A syllogism is a pattern of deductive argument of a certain limited range of form. The study of syllogisms was originated by Aristotle and later developed by his followers as well as by the Stoics. Syllogisms were central objects of study for medieval logicians.
Syllogisms involve two premises and a conclusion, each expressing a statement of the form that either all, none or some members of a class do or do not have a certain property. For example,
The definition by Aristotle that has received the most attention (out of several variants) is from Prior Analytics 24b18-22 (here taken from translation by Striker (2009)):
A ‘syllogismos’ is an argument (logos) in which, (i) certain things being posited (tethentôn), (ii) something other than what was laid down (keimenôn) (iii) results by necessity (eks anagkês sumbainei)(iv) because these things are so. By ‘because these things are so’ I mean that it results through these, and by ‘resulting through these’ I mean that no term is required from outside for the necessity to come about.
Stanford Encyclopedia of Philosophy, Aristotle’s logic
Gisela Striker, Aristotle’s Prior Analytics Book I: Translated with an Introduction and Commentary, OUP, Oxford (2009) PhilPapers entry
After J. Lukasiewicz’s classic monograph on Aristotelian logic from a modern point of view, the 1970s saw influential interpretations of it as systems of natural deduction:
J. Bacon, Natural-deduction rules for syllogistic , JSL 31 (1966) pp.686-7.
J. Corcoran, Completeness of an ancient logic , JSL 37 (1972) pp.696-702.
T. Smiley, What is a syllogism , JPL 2 (1973) pp.136-154.
An approach to Aristotelian logic using category theory is in
M. La Palme Reyes, J. Macnamara, G. E. Reyes, H. Zolfaghari, The non-Boolean logic of natural language negation , Phil. Math. 2 no.1 (1994) pp.45-68.
A proposal for formalization of syllogisms within linear logic is in
Ruggero Pagnan, A diagrammatic calculus of syllogisms, Journal of Logic, Language and Information July 2012, Volume 21, Issue 3, pp 347-364 (arXiv:1001.1707)
Ruggero Pagnan, Syllogisms in Rudimentary Linear Logic, Diagrammatically (arXiv:1302.7111)