in Aristotle’s words, ‘a discourse in which, a certain thing being stated, something other than what is stated follows of necessity from being so’ (Prior Analytics, 24b 18). Three types of syllogism were usually distinguished: categorical, hypothetical, and disjunctive. Each will be treated in that order.
The categorical syllogism. This is an argument consisting of three categorical propositions, two serving as premises and one serving as conclusion. E.g., ‘Some college students are happy; all college students are high school graduates; therefore, some high school graduates are happy’. If a syllogism is valid, the premises must be so related to the conclusion that it is impossible for both premises to be true and the conclusion false. There are four types of categorical propositions: universal affirmative or A-propositions – ‘All S are P’, or ‘SaP’; universal negative or E-propositions – ‘No S are P’, or ‘SeP’; particular affirmative or I-propositions – ‘Some S are P’, or ‘SiP’; and particular negative or O-propositions: ‘Some S are not P’, or ‘SoP’. The mediate basic components of categorical syllogism are terms serving as subjects or predicates in the premises and the conclusion. There must be three and only three terms in any categorical syllogism, the major term, the minor term, and the middle term. Violation of this basic rule of structure is called the fallacy of four terms (quaternio terminorum); e.g., ‘Whatever is right is useful; only one of my hands is right; therefore only one of my hands is useful’. Here ‘right’ does not have the same meaning in its two occurrences; we therefore have more than three terms and hence no genuine categorical syllogism.
The syllogistic terms are identifiable and definable with reference to the position they have in a given syllogism. The predicate of the conclusion is the major term; the subject of the conclusion is the minor term; the term that appears once in each premise but not in the conclusion is the middle term. As it is used in various types of categorical propositions, a term is either distributed (stands for each and every member of its extension) or undistributed. There is a simple rule regarding the distribution: universal propositions (SaP and SeP) distribute their subject terms; negative propositions (SeP and SoP) distribute their predicate terms. No terms are distributed in an I-proposition. Various sets of rules governing validity of categorical syllogisms have been offered. The following is a ‘traditional’ set from the popular Port-Royal Logic (1662). R1: The middle term must be distributed at least once. Violation: ‘All cats are animals; some animals do not eat liver; therefore some cats do not eat liver’. The middle term ‘animals’ is not distributed either in the first or minor premise, being the predicate of an affirmative proposition, nor in the second or major premise, being the subject of a particular proposition; hence, the fallacy of undistributed middle. R2: A term cannot be distributed in the conclusion if it is undistributed in the premises. Violation: ‘All dogs are carnivorous; no flowers are dogs; therefore, no flowers are carnivorous’. Here the major, ‘carnivorous’, is distributed in the conclusion, being the predicate of a negative proposition, but not in the premise, serving there as predicate of an affirmative proposition; hence, the fallacy of illicit major term. Another violation of R2: ‘All students are happy individuals; no criminals are students; therefore, no happy individuals are criminals’. Here the minor, ‘happy individuals’, is distributed in the conclusion, but not distributed in the minor premise; hence the fallacy of illicit minor term. R3: No conclusion may be drawn from two negative premises. Violation: ‘No dogs are cats; some dogs do not like liver; therefore, some cats do not like liver’. Here R1 is satisfied, since the middle term ‘dogs’ is distributed in the minor premise; R2 is satisfied, since both the minor term ‘cats’ as well as the major term ‘things that like liver’ are distributed in the premises and thus no violation of distribution of terms occurs. It is only by virtue of R3 that we can proclaim this syllogism to be invalid. R4: A negative conclusion cannot be drawn where both premises are affirmative. Violation: ‘All educated people take good care of their children; all who take good care of their children are poor; therefore, some poor people are not educated’. Here, it is only by virtue of the rule of quality, R4, that we can proclaim this syllogism invalid.
R5: The conclusion must follow the weaker premise; i.e., if one of the premises is negative, the conclusion must be negative, and if one of them is particular, the conclusion must be particular.
