Probably this is a very naïve question, but I could not find a correct answer to this question when surfing the internet. That's why I'm asking this question here. In this book, the authors write:
The concept of consequence, however, belongs to the metalanguage of logic
Discourse. If the couple $ langle F, vdash rangle $ is taken as the definition of a logic that elements in $ F $The sentences (or formal strings belonging to the alphabet, the well-formed formulas) fall within the object-level entities during the relationship $ vdash $This is a relationship between a subset $ X $ from $ F $ and an element $ α $ from $ F $, belongs to the metalanguage. $ X vdash α $if spelled correctly, will $ “ X " vdash“ α "$What an assertion is, namely. the wff
called $ “ alpha "$ is a consequence of the amount of said WFFs $ “ X $ ', This claim speaks a little about certain object language elements $ X $ and $ α $, So that's a metalinguistic claim.
I agree with the analysis here, but the analysis here is specific to the way we see it $ vdash $, I wonder, is it Got to Is it the case that the concept of logical consequence can only be expressed in the metalanguage? Why so? More specifically, how can we be sure that the concept of logical consequence can not be expressed in the object language?