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Mathematical logic

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Mathematical logic emerged in the mid-19th century as a subfield of mathematics independent of the traditional study of logic ([[#CITEREFFerreirós2001|Ferreirós 2001]], p.&nbsp;443). "Mathematical logic, also called 'logistic', 'symbolic logic', the '[[Boolean algebra|algebra of logic]]', and, more recently, simply 'formal logic', is the set of logical theories elaborated in the course of the last [nineteenth] century with the aid of an artificial notation and a rigorously deductive method."<ref>[[Jozef Maria Bochenski]], ''A Precis of Mathematical Logic'' (1959), rev. and trans., Albert Menne, ed. and trans., Otto Bird, Dordrecht, South Holland: Reidel, Sec. 0.1, p. 1.</ref> Before this emergence, logic was studied with [[rhetoric]], with ''calculationes'',<ref>[[Richard Swineshead]] (1498), ''Calculationes Suiseth Anglici'', Papie: Per Franciscum Gyrardengum.</ref> through the [[syllogism]], and with [[philosophy]]. The first half of the 20th century saw an explosion of fundamental results, accompanied by vigorous debate over the foundations of mathematics.
But even today mathematical logic is being used in sciences such as [[earth science]]
=== Early history ===
Numerous results in recursion theory were obtained in the 1940s by [[Stephen Cole Kleene]] and [[Emil Leon Post]]. Kleene ([[#CITEREFKleene1943|1943]]) introduced the concepts of relative computability, foreshadowed by Turing ([[#CITEREFTuring1939|1939]]), and the [[arithmetical hierarchy]]. Kleene later generalized recursion theory to higher-order functionals. Kleene and Kreisel studied formal versions of intuitionistic mathematics, particularly in the context of proof theory.
<!-- Perhaps it is better to stop this history around 1950 -->
But even today mathematical logic is being used in sciences such as [[earth science]]
== Formal logical systems {{anchor|Formal logic}} ==
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