This sort of argument is called _à fortiori_ argument, because the premises are more than sufficient to prove the conclusion, and the extent of the conclusion is thereby greater than its mere form would indicate. Thus, ‘A is less than B, B is less than C, therefore, _à fortiori_, A is less than C,’ means that the extent to which A is less than C must be greater than that to which A is less than B, or B than C. In the syllogism last written, either of the bracketed premises might be struck out without destroying the conclusion; which last would, however, be weakened. As it stands, then, the part of the conclusion, ‘Some C is not A,’ follows it _à fortiori_. The argument _à fortiori_, may then be defined as a universally affirmative syllogism, in which both of the premises are shewn to be less than the whole truth, or greater. Thus, in ‘Every A is X, Every X is B, therefore Every A is B,’ we do not certainly imply that there are more Xs than As, or more Bs than Xs, so that we do not know that there are more Bs than As. But if we are at liberty to state the syllogism as follows, All the As make up part (and part only) of the Xs Every X is B; then we are certain that All the As make up part (and part only) of the Bs. But if we are at liberty further to say that All the As make up part (and part only) of the Xs All the Xs make up part (and part only) of the Bs then we conclude that All the As make up _part of part_ (only) of the Bs and the words in Italics mark that quality of the conclusion from which the argument is called _à fortiori_. Most syllogisms which give an affirmative conclusion are generally meant to imply _à fortiori_ arguments, except only in mathematics. It is seldom, except in the exact sciences, that we meet with a proposition, ‘Every A is B,’ which we cannot immediately couple with ‘Some Bs are not As.’ When an argument is completely established, with the exception of one assertion only, so that the inference may be drawn as soon as that one assertion is established, the result is stated in a form which bears the name of an _hypothetical_ syllogism. The word hypothesis means nothing but supposition; and the species of syllogism just mentioned first lays down the assertion that a consequence will be true if a certain condition be fulfilled, and then either asserts the fulfilment of the condition, and thence the consequence, or else denies the consequence, and thence denies the fulfilment of the condition. Thus, if we know that When A is B, it follows that P is Q;