THIRD BOOK: On Syllogism.
First Chapter: Definition and division of Syllogism
§76.
Syllogism is a speech composed of propositions, [of such a nature, that] if they are admitted, there follows, from them, taken in themselves, another speech.
§77.
A Syllogism is [called] interpellative (hypothetical), if the conclusion itself or its contrary is actually mentioned in it, as “if this be a body, it is spacial.” Here the very conclusion is mentioned in it. And if we say “but it is not spacial” it follows that it is not a body. In this instance the contradictory is mentioned in it. A Syllogism is called conjugate if it is not like the preceding, e.g. “every body is composed of parts, every thing composed of parts is temporal,” it follows “every body is temporal.” Neither the conclusion nor its opposite are actually mentioned in it.
§78.
The subject of the question is called minor [term,] and its predicate is called major, and a proposition which forms part of a Syllogism is called premiss, and the premiss which contains the minor [term] is called minor [premiss], and that which contains the major [term] major [premiss], and the repeated intermediate termis called themiddle term, the conjugation (connexion) between the minor and major premisses is called the mood, and the shape resulting from the manner in which the middle term is placed in regard the other two terms is called figure. There are four figures: in the first figure the middle term is the predicate in the minor premiss and the subject in the major premiss; in the second figure it is the predicate in both; in the third figure it is the subject in both; and in the fourth figure it is the subject in the minor premiss and the predicate in the major premiss.
§79.
In the first figure theminor premissmust be affirmative, for else the minor term is not contained in the middle term. The major premiss must be a universal (proposition), else it may be that some [things] predicated by themajor termare not the same which are predicated of theminor term.
It [this figure] admits of four conclusive moods. First, from two affirmative universals an affirmative universal conclusion is derived, as “every C is B; and every B is A; therefore every C is A.”
Secondly. - From two universals, the minor premiss being affirmative and the major negative, a universal negative conclusion results as every C is B, no B is A, therefore no C is A.
Thirdly. - From two affirmatives, the minor premiss being a particular, results a particular affirmative conclusion, as some C is B, every B is A; therefore some C is A.
Fourthly. - From an affirmative particular minor premiss and a negative universal major premiss results a negative particular conclusion, as some C is B, no B is A; therefore some C is not A.
The conclusions of this figure are self-evident.
§80.
In the second figure the two premisses must be different in quale (one must be affirmative and the other negative;) and the major premiss must be a universal: else (if either of these two conditions is not fulfilled) we get a non-identity which warrants no inference, i.e. from correct premisses, sometimes, you obtain a conclusion which you are able to affirm, and, at another, one which you are obliged to deny.
The conclusive moods are again four. Firstly, - From two universals, the minor premiss being affirmative, a negative universal conclusion is obtained, e.g., every C [man] is B [animal;] no A [stone] is B [animal;] therefore no C [man] is A [a stone.] This can be shown by reductio ad impossible, i.e., the contradictory of the conclusion is attached to the major premiss, producing the contradictory of the minor premiss as conclusion, [e.g., if you deny that no man is a stone, let us suppose, some men are stones; under this supposition we have: some men are stones; no stone is an animal; therefore some men are not animals - this is contrary to the admission, that everyman is an animal.] [It can also be demonstrated] by conversion of the major premiss, [e.g., every animal is a not-stone,] whereby it is reduced to the first figure.
Secondly. - From two universals, the major premiss being affirmative a negative universal conclusion is obtained, e.g., no C is B; and every A is B; therefore no C is A. This can be demonstrated by reductio ad impossibile; and also by converting the minor premiss, putting it into the place of the major [taking the major as the minor and converting of the conclusion].
Thirdly. - From an affirmative particular minor premiss and negative universal major a negative particular conclusion is deduced, as: some C [men] are B [fair]; no A [negro] is B; therefore some C are not A. This can be demonstrated by reductio ad impossibile and conversion of the major whereby it is reduced to the first figure. [It can also be demonstrated by supposition:] let us suppose for this purpose that the exact subject of the particular proposition be D [Caucasians], then every D is B, no A is B; therefore no D is A. Hence we say, some C is D; and no D is A; therefore some C is not A.
Fourthly. - From a negative particular minor and an affirmative universal major a negative particular conclusion is deduced, as: some C is not B; and every A is B; therefore some C is not A. It can be demonstrated by reductio ad impossibile; and by supposition, if the negative be compound, (i.e. not indivisible, otherwise the subject might have no assignable significates; see §46.)
§81.
