User:IssaRice/Computability and logic/Intended interpretation versus all interpretations

Something I have found tricky in mathematical logic is that some theorems/propositions apply to just the intended/standard interpretation (structure), while others are about all possible interpretations. Texts also don't necessarily emphasize this point each time, so you have to figure it out.

Normally when we say a mathematical statement is true, we mean that it is true in the standard (or intended) interpretation. For instance, we say that $2+2=4$ is true. But $2+2=4$ could also be false if we adopt a non-standard interpretation. For example, we could say "$2$" is the number zero, "$4$" is the number one, and "$+$" is the usual multiplication. In other words, we still keep the same signature ($2$ and $4$ are still constants, $+$ is still a binary function) but we assign different meanings to these non-logical symbols.

When we talk about semantic consequence (a.k.a. logical consequence, logical implication) and write things like $\Gamma \models \phi$ (where $\Gamma$ is a set of sentences and $\phi$ is a sentence) then this doesn't just mean that "if all sentences in $\Gamma$ are true, then the sentence $\phi$ also is true". Rather, it means that this "if-then" is true in every possible interpretation. In other words, when we write $\Gamma \models \phi$ we mean that there is no interpretation in which every sentence in $\Gamma$ is true but $\phi$ is false. This is also sometimes expressed by saying that $\phi$ is true in all models of $\Gamma$.

Similarly when we say that a sentence is valid, this is much stronger than saying it is true in our intended interpretation. To say a sentence is valid means that no matter what interpretation we assign, the sentence ends up true. Thus, $2+2=4$ is not a valid sentence. But $2+2=4{\text{ or }}2+2\neq 4$ is a valid sentence; no matter what meanings we assigned to the non-logical symbols, the sentence would come out true.

Expresses/captures: these are also relative to an interpretation.