structural:summarization, linguistic:translation
Introduction

While propositional logic deals with simple declarative propositions, first-order logic additionally covers predicates and quantification. A predicate evaluates to true or false for an entity or entities in the domain of discourse.

Consider the two sentences " Socrates is a philosopher" and " Plato is a philosopher". In propositional logic, these sentences themselves are viewed as the individuals of study, and might be denoted, for example, by variables such as p and q. They are not viewed as an application of a predicate, such as isPhil {\displaystyle {\text{isPhil}}}, to any particular objects in the domain of discourse, instead viewing them as purely an utterance which is either true or false. However, in first-order logic, these two sentences may be framed as statements that a certain individual or non-logical object has a property. In this example, both sentences happen to have the common form isPhil (x) {\displaystyle {\text{isPhil}}(x)} for some individual x {\displaystyle x}, in the first sentence the value of the variable x is "Socrates", and in the second sentence it is "Plato". Due to the ability to speak about non-logical individuals along with the original logical connectives, first-order logic includes propositional logic.

The truth of a formula such as " x is a philosopher" depends on which object is denoted by x and on the interpretation of the predicate "is a philosopher". Consequently, " x is a philosopher" alone does not have a definite truth value of true or false, and is akin to a sentence fragment. Relationships between predicates can be stated using logical connectives. For example, the first-order formula "if x is a philosopher, then x is a scholar", is a conditional statement with " x is a philosopher" as its hypothesis, and " x is a scholar" as its conclusion, which again needs specification of x in order to have a definite truth value.

Quantifiers can be applied to variables in a formula. The variable x in the previous formula can be universally quantified, for instance, with the first-order sentence "For every x, if x is a philosopher, then x is a scholar". The universal quantifier "for every" in this sentence expresses the idea that the claim "if x is a philosopher, then x is a scholar" holds for all choices of x.

The negation of the sentence "For every x, if x is a philosopher, then x is a scholar" is logically equivalent to the sentence "There exists x such that x is a philosopher and x is not a scholar". The existential quantifier "there exists" expresses the idea that the claim " x is a philosopher and x is not a scholar" holds for some choice of x.

The predicates "is a philosopher" and "is a scholar" each take a single variable. In general, predicates can take several variables. In the first-order sentence "Socrates is the teacher of Plato", the predicate "is the teacher of" takes two variables.

An interpretation (or model) of a first-order formula specifies what each predicate means, and the entities that can instantiate the variables. These entities form the domain of discourse or universe, which is usually required to be a nonempty set. For example, consider the sentence "There exists x such that x is a philosopher." This sentence is seen as being true in an interpretation such that the domain of discourse consists of all human beings, and that the predicate "is a philosopher" is understood as "was the author of the Republic." It is true, as witnessed by Plato in that text.

There are two key parts of first-order logic. The syntax determines which finite sequences of symbols are well-formed expressions in first-order logic, while the semantics determines the meanings behind these expressions.