What distinguishes logical necessity, logical consequence, logical truth, and tautology from one another?

Question

The text I'm reading distinguishes logical necessity, logical consequence, logical truth, and tautology from one another; however it doesn't make their distinctions especially perspicuous.

As far as I interpreted it,

• Logical consequence: truth of the antecedent or premises guarantees the truth of the consequent or conclusions.
• Logical truth: Any statement that must be valuated true, even if the set of premises is the empty set. (3=3, p v ~p, etc)
• Tautology: Any statement that must be valuated true, but only when the statement is stated. (as opposed to logical truths)
• Logical necessity: for any set of statements containing the logically necessary statement in question, the logically necessary statement can be evaluated as true under at least one valuation.

Answer 1

Tautologies are logical truths in the context of propositional logic:

φ is a tautology       =_def   φ is assigned ⊤ by all rows of the truth-table for φ.

Logical truths are something more general, and can be defined as follows:

φ is a logical truth   =_def   a true interpretation of the logical constants occurring in φ makes φ true.

Logical consequence is similar to that:

φ is a logical consequence of ψ   =_def   every true interpretation of ψ makes φ true.

Logical necessity is a modal notion, and can be defined using state-descriptions:

φ is logically necessary   =_def   φ is true in all state-descriptions.

All of these definitions are inspired by Carnap, but may differ from his actual definitions.