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Two years ago, when I was a super beginner to logic, my logic teacher, who is famous in our Country, told me that $1+1=2$ is not a tautology, but a logical truth. However, after reading some truly rigorous logic books(not mathematical logic books yet), I think that $1+1=2$ is neither a tautology, nor a logical truth; my teacher was wrong.

What I think is that $1+1=2$ is followed by the axioms of natural numbers or real numbers. It may be transfered to a predicate logic sentence like $f(a,a)=b$, so it is not automatically true under every model. Instead, $1+1=2$ is a logical consequence of the axioms of numbers, only the models that satifies the axioms of numbers will satisfy $1+1=2$. Am I right? Or is there somewhere I should revise?

Asaf Karagila
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Eric
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    I would think it a definition. $1+1$ is certainly a number, and for brevity we give that number the name $2$. – Arthur Jul 19 '17 at 16:23
  • See the picture in https://en.wikipedia.org/wiki/Principia_Mathematica. – lhf Jul 19 '17 at 16:24
  • @Arthur Maybe I should use an example like $2+3=5$ or $2\times 3=6$ to clarify the question? – Eric Jul 19 '17 at 16:25
  • Sometimes I found question like this at this site not for solving but for thinking ... +1 – Atul Mishra Jul 19 '17 at 16:26
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    Not so sure that applies here, @Parcly. I think there are similarities, but not necessarily a duplicate is about first order logic, and the questions concerns whether $1+1=2$ is a tautology? Not about whether it's provable. Any way; just unsure myself as to whether the questions are distinctly different questions. I'd rather let those with strong arguments one way or another determine that status. – amWhy Jul 19 '17 at 17:08
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    Correct; it is not a tautology (better: a valid formula) i.e. a formula that is true in very interpretation. It is a logical consequence of the axioms for arithmetic (see Peano axioms): call $\mathsf {PA}$ the "conjunction" of the arithmetical axioms; we have that (unwindimg the definitional abbreviations): $\vDash \mathsf {PA} \to S(0)+S(0)=S(S(0))$. – Mauro ALLEGRANZA Jul 20 '17 at 06:15

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You are correct: $1$, $2$, and $+$ are non-logical symbols, meaning that they can be interpreted in any way you want. So we could do something silly like $1$ stands for some apple, $2$ for some orange, and $x+y$ for 'stack $x$ on top of $y$'. Under that interpretation, $1+1=2$ is false, for stacking an apple on top of an apple (technically, it would have to be the same apple we stack on itself) doesn't give you an orange. So it is indeed not a logical truth.

A logical truth would be something like $1+1=1+1$ or $2=2$, because the $=$ is a logical symbol, and its meaning is fixed (and is exactly what you think it is)

And yes, $1+1=2$ could be a logical consequence of some axioms involving numbers. In fact, we could just have $1+1=2$ as an axiom. But we can also infer it from more basic principles, e.g. the Peano axioms. Now, actually, the typical Peano axioms don't immediately work, since they only use $0$ and the successor function $s$. However, we could say that we use $1$ as short-hand for the expression $s(0)$, and $2$ for $s(s(0))$, and now you can prove '1+1=2' as an arithmetical truth that logically follows from the Peano Axioms:

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Russell and Whitehead defined $1$, $2$, and $+$ in terms of sets, where things get more complicated, which is why it took them so long to prove that $1+1=2$

But in the end, all this proof shows is that $1+1=2$ is an arithmetical/mathematical truth, or an arithmetical/mathematical theorem: something that can be proven on the basis of certain assumptions/definitions made in the domain of arithmetic/mathematics. But it is not a truth on the basis of purely logical principles, and so it is not a logical truth.

Bram28
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