SECOND BOOK: On propositions and rules regarding them
This book is divided into an introduction and three chapters.
INTRODUCTION
Definition of proposition and its primary division
§38.
Proposition (literally a decision) is a speech, which allows that he who utters it be told that he is true or false (right or wrong). It is called categorical, if its two extremities (terms) are resolvable into two simple [ideas], as Zayd is informed, or Zayd is not informed, [or from “the Sun is rising” follows “the day is approaching,”] and it is hypothetical, if they are not thus resolvable, [e.g. if the Sun rises day will approach].
§39.
The hypothetical [proposition] is either conjunctive (conditional), or disjunctive. It is called conjunctive, if we pronounce in it a proposition (i.e. one of the two propositions of which it consists) to be true or untrue, under the assumption that another (the other) proposition be true. [Example of an affirmative conjunctive] “if this is a man, it is an animal.” [Example of a negative conjunctive], “if this is a man, it cannot be a mineral.”
A hypothetical proposition is called disjunctive if we pronounce in it that two propositions exclude (literally deny or refute) each other, either both in [case of] truth and [in case of] falsity or in one of the two only, or that their mutual exclusion is denied, e.g. “this number is either even or odd.” “That this man is either a writer or a negro, is not admissible.”
First Section: On the categorical (proposition)
First Inquiry: Its parts and kinds
§40.
The categorical proposition consists of three parts: the part on which judgment is passed, - which is called subject; the one by which judgment is passed, - which is called predicate; the relation between the two, showing the bearing of the predicate to the subject - which is called judicial relation; and the word which expresses it is called copula, as “is” in the sentence “Zayd is informed.” Such a proposition is called ternary.
In some cases, which are very easily intelligible, the copula is omitted, and the proposition is called binary.
§41.
If the relation is of such a description that you can say that the subject is in agreement [with the predicate], the proposition is called affirmative, as man is an animal; and if it is of such a description that you can say that it is not in agreement it is negative, as a man is not a horse.
§42.
If the subject of a categorical proposition is a definite individuum, it (the proposition) is called peculiarized or individual (singular). If the subject is a universal, and if the quantity of the singulars (or individua) of which the judgment is true is shown in it, the word expressing the quantity is called wall and the proposition is called fenced or walled-in. It is of four kinds: if it is shown in it that the judgment [applies] to all the singulars, it is [called] an universal [categorical proposition]. This again is either affirmative [or negative: in the affirmative] the wall is “every one,” e.g. every fire is hot. In the negative the wall is “no,” “none,” “not one,” e.g. every fire is hot. In the negative the wall is “no,” “none,” “not one,” e.g. no man is a mineral. If it is shown in it that the judgment [applies] to some things, it is particular, and [again it is] either affirmative, and [in this case] the wall is “some” “one,” e.g. some animals are men, or one animal is a man; or it is negative and the wall is “not all,” “some (are) not,” e.g. not all animals are men, or some animals are not men.
§43.
If the quantity of the singulars is not shown in it and if we can neither say that it is a universal nor that it is a particular proposition it is called a physical proposition, e.g. “animal” is the genus and “homo” is the species. But if we can say it is universal or particular [but it is not stated] it is called ambiguous, e.g. man is at a loss, or man is not at a loss. Such a proposition is virtually a particular proposition, for if it predicates that man is at a loss, it predicates that some men are at a loss and vice versa.
Second Inquiry: On the four fenced Propositions.
§44.
The expression every C is B, is sometimes employed in reference to the verity, and its meaning is that every possible (imaginable) singular which may exist and is C, is B by reason of its existence; i.e. whatever is the substrate of C is also the substrate of B. [Such a proposition is called verity-proposition القضیه الحقیقه] Sometimes the expression is used in reference to actual existence, and it means that every C in actual existence, be it at the time of the judgment or before or after it, is B in actualexistence, [such a proposition is called actuality-proposition القضیه الخارجیه].
