_Rules of Opposition._--The following rules will be found universally applicable to propositions as opposed to each other. If the universal is true, so is the particular. If the particular is false, so is the universal. Contraries are never both true, but may be both false. Sub-contraries are never both false, but may be both true. Contradictories are never both true, or both false, but always one is true, the other false. The truth of these maxims will be evident on applying them to any proposition and its opposites, as for example, to the affirmation, Every man is mortal. _Categorical and hypothetical Propositions._--Propositions may be further distinguished as categorical or hypothetical; the one asserting or denying directly, as, _e. g._, The earth is round; the other conditionally,--as, If the earth is round, it is not oblong. _Pure, and Modal._--The proposition, moreover, may be either pure or modal, the former asserting or denying without qualification,--as, Man is liable to err; the latter qualifying the statement,--as, Man is extremely or unquestionably liable to err. II. ANALYSIS OF THE SYLLOGISM. _Proposition the Link, Syllogism the Chain._--All reasoning admits of being reduced to the form of a syllogism. Having discussed the proposition which forms the material or groundwork of every connected chain of argument, we are prepared now to examine the syllogism, or chain itself, into which the several propositions, as so many links, are wrought. _Syllogism defined._--A syllogism is an argument so expressed that the conclusiveness of it is manifest from the mere form of expression. When, for example, I affirm that all A is B, that all B is C, and that, consequently, all A is C, it is impossible that any one who is able to reason at all, and who comprehends the force of these several propositions taken singly, should fail to perceive that the conclusion follows inevitably from the premises. That which is affirmed, may or may not be _true_, but it is _conclusive_. If the premises are true, so is the conclusion; but whether they are true or not, the argument, as such, is conclusive; nay, even if they are false, the conclusion may possibly be true. For example, Every tyrant is a good man; Washington was a tyrant; therefore, Washington was a good man, Both the premises are false, but the argument, as regards the form, is valid, and the conclusion is not only correctly drawn, but is, moreover, a true proposition. In a word, the syllogism concerns itself not at all with the truth or falsity of the thing stated, but only with the form of stating, and that form must be such, that the premises being conceded, the conclusion shall be obvious and inevitable. All valid reasoning admits of such statement.