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The logic $L_{{\omega }_1,\omega }$ extends first-order logic by countable logical conjunctions and logical disjunctions. Each formula has finitely many free variables, and quantifier strings are finite. In a countable first-order language, every countable structure has a Scott sentence in this logic that characterizes its isomorphism class among countable structures.

A first-order theory is a set of sentences in one first-order language. Depending on convention, the term may instead mean its deductive closure.

Second-order quantifiers range over specified collections of relations, which can be proper subcollections of all external relations. With these collections as additional sorts, this is a form of first-order logic. Full-class conclusions cannot be inferred merely from truth with Henkin semantics.

In first-order logic with logical equality, $s=t$ asserts that the terms denote the same domain element. Logical equality is interpreted as identity rather than as an arbitrary binary predicate. Its natural deduction rules include reflexivity and substitution: equal terms can replace one another in a first-order formula, subject to capture-avoiding substitution. Double-negating logical equality atoms in a Gödel-Gentzen negative translation preserves these rules when the transported translated first-order formula is stable.