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\begin{document}

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\section*{syllogism}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{deduction_and_induction}{}\paragraph*{{Deduction and Induction}}\label{deduction_and_induction}

[[!include deduction and induction - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{original_definition}{Original definition}\dotfill \pageref*{original_definition} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \textbf{syllogism} is a pattern of [[deduction|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,

\begin{itemize}%
\item No fish is a mammal.
\item Some sea creatures are fish.
\item Therefore, some sea creatures are not mammals.

\end{itemize}
In the nineteenth century, syllogism became seen to be insufficiently expressive as more complicated relations and [[quantification]] were encountered. [[Predicate logic]] came to take its place.

\hypertarget{original_definition}{}\subsection*{{Original definition}}\label{original_definition}

The definition by Aristotle that has received the most attention (out of several variants) is from \emph{Prior Analytics} 24b18-22 (here taken from translation by \hyperlink{Striker}{Striker (2009)}):

\begin{quote}%
A `syllogismos' is an argument (logos) in which, (i) certain things being posited (tethent\^o{}n), (ii) something other than what was laid down (keimen\^o{}n) (iii) results by necessity (eks anagk\^e{}s 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.


\end{quote}
\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[natural deduction]]


\item [[co-Heyting negation]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Stanford Encyclopedia of Philosophy, \emph{\href{http://plato.stanford.edu/entries/aristotle-logic/}{Aristotle's logic}}


\item Gisela Striker, \emph{Aristotle's Prior Analytics Book I: Translated with an Introduction and Commentary}, OUP, Oxford (2009) \href{http://philpapers.org/rec/STRAPA-2}{PhilPapers entry}



\end{itemize}
\begin{itemize}%
\item [[Georg Hegel]], \emph{[[Lectures on the History of Philosophy]] -- \href{Lectures+on+the+History+of+Philosophy#AristotleLogic}{Aristotle -- Logic}}.


\item [[Georg Hegel]], \emph{[[Science of Logic]] -- \href{Science+of+Logic#SchlussSyllogism}{Subjective logic -- Subjectivity -- Syllogism}}



\end{itemize}
An interpretation of syllogistic term logic in modern [[predicate logic]] is discussed in detail in the textbook on [[mathematical logic]]:

\begin{itemize}%
\item [[David Hilbert]], [[Wilhelm Ackermann]], \emph{The calculus of classes (monadic predicate calculus)} (\href{http://books.google.it/books?id=45ZGMjV9vfcC&pg=PA44#v=onepage&q&f=false}{web}), chapter II of \emph{Grundz\"u{}ge der Theoretischen Logik} , 4th ed. Springer Heidelberg 1959 1928

\end{itemize}
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]]:

\begin{itemize}%
\item J. Bacon, \emph{Natural-deduction rules for syllogistic} , JSL \textbf{31} (1966) pp.686-7.


\item J. Corcoran, \emph{Completeness of an ancient logic} , JSL \textbf{37} (1972) pp.696-702.


\item T. Smiley, \emph{What is a syllogism} , JPL \textbf{2} (1973) pp.136-154.



\end{itemize}
An approach to Aristotelian logic using [[category theory]] is in

\begin{itemize}%
\item [[Marie La Palme Reyes]], John Macnamara, [[Gonzalo Reyes]] , \emph{Functoriality and Grammatical Role in Syllogisms} , Notre Dame J. Formal Logic \textbf{35} no.1 (1994) pp.41-66. (\href{http://projecteuclid.org/euclid.ndjfl/1040609293}{Euclid}, \href{http://projecteuclid.org/download/pdf_1/euclid.ndjfl/1040609293}{pdf})


\item M. La Palme Reyes, J. Macnamara, [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari, \emph{The non-Boolean logic of natural language negation} , Phil. Math. \textbf{2} no.1 (1994) pp.45-68.



\end{itemize}
A proposal for formalization of syllogisms within [[linear logic]] is in

\begin{itemize}%
\item Ruggero Pagnan, \emph{A diagrammatic calculus of syllogisms}, Journal of Logic, Language and Information July 2012, Volume 21, Issue 3, pp 347-364 (\href{http://arxiv.org/abs/1001.1707}{arXiv:1001.1707})


\item Ruggero Pagnan, \emph{Syllogisms in Rudimentary Linear Logic, Diagrammatically} (\href{http://arxiv.org/abs/1302.7111}{arXiv:1302.7111})



\end{itemize}
[[!redirects syllogisms]]

[[!redirects Aristotle's logic]] [[!redirects Aristotelian logic]]



\end{document}