# Is $\forall I: I(f)=1$ where $f$ stands for a formula the meaning / definition of a tautology?

I have a question about the following formula. Is $\forall I: I(f)=1$ where $f$ stands for a formula and I for an interpretation the meaning / valid definition of tautology?

• Follow the definition in your textbook. I an imagine such a thing in a textbook, although I would guess most textbooks do not write it that way, quantifying over interpretations, using $1$ for "true" and so on. – GEdgar Jul 26 '18 at 12:55
• This is a question that really needs some context. Is this supposed to be first order logic? If it is I'd address the quantification over a function symbol. Is the quantifier happening in some meta-language with $f$ a formula in some object language? If so, what are the existing assumptions about the object language, the interpretation function, etc.? Is 1 here the top element of a Boolean algebra, a natural number, a real number, something else? – Malice Vidrine Jul 26 '18 at 17:07

• Syntactic tautologies: $f$ is a theorem of the empty theory, with respect to whatever proof system is being used.
• Semantic tautologies: $f$ is true in every structure in the language of $f$ (or, every interpretation).