IV. From O "Some A is not B" follow: IV. A B A-NOT-B B=NOT-A (1) Some A is not-B (I, obverse). (7) Some not-B is not not-A (0). The other forms in the case of I and O are wanting. be Of the seven forms given above, three—(1), (4), and (5)—have, as we have already stated, special names: obverse, converse, and contrapositive respectively; the others—(2), (3), (6), and (7)— have no special names. That these inferences are valid may easily proved also by the older method. For example, of the inferences drawn from A, (7) is the obverse of its contrapositive, (6) is the obverse of its converse, (3) is the converse of the obverse of its contrapositive, and (2) is the obverse of the last. Of the inferences drawn from E, (2) is the contrapositive of its converse, (3) is the obverse of (2), (6) is the obverse of its converse, and (7) is the obverse of its contrapositive. Thus the four additional forms may be inferred by the older method as well as by the method adopted in this work,-by the former as an inference from an inference, and by the latter as an immediate inference from the given proposition. § 10. Miscellaneous Exercises. I. Give the obverse of the converse of the following propositions:(1) The useful is not the beautiful. (2) Beauty is unity in variety. (3) Wise men are few. (4) A touches B. (5) (a) I know, (b) I am, (c) He is. (6) A is equal to B. (7) A lies above B. (8) The number of substances containing more than four ele ments is very small. (9) Where no object is distinguished, we are not conscious of any. 5. If the rays of light fall upon the eye, they will produce the sensation of vision; .. If the sensation of vision is not produced, the rays of light have not fallen upon the eye. 6. All A is B. .. Some not-A is not-B. III. Give the converse of the contradictory of each of the following propositions:— 1. Every man is not learned. 2. Only animals are sentient beings. 3. Nothing is annihilated. 4. If A is B, C is not D. IV. Give the contrapositive of the contrary of each of the following propositions: 1. Every phenomenon has a cause. 2. No man is perfect. 3. If A is B, C is D. 4. If A is B, C is not D." 7 V. Give the converse of the contrapositive of the contrary or subcontrary of the contradictory of each of the following propositions:— 1. All sensations are feelings. 2. No man is immortal. 3. Some men are wise. 4. Some elements are not metals. VI. Given the proposition 'Some men are not selfish' as true: state the propositions that can be inferred from it, (1) as true, (2) as false, and (3) as doubtful or unknown. VII. Given the proposition 'All virtuous men are happy' as true: state the propositions that can be inferred from it, (1) as true, (2) as false, and (3) as doubtful or unknown. VIII. Given the proposition 'Some men are unjust' as true: state the propositions that can be inferred from it, (1) as true, (2) as false, and (3) as doubtful or unknown. IX. Given the proposition 'No man is infallible' as true: state the propositions that can be inferred from it, (1) as true, (2) as false, and (3) as doubtful or unknown. X. Infer as many verbal or analytical propositions as you can from each of the following terms:-(1) animal, (2) matter, (3) triangle, (4) circle, (5) square, (6) man, (7) plant, (8) metal, (9) force, (10) book, (11) table, (12) horse, (13) mammal, (14) mind, (15) perception, (16) sensation, (17) house, (18) philosopher, (19) poet, (20) king, (21) nation, (22) society, (23) paper, (24) chair, (25) examination. XI. Draw as many inferences as you can from the truth and also from the falsity of each of the following propositions: (1) All S is P. (2) No S is P. (3) Some S is P. (4) Some S is not P. XII. Infer as many propositions as you can from each of the following propositions being given as true:— (1) Every phenomenon has a cause. (2) The invariable antecedent of a phenomenon is the cause of the phenomenon. (3) The absolute commencement of a phenomenon is not conceivable. (4) The infinite non-commencement of a phenomenon is not conceivable. (5) At least one substance has no cause. CHAPTER III. OF SYLLOGISMS. § 1. A Syllogism is the inference of a proposition from two given propositions, the inferred proposition being less general than either of the two given propositions. As an argument fully expressed in language, it consists of three propositions, one of which, the conclusion, follows necessarily from the other two, called the Premisses, and thus differs from Immediate Inference, which, as the simplest and most elementary form of argument, consists of two propositions, the conclusion and the proposition from which the conclusion necessarily follows. From the proposition All men are mortal' follows 'Some mortal beings are men' by immediate inference,-i. e., the latter is a conclusion derived from the former without the aid of any other proposition. In a Syllogism such aid is necessary, that is, a conclusion is drawn not from one proposition but from at least two propositions. For example, from the two propositions 'All men are mortal' and 'Philosophers are men,' I infer the proposition 'Philosophers are mortal.' Here (1) the conclusion follows from the two propositions taken jointly, and not from either of them singly. The two propositions must be brought together before I can legitimately infer the third which is involved in them, and yet is distinct from either. The conclusion Philosophers are mortal' is not the same as either of the two propositions 'All men are mortal' and 'Philosophers are men'; nor does it follow from one of them. By this character a syllogism is distinguished from an immediate inference. Again, (2) the two propositions being true, the conclusion must be true. The one conjointly with the other makes the conclusion necessarily admissible, legitimate, or valid. By this character, a syllogism, that is, a correct or valid syllogism, is distinguished from an apparent one or a mere combination of three propositions in which the conclusion does not follow from the premisses. And (3) the conclusion can not be more general than either of the two propositions from which it is inferred. The proposition 'Philosophers are mortal' is less general than the proposition 'All men are mortal,' the latter being applicable to a much larger number of individual things than the former. By this character, a syllogism is distinguished from an induction, in which we pass from the less general to the more general, from the particular to the universal1. A syllogism is either pure or mixed. It is pure when both its premisses have the same relation, that is, when they are both categorical or both hypothetical; and mixed when they have different relations, that is, when one of them is hypothetical and the other categorical, or one disjunctive and the other categorical. These distinctions will be referred to more fully in a subsequent chapter2. § 2. Of Categorical Syllogisms. A Categorical Syllogism is a syllogism consisting of two categorical premisses and a categorical conclusion necessarily following from them. It is a reasoning in which a term is affirmed or denied of another by means of a third. Given two terms: if I affirm or deny one of the other, I get a categorical proposition 'A is B' or 'A is not B.' In this act there is no reasoning, mediate or immediate; there is merely an act of judgment, the direct comparison of one term with the other. If every term could be thus directly affirmed or denied of every other, there would be no such mental act as reasoning; there would be no need of it. But constituted and circumstanced as we are, we can not directly affirm or deny every term of every other. We have often to establish a relation between two terms 1 See above, Part III, Chap. I. 2 See below, Part III, Chap. v. |