Cursus Philosophicus Thomisticus, Prima Pars Artis Logicae

The third operation of the intellect is carried out in reasoning, which necessarily requires inference and consequence, whereby one thing is deduced from another. Consequence in general is defined as an argumentation, that is, “an utterance in which, one thing being given, another follows.” Since every inference is grounded in the connection of one truth with another, when the term follows occurs in the definition of consequence, it does not mean “is connected.” If it did, only a good consequence would be defined, since only that has a connection and inference of one thing from another. But if follows is taken as common to good and bad consequence, it means “is signified to follow”: wherever utterances are joined by a sign of inference, there is a consequence, even if in reality the inference is not good and fitting.

What a good and bad consequence is—the first division of consequence in general—we explained in the preceding book, chapter 5. If someone asks why consequence is divided into good and bad, rather than true and false, I answer that consequence is not a proposition, to which truth and falsity alone pertain because it affirms or denies. Rather, it is an inferential connection of propositions, to which the proper arrangement and fittingness of the connection pertain. Fitting and unfitting, then, make a consequence good or bad, not true or false.

Consequence is divided secondly into material and formal. A material consequence is good only by reason of some matter; a formal consequence is good in any matter, provided its terms have the same form. Therefore, a formal consequence that is once good is always good. For example, if you say, “Some human being is rational; therefore every human being is rational,” this consequence holds in that matter, because it is necessary matter in which a universal can be inferred from a particular. But it does not hold by force of the form, since it fails in another matter of similar form: “Some human being is white; therefore every human being is white.” From this you will understand what the form of a consequence is, by reason of which it is said to be formally good. It is the arrangement of propositions and terms according to quantity, quality, and the other logical properties, ordered to inferring one thing from another. When one thing follows from another in any matter not by reason of the matter arranged, but by reason of the arrangement itself, the consequence is called formal. Similarity of form is the agreement of the arrangement of propositions in the same quantity, quality, and logical properties. We said above that logical properties include supposition, ampliation, and the like; they will be recognized more clearly from the delineation of the defects in consequences that we shall set out below.¹

In the preceding book, chapter 5, on argumentation, we said that all the species of argumentation could be reduced to two: induction and syllogism. For an enthymeme is a truncated syllogism, and an example is an imperfect induction, as St. Thomas says at the end of 1 Poster., lect. 1.¹ Hence, finally, in lecture 30 of the same book,² St. Thomas lists only two ways of acquiring knowledge: demonstration and induction. Demonstration is a syllogism that proceeds from universals, while induction proceeds from singulars, since all our cognition originates from singulars, which are perceived through the senses.

Induction is therefore defined as “a progression from singulars sufficiently enumerated to the universal”; for example, if you say, “This fire heats, and that one, and that one, etc.; therefore every fire heats.” And since opposites have the same account, under this definition of induction by ascent is understood its opposite, descent, that is, a progression from the universal to singulars. Induction, insofar as it proceeds by ascent, is ordered to discovering and proving universal truths as universal—that is, insofar as they consist of the singulars contained under them. For something cannot be proved to be such universally except because its singulars are such. Descent, however, is ordered chiefly to showing the falsity of a universal as universal: the falsity of a universal is best shown by descending under it and showing that some singular is not such. Still, once the truth of the universal has been established and discovered through ascent, descent also serves to show the universal’s correspondence to the singulars contained under it. The principal function of induction, however, is to bring proof of a universal to the senses by ascending from singulars.

It is therefore clear that Aristotle rightly set out the difference between syllogism and induction in the second book of the Prior Analytics, chapter 23.³ A syllogism uses a middle term to show that the extremes are united with one another; induction, by contrast, proves the extreme of the middle by means of a third. That is to say, a syllogism uses premises in which the middle is assumed to be united with the extremes, so that through it the union of the extremes with one another may be proved. Induction, however, does not assume a middle uniting the extremes in order to prove that the extremes are united with one another. Rather, it proves that an extreme, or some predicate, belongs to a common subject because it belongs to the singulars, or that it belongs to the singulars because it belongs to the common subject.

You may ask how many kinds of ascent and descent there are and what they are. I answer that there are four: conjunctive, disjunctive, compounded, and disjoined. Conjunctive ascent and descent occurs when singulars are enumerated through conjunctive propositions, as in “Every man is an animal; therefore this man is an animal, and that man is an animal,” etc. Disjunctive occurs when they are enumerated through disjunctive propositions, as in “Some man is white; therefore either this man is white, or that man is white,” etc. Compounded ascent and descent occurs when a term is resolved by singulars enumerated together in a complex proposition. For example, “All the apostles of God are twelve” is resolved as “This one and that one, etc., are twelve,” enumerating them all in one proposition. Disjoined occurs when a term is resolved through its singulars, enumerated separately in a single proposition. Thus, if you say, “Every man is an animal,” the predicate “animal” is resolved disjointly as “Every man is this animal or that animal or that other animal,” so that the whole disjunction is the predicate. Whether induction, by its formal account and its own proper force, is a formal consequence, we shall say below in question 8, article 2.

Since every ascent and descent proceeds from singulars to common terms and conversely, according to the different quantity and supposition of a term—whether it is universal or particular, supposing determinately or confusedly—ascent and descent are applied in different ways.

Two things must therefore be observed. First, which kind of ascent and descent belongs to this or that supposition. Second, what order of descending or ascending under a term is immediate or mediate; this is to resolve the term.

As to the first, there are three rules. First: A term supposing distributively requires conjunctive ascent and descent. Second: A term supposing determinately requires disjunctive ascent and descent. Third: A term supposing confusedly only in a conjunctive way requires compounded ascent and descent; a term supposing confusedly in a disjunctive way requires disjoined ascent and descent. Nevertheless, this compounded or disjoined ascent is not properly resolutory but manifests the truth; for it remains equally unknown to posit a term that supposes confusedly or is enumerated disjunctively. Therefore such a term remains simply unresolved and fixed—that is, it cannot be moved or proceeded through by resolution from singulars to universals or conversely.

