Editing Mathematical logic (section)

=== Nonclassical and modal logic ===
{{main|Non-classical logic}}
[[Modal logic]]s include additional modal operators, such as an operator which states that a particular formula is not only true, but necessarily true. Although modal logic is not often used to axiomatize mathematics, it has been used to study the properties of first-order provability{{sfnp|Solovay|1976}} and set-theoretic forcing.{{sfnp|Hamkins|Löwe|2007}}

[[Intuitionistic logic]] was developed by Heyting to study Brouwer's program of intuitionism, in which Brouwer himself avoided formalization. Intuitionistic logic specifically does not include the [[law of the excluded middle]], which states that each sentence is either true or its negation is true.  Kleene's work with the proof theory of intuitionistic logic showed that constructive information can be recovered from intuitionistic proofs. For example, any provably total function in intuitionistic arithmetic is [[computable]]; this is not true in classical theories of arithmetic such as [[Peano arithmetic]].