_Rejects Modality._--He strenuously and consistently rejects the modality of the proposition and the syllogism, on the ground that logic is not concerned with the character of the matter, whether it be true or false, necessary or contingent, but only with the form of statement, and consequently, all distinctions founded on the truth or falsity, the necessity or contingence of the matter, are utterly irrelevant to the science--a principle admitted by others, but not previously carried out to its true results. _Doctrine of Figure._--He shows that the figure of the syllogism is a matter accidental, rather than essential, that it may be even entirely _unfigured_; abolishes the fourth figure as superfluous; and sets aside, as quite useless and unnecessary, the old laborious processes of reducing and connecting the several figures to the first. _Rejects hypothetical Syllogism._--He throws out of the syllogism entirely, the so-called _hypothetical_ forms, both conjunctive and disjunctive, as reducible to _immediate_ inference, and not, therefore, to be included under syllogistic reasoning, which is always mediate. _The single Canon._--He reduces the several laws and canons of the figured syllogism to a single comprehensive canon. _Quantification of the Predicate._--But the most important discovery made by Hamilton in this science, is the quantification of the predicate. The predicate is always a given quantity in relation to the subject, and that quantity should be stated. This, logicians have always overlooked, quantifying only the subject, as, All men, Some men, etc., but never the predicate. Fully quantified, the proposition reads, All man is _some_ animal, no animal, etc., _i. e._, some sort or species of animal. This doubles the number of possible propositions, giving eight in place of four, and gives a corresponding increase in the number of words. These eight propositions are shown to be, not only possible, but admissible and valid. They are thus enumerated and named: AFFIRMATIVE. NEGATIVE. I. _Toto-total_: All A is all B. Any A is not any B. II. _Toto-partial_: All A is some B. Any A is not some B. III. _Parti-total_: Some A is all B. Some A is not any B. IV. _Parti-partial_: Some A is some B. Some A is not some B.