As to the second, when a proposition contains several terms supposing in different ways, one cannot descend or ascend under each of them equally immediately. Three rules therefore help us determine this. First: Under a term supposing determinately with respect to a distributed term, one must ascend before descending under the distributed term, although one can descend immediately under the distributed term. The contrary is a fallacy called “from several determinates to a single determinate.” Thus, for ascending, a term supposing determinately is always more suitable; for descending, a distributed term is more suitable. For example, if you say, “Some man is not white,” you cannot rightly descend immediately under the distributed term by saying, “Some man is not this white thing, nor that white thing; therefore some man is not white.” Yet under the particular term supposing determinately you can descend immediately under the distributed term; thus, as to ascent, the resolution depends on that prior step, but not as to descent.

Second rule: Under a term supposing confusedly in a disjunctive way with respect to an antecedent universality, a disjunctive descent is not permitted before descending under the universal term; ascent, however, is valid. For example, if you say, “Every man is an animal,” “Therefore every man is this animal or that animal,” etc., does not follow; but the ascent does follow: “Every man is this animal, or every man is that animal; therefore every man is an animal.” Thus, in descending, the resolution of the confused term depends on resolving the antecedent distributed term. The contrary is the fallacy from one distribution to several, discussed below in chapter 13.

Third rule: If all terms suppose either distributively or determinately, resolution can immediately begin with any term in ascent or descent, insofar as this follows from the force of supposition, unless an impediment arises from another source. For example, in the distributive case, “No man is a stone,” or in the determinate case, “Some man is white”: you can begin with any term. Accordingly, in the ascent of terms, which serves to prove truth, the order is this: first resolve the term standing determinately, second the distributed term. In descent, however, we can begin immediately with a distributed term; with a confused term, only after descending under the distributed term. Ascent and descent are always understood to occur with logical properties unchanged, including kind of supposition, restriction, ampliation, appellation, etc. This is especially to be observed in complex terms, particularly those consisting of a determination and what is determined; differences in restriction are a major obstacle to immediate or mediate resolution, as we shall touch on below when treating the defects of consequences¹ and below in question 7, article 2.

Finally, there are two rules for relatives. First: With a reciprocal relative, one may not perform resolution by ascent or descent before resolving under the antecedent, because it supposes with the same supposition as its antecedent; this is called a sealed supposition. Second: A non-reciprocal relative does not necessarily depend on its antecedent for its resolution. For example, if you say, “Every man is an animal, and that is rational,” one may descend or ascend under “that” before doing so under “every man” or “animal,” just as with a term placed in a separate proposition.

A syllogism is defined as “a discourse in which, certain things having been posited and granted, something other than them necessarily follows because those things have been posited.” We should know that a syllogism aims to order reasoning according to the connection of terms with one another, not, as induction does, through resolution and reduction to singulars. It must therefore regard extremes capable of being united with one another, and a rationale or middle term on account of which they are united. Hence, from a middle term applied or united to the extremes, it unites them and infers that the extremes are united with one another. This is the whole order and progression of a syllogism: to infer and deduce a proposition in which two terms, or extremes, are united, from another proposition or propositions in which the middle term, or rationale by which they can and ought to be united with one another, is made explicit. That is why the definition says that it is “a discourse in which, certain things having been posited”—namely, the premises, in which the middle term is applied to the extremes—“something else necessarily follows,” that is, the conclusion is inferred, in which the extremes are joined to one another, “because those things have been posited,” that is, by the force and positing of the premises. An example is this syllogism: “Every body is a substance; every man is a body; therefore every man is a substance.” Here you see three propositions: two premises that infer, and a third that is the conclusion. In the third, the two extremes “man” and “substance” are united; in the premises, the middle term, on account of which those extremes are united, is applied to the extremes. The middle term is “body,” which, because it belongs to man and to substance, makes man and substance belong together.

A syllogism, then, consists of matter and form, like any artifact. Its FORM is the very arrangement and art that orders and disposes the matter so that it can properly infer and conclude. Its matter is twofold: one is proximate, the other remote. The proximate matter consists of the propositions from which the syllogism is constructed; the remote matter consists of the terms or extremes that compose the proposition and into which the syllogism is ultimately resolved. Therefore Aristotle defines a term in this part of logic as that into which a proposition is resolved, as into subject and predicate.¹ It is defined by resolution, as St. Thomas says in Opusculum 48, treatise on the Syllogism, chapter 1,² because our knowledge and judgment proceed by resolving a thing into its causes. Thus, insofar as a term serves the judging or resolving part, it is defined by resolution, not by composition. The definition says “as into predicate and subject” because the verb is not a term with respect to the syllogism, but a union; a term is only what functions as an extreme that infers or is inferred. Through the union expressed by the verb, then, there is composed not only what pertains to the truth of an enunciation, but also what pertains to the inferential connection. Thus the union, or copula, of the verb is not that into which the syllogism is resolved, but that which is dissolved when the syllogism is resolved.

Therefore, there are only three propositions in a syllogism: two premises, which infer, and one conclusion, which is inferred. The first premise is called the major proposition, the second the minor, and the third the conclusion or consequent. Similarly, there can be only three terms, although they are repeated in the three propositions. Two are the extremes, which are to be united in the conclusion; the other is the middle, which is united to the extremes in the premises but does not enter the conclusion, for it infers it. One extreme is called major, the other minor. The major extreme is the one taken in the major premise and placed in the conclusion. The minor extreme is the one taken in the minor premise and placed in the conclusion. The middle is the term posited twice in the premises but not in the conclusion. By “term” we mean not only something simple or uncomplex, but also something complex, provided that it functions as one extreme or middle.

