In the context of first-order logic, a distinction is maintained between logical validities, sentences that are true in every model, and tautologies, which are a proper subset of the first-order logical validities.
https://en.wikipedia.org/wiki/Tautology_(logic)
First order logic is undecidable, which means (again, I think) that given a set of sentences A and a sentence B, there is no procedure for determining whether A implies B (i.e. it's not the case that A are true and B is false) in all interpretations.
https://philosophy.stackexchange.com/questions/15525/how-is-first-order-logic-complete-but-not-decidable
For example, because ∨ ¬ is a tautology of propositional logic, (∀ (=)) ∨ (¬ ∀ (=)) is a tautology in first order logic. Similarly, in a first-order language with a unary relation symbols R,S,T, the following sentence is a tautology:
https://en.wikipedia.org/wiki/Tautology_(logic)
A tautology in first-order logic is a sentence that can be obtained by taking a tautology of propositional logic and uniformly replacing each propositional variable by a first-order formula (one formula per propositional variable).
https://en.wikipedia.org/wiki/Tautology_(logic)
In the context of predicate logic, many authors define a tautology to be a sentence that can be obtained by taking a tautology of propositional logic and uniformly replacing each propositional variable by a first-order formula (one formula per propositional variable).
https://en.wikipedia.org/wiki/Tautology_(logic)
In logic, a tautology (from the Greek word ταυτολογία) is a formula that is true in every possible interpretation. Philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921.
https://en.wikipedia.org/wiki/Tautology_(logic)
This page defines the syntax and semantics of Herbrand logic, and for the purpose of comparison, first-order logic. It finishes with a couple of quick examples illustrating the difference between first-order semantics and Herbrand semantics.
https://www.cs.uic.edu/~hinrichs/herbrand/html/herbrandlogic.html
The fundamental definition of a tautology is in the context of propositional logic. The definition can be extended, however, to sentences in first-order logic (see Enderton (2002, p. 114) and Kleene (1967 secs. 17–18)). These sentences may contain quantifiers, unlike sentences of propositional logic.
https://en.wikipedia.org/wiki/Tautology_(logic)
First order logic is complete, which means (I think) given a set of sentences A and a sentence B, then either B or ~B can be arrived at through the rules of inference being applied to A. If B is arrived at, then A implies B in every interpretation. If ~B is arrived at, then A implies ~B in every interpretation.
https://philosophy.stackexchange.com/questions/15525/how-is-first-order-logic-complete-but-not-decidable
Herbrand Semantics. Herbrand logic differs from first-order logic solely in the structures it considers to be models. The semantics of a given set of sentences is defined to be the set of Herbrand models that satisfy it, for a given vocabulary. Definition (Herbrand Model): A Herbrand model for vocabulary V is any set of ground atoms in V.
https://www.cs.uic.edu/~hinrichs/herbrand/html/herbrandlogic.html
define first order sentence in logic
A first order sentence in the context of first-order logic is a distinction maintained between logical validities, sentences that are true in every model, and tautologies, which are a proper subset of the first-order logical validities.