The difference between these two views is evident, for if no square exists in reality, still we are correct in saying every square is a figure in regard to the first view, but not in regard to the second. And if no figure did exist but squares, we would be correct in saying every figure is a square by the second view. From this you can deduce rules regarding the other fenced propositions.
Third Section: On Privatives and Attributes
§45.
If a negative particle is part of the subject, e.g. an inanimate being is a mineral; or of the predicate e.g. minerals are without intellect (unintellectual); or of both; the proposition is called privative whether it be affirmative or negative. But if no particle forms part of either extremity then the proposition, if it be affirmative, is called attributive and if it be negative indivisible.
§46.
A proposition is affirmative or negative by reason of its affirmative or negative relation (copula) and not by reason of its extremities. If we say “every thing that is not living is without intellect,” it is an affirmative proposition though both extremities are nonentities, and if we say “a moving being is not at rest” it is a negative proposition though both extremities have [positive] existence.
§47.
The indivisible negative proposition [e.g. the partner of God is not omnipotent] is more general (contains more) than the affirmative with privative predicate, [e.g. the partner of God is impotent], for the negation may be true though the subject is a nonentity (i.e. though there is no such thing as a partner of God, we can still say if there were one he could not be omnipotent), but the affirmation cannot be true (i.e. if we say the partner of God is impotent, we admit that there is a partner): because affirmation is admissible only in regard to a thing of ascertained (or acknowledged) existence, as for instance in propositions whose subject is an actually existing individuum or in regard to a thing of assumed existence as for instance in propositions whose subject is a verityl. If the subject does exist the indivisible negative and affirmative privative propositions are equivalent. The difference in the expression [between the indivisible negative and the affirmative with a privative predicate] is this: in the ternary, if it is affirmative, the copula stands before the negative particle, and, if it is negative, it stands after the particle, [as there are no binary propositions in English; the following sentence, of the text which refers to a peculiarity of the Arabic language is omitted].
Fourth Inquiry: On Modal Propositions
§48.
The relation of the predicates to the subjects, be they affirmative or negative, must have a certain qualification as “necessarily,” “perpetually,” “not-necessarily” “not-perpetually.” Such a qualification is called the materia of the proposition, and the word expressing it, is called the mode of the proposition.
§49.
There are thirteen modal propositions into which it is usual to inquire. Some of them are simple, that is to say, their verity is simply an affirmation or negation; and some are compound, that is to say, their verity is composed at the same time of an affirmation and a negation.
§50.
There are six simple modal propositions.
1. The absolute necessary [proposition]. It pronounces that the predicate is affirmed or denied of the subject of necessity as long as the essence of the subject exists, as if we say, “every man is of necessity an animal” and “of necessity no man is a stone”.
2. The absolute perpetual [proposition]. It pronounces that the predicate is affirmed or denied of the subject in perpetuity as long as the essence of the subject exists. The preceding affirmative and negative examples apply to this case.
3. The general conditioned [proposition]. It pronounces that the predicate is affirmed or denied of necessity under the condition of [the continuance of] a certain attribute of the subject, as if we say “every writer is of necessity moving the fingers as long as he writes.” “A writer does not keep his fingers at rest as long as he writes.”
4. The general conventional [proposition]. It pronounces that the predicate is affirmed or denied of the subject in perpetuity under the condition of [the continuance of] a certain attribute of the subject. The preceding affirmative and negative examples illustrate this case.
5. The general absolute [proposition]. It pronounces that the predicate is actually affirmed or denied of the subject, as if we say “every man without exception (literally with general absoluteness) is breathing.” “Everyman without exception (literally with general absoluteness) is not breathing.”
6. The general possible [proposition]. It pronounces that there is no absolute necessity that what is contrary to the judgment should not be the case, as “by a general possibility fire may be hot.” “By a general possibility what is warm is not cold.”
§51.
The compound modal propositions are seven in number.
1. The special conditioned. It is the same as the general conditioned with the restriction that the relation of the subject to the predicate is not [enounced to be] perpetual in regard to the essence [of the subject].