In the third figure the minor must be affirmative, else there will be non-identity, and one of the two premisses must be universal, else some of the things of which the minor term is predicated may be different from some of the things of which the major is predicated, and consequently it leads to no result.
The conclusive moods of this figure are six: First. - From two universal affirmative premisses an affirmative particular conclusion is derived, as, every B is C; and every B is A; therefore some C is A. It can be demonstrated by reductio ad impossibile, i.e. the contradictory of the conclusion is [taken as major premiss and] added to the minor premiss to deduce the contradictory of the major; and [it can also be demonstrated by reduction to the first figure,] which is effected by the conversion of the minor.
Secondly. - From two universals the minor premiss being negative, a negative particular conclusion is deduced, as: every C is B, and no B is A; therefore some C is not A. [It can be demonstrated by reductio ad impossibile and] by conversion of the minor premiss.
Thirdly. - From two affirmative premisses, the major being a universal, an affirmative particular conclusion is deduced, as, some B is C, and every B is A; therefore some C is A. [This can be demonstrated] by reductio ad impossibile and by conversion of the minor, and by supposing the [exact] subject of the particular premiss to be D. Then: every D is B, and every B is A; therefore every D is A, then we say: D is C and every D is A; therefore some C is A; and this was to be demonstrated.
Fourthly. - From an affirmative particular minor premiss and a negative universal major a particular negative conclusion is deduced, as some B is C, and no B is A; therefore some C is not A. This can be demonstrated by reductio ad impossibile and by conversion of the minor and by supposition.
Fifthly. - From two affirmative premisses the minor being universal an affirmative particular is derived, as, every B is C, and some B is A; therefore some C is A. This can be shown by reductio ad impossibile and by using the converted major as minor and then converting the conclusion. It can also be shown by supposition.
Sixthly. - From an affirmative universal minor premiss and a negative particular major a negative particular conclusion is derived, as, every B is C, and some B is not A, therefore some C is not A. This can be shown by reductio ad impossibile and by supposition if the negative be compound [see §46].
§82.
Fourth figure. In regard to the quality, and quantity, it is necessary that the two premisses be affirmative and the minor premiss a universal; or the two premisses must differ from each other in quality and one of them must be a universal. If this be not the case there will be non-identity which renders it impossible to come to a conclusion. This figure has eight conclusive moods: -
First. - From two affirmative universal premisses an affirmative particular conclusion is deduced, as, every B is C, and every A is B; therefore some C is A. It is demonstrated by conversion of the arrangement which gives a converted conclusion, [i.e. every A is B, and every B is C; therefore every A is C.]
Secondly. - From two affirmative premisses, the major being a particular, follows an affirmative particular conclusion, as, every B is C, and some A is B; therefore some Cis A; the demonstration is the same as in the preceding mood.
Thirdly. - From two universal premisses, the minor being negative, follows a negative universal conclusion, as, no B is C, and every A is B; and therefore no C is A. The demonstration is the same as above.
Fourthly. - From two universal premisses, the minor being affirmative, follows a negative particular conclusion, as, every B is C, and no A is B; therefore some C is not A. It is demonstrated by the conversion of the two premisses; [viz. some C is B, and no B is A; therefore some C is not A.]
Fifthly. - From an affirmative particular minor and a negative universal major follows a negative particular conclusion, as, some B is C, and no A is B; therefore some C is not A. It is demonstrated like the preceding.
Sixthly. - From a negative particular minor and an affirmative universal major follows a negative particular conclusion, as, some B is not C, and every A is B; therefore some C is not A. By conversion of the minor it is reduced to the second [figure].
Seventhly. - From an affirmative universal minor and a negative particular major follows a negative particular conclusion, as, every B is C, and some A is not B; therefore some C is not A. By conversion of the major it is reduced to the third figure.
Eighthly. - From a negative universal minor and an affirmative particular major follows a negative particular conclusion, as, no B is C, and some A is B; therefore some C is not A. It is demonstrated by conversion of the arrangement whereby a converted conclusion is arrived to; the first five moods can also be demonstrated by reductio ad impossibile, that is to say, the contradictory of the conclusion is added to one of the two premisses in order that a conclusion may be come to, which is the converse of the contradictory of the other premiss [e.g., supposing it be not true that some C is A, then it must be true that no C is A; then let us take this as the major premiss and add, every B is C, as the minor; and it follows, no B is A, and by conversion no A is B]. The second and fifth mood can be demonstrated by supposition. We employ supposition for demonstrating the second mood, and the fifth can then be treated in the same manner. Let some individua of A be D, then it follows that every D is A and every D is B, therefore we say, every B is C, and every D is B, and some C is D, and every D is A, and some C is A; this was to be demonstrated.