The FORM of a syllogism can also be said to be twofold, corresponding to this twofold proximate and remote matter. Since the form of a syllogism is the ordering or arrangement of its matter, the arrangement that orders the remote matter, namely, the terms, is called a figure; the arrangement that orders the proximate matter, namely, the propositions, is called a mode. A figure in a syllogism, then, is the arrangement of terms according to the subject and predicate positions. A mode is the ordering of premises or propositions according to their proper quantity and quality. The proper quality is that at least one premise be affirmative, and that not all be negative. The proper quantity is that at least one be universal, and that not all be particular. Figures are determined according to the predication and subjection of the middle term with respect to the extremes; different figures result from different combinations of predication and subjection, as we shall say in the next chapter.

The division of syllogism can be taken either from the side of matter or from the side of form. First, on the side of matter, there is the division belonging to the books of the Posterior Analytics: a syllogism is fourfold—demonstrative, probable, sophistical, or erroneous. A demonstrative syllogism generates knowledge and consists of necessary matter; a probable syllogism consists of contingent matter and generates opinion; an erroneous syllogism consists of remote matter and generates error; a sophistical syllogism does not observe correct rules but fallacious ones, and generates deception.

Second, a syllogism is divided into common and expositional. A common syllogism is one in which a common term is posited as the middle; an expositional syllogism is one in which a singular term is posited as the middle. This division, then, is taken from the side of matter, since it is based on the terms.

It can also be divided into hypothetical and categorical, and into affirmative and negative, according to the propositions of which it consists.

On the side of form, a syllogism is first divided by the three figures. In each figure there are several modes, namely sixteen, both useful and useless, perfect and imperfect; but all are reduced to nineteen useful modes, which are contained in these verses:

First: Barbara, Celarent, Darii, Ferio, Baralipton. Celantes, Dabitis, Fapesmo, Frisesomorum. Second: Cesare, Camestres, Festino, Baroco. Third: Darapti, Felapton, Disamis, Datisi, Bocardo, Ferison.

A perfect mode is one in which the regulative principles of syllogisms are perfectly preserved; these are “Dici de omni” and “Dici de nullo,” as will be explained below. An imperfect mode is one in which these principles are not perfectly preserved, and which can therefore be reduced to a perfect mode. A useful mode is one in which the rules necessary for a good consequence are preserved; a useless mode is one in which those rules are not preserved. A direct mode concludes directly by predicating the major extreme of the minor in the conclusion; an indirect mode concludes conversely, by predicating the minor extreme of the major extreme in the conclusion.

THAT THERE ARE THREE FIGURES, and no more, is clear, because there can be only four combinations: either the middle is predicated in both premises, or is subject in both, or is subject in the major and predicate in the minor, or finally is predicate in the major and subject in the minor. It must nevertheless be observed that this last of the four combinations concludes indirectly. Since syllogisms that conclude indirectly are reduced to direct ones, the fourth figure, which always concludes indirectly, is therefore always reduced to the first by transposing the premises; it is consequently not counted separately, as though it were not distinct from the first. A syllogism is said to conclude directly when the major extreme is predicated in the conclusion of the minor extreme; it concludes indirectly when the reverse occurs, because the major extreme is the higher extreme, and it belongs to the higher to be predicated directly, while it belongs to the lower to be made subject. Thus, for it to be predicated in the conclusion of the minor extreme, it must also be predicated in the major premise of the middle as something higher. Therefore the direct figure is one in which the middle is subject in the major premise and the major extreme is predicated of it: this is the first figure. If, however, the middle is predicated in the major premise and the major extreme is made subject, the conclusion will be indirect; the syllogism must therefore be reduced to the first figure by transposing the premises. Accordingly, the fourth figure is not admitted as distinct but is reduced to the first, because figures are divided only by a different, natural, and direct way of concluding, not by an indirect one, since the latter belongs to and is reduced to the direct way.

Thus there remain three distinct figures: the first, in which the middle is subject in the major premise and predicate in the minor; the second, in which the middle is predicate in both premises; and the third, in which the middle is subject in both. We shall give examples below, enumerating all the modes of the figures.

Since figures, as was said, are nothing other than different combinations according to the predication and subjection of the middle term, while moods are combinations according to the quantity and quality of propositions, sixteen combinations of moods can be made in each figure with respect to quantity and quality. For the quantity of propositions can vary in four ways: both premises may be universal, both particular, the first universal and the second particular, or conversely the first particular and the second universal. I say similarly of quality: both premises may be affirmative, both negative, the first affirmative and the second negative, or conversely the first negative and the second affirmative.

Thus, given four combinations with respect to quality and four with respect to quantity, quantity and quality can be combined with one another in sixteen ways, because each mood or combination of quantity admits four modes of quality, and conversely. For example, if both premises are universal—which is the first combination with respect to quantity—four modes of quality are possible: both universals may be affirmative, both negative, the first affirmative and the second negative, or conversely. Exactly the same can be done with every other combination. Since there are four combinations of quantity and four of quality, and each of the four modes can be combined with the others, sixteen moods result in each figure according to the different quantities and qualities.

But not all moods are useful, that is, valid and legitimate inferences. To discern which are useful and which useless in each figure, we must consider the rules required for validity in these syllogisms. A mood that observes all of them is useful; one that does not is useless.

Some rules required for a good syllogism are therefore general to all figures, and some are proper to each.

There are four COMMON RULES. First: Nothing follows from purely particular or indefinite premises. Second: Nothing follows from purely negative premises. The reason for both conditions is drawn from the general principle on which the whole structure of syllogism rests, as will be explained below: “Whatever things are the same as one third thing are the same as one another”; that is, because the extremes are united in something, they are united with one another. Therefore they must be united in at least one premise, and so they must not both be divided or denied in both premises. Likewise, the third thing in which they are united must include the extremes and be distributed with respect to them; otherwise it will not be said of all or of none. Therefore at least one premise must have distribution and affirmation. The third rule is that no term may be distributed in the conclusion unless it is distributed in the premises; otherwise one would argue from an undistributed term to a distributed one. Fourth: The middle term must not enter the conclusion, as was said. Thus every syllogism that yields a negative conclusion is negative, and one that yields a particular conclusion is particular.