If it is affirmative, as “every writer of necessity moves his fingers as long as he writes, but not perpetually,” it is composed of the affirmative general conditioned and of the negative general absolute propositions. And if it is negative, as “the fingers of a writer are necessarily not at rest as long as he writes, but not perpetually,” it is composed of the negative general conditioned and of the general affirmative absolute.
2. The special conventional [proposition] is the same as the general conventional with the restriction that [the relation do] not [take place] perpetually in reference to the essence. If it is affirmative it is composed of the affirmative general conventional and of the negative general absolute, and if it is negative it is composed of the negative general conventional and of the affirmative general absolute. The preceding affirmative and negative examples illustrate this case.
3. The not-necessary existencial. It is the same as the general absolute with the restriction that [the relation do] not [take place] of necessity in reference to the essence. If it is affirmative, as “man is actually risible (or it happens that man is risible) but not of necessity (he would be man without that property;)” it is composed of the affirmative general absolute and the negative general possible. And if it is negative, as “man is not actually risible but not necessarily” it is composed of the negative general absolute and the affirmative general possible.
4. The non-perpetual existencial. It is the same as the general absolute with the restriction of non-perpetuity in reference to the essence [of the subject]. Whether it be affirmative or negative it is composed of two general absolute [propositions] one of which is affirmative and the other negative. The preceding affirmative and negative examples explain this case.
5. The temporal. It pronounces that the predicate is affirmed or denied of the subject of necessity during a definite period of the existence of the subject, under the restriction of non-perpetuity in regard to the essence [of the subject]. If it is affirmative, as “an eclipse of the moon takes of necessity place during the time the earth is placed between the sun and themoon but not perpetually,” it is composed of the affirmative absolute temporal and the negative general absolute. And if it is negative, as “of necessity no eclipse of the moon takes place when the earth, moon and sun are at right angles but not perpetually,” it is composed of the negative absolute temporal and the affirmative general absolute.
6. The spread [proposition]. It pronounces that the predicate is affirmed or denied of the subject of necessity and during an indefinite period of the existence of the subject, under the restriction of non-perpetuity in reference to the essence [of the subject]. If it is affirmative, as “every man is of necessity breathing at times but not perpetually,” it is composed of the affirmative absolute spread [proposition] and the negative general absolute. And if it is negative, as “man is of necessity not breathing at times but not perpetually,” it is composed of the negative absolute spread [proposition] and the affirmative general absolute.
7. The particular possible [or contingent proposition]. It pronounces that there is no absolute necessity either for the existence or nonexistence of the thing (or relation). It makes no difference whether it is affirmative, as “by peculiar possibility every man is a writer (i.e.
every man can or may be a writer,)” or negative, as “by peculiar possibility every man is not a writer.” It is composed of two general possible propositions, one of which is affirmative and the other negative.
The general rule is that, if a proposition is restricted by non-perpetuity, it indicates that it is a general absolute proposition, and if it is restricted by non-necessity, that it is a general possible proposition disagreeing in mode but agreeing in quantity.
Second Section: On the different kinds of hypothetical Propositions
§52.
The first part (or the first proposition) of a hypothetical is called antecedent and the second consequent.
It (the hypothetical proposition) is either conjunctive or disjunctive. [See §39.]
The conjunctive (conditional) is either cogent (literally adhesive) [or contingent.] In the cogent the consequent is true under the supposition that the antecedent be true on account of the connexion between them, which is the cause thereof, as for instance, if the two propositions be connected by causation [e.g. if the sun rises day approaches, if day approaches the sun rises; if day approaches the world becomes illuminated - the cause of both phenomena being the rising of the sun;] or correlation [e.g. if Zayd is the father of Bakr, Bakr is his son]. In the contingent [the consequent is true if the antecedent is true] by merely accidental agreement of the two parts (or of the two propositions of which the hypothetical consists) in being true, e.g.
if man is endowed with reason, the donkey is endowed with the faculty of braying.
