Editing Philosophical logic (section)

=== Free ===
[[Free logic]] rejects some of the existential presuppositions found in classical logic.<ref name="Nolt1"/><ref name="Morscher">{{cite book |last1=Morscher |first1=Edgar |last2=Simons |first2=Peter |title=New Essays in Free Logic: In Honour of Karel Lambert |date=2001 |publisher=Springer Netherlands |isbn=978-94-015-9761-6 |pages=1–34 |url=https://link.springer.com/chapter/10.1007/978-94-015-9761-6_1 |language=en |chapter=Free Logic: A Fifty-Year Past and an Open Future|doi=10.1007/978-94-015-9761-6_1 }}</ref><ref name="Lambert">{{cite book |last1=Lambert |first1=Karel |title=The Blackwell Guide to Philosophical Logic |date=2017 |publisher=John Wiley & Sons, Ltd |isbn=978-1-4051-6480-1 |pages=258–279 |url=https://onlinelibrary.wiley.com/doi/10.1002/9781405164801.ch12 |language=en |chapter=Free Logics|doi=10.1002/9781405164801.ch12 }}</ref> In classical logic, every singular term has to denote an object in the domain of quantification.<ref name="Nolt1">{{cite web |last1=Nolt |first1=John |title=Free Logic: 1. The Basics |url=https://plato.stanford.edu/entries/logic-free/#1 |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=11 December 2021 |date=2021}}</ref> This is usually understood as an ontological commitment to the existence of the named entity. But many names are used in everyday discourse that do not refer to existing entities, like "Santa Claus" or "Pegasus". This threatens to preclude such areas of discourse from a strict logical treatment. Free logic avoids these problems by allowing formulas with non-denoting singular terms.<ref name="Morscher"/> This applies to [[proper names]] as well as [[definite descriptions]], and functional expressions.<ref name="Nolt1"/><ref name="Lambert"/> Quantifiers, on the other hand, are treated in the usual way as ranging over the domain. This allows for expressions like {{nowrap|"<math>\lnot \exists x (x = santa)</math>"}} (Santa Claus does not exist) to be true even though they are self-contradictory in classical logic.<ref name="Nolt1"/> It also brings with it the consequence that certain valid forms of inference found in classical logic are not valid in free logic. For example, one may infer from {{nowrap|"<math>Beard(santa)</math>"}} (Santa Claus has a beard) that {{nowrap|"<math>\exists x (Beard(x))</math>"}} (something has a beard) in classical logic but not in free logic.<ref name="Nolt1"/> In free logic, often an existence-predicate is used to indicate whether a singular term denotes an object in the domain or not. But the usage of existence-predicates is controversial. They are often opposed, based on the idea that existence is required if any predicates should apply to the object at all. In this sense, existence cannot itself be a predicate.<ref name="Britannica">{{cite web |title=Philosophy of logic |url=https://www.britannica.com/topic/philosophy-of-logic |website=www.britannica.com |access-date=21 November 2021 |language=en}}</ref><ref>{{cite journal |last1=Moltmann |first1=Friederike |title=Existence Predicates |journal=Synthese |date=2020 |volume=197 |issue=1 |pages=311–335 |doi=10.1007/s11229-018-1847-z |s2cid=255065180 |url=https://philpapers.org/rec/MOLEP}}</ref><ref>{{cite journal |last1=Muskens |first1=Reinhard |title=Existence Predicate |journal=The Encyclopedia of Language and Linguistics |date=1993 |pages=1191 |url=https://philpapers.org/rec/MUSEP |publisher=Oxford: Pergamon}}</ref>

[[Karel Lambert]], who coined the term "free logic", has suggested that free logic can be understood as a generalization of classical predicate logic just as predicate logic is a generalization of Aristotelian logic. On this view, classical predicate logic introduces predicates with an empty extension while free logic introduces singular terms of non-existing things.<ref name="Nolt1"/>

An important problem for free logic consists in how to determine the truth value of expressions containing empty singular terms, i.e. of formulating a [[semantics of logic|formal semantics]] for free logic.<ref name="Nolt3">{{cite web |last1=Nolt |first1=John |title=Free Logic: 3. Semantics |url=https://plato.stanford.edu/entries/logic-free/#3 |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=11 December 2021 |date=2021}}</ref> Formal semantics of classical logic can define the truth of their expressions in terms of their denotation. But this option cannot be applied to all expressions in free logic since not all of them have a denotation.<ref name="Nolt3"/> Three general approaches to this issue are often discussed in the literature: ''negative semantics'', ''positive semantics'', and ''neutral semantics''.<ref name="Lambert"/> ''Negative semantics'' hold that all atomic formulas containing empty terms are false. On this view, the expression {{nowrap|"<math>Beard(santa)</math>"}} is false.<ref name="Nolt3"/><ref name="Lambert"/>  ''Positive semantics'' allows that at least some expressions with empty terms are true. This usually includes identity statements, like {{nowrap|"<math>santa = santa</math>"}}. Some versions introduce a second, outer domain for non-existing objects, which is then used to determine the corresponding truth values.<ref name="Nolt3"/><ref name="Lambert"/>  ''Neutral semantics'', on the other hand, hold that atomic formulas containing empty terms are neither true nor false.<ref name="Nolt3"/><ref name="Lambert"/>  This is often understood as a [[three-valued logic]], i.e. that a third truth value besides true and false is introduced for these cases.<ref>{{cite web |last1=Rami |first1=Dolf |title=Non-Standard Neutral Free Logic, Empty Names and Negative Existentials |url=https://philpapers.org/archive/RAMNNF.pdf}}</ref>