§83.
The ancients considered only the first five moods of this figure as conclusive and they held that owing to non-identity in the conclusion the remaining three were not conclusive, this is the case if both premisses are simple, we therefore make it a condition that the negative premiss be of one of the two kinds of peculiar propositions [i.e. the conditioned or the conventional]. This obviates non-identity.
Third Section: Conjugate Syllogism containing hypothetical premisses
§87.
These are of five kinds. - The first is composed of conjunctive premisses.
The norm of this class is a syllogism in which the two premisses have a complete part (term) in common and in reference to this term syllogisms of this kind are classed under the four figures. If the common term is the consequent in the minor premiss and the antecedent in the major, we have the first figure. If it is the consequent in both we have the second. If it is the antecedent in both we have the third figure. If it is the antecedent in the minor premiss and the consequent in the major we have the fourth figure. The conditions of arriving at conclusions, the number ofmoods and the quantity and quality of the conclusion of every figure are exactly the same as in the categorical. Example of the first mood: whenever A is B, C is D, and whenever C is D, E is Z, consequently whenever A is B, E is Z.
§88.
Second kind. It is composed of two disjunctive premisses; the norm of this class is a syllogism in which the two premisses have not a complete part in common, as: invariably either every A is B or every C is D; again, either every D is E, or every D is Z, consequently, either every A is B or every C is E or every D is Z. [This conclusion is correct,] on account of the exclusiveness which there exists between the two premisses of the composition [i.e., every C is D and every D is E] and one of the other two premisses [i.e. every A is B and every E is Z].
§89.
Third kind. It is composed of a categoric and conjunctive premiss.
The norm of this class is a syllogism in which the categorical proposition is themajor and has a termin commonwith the consequent of the conjunctive [minor]. The conclusion of the syllogism is a conjunctive proposition, the antecedent of which is the antecedent of the conjunctive premiss, and the consequent is the conclusion of the composition between the consequent [in the minor] and the categorical [premiss], e.g., whenever A is B; C is D; farther D is E; therefore, whenever A is B, every C is E.
§90.
Fourth kind. It is composed of a categorical and a disjunctive premiss and it is of two descriptions. First. - The number of categorical propositions is the same as the number of disjunctions, and each categorical proposition has one term with the parts of the disjunction in common, and the composition is either identical or there is a difference of composition in the conclusion. Example of a case in which the composition is identical:
Every C is either B orD or E, and every B is T and everyD is T and every E is T; hence it follows that every C is T, because the parts of the disjunction [B, D, E] are true of that term of the categoric premiss which it has in common with the disjunctive premiss. Example in which there is a difference of composition in the conclusion, every C is either B or D or E; but every B is C and every D is T and every E is Z, hence it follows that every C is either C or T or Z, for the reasons just mentioned.
Secondly. - If there are fewer categoric propositions than there are parts of the disjunction, let us suppose there be a categorical proposition of one part and a disjunctive one of two parts, and the categoric proposition have a term in common with the latter, e.g. either, every A is T, or every C is B, but every B is D, hence it follows that either every A is T, or every C is D, on account of the exclusivenesswhich there is between the premisses of the composition and the term which they have not in common. [If there is no such exclusiveness, the conclusion is not of necessity correct.]
§91.
Fifth kind. It is composed of a conjunctive and of a disjunctive proposition, and the two premisses have either a complete part in common or an incomplete part. In either case only a syllogism in which the conjunctive proposition forms the minor and the disjunctive, the major, is conclusive. Example of the first case: Whenever A is B, C is D, but invariably either every C is D, or E is Z, hence it follows that invariably either, every A is B or E is Z. If the disjunctive proposition, [either C is D or E is Z] is incompatible, the conclusion is equally incompatible, because if a thing is incompatible with the adherent, either perpetually, or only now and then, it follows of necessity that it be also incompatible with the substrate either perpetually or now and then, (i.e. under certain circumstances;) and if the disjunctive is exclusive, the conclusion is “it happens sometimes;” for if Ais not B, then E is Z, for the contradictory of the middle term[C is D] requires the two terms [of the conclusion to be “E is Z” and “the contradictory of A is B.”] The question is demonstrated by the third figure.
Secondly. - [If the two premisses have an incomplete part in common, we say] whenever A is B, every C is D, and perpetually either, every D is E, or D is Z; if the disjunctive proposition is exclusive, the conclusion is, whenever A is B, either every C or E, or D is Z.