§53.
The disjunctive [hypothetical proposition] is divided into the veritable disjunctive proposition [the incompatible and the exclusive]. The veritable disjunctive proposition pronounces that its two parts exclude each other (literally deny or refute each other) both in [case of] truth and [in case of] falsity, [i.e. if the one is true the other must be false and also if the one is false the other must be true,] as “this number is either even or odd.”
The incompatible disjunctive (literally the hypothetical which excludes coexistence) pronounces that the two parts are opposed to each other in truth only, e.g. this thing is either a stone or a tree, [if it is a stone it cannot be a tree, but it may be neither of the two, and therefore if it is not a stone it does not follow that it is a tree]. The exclusive hypothetical (literally the hypothetical which leaves no vacuum) pronounces that the two parts are opposed to each other in falsity only, as “either Zayd is at sea or else he will not be drowned.”
Each of these three kinds [of disjunctives] is either antagonistical [or coincidental].
A disjunctive is called antagonistical if the two parts exclude each other in their nature, as in the above examples; and it is called coincidental, if this exclusion is a mere coincidence as if we say “non-writer” of a black man. But if we say the man is either black or a writer it is a veritable disjunctive proposition; if, he is a not-black or a writer, it is an incompatible proposition; and if, he is either black or a not-writer, it is an exclusive proposition.
§54.
Any of these eight [hypothetical] propositions is called negative if that [connexion or exclusion] which is pronounced [to exist] in the affirmative, is denied. If it negatives the cohesion, it is called negative-cogent, if it negatives antagonism it is called negative-antagonistic, and if it denies coincidence it is called negative-coincidental.
§55.
The affirmative conjunctive proposition is true (i.e. the inference is correct) of two true and of two false [propositions, e.g. if Zayd is a man he is an animal; if Zayd is a stone he is a mineral]; and of one whose truth and falsity is not known [e.g. if Zayd be writing he is moving his fingers], and of a false antecedent and true consequent, [e.g. if Zayd be a donkey he is an animal,] but not the revers, because from a true [propositiion] does not follow a false one.
The affirmative conjunctive is false (nugatory) of two false parts (propositions) and of a false antecedent and true consequent and vice versa, and if it be cogent also of two true [propositiions], but if it is coincidental, it is impossible that it be false of two true [propositions].
The veritable affirmative disjunctive proposition is true of one true and one false [proposition], e.g. this number is either even or odd; and it is false (nugatory) of two true and of two false [propositions, e.g. four is either even or divisible by two; three is either pair or divisible by two]. The incompatible is true (holds) of two false [propositions, e.g. Zayd may be a tree or a stone]; and it is false (nugatory) of two true ones [e.g. Zayd may be a man or rational]. The exclusive is true of two true [propositions] and of a true one and a false one and it is false (nugatory) of two false ones. The negative is true of what the affirmative is false and it is false of what the affirmative is true.
§56.
The universality of a hypothetical proposition consists in this, that (or a hypothetical proposition is called universal if) the consequent be adherent or antagonistic to the antecedent [at all times] and under all circumstances under which the antecedent can be, that is to say, such circumstances under which the antecedentmay be placed by reason of its connexion with things which are compatible with it. The hypothetical proposition is particular if this is the case under some of those circumstances, and it is peculiarized if it is the case under a definite circumstance. The walls (terms indicative) of the affirmative universal are “whenever,” “whatever,” “when,” [e.g. whenever the sun rises it is day], and of the disjunctive “always” [or “at any time,” e.g. at any time either the sun is up or it is not day].
The wall of the negative universal is in both cases, (i.e. in the conjunctive and disjunctive) “certainly not” [e.g. when the sun is up it is certainly not night]. The wall of the affirmative particular is in both cases “it will then be,” [e.g. it will then be day when the sun rises] and of the negative particular in both cases “it will then not be.” An affirmative universal can be rendered negative by the introduction of the negative particle into the wall.
