Aquinas Day by Day: Logic of Arguments: Figures of Syllogisms

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Aquinas’ Topic: Logic of Arguments: Figures of Syllogisms

Scripture:

Aquinas’ text:  Expositio libri Posteriorum Bk 1, lec 31 and 26

While inductive arguments are exceedingly important because human knowledge begins in sensation, Aristotle had devoted most of his logic to deductive arguments. His Prior Analytics was a study of the form or structure of deductive reasoning, one that focused on the categorical syllogism.  Br. Thomas wrote no commentary on this work. In these two passages from his commentary on the Posterior Analytics, he summarizes the essentials of Aristotle’s syllogistic: all categorical syllogisms contain three propositions—a conclusion that follows from two premises; they are limited to three terms—major, minor, and middle terms; and in structure they fall into one of three “figures,” which can be outlined as follows (where A is the major term, B the middle term, and C the minor term):

 

Neither Aristotle nor Aquinas recognized the fourth figure, because it is simply the inverse of the first.

From lectio 31:

With respect to the first (81b10) [Aristotle] touches on three points.  The first of them is common to every syllogism, namely, that every syllogism is formed from three terms, as is indicated in Prior Analytics I. The second pertains to the affirmative syllogism.  Its form is such that it concludes A to be in C, because A is in B and B in C. [All B is A, All C is B, therefore All C is A.]. Now this is the form of a syllogism in the first figure, and in this figure alone can one draw a universal affirmative conclusion, which is the chief goal of demonstrations. The third pertains to the negative syllogism, which necessarily has one affirmative proposition and the other proposition negative, though this happens differently in the first figure and in the second, as is clear from what is shown in the Prior Analytics.

In lectio 26, Aquinas provides some examples of negative syllogisms in the first and second figures:

[Aristotle] therefore first says that when A, the major term, or B, the minor term, is in some shole as a species is in its genus, or both are under diverse genera, it does not occur that A is not in B first, that is, it does not occur that this proposition “No B is A” is immediate.  He therefore first manifests this when A is in some whole, as in C, and B is in no whole. For example, if A is “human,” C is “substance,” and B is “quantity,” it is possible to form a syllogism to prove that A is in no B on the ground that C is in every A and in no B. In this way we get a syllogism in the second figure:

Every human is a substance.

No quantity is a substance.

Therefore, no quantity is a human.

Likewise, if B, the minor term, is in some whole, say D, but A is not in that whole, it is possible to syllogize that A is in no B.  For example, if A is “substance,” B is “line,” and D is “quantity,” we get a syllogism in the first figure:

No quantity is a substance.

Every line is a quantity.

Therefore, no line is a substance.

Likewise, a negative conclusion can be demonstrated if either is in some whole.  For example, if A is “line,” C is “quantity,” B is “whiteness,” and D is “quality,” it is possible to form a syllogism in the first and in the second figure.  In the second figure:

Every line is a quantity.

No white is a quantity.

Therefore, no white is a line.

And in the first figure:

No quality is a line.

Every white is a quality.

Therefore, no white is a line.

Circa primum tria tangit. Quorum primum est commune omni syllogismo, scilicet quod omnis syllogismus est per tres terminos; ut manifestum est in libro priorum. Secundum autem pertinet ad syllogismum affirmativum; cuius forma est talis quod concludit a esse in c propter id, quod a est in b, et b est in c; et haec est forma syllogistica in prima figura, in qua sola potest concludi affirmativa universalis, quae maxime quaeritur in demonstrationibus. Tertium est quod pertinet ad syllogismum negativum, qui de necessitate unam propositionem habet affirmativam, aliam autem negativam; differenter tamen in prima figura et in secunda, ut patet per ea, quae in libro priorum ostensa sunt.

Dicit ergo primo quod cum a, idest maior terminus, aut b, idest minor terminus, sunt in quodam toto, sicut species in genere, aut etiam ambo sunt sub aliquo genere, non contingit a non esse in b primo, idest non contingit quod haec propositio, nullum b est a, sit immediata. Et primo manifestat hoc quando a est in quodam toto, scilicet c; b autem in nullo; ut puta, si a sit homo, c substantia, b quantitas: potest enim syllogismus fieri ad probandum quod a nulli b insit per hoc, quod c omni a inest, b autem nulli; ut si fiat syllogismus in secunda figura, talis: omnis homo est substantia; nulla quantitas est substantia; ergo nulla quantitas est homo.

Et similiter est, si b, idest minor terminus, sit in quodam toto, ut in d, a autem non sit in aliquo toto; syllogizari poterit quod a sit in nullo b. Ut sit a substantia, b linea, d quantitas; et fiat syllogismus in prima figura sic: nulla quantitas est substantia; omnis linea est quantitas; ergo nulla linea est substantia.

Eodem autem modo poterit demonstrari conclusio negativa, si utrumque sit in quodam toto; ut si sit a linea, c quantitas, b albedo, et d qualitas; potest syllogizari in secunda figura, et in prima. In secunda figura sic: omnis linea est quantitas; nulla albedo est quantitas; ergo nulla albedo est linea. In prima figura sic: nulla qualitas est linea; omnis albedo est qualitas; ergo nulla albedo est linea.

[Introductions and translations © R.E. Houser]