From these general rules, four further PARTICULAR RULES for the individual figures are inferred. First: In the first figure, in moods that conclude directly, the major premise cannot be particular and the minor cannot be negative. The reason is that if the major is particular, the middle term is not distributed, since the middle must be subject in the major and predicate in the affirmative minor. If the minor is negative, it yields a negative conclusion in which some term would be distributed although it was not distributed in the affirmative premise. For example: “Every man is an animal; no horse is a man; therefore no horse is an animal.” Here “animal” is distributed in the conclusion but not in the premises. In moods that conclude indirectly, however, the major can be particular, because the middle is not subject in them, and the minor can be negative, as in Fapesmo and Frisesomorum. The second rule, for the second figure, is: If both premises are affirmative, nothing is concluded. The reason is that the middle term is predicated in both premises in the second figure and is therefore not distributed if both are affirmative. The third rule, for the same figure, is: If the major premise is particular, nothing follows. For in that case some term would be distributed in the conclusion but not in the premises. For example: “Some animal is a man; no horse is a man; therefore no horse is an animal.” Here “animal” is distributed in the conclusion but not in the premises. The fourth rule, for the third figure, is: If the minor is negative, nothing is concluded. For then the major must be affirmative, and its predicate will not be distributed, whereas it will be distributed in the conclusion. For example: “Every man is an animal; no man is a horse; therefore no horse is an animal.” Here “animal” is distributed in the conclusion but not in the premises.

Once these rules are seen, you can easily reject all useless moods in each figure. In the first figure, out of sixteen combinations, all four particular combinations and all four purely negative ones must be rejected by the first two general rules. Next, all those with a particular major and those with a negative minor must be rejected. Thus only four moods remain, in which either both premises are affirmative and the major universal, or one premise is negative and the minor affirmative. These are Barbara, Celarent, Darii, and Ferio. The vowels in those words, as we said in the preceding book, chapter on Conversion,¹ signify: A, a universal affirmative; E, a universal negative; I, a particular affirmative; O, a particular negative.

Nevertheless, since, as we said, the first two particular rules for the first figure are required only for direct, not indirect, conclusions, other indirectly concluding moods can be admitted in the first figure in which either the major is particular or the minor negative, provided they conclude indirectly, so that the major extreme is not predicated of the minor in the conclusion but is made subject. There are five such moods, named by these five words: Baralipton, Celantes, Dabitis, Fapesmo, Frisesomorum. In all of them an indirect conclusion is preserved, namely, the minor extreme is predicated of the major; yet not all of them have a particular major or negative minor, though either may occur. An example of a syllogism concluding directly in the first figure is: “Every animal is living; every man is an animal; therefore every man is living.” An example of one concluding indirectly, in Celantes, is: “No stone is an animal; every animal is sensitive; therefore no stone is sensitive.”

Likewise, in the second figure twelve of the sixteen possible combinations are excluded. Eight are excluded by the first two general rules, namely, those with purely particular premises and those with purely negative premises. The second and third rules also exclude the combination in which both premises are affirmative, and the combination in which the major is particular. Each of these combinations can occur twice: both premises affirmative with the first particular and second universal, or conversely; and both premises particular with the first affirmative and second negative, or conversely. Thus only four useful moods remain: Cesare, Camestres, Festino, and Baroco. An example is Cesare: “No animal is a stone; every marble is a stone; therefore no marble is an animal.”

Finally, in the third figure ten combinations are excluded: eight by the general rules. By the fourth rule, if the minor is negative, nothing is concluded, whether the first premise is particular and the second universal or conversely; this excludes two more combinations. Thus six useful moods remain: Darapti, Felapton, Disamis, Datisi, Bocardo, and Ferison.

Thus all the useful moods are nineteen, contained in the verses cited above.²

Note that in this third figure the first two moods, Darapti and Felapton, have both premises universal but conclude particularly, not universally. If they concluded universally, they would argue from an undistributed term to a distributed one, contrary to the third rule, as is clear in this Darapti syllogism: “Every animal is sensible; every animal is a substance; therefore every substance is sensible.” Here “substance” is distributed in the conclusion but not in the premises.

The twofold power of syllogisms is commonly assigned, although Aristotle in Prior Analytics 2¹ assigned six; for the present, however, it suffices for us to explain the two principal ones. The first is the power of concluding several things. The second is the power of reducing imperfect syllogisms to perfect ones.

The power of concluding several things is nothing other than this: by the very fact that some consequence or conclusion is concluded, one can also conclude what follows from that consequence; for example, if a universal affirmative or negative is concluded, one can also conclude the particular contained under it. Similarly, one can conclude the converse of one’s conclusion and its equivalent, and universally whatever follows from such a conclusion. All this is presupposed for ostensive reduction, as we shall say at once.

The power of reducing imperfect syllogisms to perfect ones consists in this: those syllogisms in which “being said of all” and “being said of none”—the regulative principles of the syllogism, as we shall say below in ch. 10—do not appear so evidently and perfectly are reduced to, and proved through, those in which these principles are preserved perfectly and evidently. The most perfect modes of concluding are the first four of the first figure, namely Barbara, Celarent, Darii, and Ferio. For all conclusions that can be inferred are either universal affirmatives, universal negatives, particular affirmatives, or particular negatives. The universal affirmative is inferred most perfectly in Barbara, the universal negative in Celarent, the particular affirmative in Darii, and the particular negative in Ferio, as we shall show below in ch. 10.