R6: From two particular premises nothing follows. Let us offer an indirect proof for this rule. If both particular premises are affirmative, no term is distributed and therefore the fallacy of undistributed middle is inevitable. To avoid it, we have to make one of the premises negative, which will result in a distributed predicate as middle term. But by R5, the conclusion must then be negative; thus, the major term will be distributed in the conclusion. To avoid violating R2, we must distribute that term in the major premise. It could not be in the position of subject term, since only universal propositions distribute their subject term and, by hypothesis, both premises are particular. But we could not use the same negative premise used to distribute the middle term; we must make the other particular premise negative. But then we violate R3. Thus, any attempt to make a syllogism with two particular premises valid will violate one or more basic rules of syllogism. (This set of rules assumes that A- and E- propositions have existential import and hence that an I- or an O-proposition may legitimately be drawn from a set of exclusively universal premises.)
Categorical syllogisms are classified according to figure and mood. The figure of a categorical syllogism refers to the schema determined by the possible position of the middle term in relation to the major and minor terms. In ‘modern logic,’ four syllogistic figures are recognized. Using ‘M’ for middle term, ‘P’ for major term, and ‘S’ for minor term, they can be depicted as follows: Aristotle recognized only three syllogistic figures. He seems to have taken into account just the two premises and the extension of the three terms occurring in them, and then asked what conclusion, if any, can be derived from those premises. It turns out, then, that his procedure leaves room for three figures only: one in which the M term is the subject of one and predicate of the other premise; another in which the M term is predicated in both premises; and a third one in which the M term is the subject in both premises. Medievals followed him, although all considered the so-called inverted first (i.e., moods of the first figure with their conclusion converted either simply or per accidens) to be legitimate also. Some medievals (e.g., Albalag) and most moderns since Leibniz recognize a fourth figure as a distinct figure, considering syllogistic terms on the basis not of their extension but of their position in the conclusion, the S term of the conclusion being defined as the minor term and the P term being defined as the major term. The mood of a categorical syllogism refers to the configuration of types of categorical propositions determined on the basis of the quality and quantity of the propositions serving as premises and conclusion of any given syllogism; e.g., ‘No animals are plants; all cats are animals; therefore no cats are plants’, ‘(MeP, SaM /, SeP)’, is a syllogism in the mood EAE in the first figure. ‘All metals conduct electricity; no stones conduct electricity; therefore no stones are metals’, ‘(PaM, SeM /, SeP)’, is the mood AEE in the second figure. In the four syllogistic figures there are 256 possible moods, but only 24 are valid (only 19 in modern logic, on the ground of a non-existential treatment of A- and E-propositions). As a mnemonic device and to facilitate reference, names have been assigned to the valid moods, with each vowel representing the type of categorical proposition. William Sherwood and Peter of Spain offered the famous list designed to help students to remember which moods in any given figure are valid and how the ‘inevident’ moods in the second and third figures are provable by reduction to those in the first figure: barbara, celarent, darii, ferio (direct Fig. 1); baralipton, celantes, dabitis, fapesmo, frisesomorum (indirect Fig. 1); cesare, camestres, festino, baroco (Fig. 2); darapti, felapton, disamis, datisi, bocardo, ferison (Fig. 3). The hypothetical syllogism. The pure hypothetical syllogism is an argument in which both the premises and the conclusion are hypothetical, i.e. conditional, propositions; e.g., ‘If the sun is shining, it is warm; if it is warm, the plants will grow; therefore if the sun is shining, the plants will grow’. Symbolically, this argument form can be represented by ‘A P B, B P C /, A P C’. It was not recognized as such by Aristotle, but Aristotle’s pupil Theophrastus foreshadowed it, even though it is not clear from his example of it – ‘If man is, animal is; if animal is, then substance is; if therefore man is, substance is’ – whether this was seen to be a principle of term logic or a principle of propositional logic. It was the Megaric- Stoic philosophers and Boethius who fully recognized hypothetical propositions and syllogisms as principles of the most general theory of deduction.
Mixed hypothetical syllogisms are arguments consisting of a hypothetical premise and a categorical premise, and inferring a categorical proposition;