The walls of the ambiguous conjunctive are simply “if” “when” and of the ambiguous disjunctive “either - or.”
§57.
The hypothetical [proposition] may be composed [1] of two categorical propositions or [2] of two conjunctive ones or [3] of two disjunctive ones or [4] of a categorical and of a conjunctive one or [5] of a categorical and disjunctive one or [6] of a conjunctive and a disjunctive one. Each of the last three kinds if it be conjunctive is sub-divided into two sorts on account of the natural distinction between their antecedent and consequent.
But the disjunctives are not thus subdivided because their antecedent is distinguished from the consequent by appointment only. There are therefore nine divisions (or kinds) of conjunctive hypotheticals and six of disjunctive hypotheticals. You will be able to form examples yourself.
Third Section: Rules concerning propositions
First Inquiry: On Contradiction
§58.
Contradiction is defined as a difference between two propositions in affirming and denying of such a description, that it follows from the difference itself [without medium,] that the one be true and the other false, [e.g. Zayd is a man, Zayd is not a man. But, Zayd is a man, Zayd is irrational, are not included in this definition, because they are contradictory by a medium.]
§59.
The contradiction of two peculiar (singular) propositions is not ascertained (established), unless the subject and predicate are identical, [example of the contrary: Zayd stands, Amr does not stand.] The identity of the former (subject) comprizes the unity of the condition, [example of the contrary: a body is visible, if it be white, a body is not visible, if it be black;] and the unity of “part” and “all” (quantity of the proposition,) [example of the contrary: Africans are black, that is to say some of them; the Africans are not black, that is to say not all of them.] The identity of the predicate comprizes unity of time and place, [example of the contrary: Zayd sleeps at night or in bed, Zayd wakes at day time or in the b´az´ar,] unity of relation, [example of the contrary: Zayd is father, i.e. of ’Amr; Zayd is not father, i.e. of Bakr,] unity of possibility and reality, [example of the contrary: wine inebriates in a basin, i.e. it may inebriate; wine does not inebriate in a basin, it does not do so actually.]
If the two propositions be fenced, it is requisite, in addition to the above, that there be a difference in quantity, for two particulars are true, [e.g. some animals are men, some animals are not men,] and two universals are false [e.g. every animal is a man, no animal is a man,] in every matter in which the subject is more general (more extensive) than the predicate. In the “all” it is requisite that there be a difference in the mode; for two possible (contingent) propositions are true and two necessary propositions false in matter of possibility (contingency).
§60.
The contradictory of the absolute necessary proposition is the general possible, for if the necessity is of necessity negatived, the two propositions will surely be contradictory. The contradictory of the absolute perpetual proposition is the general absolute; because the contradiction of the negative “at no time” is the affirmation “at some times”, and vice versa.
The contradictory of the general conditioned is the possible temporal, that is to say, the proposition which pronounces that necessity in reference to the attribute [see §50] is not applicable to the converse, e.g. every body affected with pleurisy will cough at times on account of his illness. The contradictory of the general conventional is the absolute temporal, i.e. the proposition which pronounces that the predicate is affirmed or denied of the subject at some times when the subject is under certain circumstances.
The preceding examples illustrate this case.
§61.
The contradictory of a compound proposition is the contradiction of its two parts. This will be evident to you after you have comprehended the verities of compound propositions and the contradictories of simple propositions, for after you have ascertained that the non-perpetual existential proposition is composed of two general absolute propositions, one of which is affirmative and the other negative, and that the contradictory of the absolute is the perpetual, you will understand, that its opposite is the opposite perpetual or the agreeing perpetual.
§62.
If [the compound proposition] is particular, what we have mentioned will not be sufficient to contradict it, for it would be false, were we to say “some bodies are animals but not always.” And it would be equally wrong, were we to employ the contradictory of either of the two parts [e.g. no body is ever an animal]. The correct way of forming the contradictory is to place the contradictories of the two parts universally into a dilemmatic sentence, that is to say, every one must be the contradictory of one of the two parts, e.g. every single individuum of the genus ‘body’ is ever either an animal or not an animal.