Therefore all the other modes, whether concluding indirectly or directly, can be reduced to these four and proved through them. Reduction is twofold: one kind is ostensive, the other is by impossibility. OSTENSIVE reduction uses two principles, namely the conversion of a proposition and transposition, that is, changing the major into the minor or conversely. For this reduction shows only that the premises can be arranged in a clearer way, so that the regulation “being said of all” and “being said of none” becomes more evident there. The whole procedure is grounded in the principle stated above: “Whatever follows from the consequent follows from its antecedent.”

REDUCTION BY IMPOSSIBILITY proves its case by deriving two contradictories as simultaneously true, which is impossible. For this purpose it assumes the contradictory of the proposition denied and, together with one proposition that has been granted, infers the contradictory of the other proposition granted, and consequently infers two contradictories as simultaneously true. It is grounded in this principle: “In a good consequence, from the contradictory of the consequent follows the contradictory of the antecedent.”

By this reduction by impossibility all the imperfect modes can be reduced to the perfect ones, because in every useful mode in which the consequence is denied, one must grant the contradictory of what is denied; from that granted contradictory and the other granted premise follows the contradictory of the other. Nevertheless, two modes in particular are said to be reduced by impossibility, namely Baroco and Bocardo, because they cannot be reduced in another way.

We shall give examples of both kinds of reduction. For example, I wish to reduce to the perfect mode Ferio the imperfect mode Fapesmo of the same first figure. I form a syllogism in Fapesmo as follows: “Every body is quantified; no soul is a body; therefore some quantified thing is not a soul.” I reduce it to Ferio by transposing the premises—that is, putting the major in place of the minor and the minor in place of the major—and by converting the affirmative proposition by accident and the negative simply. I thus form the syllogism: “No body is a soul; some quantified thing is a body; therefore some quantified thing is not a soul.” I wish to reduce Darapti, the first mode of the third figure, to Darii. I form the syllogism in Darapti thus: “Every animal is a substance; every animal is sentient; therefore some sentient thing is a substance.” I reduce it to Darii by converting the minor by accident, and say: “Every animal is a substance; some sentient thing is an animal; therefore some sentient thing is a substance.”

Likewise, for reduction by impossibility, I give an example in Baroco, the fourth mode of the second figure, which I reduce by impossibility to Barbara. I form a syllogism in Baroco thus: “Every animal is sentient; some stone is not sentient; therefore some stone is not an animal.” If you deny the consequence, I assume its contradictory and say: therefore the contradictory is true, namely, “Every stone is an animal.” Once this is granted, I take the major of the first syllogism, which had been granted, and this contradictory as the minor, and infer the contradictory of the granted minor. Thus I compel you to grant two contradictories in Barbara as follows: “Every animal is sentient; every stone is an animal; therefore every stone is sentient.” But you had granted that some stone is not sentient; therefore you grant two contradictories.

Likewise, I wish to reduce Bocardo, the fifth mode of the third figure, to Barbara by impossibility. I form a syllogism in Bocardo: “Some virtue is not natural; every virtue is a habit; therefore some habit is not natural.” If, after granting the major and minor, you deny the consequence, I take the contradictory of what was denied and say: therefore the contradictory is true, namely, “Every habit is natural.” I take this as the major while retaining the granted minor, and form a syllogism in Barbara that infers the contradictory of the major thus: “Every habit is natural; every virtue is a habit; therefore every virtue is natural”—which is the contradictory of the major, “Some virtue is not natural.”

To help you remember easily which modes are reduced to which others, and how, observe the device by which the verses given above—Barbara, Celarent, and so on—were formed. The first four begin with the initial consonants B, C, D, F; and all the other modes, whether in the first, second, or third figure, begin with one of these letters to signify that each mode is reduced ostensively to the mode beginning with the same letter. Thus Baralipton is reduced ostensively to Barbara. But Bocardo and Baroco are not reduced ostensively to Barbara, but by impossibility, because one must assume the contradictory of the conclusion, which is a universal affirmative, in order to put it into Barbara. This is reduction by impossibility, namely, by assuming the contradictory of the conclusion. Celantes, Cesare, and Camestres are reduced ostensively to Celarent. Dabitis, Darapti, Disamis, and Datisi are reduced ostensively to Darii. Fapesmo, Frisesomorum, Festino, Felapton, and Ferison are reduced to Ferio. Thus, from the letter with which each expression begins, one must reduce ostensively to the corresponding expression beginning with the same letter in one of the first four modes.

You will also find other consonants in these expressions, deliberately placed because they indicate either a conversion of a proposition to be made or a transposition of the premises. These are the three letters P, S, and M. Wherever you find the letter S, it signifies that the proposition is to be converted simply; wherever you find P, that it is to be converted by accident; wherever you find M, that the premises are to be transposed. For example, in Fapesmo you will find the letter P after the first A: it signifies that the universal affirmative is to be converted by accident. After the E you will find the letter S, signifying that the universal negative is to be converted simply. The letter M signifies that the premises are to be transposed, namely, the major into the minor. And so it is reduced ostensively to Ferio. You can readily observe the same in the other cases. When you find the letter C in the middle of an expression, it signifies that the mode cannot be reduced ostensively, but only by impossibility; this occurs only in Baroco and Bocardo.

In the ostensive conversion of Baralipton in particular, note that it yields an indirect particular affirmative, and that it has the letter P not beside either premise but beside the conclusion. This indicates that it is reduced to Barbara by retaining the very same premises of Baralipton and immediately and directly inferring the conclusion of Barbara, which is a universal affirmative. Since this universal affirmative is converted by accident through a good consequence, whoever immediately infers it in Barbara can also infer, by adding conversion by accident, the indirect particular affirmative that was the conclusion in Baralipton. Thus Baralipton is not reduced by converting or transposing it into Barbara, but rather Baralipton is derived from Barbara by conversion by accident of its conclusion, which is a universal affirmative. For example, this syllogism is in Baralipton: “Every animal is a substance; every human being is an animal; therefore some substance is a human being.” I reduce it to Barbara by proving that such a conclusion is inferred mediately from Barbara, as follows: “Every animal is a substance; every human being is an animal; therefore every human being is a substance.” And by conversion by accident: “Therefore some substance is a human being,” which was the conclusion of Baralipton.

