Extension of propositional logic with terms, predicates, quantifiers, structures, satisfaction, models, validity, and entailment.
First-order logic extends propositional logic by allowing formulas to speak about objects, properties of objects, and relations between objects, rather than treating every statement as an indivisible unit.
The language contains variables, constants, function symbols, predicate symbols, logical connectives, and quantifiers, and these symbols allow mathematical statements to be expressed with much greater internal structure than in propositional logic.
The first part of the chapter defines terms, predicates, and formulas, where terms are expressions that denote objects and formulas are expressions that can become true or false once an interpretation has been fixed.
The second part studies quantifiers and scope, since formulas such as and depend not only on the symbols used but also on which part of the formula is controlled by each quantifier.
The chapter then introduces structures and interpretations, where a structure supplies a domain of objects together with meanings for constants, functions, and predicates, so that the same formal sentence may be true in one structure and false in another.
Satisfaction and models are then defined to make truth in a structure precise, and this gives a formal way to say that a structure satisfies a formula or that a structure is a model of a collection of sentences.
Finally, validity and entailment describe truth that follows from logical form alone or from a given set of assumptions, and these notions prepare the way for later results about completeness, compactness, and model theory.