Me: Imagine a chair. Now imagine another chair - this chair looks just like the first one. Using 'number' we can say 'there are two chairs'. But we can equally say 'both chairs are the same'. Both descriptions are true.
How do you "know" both descriptions are true? You see, even at your first utterance, you invoke the logic embedded in your language.
No, I will know they are true not through logic but because I look at the chairs and see that there really are two of them and also that they are alike.
If I was suffering from double vision, i.e. there was really only one chair, then my description would be wrong. Logic would not come into it.
These are not empirical rules that you can take or leave; they are apriori, and are forced to use them. Take a simple assertion: the cat is on the mat. Certainly, there are features of this that are matters of convention. That 'cat' is the noise made to designate those fluffy four-legged creatures, or that the verb follows the subject (unlike the Korean counterpart), or the prepositional phrases are constructed in such a manner, etc. But underlying these conventions there is a foundation of logic, as is found in symbolic logic: modus ponens, disjunctions, hypotheticals and the rest. Unless you can use these, you simply cannot speak.
'
The cat is on the mat' can be reduced to my pointing at the two objects.
But logic etc. is entirely unconnected to objects like cats and mats.
In symbolic logic we use symbols rather than words precisely because it is unspecific; any proposition would do, whether it is
really true or false doesn't matter. In logic we are only interested in the connections, the symbols are simply place markers for whatever truth values we decide to give them.
It doesn't assert any fact about the world; '
1 + 1 = 2' does not assert
'there are two apples'. If it did then it would be wrong! There would still be
this green apple over here, and
that red apple over there in the identical state to before they got 'added'. Maths only works if we move away from actual objects and deal with abstractions.
'John plus Jane' does not make 'two'. We first have to abstract them into
'people', i.e. we must abandon any interest in the truth of any fact about John or Jane, i.e. whether these people exist etc..
The question we are discussing is '
Is there a way to refute '1+1=2'?'My answer is that there is nothing to refute, because it doesn't refer to anything. We might as well be asking;
'Is there any way to refute concept'? or
'Does it exist?'