As for the reduction of syllogisms by impossibility, it is certain that all the imperfect modes, whether of the first, second, or third figure, can be reduced by impossibility to one of the first four. But they are not all reduced in the same way as in ostensive reduction, where each expression is reduced to a mode among the first four that begins with the same letter—except Baroco and Bocardo, which are reduced to Barbara by taking the contradictory of the inferred conclusion, which is a universal affirmative. To reduce all the other modes by impossibility, always attend to the inferred conclusion, take its contradictory, put it with the other premise in place of one premise, and infer the contradictory or contrary of the other granted premise. In this way you derive an impossibility, namely, the granting of two contradictories or contraries.

To do this quickly and readily, note that in the five imperfect modes of the first figure—Baralipton, Celantes, Dabitis, Fapesmo, and Frisesomorum—the contradictory of the conclusion must be taken and put in place of the major, the granted major put in place of the minor, and the contrary of the granted minor inferred. But make an exception for Celantes: because the contradictory of its conclusion is particular, it must be put in place of the minor, and the minor of Celantes put in place of the major; thus in Darii one infers the contradictory of the granted major. In the modes of the second figure, if one forms a syllogism and wishes to reduce it by impossibility, take the contradictory of the conclusion, put it in place of the minor, and with the major of the syllogism infer the contrary or contradictory of the granted minor. In the modes of the third figure, take the contradictory of the conclusion, put it in place of the major, and with the same minor infer the contradictory or contrary of the granted major.

Although these rules alone, in their actual application, lead to the reduction by impossibility of each syllogism to its corresponding mode among the four perfect ones, a verse has nonetheless been devised to correspond to all the imperfect modes and to contain the four vowels A, E, I, O. A signifies reduction by impossibility to Barbara, E to Celarent, I to Darii, and O to Ferio. The verse is this:

Febifecaxis obit terras spherámque quotannis.

Others set down these expressions in this order: Nesciebatis, Odiebam, Laetare, Romanis; they amount to the same thing.

Febifecaxis has five vowels, corresponding to the five imperfect modes of the first figure. The first vowel, E, signifies that the first imperfect mode, namely Baralipton, is reduced to Celarent. The second vowel, I, signifies that the second mode, Celantes, is reduced to Darii. The third vowel, E, that the third mode, Dabitis, is reduced to Celarent. The fourth vowel, A, that the fourth mode, Fapesmo, is reduced to Barbara, and so on. Obit terras corresponds to the four modes of the second figure, and spherámque quotannis to the six modes of the third figure. The first vowel, O, signifies that the first mode of the second figure is reduced to Ferio. The second, I, that the second is reduced to Darii, and so on.

An expository syllogism is said to be one “which has a singular term for its middle.” It is called expository, as St Thomas says in Opusc. 47,¹ because it is a kind of sensible resolution that, as it were, exposes the thing to the senses, as when you say: “Peter is white, Peter is running, therefore what is running is white.” And for this reason nearly all expository syllogisms are made in the third figure, because in it the middle is the subject in both premises, and a singular term is more suitably made subject than predicate. Yet it can also be made in the other figures. Since, however, the other extremes are common terms and not singular, all the conditions observed in other syllogisms must be observed in them, and the defects that may be found must be avoided. But since the middle itself is not a common term but a singular one, it can neither be distributed nor universalized; rather, it must be perfectly singularized. Hence, if it has not been perfectly and completely singularized, the syllogism will be defective. This must be observed especially with divine terms, in which the term “God,” “divine essence,” and others pertaining to the absolutes are singular in such a way that they are equivalently common, because in reality they are communicated to several persons. Hence they are not suited to an expository syllogism, as this is invalid: “This God is the Father, this God is the Son, therefore the Son is the Father,” because the middle is not perfectly singularized. Therefore one must use it as a common term and, consequently, if it is posited as the middle, it must be perfectly distributed, by saying: “Everything that is God is the Father; everything that is God is the Son,” etc.; but the premises are false.

Second, it is required that the middle, which is a singular term, be taken for the same individual; otherwise there is an equivocation of the middle, or a different sense, or the argument proceeds with four terms. Thus this is invalid: “This animal is a horse” (pointing to a horse), “this animal is a human being” (pointing to Peter), “therefore a human being is a horse,” because the middle is not taken for the same thing.

Third, when expository syllogisms are made in the third figure, care must be taken that the minor premise not be negative, in accordance with the rules of the third figure. For this reason the following is invalid: “Peter is a human being, Peter is not Paul, therefore Paul is not a human being.” Conversely, this is valid: “Peter is not Paul, Peter is a human being, therefore some human being is not Paul,” and it is in Ferison.

In the past, it was considered very difficult to understand perfectly and clearly the discovery of a middle term. Now this matter is readily explained—not, as some think, by merely assigning as the middle term that which is the cause and rationale for the predicate belonging to the subject, or, as others put it, by taking as the middle term in order to conclude an affirmative conclusion that with which the things are identified, and in order to conclude a negative one that with which one extreme is identified and from which the other is separated. For what we are asking is precisely what that is in which the extremes are identified or one of them is separated, and what is the rationale and cause for the predicate belonging to the subject. One may consult the brief work of St. Thomas on this, the fiftieth among his opuscula.¹

We should therefore observe that every conclusion to be inferred is either a universal affirmative, a universal negative, a particular affirmative, or a particular negative. Since the term taken as the middle must be communicated and united to the extremes, or separated from one of them, so that the extremes themselves may be united or separated—and since the extremes capable of being united with one another are the subject and predicate in the conclusion—we must take as a middle term that unites or divides them either something that is a higher or lower predicate, something equal, or something incompatible or extraneous. The ancients called a higher or equal predicate a consequent term, because it follows from another: for example, from “man” one infers “animal,” which is higher, and “capable of laughter,” which is equal, since both are contained in man. They called a lower predicate an antecedent term, because it entails but is not entailed by what is higher: “man” does not follow from “animal,” but the reverse does. They called an incompatible predicate an extraneous term.

