Editing Philosophical logic (section)

=== Intuitionistic ===
[[Intuitionistic logic]] is a more restricted version of classical logic.<ref name="Moschovakis">{{cite web |last1=Moschovakis |first1=Joan |title=Intuitionistic Logic: 1. Rejection of Tertium Non Datur |url=https://plato.stanford.edu/entries/logic-intuitionistic/#RejTerNonDat |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=11 December 2021 |date=2021}}</ref><ref name="Burgess6">{{cite book |last1=Burgess |first1=John P. |title=Philosophical Logic |date=2009 |publisher=Princeton, NJ, USA: Princeton University Press |url=https://philpapers.org/rec/BURPL-3 |chapter=6. Intuitionistic logic}}</ref><ref name="MacMillanNonClassical"/> It is more restricted in the sense that certain rules of inference used in classical logic do not constitute valid inferences in it. This concerns specifically the [[law of excluded middle]] and the [[double negation elimination]].<ref name="Moschovakis"/><ref name="Burgess6"/><ref name="MacMillanNonClassical"/> The law of excluded middle states that for every sentence, either it or its negation are true. Expressed formally: <math>A \lor \lnot A</math>. The law of double negation elimination states that if a sentence is not not true, then it is true, i.e. {{nowrap|"<math>\lnot \lnot A \to A</math>"}}.<ref name="Moschovakis"/><ref name="MacMillanNonClassical"/> Due to these restrictions, many proofs are more complicated and some proofs otherwise accepted become impossible.<ref name="Burgess6"/>

These modifications of classical logic are motivated by the idea that truth depends on verification through a [[Formal proof|proof]]. This has been interpreted in the sense that "true" means "verifiable".<ref name="Burgess6"/><ref name="MacMillanNonClassical"/> It was originally only applied to the area of mathematics but has since then been used in other areas as well.<ref name="Moschovakis"/> On this interpretation, the law of excluded middle would involve the assumption that every mathematical problem has a solution in the form of a proof. In this sense, the intuitionistic rejection of the law of excluded middle is motivated by the rejection of this assumption.<ref name="Moschovakis"/><ref name="MacMillanNonClassical"/> This position can also be expressed by stating that there are no unexperienced or verification-transcendent truths.<ref name="Burgess6"/> In this sense, intuitionistic logic is motivated by a form of metaphysical idealism. Applied to mathematics, it states that mathematical objects exist only to the extent that they are constructed in the mind.<ref name="Burgess6"/>