Editing First-order logic (section)

==Introduction==
{{Logical connectives sidebar}}
While [[propositional logic]] deals with simple declarative propositions, first-order logic additionally covers [[Predicate_(mathematical_logic)|predicate]]s and [[Quantification (logic)|quantification]]. A predicate evaluates to [[True (logic)|true]] or [[False (logic)|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 calculus|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 <math>\text{isPhil}</math>, to any particular objects in the domain of discourse, instead viewing them as purely an utterance which is either true or false.<ref>[[Harvey Friedman (mathematician)|H. Friedman]], "[https://bpb-us-w2.wpmucdn.com/u.osu.edu/dist/1/1952/files/2022/07/RossProg1072322pdf-1-2.pdf Adventures in Foundations of Mathematics 1: Logical Reasoning]", [[Arnold_Ross#Ross Mathematics Program|Ross Program]] 2022, lecture notes. Accessed 28 July 2023.</ref> 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 <math>\text{isPhil}(x)</math> for some individual <math>x</math>, 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.<ref>[[Ben Goertzel|Goertzel, B.]], Geisweiller, N., Coelho, L., Janičić, P., & Pennachin, C., ''Real-World Reasoning: Toward Scalable, Uncertain Spatiotemporal, Contextual and Causal Inference'' ([[Amsterdam]] & [[Paris]]: [[Springer Science+Business Media|Atlantis Press]], 2011), [https://books.google.com/books?id=g7UAIhnmJpsC&pg=PA29&redir_esc=y#v=onepage&q&f=false pp. 29–30].</ref>{{rp|29–30}}

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.<ref name="Quine81" /> Relationships between predicates can be stated using [[logical connective]]s. For example, the first-order formula "if ''x'' is a philosopher, then ''x'' is a scholar", is a [[material conditional|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 (Plato)|Republic]]''." It is true, as witnessed by Plato in that text.{{clarification needed|date=March 2023}}

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.