Therefore, to infer a universal affirmative, which is concluded only in Barbara, we take as middle term something consequent upon the subject, whether the predicate is higher or equal, and antecedent to the predicate, that is, something lower than it. For example, if you want to prove that every man is an animal, take “capable of laughter” as the middle term: it follows from “man” and entails “animal,” or is antecedent to “animal.” Thus: “Everything capable of laughter is an animal; every man is capable of laughter; therefore every man is an animal.”

To infer a universal negative, take as middle term what follows from the subject and is incompatible with the predicate. This is done in Celarent, the second mode of the first figure, and in Cesare, the first mode of the second. In Camestres, however, where the minor premise is negative, the middle must be sought in the reverse way: something incompatible with the subject and consequent upon the predicate, since it must be removed from the minor extreme and follow from the major. For example, to conclude that no man is a stone, “rational” or “capable of laughter” is a good middle term in Celarent and Cesare, since it follows from man and is incompatible with stone; in Camestres, the middle is “insentient,” which belongs to stone and is incompatible with man.

To infer a particular affirmative in the first figure, as in Darii, a suitable middle term is one that follows from the subject and is antecedent to the predicate. Thus, to infer that some animal is a substance, a suitable middle term is “sensible,” which follows from animal and entails substance. If the conclusion is inferred in the third figure (for a particular affirmative is not inferred in the second), as in Darapti or Disamis, the suitable middle term is one that is antecedent—that is, lower than both the predicate and subject—and that must be made subject in both premises. For example, “man” serves to infer that some animal is a substance, because man is an animal and is a substance.

Finally, to infer a particular negative, which can be concluded in all three figures, if it is inferred in the first figure, take as middle term something that follows from the subject and is incompatible with the predicate. For example, to conclude that some animal is not a stone, I take “man” as the middle term, saying: “No man is a stone; some animal is a man.” Or, if I want to conclude “Some animal is not a horse,” I take “rational,” which follows from some animal and is incompatible with horse, and say: “Nothing rational is a horse; some animal is rational,” etc. If a particular negative is inferred in the second figure, in the third mode, Festino, the same middle term is taken as in the first figure. In Baroco, however, the middle term taken is something extraneous to the subject and consequent upon the predicate. Thus, in the conclusion given above, “Some animal is not a stone,” we take “insentient” as the middle term, which is incompatible with animal and follows from stone. Finally, if the conclusion is inferred in the third figure, take as middle term something antecedent—that is, lower than the subject—and extraneous or incompatible with the predicate; for the conclusion given above, take “rational,” which is lower than animal and incompatible with stone.

The whole account of the syllogistic art rests on three principles known per se:

The first is: “Whatever things are identical with one third are identical with one another.” The force of discursive proof rests on this. Since reasoning is ordered to inferring a conclusion in which two extremes are joined through the force of some middle term, by which their union is shown, it is necessary that things be united with one another because they are united in some middle term. The opposite holds in like manner when a negative conclusion is inferred, in which one thing is denied and separated from another: one of them must be denied and separated from a third; “for things that are identical with one another, if one of them is distinguished from some third thing, the other too is distinguished and separated from that third.” Therefore, from the denial of one thing with some third thing follows the denial of what is identified with it in relation to that same third thing.

Note that St. Thomas, Summa theologiae I, q. 28, art. 3, ad 1, explains that principle as follows: “Things that are identical with one third are identical with one another, if they are identical with one third in reality and in account.” This does not mean that they are distinguished neither in reality nor in account. Rather, “in reality and in account” must be construed with “with one third,” so that they are identical with one third that is one in reality and in account, that is, not virtually or formally multiple. For in that case something can be identical with one extreme under one formal aspect and with another under a different aspect, and consequently it does not follow that those extremes are identical with one another. Indeed, many paralogisms arise from this, because the middle term is not perfectly and simply one, but multiple, at least virtually or formally. Hence, from the union of such a middle term with the extremes, unless it is under the same formal aspect, no union of the extremes with one another follows. For example, this does not follow: “Whiteness is a relation; whiteness is intense; therefore a relation is intense.” Relation is identified with whiteness in one way, intensity in another, and so they are not identified with one another. Nor does this hold in divine terms: “The Father is the divine essence; the Son is the divine essence; therefore the Son is the Father.” The divine essence in which they are united is, because of its infinite perfection, one most simple reality and virtually multiple; it is modified under different concepts by the Persons, each inadequately.

The second principle is “Dici de omni,” that is: “Whatever is said universally of a subject is said of everything contained under that subject”; for example, if it is affirmed universally of man that he is capable of laughter, it must be affirmed of every individual man.

The third principle is “Dici de nullo,” that is: “Whatever is denied universally of some subject is also denied of everything contained under that subject”; for example, if you deny universally of man that he is a stone, you must deny this of every individual contained under man.

These principles are known per se because the nature of a universal consists in being said of all the things with respect to which it is universal. These principles do not serve an expositional syllogism, which has a singular middle term, but a syllogism with a common middle term. In the latter, reasoning must proceed through the force of some distributed term, because if none is distributed, the argument proceeds from pure particulars, which is a defective form of consequence: particular propositions do not provide a middle term that fully embraces and contains what is to be inferred and united in the conclusion. Once a term is distributed, however, whatever is said or denied of it is also said or denied of what is contained under it through that distribution; thus what was denied or affirmed of the distributed term in the antecedent is rightly concluded of what is contained under it. These two principles therefore serve to establish the necessity of inference, not by reasoning from pure particulars but by distribution.

