་ lesson point out all the ways in which we can from a single proposition of the forms A, E, I or O, pass to another proposition. We are said to convert a proposition when we transpose its subject and predicate; but in order that the converse or converted proposition shall be inferred from the convertend, or that which was to be converted, we must observe two rules (1) the quality of the proposition (affirmative or negative) must be preserved, and (2) no term must be distributed in the Converse unless it was distributed in the Convertend. If in "all metals are elements" we were simply to transpose the terms, thus—“ all elements are metals,” we imply a certain knowledge about all elements, whereas it has been clearly shewn that the predicate of A is undistributed, and that the convertend does not really give us any information concerning all elements. All that we can infer is that ". some elements are metals;" this converse proposition agrees with the rule, and the process by which we thus pass from A to I is called Conversion by Limitation, or Per accidens. When the converse is a proposition of exactly the same form as the convertend the process is called simple conversion. Thus from " some metals are brittle substances" I can infer some brittle substances are metals," as all the terms are here undistributed. Thus I is simply converted into I. Again, from "6. no metals are compounds," I can pass directly to "no compounds are metals," because these propositions are both in E, and all the terms are therefore distributed. Euler's diagram (p. 73, Fig. 3) clearly shows, that if all the metals are separated from all the compounds, all the compounds are necessarily separated from all the metals. The proposition E is then simply converted into E. 3x [gxi fx] But in attempting to convert the proposition O we encounter a peculiar difficulty, because its subject is undistributed; and yet the subject should become by conversion the predicate of a negative proposition, which distributes its predicate. Take for example the proposition, "some existing things are not material substances.” By direct conversion this would become "all material substances are not existing things;" which is evidently absurd. The fallacy arises from existing things being distributed in the converse, whereas it is particular in the convertend; and the rules of the Aristotelian logic prevent us from inserting the sign of particular quantity before the predicate. The converse would be equally untrue and fallacious were we to make the subject particular, as in “ some material substances are not existing things." We must conclude, then, that the proposition O cannot be treated either by simple conversion of conversion by limitation. It is requisite to apply a new process, which may be called Conversion by Negation, and which consists in first changing the convertend into an affirmative proposition, and then converting it simply. If we attach the negation to the predicate instead of to the copula, the proposition becomes "some existing things are immaterial substances," and, converting simply, we have "some immaterial substances are existing things," which may truly be inferred from the convertend. The proposition O, then, is only to be converted by this exceptional method of negation. Another process of conversion can be applied to the proposition A, and is known as conversion by contraposition. From "all metals are elements," it necessarily follows that "all not-elements are not metals.” If this be not at the first moment apparent, a little reflection will render it so, and from fig. 5 we see that if all the metals be among the elements, whatever is not ele ment, or outside the circle of elements, must also be outside the circle of metals. We may also prove the truth Fig. 5. Elements Metals of the contrapositive proposition in this way, if we may anticipate the contents of Lesson XXIII.:—If what is notelement should be metal, then it must be an element by the original proposition, or it must be at once an element and not an element; which is impossible according to the Primary Laws of Thought (Lesson XIV.), since nothing can both have and not have the same property. It follows that what is not-element must be not-metal. Mistakes may readily be committed in contrapositive conversion, from a cause which will be more apparent in Lesson XXII. We are very liable to infer from a proposition of the form "all metals are elements,” that all not-metals are not-elements, which is not only a false statement in itself, but is not in the least warranted by the original proposition. In fig. 5, it is apparent that because a thing lies outside the circle of metals, it does not necessarily lie outside the circle of elements, which is wider than that of metals. Nevertheless the mistake is often made in common life, and the reader will do well to remember that the process of conversion by contraposition consists only in taking the negative of the predicate of the proposition A, as a new subject, and affirming of it universally the negative of the old subject. Contrapositive conversion cannot be applied to the particular propositions I and 0 at all, nor to the proposition E, in that form; but we may change E into A by attaching the negation to the predicate, and then the process can be applied. Thus "no men are perfect," may be changed into "all men are not-perfect," i. e. 'are imperfect,” and then we infer by contraposition "all not-imperfect beings are not-men.” But not-imperfect is really the same as perfect, so that our new proposition is really equivalent to "all perfect beings are not men," or 66 no perfect beings are men," (E) the simple converse of the original proposition. There remain to be described certain deductions which may be drawn from a proposition without converting its terms. They may be called immediate inferences, and have been very clearly described by Archbishop Thomson in his "Outline of the Necessary Laws of Thought" (pp. 156, &c.). Immediate Inference by Privative Conception consists in passing from any affirmative proposition to a negative proposition implied in it, or equivalent to it, or vice versa, in passing from a negative proposition to its corresponding affirmative. The following table contains a proposition of each kind changed by privative conception into an equivalent proposition: ~] x [fx. JA all metals are elements. some men are not untrustworthy. some men are not trustworthy. some men are untrustworthy. ? The truth of any of the above can be clearly illustrated by diagrams; thus it will be apparent that if the whole circle of metals lies inside the circle of elements, no part can lie outside of that circle or among the compounds. Any of the above propositions may be converted, but the results will generally be such as we have already obtained. Thus the simple converse of "no metals are compounds" is no compounds are metals," or "no notelements are metals," the contrapositive of "all metals are elements." From the last example we get also by simple conversion " some untrustworthy beings are men,” which is obviously the converse by negation, as before explained. Applying this kind of conversion to "some men are not untrustworthy," we have "some not-untrustworthy beings are men." Lastly, from "all men imperfect" we may obtain through conversion by limitation, some imperfect beings are men." are Immediate Inference by added determinants consists in joining some adjective or similar qualification both to the subject and predicate of a proposition, so as to render the meaning of each term narrower or better determined. Provided that no other alteration is made the truth of the new proposition necessarily follows from the truth of the original in almost all cases. From "all metals are elements," we may thus infer that "all very heavy metals are very heavy elements." From "a comet is a material body" we infer "a visible comet is a visible material body." But if we apply this kind of inference too boldly we may meet with fallacious and absurd results. Thus, from "all kings are men," we might infer "all incompetent kings are incompetent men;" but it does not at all follow that those who are incompetent as kings would be incompetent in other positions. In this case and many others the qualifying adjective is liable to bear different meanings in the subject and predicate; but the inference will only be true of |