Editing Discrete mathematics (section)

===Logic===
{{Main|Mathematical logic}}
Logic is the study of the principles of valid reasoning and [[inference]], as well as of [[consistency]], [[soundness]], and [[Completeness (logic)|completeness]]. For example, in most systems of logic (but not in [[intuitionistic logic]]) [[Peirce's law]] (((''P''→''Q'')→''P'')→''P'') is a theorem. For classical logic, it can be easily verified with a [[truth table]]. The study of [[mathematical proof]] is particularly important in logic, and has accumulated to [[automated theorem proving]] and [[formal verification]] of software.

[[Well-formed formula|Logical formulas]] are discrete structures, as are [[Proof theory|proofs]], which form finite [[tree structure|trees]]<ref>{{cite book| first1 = A.S. |last1=Troelstra|first2=H. |last2=Schwichtenberg| title = Basic Proof Theory| url = https://books.google.com/books?id=x9x6F_4mUPgC&pg=PA186| date = 2000-07-27| publisher = Cambridge University Press| isbn = 978-0-521-77911-1| page = 186 }}</ref> or, more generally, [[directed acyclic graph]] structures<ref>{{cite book| first = Samuel R. |last=Buss| title = Handbook of Proof Theory| url = https://books.google.com/books?id=MfTMDeCq7ukC&pg=PA13| year = 1998| publisher = Elsevier| isbn = 978-0-444-89840-1| page = 13 }}</ref><ref>{{cite book| first1 = Franz |last1=Baader|first2=Gerhard |last2=Brewka|first3=Thomas |last3=Eiter| title = KI 2001: Advances in Artificial Intelligence: Joint German/Austrian Conference on AI, Vienna, Austria, September 19-21, 2001. Proceedings| url = https://books.google.com/books?id=27A2XJPYwIkC&pg=PA325| date = 2001-10-16| publisher = Springer| isbn = 978-3-540-42612-7| page = 325 }}</ref> (with each [[Rule of inference|inference step]] combining one or more [[premise]] branches to give a single conclusion). The [[truth value]]s of logical formulas usually form a finite set, generally restricted to two values: ''true'' and ''false'', but logic can also be continuous-valued, e.g., [[fuzzy logic]]. Concepts such as infinite proof trees or infinite derivation trees have also been studied,<ref>{{cite journal | title = Cyclic proofs of program termination in separation logic | first1 = J. | last1 = Brotherston | first2 = R. | last2 = Bornat | first3 = C. | last3 = Calcagno | journal = ACM SIGPLAN Notices | volume = 43 | issue = 1 |date=January 2008 | pages = 101–112 | doi = 10.1145/1328897.1328453 }}</ref> e.g. [[infinitary logic]].