Is $1+1=2$ a logical truth? 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?
 A: 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:

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 $1+1=2$
