non-contradiction. I refuted this claim by showing that the derivation itself depended on assuming the very law the OP meant to derive. You can, of course, take non-contradiction and double negation elimination as your two axioms. That's what the word "equivalent" means, after all. But I never disagreed with this anywhere in the answer, so I'm puzzled why you think there's something circular in my own reasoning. – probably_someone Sep 20 '19 at 20:41
• I would actually disagree that they were claiming there only needed to be 1; rather, they were questioning why there are 2 if equivalent (not necessarily implying to the extent suggested, due to it being a "seeming" question). Anyways... – Andrew Sep 20 '19 at 20:52
• Well, basically, I meant that I think your claim that his reasoning was circular because the two laws are equivalencies was itself circular reasoning. If LEM can be concluded from DN (and vice versa), yet DN is what's truly distinct from LNC, then doesn't it stand to reason that LEM is in fact the unnecessary equivalency, inverting your logic on OP's reasoning being circular? :) – Andrew Sep 20 '19 at 20:52

`~(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.

• A statement that is neither true nor false is allowed by the Law of Non-Contradiction, when it would be forbidden by the Law of Excluded Middle, so the two laws cannot be equivalent even in classical logic. – probably_someone Sep 20 '19 at 11:29
• To be clear, in classical logic the statements are both tautologies. – Graham Kemp Sep 20 '19 at 13:51
• Ahh, so this really all has to do moreso with the semantics of logic itself; whereas, when I look at `~(A & ~A)`, I presume that this means you can only ever have just `A` or `~A`. To me, that is therefore kind of misleading to state the LNC with such symbology, unless you simultaneously make this clear. – Andrew Sep 20 '19 at 20:56

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

• I don't really get your point here. Saying that e.g., `!(!P)` and its equivalent, `P` cover both the True (`P`) and False (`!(!P)`) values isn't really useful. – Andrew Sep 20 '19 at 20:39
• @Andrew It isn't useful until it is applied to a domain where the law of bivalence semantically holds. For finite mathematical domains there should be no problem with it. For infinite mathematical domains one would find an intuitionist objection. For other domains anti-realists would have other reasons to reject bivalence. But in terms of understanding a difference in meaning between the law of the excluded middle and the law of non-contradiction, Wikipedia's suggestion to view these as two parts of a dichotomy seems helpful. – Frank Hubeny Sep 20 '19 at 21:11

To expand on @conifold's comment:

In the process of deriving the law of excluded middle from the law of non contradiction, you used double negation removal.

But double negation removal (NOT NOT A equivalent to A) is actually just the law of excluded middle in disguise!

So while you showed that they are indeed equivalent in classical logic (as are any two true propositions), your proof does not show that they are equivalent in logic without the law of the excluded middle (intuitionistic logic).

Whatever the laws of logic, they're all going to be equivalent, because they're all going to be tautologies. Any two statements that are always true are equivalent. Suppose that two laws of logic were not equivalent: then there would be some cases in which one is true and the other false. But then how can they both be laws of logic?

So the question might be: Why is there focus on Non-Contradiction and Excluded Middle among all the tautologies of classical logic? The answer is that some interesting logics reject one of these, or even both. Intuitionistic logic rejects Excluded Middle, and various kinds of Paraconsistent logics reject Non-Contradiction. In those kinds of logics, Non-Contradiction and Excluded Middle are not equivalent.

• I disagree but upvoted because you explained your position well. One ought not to use contrary systems as the basis for depicting what the fundamentals for a system are to be, because if those contrary systems are in fact incorrect, then they have no bearing on what is true under the original, true system. That an aspect is rejected by contrary systems does not necessarily imply that it is a fundamental of the original system. – Andrew Sep 20 '19 at 21:10

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.