Hence those syllogisms that are immediately regulated by these two principles are called perfect, because there the application of what is said or denied universally to what is contained under those universals is made evident. Other modes, by contrast, are called imperfect: they are not immediately regulated by these principles, nor is that application made so evident in them; rather, they are regulated mediately, and are therefore reduced to those perfect modes that are regulated immediately and evidently per se by such principles. The four modes of the first figure are regulated immediately by these principles, because in them the middle term is perfectly distributed in the major premise, where it stands universally as subject. When it is then predicated in the minor premise, this very fact makes it evident that what it is predicated of is contained under its universality, and consequently that what was predicated or denied of the middle term, posited universally in the major, also belongs to that thing. This is an immediate manifestation of “Dici de omni” and “Dici de nullo,” because it is shown immediately and per se that something is contained under or denied of a subject taken universally, and that subject is the middle term.¹ See below, question 8, final article.

But for all this to be understood more clearly, and for all the defects of bad consequence and syllogisms to be recognized at the same time, we must treat the goodness and defects of consequences both in general and in particular.

In regulating the goodness of any consequence, there is one most universal principle from which the others are derived and to which they are reduced, namely: “In a good consequence, it cannot be that the antecedent is true and the consequent false; rather, if the antecedent is true, so is the consequent.” This principle is immediately reduced to the highest of all principles: “Everything either is or is not,” or “It is impossible for the same thing both to be and not to be at the same time.” For if the antecedent is true, then things are already as it signifies. But if the consequent is false, it draws the antecedent after it by reason of the connection that the truth of the consequent has with the truth of the antecedent. Therefore, to posit a true antecedent and a false consequent is the same as positing a true antecedent and something false on its part, because the consequent is as it were a part of, and something connected with, the antecedent, and the consequence itself is a kind of connection. But what is false in part is not true absolutely. Therefore, if a true antecedent and a false consequent are given, a true antecedent and a non-true one are given, and these are contradictories.

The goodness of consequence is therefore grounded in these principles. Hence the whole effort in assigning defects of consequence consists in producing a true antecedent and a false consequent. Thus, whatever more readily tends to falsify the consequent than the antecedent contributes to a bad consequence; whatever more readily contributes to the truth of the consequent than of the antecedent builds a good consequence. The reason is that the antecedent must contain in itself the truth of the consequent, since the latter is inferred from it. Hence Aristotle said in Posterior Analytics I,¹ “the premises must be more true than the conclusion,” that is, more certain. More, then, must be required for the truth of the antecedent than for the truth of the consequent, since the antecedent is truer than the consequent. Therefore, the antecedent must always be harder to make true than the consequent; for that is harder to obtain which requires more. Since a consequence is nothing other than an inferential connection from one thing to another, and the truth of an antecedent is not well connected with the falsity of what is inferred, the connection itself, or consequence, becomes bad whenever the antecedent is true and the consequent false.

From this principle several rules that dialecticians give concerning the goodness of consequence are made clear. They are usually six, which we reduce to three.

First: If the antecedent is true, so is the consequent; and conversely, if the consequent is false, so is the antecedent. This is the very principle we have stated.

Second: If the antecedent is possible, so is the consequent; and conversely, if the consequent is impossible, so is the antecedent. This follows from the first rule: if the antecedent is possible, then it can be true, and therefore its consequent can also be true. For if it were impossible for the consequent to be true, it could not be true in any case. But since the antecedent is possible, it can be true in some case; then, in some case, the antecedent would be true and the consequent false.

Third: If the antecedent is necessary, so is the consequent; and conversely, if the consequent is contingent, so is the antecedent. This too follows from the same principle: if the antecedent is necessary, it is always true, and therefore the consequent must always be true as well; otherwise there could be a case in which the antecedent is true and the consequent false. Hence, if the consequent is contingent and can be false, the antecedent cannot be necessary, since what is necessary is always true.

Understand, however, that a necessary consequent always follows from a necessary antecedent, but it is not required that the consequent be necessary in every respect. It suffices that it be necessary under the respect and account in which the consequent is contained in the antecedent, and insofar as it is inferred from the antecedent, though not under some other account. For example, in the consequence “God knows that Peter will sin; therefore he will sin,” the consequent is contingent in itself and in its relation to the proximate cause, but it is infallible as something known by God, and consequently as inferred from the antecedent. This is because God knows not only that Peter will sin, but that he will sin freely and contingently. Thus, from that antecedent it is inferred both that he will sin contingently and that it is infallible that he will sin contingently.

Finally, two other rules follow from that principle, and these too serve the goodness of consequence.

First: Whatever follows from the consequent of a good consequence follows from its antecedent. In other words, as is commonly said: “The consequence holds from the first to the last.” This is self-evident.

Second: Whatever is incompatible with the consequent in a good consequence is also incompatible with the antecedent. In other words, if the consequent follows from the antecedent, the opposite of the consequent follows from the opposite of the antecedent. The reason is that if the opposite of the consequent does not entail the opposite of the antecedent, then the opposite of the consequent could stand together with this antecedent, since it does not entail the opposite of the antecedent. But the consequent posited in the consequence is false, given that it was true—that is, inferred from a true antecedent. Therefore, the falsity of the consequent could stand together with the antecedent, and thus a true antecedent and a false consequent would be given. We very often use this rule to establish the goodness of a consequence by proceeding from the contradictory of the consequent to the contradictory of the antecedent. We also use it in reducing syllogisms by impossibility; and counterexamples that serve to show the defect of a consequence also serve in the case where we proceed from the opposite to the opposite.