# Is A∨¬A a tautology when there is a proof (by contradiction)?

A∨¬A is stated as a "tautology" but is it really a tautology? It can be proven by counterposition. And therefore it is not a tautology when it can be proven(?)

## Update

Here's the proof (by contradiction) I mean:

¬(A∨¬A) (assumption)
A      (assumption)
A∨¬A  (rule of introduction)
¬A
A∨¬A   (rule of introduction)
¬¬(A∨¬A)
A∨¬A

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I'm having a hard time understanding your question.Why do you think $A \vee \neg A$ isn't a tautology? – Git Gud Feb 3 at 8:59
Make a truth table, it cannot be proven otherwise as you will see – Math_Illiterate Feb 3 at 9:13
I'm not sure what you mean, but $1=1$ isn't a tautology. – Git Gud Feb 3 at 9:29
@GitGud Why isn't "1=1" a tautology? I could've misunderstood. – Nick Rosencrantz Feb 3 at 9:31
Um, I hate to wade into this deep philosophical discussion, but why do you have a couple of Han characters for people in your proof? The usual symbol for a contradiction is the up tack, $\bot$. – Rahul Narain Feb 3 at 10:50
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$A\vee \neg A$ is a tautology in classical (i.e., Aristotelian) logic because you can prove that using the deduction rules of the classical proposition calculus no matter what the truth value of $A$ is, the truth value of $A\vee \neg A$ is always true. That is the meaning of tautology.

In non-classical logical systems, such as intuitionism or constructivism, $A \vee \neg A$ is not a tautology. There the interpretation of $P \vee Q$ is not "either P or Q is true" but rather the more constructive "Either I have a proof of P or I have a proof of Q". A famous example to illustrate this is the following: Theorem: There exist two irrational numbers $a,b$ such that $a^b$ is rational. A classical proof can go like this: if $\sqrt2 ^\sqrt2$ is rational we are done. Else, consider $(\sqrt2^{\sqrt2})^{\sqrt2}=\sqrt2^2=2$, a rational. Classically this finishes the proof but constructively it is not a valid proof since it does not actually show which one of the two candidates works.

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 Thank you for the answer. I added the proof I was thinking of. – Nick Rosencrantz Feb 3 at 9:25 @Ittay Weiss don't you mean $\displaystyle \bigl(\sqrt{2}^\sqrt{2} \bigr)^\sqrt{2}=\sqrt{2}^{\sqrt{2}\cdot \sqrt{2}}$? – Git Gud Feb 3 at 12:29 This is why I would call $A\ \or\ \not A$ an axiom rather than a tautology in most uses of first order predicate calculus, but not all. For example, the Fundamental Theorem of Algebra holds for the constructive complex numbers. – Barbara Osofsky Feb 3 at 14:52 @BarbaraOsofsky \neg to get $\neg$ and \vee to get $\vee$. – Git Gud Feb 3 at 20:41 I prefer $\\lor$ for $\lor$ so I was willing to accept that missing letter, but thanks for pointing out the $\\neg$. I find inputting the LaTeX the hardest part of commenting expecially since in commenting I cannot see the mathjax and there is a strict time limit on editing and only 2 edits. – Barbara Osofsky Feb 4 at 2:10

Try constructing a truth table and you will see that it is in fact a tautology.

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But what's the difference then between tautologies and truths? – Nick Rosencrantz Feb 3 at 9:25
@NickRosencrantz A tautology is a statement which truth table is always true. A logical truth is a statement that is always true after you interpret its meaning. For instance, if I say that "every goba is esdel or not every goba is esdel" you know that this is true even though you don't know what a goba or an esdel are. The statement $1=1$ is a logical truth and you know that only after knowing the meaning of $1$, $+$ and $=$. – Git Gud Feb 3 at 10:13
@GitGud 'If I say that "every goba is esdel or not every goba is esdel" you know that this is true even though you don't know what a goba or an esdel are.' Not so. The most you know is that if it is a contentful English sentence, then it expresses a truth. But if you don't know whether it is contentful, you can't know whether it is true. – Peter Smith Feb 3 at 12:16
@PeterSmith I get what you're saying and I love such detail. However I chose to keep things simpler by taking a "statement" in english, I think it's an easier way to make things clearer. I'm going to e-mail one of these days, by the way. – Git Gud Feb 3 at 12:19
@PeterSmith *e-mail you – Git Gud Feb 3 at 12:25
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Part of the problem here may be that "tautology" has a far more specific meaning in mathematical logic than in ordinary usage. The more specific meaning is "a statement S that always true, solely on the basis of how S is constructed from smaller statements by means of propositional connectives and the meanings (truth tables) of the connectives". So $A\lor\neg A$ is a tautology because it is true solely because of the meanings of $\lor$ and $\neg$. But $1=1$ is not a tautology because its truth depends on the meaning of $=$, which is not a propositional connective. Similarly, if $P$ is a unary predicate, then $P(a)\to(\exists x)\,P(x)$, though logically valid, is not a tautology because its validity depends on the meanings of both $\to$ and $\exists$, the latter of which is not a propositional connective.

In ordinary, non-technical usage, "tautology" means (according to my dictionary) saying the same thing in different words; I've heard it used more generally to mean anything that is obviously true. So all of the examples in my first paragraph would be tautologies in this sense.

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