§63.
The contradictory of the universal hypothetical is the particular which agrees with it in genus and species, but which is opposed to it in “quale” (quality) and “quantum” (quantity,) and vice versa.
Second Inquiry: On even Conversion (Conversio simplex)
§64.
Even conversion is an expression which means that the first part of a proposition be put second and the second part first, and that the truth and quale remain unaltered, (i.e. that the converted proposition remain true, if the original proposition is true, and that it remain affirmative, if the original one is affirmative, and negative if (negative,) (e.g. everyman is an animal -some animals are men; or no man is a stone, no stone is a man.)
§65.
There are seven [modal] forms of negative universal propositions, which cannot be converted, viz., the two temporals, the two existentials, the two possibles and the general absolute; because the most peculiar among them, the temporal, does not admit of conversion, and if the most peculiar cannot be converted the more general ones cannot be converted, for if the more general can be converted, surely the more peculiar can also be converted; for an adhaerens of the more general thing, of necessity, also adheres to the more peculiar. We are correct in saying, the moon can by no means be eclipsed, when she, the sun, and earth form a right angle, but not always; and we are wrong in saying, by general possibility some lunar eclipses may happen to [another celestial body and] not to the moon.
In this example we have chosen the most general mode; for every lunar eclipse operates of necessity on the moon.
§66.
The [negative] absolute necessary and absolute perpetual, become by conversion [negative] universal perpetual, for if it is of necessity, or always true, that no C is B, it is always true that no B is C, else some B would, by general absoluteness, be C, and this, together with the original proposition, would prove that some B is necessarily not B - in necessary propositions, and that some B is always not B - in perpetual propositions. This is absurd.
§67.
The general conditioned and the general conventional become by conversion universal general conventional, for if it is of necessity or perpetually true that no C is B, as long as C exists; no B can ever be C, as long as B exists, else let us suppose that some B is C, whilst it is B, and it follows, if this is taken in connexion with the original proposition, that some B is not B whilst it is B. This is absurd.
The peculiar conditioned and the peculiar conventional are converted into the peculiar non-perpetual conventional. The reason of this process in reference to the general conventional is, that it is an adherent of both kinds of general propositions, (i.e. the general conventional and the general conditioned.) The reason why the converted proposition is peculiar nonperpetual, is, because it is not true that some B is absolutely and generally C, because it is true that no B is always C, and therefore it is converted into “no C is always B,” but the original proposition was that every C is B. We have therefore proved our thesis by reductio ad absurdum.
§§68–70.
Paragraphs 68, 69 and 70, and again 72, 73 and 74, and again 84, 85 and 86, are omitted in the translation, because they contain details on modals which are of no interest. The last named four paragraphs are also omitted in most Arabic text books on Logic, and not studied in Mohammedan Schools.
Third Inquiry: On Conversion by Contradiction
§71.
This expressionmeans to place the contradictory of the second part of a proposition first, and the first part unaltered second. The quale of the new proposition will be the opposite of the original proposition, but it will be equally true, [e.g. every man is an animal, and no not-animal is a man.]
Fourth Inquiry: On the Cohesion of Hypotheticals
§75.
The affirmative universal conjunctive must be convertible into an incompatible proposition, consisting of the antecedent unaltered and of the contradictory of the consequent, and into an exclusive proposition consisting of the contradictory of the antecedent and of the unaltered consequent, and should it not be thus convertible the adhesion and conjunction are unsound.
The veritable disjunctive propositionmust be convertible into four conjunctive propositions. The antecedent of two of them is one of the parts [of the original proposition] unaltered and the consequent is the contradictory of the other part. The antecedent of the other two is the contradictory of one of the two parts and the consequent is the other part unaltered. Every other hypothetical proposition than the veritable must be convertible into another, composed of the contradictories of the two parts.
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Author: Najm al-Din ’Umar al-Qazwni al-Katibi
