Editing Existence (section)

=== Formal logic ===
{{main|Logic}}

Formal logic studies [[Deductive validity|deductively valid arguments]].<ref>{{multiref |1={{harvnb|MacFarlane|2017}} |2={{harvnb|Corkum|2015|pp=753–767}} |3={{harvnb|Blair|Johnson|2000|pp=93–95}} |4={{harvnb|Magnus|2005|loc=§ 1.6 Formal Languages|pp=12–14}} }}</ref> In [[first-order logic]], which is the most-commonly used system of formal logic, existence is expressed using the [[existential quantifier]] (<math>\exists</math>). For example, the formula <math>\exists x \text{Horse}(x)</math> can be used to state horses exist. The variable ''x'' ranges over all elements in the [[Domain of discourse|domain of quantification]] and the existential quantifier expresses that at least one element in this domain is a horse. In first-order logic, all singular terms like names refer to objects in the domain and imply the object exists. Because of this, one can deduce <math>\exists x \text{Honest}(x)</math> (someone is honest) from <math>\text{Honest}(Bill)</math> (Bill is honest).<ref>{{multiref |1={{harvnb|Shapiro|Kouri Kissel|2022|loc=§2.1 Building Blocks}} |2={{harvnb|Cook|2009|p=[https://books.google.com/books?id=JfaqBgAAQBAJ&pg=PA111 111]}} |3={{harvnb|Montague|2018|p=214}} |4={{harvnb|Casati|Fujikawa|loc=Lead Section, §1. Existence as a Second-Order Property and Its Relation to Quantification}} }}</ref> If only one object matching the description exists, the [[Uniqueness quantification|unique existential quantifier]] <math>\exists !</math> can be used.<ref>{{multiref | {{harvnb|Roberts|2009|p=[https://books.google.com/books?id=NjBLnLyE4jAC&pg=PA52 52]}} | {{harvnb|Johar|2024|p=[https://books.google.com/books?id=JnPsEAAAQBAJ&pg=PA38 38]}} }}</ref>

Many logical systems that are based on first-order logic also follow this idea. [[Free logic]] is an exception because it allows the presence of empty names that do not refer to an object in the domain.<ref>{{harvnb|Nolt|2021|loc=Lead Section, §1. The Basics}}</ref> With this modification, it is possible to apply [[logical reasoning]] to fictional objects instead of limiting it to regular objects.<ref>{{harvnb|Nolt|2021|loc=§5.4 Logics of Fiction}}</ref> In free logic one can express that Pegasus is a flying horse using the formula <math>\text{Flyinghorse}(Pegasus)</math>. As a consequence of this modification, one cannot infer from this type of statement that something exists. This means the inference from <math>\text{Flyinghorse}(Pegasus)</math> to <math>\exist x \text{Flyinghorse}(x)</math> is invalid in free logic, even though it is valid in first-order logic. Free logic uses an additional existence predicate (<math>E!</math>) to say a singular term refers to an existing object. For example, the formula <math>E!(Homer)</math> can be used to say [[Homer]] exists while the formula <math>\lnot E!(Pegasus)</math> states Pegasus does not exist.<ref>{{multiref |1={{harvnb|Lenzen|2013|p=[https://books.google.com/books?id=zn3oCAAAQBAJ&pg=PA118 118]}} |2={{harvnb|Nolt|2021|loc=Lead Section, §1. The Basics, §5.4 Logics of Fiction}} |3={{harvnb|Sider|2010|p=[https://books.google.com/books?id=GK8SEAAAQBAJ&pg=PA129 129]}} }}</ref>