Editing First-order logic (section)

====Logical symbols====
{{Main|List of logical symbols}}

Logical symbols are a set of characters that vary by author, but usually include the following:<ref>{{Cite web|title=Predicate Logic {{!}} Brilliant Math & Science Wiki|url=https://brilliant.org/wiki/predicate-logic/|access-date=2020-08-20|website=brilliant.org|language=en-us}}</ref>
* [[Quantifier (logic)|Quantifier]] symbols: {{math|[[Universal quantification|∀]]}} for universal quantification, and {{math|[[Existential quantification|∃]]}} for existential quantification
* [[Logical connective]]s: {{math|∧}} for [[logical conjunction|conjunction]], {{math|∨}} for [[disjunction]], {{math|→}} for [[material conditional|implication]], {{math|↔}} for [[logical biconditional|biconditional]], {{math|¬}} for negation. Some authors<ref>{{Cite web|title=Introduction to Symbolic Logic: Lecture 2|url=http://cstl-cla.semo.edu/hhill/pl120/notes/wffs.html|access-date=2021-01-04|website=cstl-cla.semo.edu}}</ref> use C''pq'' instead of {{math|→}} and E''pq'' instead of {{math|↔}}, especially in contexts where → is used for other purposes. Moreover, the horseshoe {{math|⊃}} may replace {{math|→}};<ref name="Quine81" /> the triple-bar {{math|≡}} may replace {{math|↔}}; a tilde ({{math|~}}), N''p'', or F''p'' may replace {{math|¬}}; a double bar <math>\|</math>, <math>+</math>,<ref>{{cite book|issn=1431-4657|isbn=3540058192|author=[[Hans Hermes]]|title=Introduction to Mathematical Logic|url=https://books.google.com/books?id=ihYPCQAAQBAJ|location=London|publisher=Springer|series=Hochschultext (Springer-Verlag)|year=1973}}</ref> or A''pq'' may replace {{math|∨}}; and an ampersand {{math|&}}, K''pq'', or the middle dot {{math|⋅}} may replace {{math|∧}}, especially if these symbols are not available for technical reasons.
* Parentheses, brackets, and other punctuation symbols. The choice of such symbols varies depending on context.
* An infinite set of ''variables'', often denoted by lowercase letters at the end of the alphabet ''x'', ''y'', ''z'', ... . Subscripts are often used to distinguish variables: {{math|1= ''x''<sub>0</sub>, ''x''<sub>1</sub>, ''x''<sub>2</sub>, ...&nbsp;.}}
* An ''equality symbol'' (sometimes, ''identity symbol'') {{math|{{=}}}} (see {{section link|#Equality_and_its_axioms}} below).

Not all of these symbols are required in first-order logic. Either one of the quantifiers along with negation, conjunction (or disjunction), variables, brackets, and equality suffices.

Other logical symbols include the following:
* Truth constants: T, or {{math|⊤}} for "true" and F, or {{math|⊥}} for "false". Without any such logical operators of valence 0, these two constants can only be expressed using quantifiers.
* Additional logical connectives such as the [[Sheffer stroke]], D''pq'' (NAND), and [[exclusive or]], J''pq''.