Whatever is affirmed or denied of a universal subject may be affirmed or denied of it it universally ... rather to give a quality that he has. For Aristotle , the relation of predicate to subject in these two sentences ... other, the predicate of both premises, or the subject of both premises. Aristotle refers to these term arrangements ... More recent scholarship has often applied the very techniques of mathematical logic to Aristotle’s theories, revealing (in the opinion of many) a number of similarities of approach and interest between Aristotle and modern logicians. This article is written from the latter perspective. As such, it is about Aristotle’s logic, which is not always the same thing as what has been called “Aristotelian” logic. 1. Introduction 2. Aristotle’s Logical Works: The Organon 3. The Subject of Logic: “Syllogisms” 3.1 Induction and Deduction 3.2 Aristotelian Deductions and Modern Valid Arguments 4. If we translate sullogismos as “syllogism”, this becomes the trivial claim “Every syllogism is a syllogism”, 4. Premises: The Structures of Assertions Syllogisms are structures of sentences each of which can meaningfully be called true or false: assertions ( apophanseis ), in Aristotle’s terminology. According to Aristotle, every such sentence must have the same structure: it must contain a subject ( hupokeimenon ) and a predicate and must either affirm or deny the predicate of the subject. Thus, every assertion is either the affirmation kataphasis or the denial ( apophasis ) of a single predicate of a single subject. In On Interpretation , Aristotle argues that a single assertion must always either affirm or deny a single predicate of a single subject. Thus, he does not recognize sentential compounds, such as conjunctions and disjunctions, as single assertions. This appears to be a deliberate choice on his part: he argues, for instance, that a conjunction is simply a collection of assertions, with no more intrinsic unity than the sequence of sentences in a lengthy account (e.g. the entire Iliad , to take Aristotle’s own example). Since he also treats denials as one of the two basic species of assertion, he does not view negations as sentential compounds. His treatment of conditional sentences and disjunctions is more difficult to appraise, but it is at any rate clear that Aristotle made no efforts to develop a sentential logic. Some of the consequences of this for his theory of demonstration are important. 4.1 Terms Subjects and predicates of assertions are terms . A term ( horos ) can be either individual, e.g. Socrates , Plato or universal, e.g. human , horse , animal , white . Subjects may be either individual or universal, but predicates can only be universals: Socrates is human , Plato is not a horse , horses are animals , humans are not horses . The word universal ( katholou ) appears to be an Aristotelian coinage. Affirmations Denials Universal \(P\) affirmed of all of \(S\) Every \(S\) is \(P\), All \(S\) is (are) \(P\) \(P\) denied of all of \(S\) No \(S\) is \(P\) Particular \(P\) affirmed of some of \(S\) Some \(S\) is (are) \(P\) \(P\) denied of some of \(S\) Some \(S\) is not \(P\), Not every \(S\) is \(P\) Indefinite \(P\) affirmed of \(S\) \(S\) is \(P\) \(P\) denied of \(S\) \(S\) is not \(P\) Whatever is affirmed or denied of a universal subject may be affirmed or denied of it it universally ( katholou or “of all”, kata pantos ), in part ( kata meros , en merei ), or indefinitely ( adihoristos ). 4.3.2 Some Convenient Abbreviations For clarity and brevity, I will use the following semi-traditional abbreviations for Aristotelian categorical sentences (note that the predicate term comes first and the subject term second ): Abbreviation Sentence \(Aab\) \(a\) belongs to all \(b\) (Every \(b\) is \(a\)) \(Eab\) \(a\) belongs to no \(b\) (No \(b\) is \(a\)) \(Iab\) \(a\) belongs to some \(b\) (Some \(b\) is \(a\)) \(Oab\) \(a\) does not belong to all \(b\) (Some \(b\) is not \(a\)) 5. The Syllogistic Aristotle’s most famous achievement as logician is his theory of inference, traditionally called the syllogistic (though not by Aristotle). Aristotle refers to these term arrangements as figures ( schêmata ): 5.1 The Figures First Figure Second Figure Third Figure Predicate Subject Predicate Subject Predicate Subject Premise \(a\) \(b\) \(a\) \(b\) \(a\) \(c\) Premise \(b\) \(c\) \(a\) \(c\) \(b\) \(c\) Conclusion \(a\) \(c\) \(b\) \(c\) \(a\) \(b\) Aristotle calls the term which is the predicate of the conclusion the major term and the term which is the subject of the conclusion the minor term. The premise containing the major term is the major premise , and the premise containing the minor term is the minor premise . A direct deduction is a series of steps leading from the premises to the conclusion, each of which is either a conversion of a previous step or an inference from two previous steps relying on a first-figure deduction. Conversion, in turn, is inferring from a proposition another which has the subject and predicate interchanged. Specifically, Aristotle argues that three such conversions are sound: \[\begin{align} Eab &\rightarrow Eba \\ Iab &\rightarrow Iba \\ Aab &\rightarrow Iba \end{align}\] He undertakes to justify these in An. Pr. I.2. From a modern standpoint, the third is sometimes regarded with suspicion. Aristotle’s Modal Logic: Essence and Entailment in the Organon , Cambridge: Cambridge University Press. Patzig, Günther, 1969. Aristotle’s Theory of the Syllogism , Jonathan Barnes (trans.), Dordrecht: D. Reidel. Peterson, Sandra, 1969. The Masker Paradox . Ph. D. Dissertation, Princeton. Primavesi, Oliver, 1996. Die aristotelische Topik , Munich: C. H. Beck. Rapp, Christopher, and Pieter Sjoerd Hasper, 2013. Logical Analysis and History of Philosophy 15 (Special Issue: Fallacious Arguments in Ancient Philosophy).