Fourth Section: On the Interpellative Syllogism
§92.
It is composed of two antecedents; one of the two is hypothetical and the other is an assertion that one of its two parts is or is not, and from this assertion follows that the other part is or is not. [In order that such a syllogism be conclusive] it is necessary: [First] that the hypotheticals be affirmative; [Secondly] that if the hypothetical is conjunctive, it be cogent (literally adhesive,) [and that, if it is disjunctive,] it be antagonistic; [Thirdly] that either the hypotheticals be universal or that the assertion that one of the parts is or is not be universal (i.e. that it be asserted it is or is not at all times and under all circumstances); unless the time of conjunction or disjunction is also the time regarding which it is asserted that the part is or is not, [e.g. whenever Zayd comes with Bakr in the afternoon, I receive him with honor, he did come with Bakr in the afternoon and therefore he was received by me with honor.]
If the hypothetical which forms part of the interpellative syllogism is conjunctive, from the interpellation of the antecedent follows the consequent as conclusion, and from the interpellation of the contradictory of the consequent follows the contradictory of the antecedent as conclusion. If this is not the case the adhesion is not established. The reverse is not admissible in either of the above two cases, for the consequent may be more general than the antecedent.
If the hypothetical is a veritable disjunctive proposition, [see §53] and if, in the interpellation any part, whichsoever, is asserted, there follows from it the contradictory of the other part on account of their incompatibility, but if the interpellation consists of the contradictory of any part, whichsoever, there follows from it the other part on account of their exclusiveness.
If the disjunctive hypothetical is incompatible, the conclusion is as in the first case only, (i.e. there follows from it the contradictory of one part, if the other is asserted with interpellation;) because the two parts are incompatible but not exclusive; and if the disjunctive is exclusive, the conclusion is as in the second case only, because the two parts are exclusive but not incompatible, (e.g. either Zayd is on the sea or he is not drowned; but he is not at sea therefore he is not drowned).
Fifth Section: Pendents of the Syllogism
§93.
These are four. First. - The compound syllogism (the Sorites). It is composed of several premisses, some (two) of which lead to a conclusion, which (conclusion) with another premiss leads to another conclusion, and so on until we arrive at the question. The conclusions are either connected, as every C is B, and every B is D, therefore every C is D; again every C is D and every D is A, and therefore every C is A; again every C is A and every A is E, therefore every C is E; or the conclusions are disconnected, as, every C is B and every B is D and every D is A and every A is E; therefore every C is E.
§94.
Second. - Reductio ad absurdum. The question is proved by disproving the contradictory thereof; e.g. If you deny that some C is not B, let every C be B and let every B be A. Now if this proposition (every B is A) is true, we say if you deny that some C is not B, you must allow that every C is A; but not every C is A, and therefore your assertion is absurd, and there follows not every C is B. This was to be demonstrated.
§95.
Third. - Induction is a judgment that, what is found inmost of the parts (dividing members) is universal, e.g. all animals move the lower jaw in eating because oxen, tiger, etc. move it. This does not enable us to arrive at certainty on account of the presumption, that not all are like those, as is the case (in regard to the above example) with the crocodile. [If a thing is found in all the dividing members, it is called القیاس المقسم enumeratio partium.]
§96.
Fourth. - Example. A judgment is affirmed of a particular (singular) which is applicable to another particular, because they have a meaning [see note 11] in common, e.g. the world is composed of parts and therefore, as in the case of a house, it does not exist from eternity. That the meaning which the two particulars have in common has the nature of a cause is demonstrated by the argument of “concomitancy” and of “division.” This last however does not amount to a dilemmatic judgment, such that if one part is false the other must be true, e.g. the cause of destructibility is either composition or such a thing or such a thing; the futility of the two latter assumptions is shown by reductio ad absurdum, and thereby the first is established.
Both these arguments are weak. The former because the last [of the four] parts of a complete cause togetherwith all the conditions is called the mad´ar of an effect, but it cannot be called its cause. Division forms a weak argument, because it is impossible to say that nothing else [than the parts enumerated] is the cause, and supposing it be admitted that, what the two things which are analogous, have in common, is the cause in the case cited, it does not follow that it is also the cause of the thing to be proved, for it may happen that a peculiarity of the case cited is the condition for the operations of the cause or that a peculiarity of the thing to be proved renders it impossible that the same cause should be in operation.
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Author: Najm al-Din ’Umar al-Qazwni al-Katibi
