I have a doubt that might seem a bit confusing so i will try to explain it the clearer i can.
Suppose we have an expression "A o B" in the meta-language, where 'o' refers to those logical connectives we are used to ( and, or, implication, equivalence ). I want to know exactly what someone means when he ( an author or maybe someone comenting about logic ) uses that expression, regardless of the context it is used.
Lets take as an example, the implication ... When someone uses the expression "A ->B".
I find that i cant find some fixed meaning or interpretations to an implication, when used in the meta-language, i get confused between 2 meanings or interpretations.
Sometiems i get the interpretation that when that expression is used in our meta-language, the person who used it, is affirming that that expression is true ( affirming that theres no interpretion that makes it false ( A cant be 1 while b is 0 ) ) .
An example would be someone using the expression "It's clear that (x is in N ) -> (x is in Z)".
But sometimes i get some different interpretation from that expression.Sometimes it seems like that when that expression is used, the person who used it, is more neutral ... just like introducing an expression, that can be both true and false, depending on which valuation or model we use.
An example would be someone using the expression "The only case in which A -> B is false is when A = 1 and B=0 ".
So, we have the same expression ( an implication ) in our meta-language that seems to have two really distinct interpretations or meanings, depending on the context it is used.
Does these 2 meanings really exist ? Or am i making some huge confusion about it ?
Thanks a lot in advance