Mathematical logic emerged in the mid-19th century as a subfield of mathematics independent of the traditional study of logic ([[#CITEREFFerreirós2001|Ferreirós 2001]], p. 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.
=== Early history ===