The _Primary Basis of Deductive Reasoning_ may be said to rest upon the logical axiom, which has come down to us from the ancients, and which is stated as follows: "_Whatever is true of the whole is true of its parts_." Or, as later authorities have expressed it: "Whatever is true of the general is true of the particular." This axiom is the basis upon which we build our Deductive Reasoning. It furnishes us with the validity of the deductive inference or argument. If we are challenged for proof of the statement that "This fungus is good to eat," we are able to answer that we are justified in making the statement by the self-evident proposition, or axiom, that "Whatever is true of the general is true of the particular." If the general "mushroom" is good to eat, then the particular, "this fungus" being a mushroom, must also be good to eat. All horses (general) being animals, then according to the axiom, Dobbin (particular horse) must also be an animal. This axiom has been stated in various terms other than those stated above. For instance: "Whatever may be affirmed or denied of the whole, may be denied or affirmed of the parts;" which form is evidently derived from that used by Hamilton who said: "What belongs, or does not belong, to the containing whole, belongs or does not belong, to each of the contained parts." Aristotle formulated his celebrated Dictum as follows: "Whatever can be predicated affirmatively or negatively of any class or term distributed, can be predicated in like manner of all and singular the classes or individuals contained under it." There is another form of Deductive Reasoning, that is a form based upon another axiom than that of: "Whatever is true of the whole is true of the parts." This form of reasoning is sometimes called Mathematical Reasoning, because it is the form of reasoning employed in mathematics. Its axiom is stated as follows: "Things which are equal to the same thing, are equal to one another." It will be seen that this is the principle employed in mathematics. Thus: "x equals y; and y equals 5; therefore, x equals 5." Or stated in logical terms: "A equals B; B equals C; therefore, A equals C." Thus it is seen that this form of reasoning, as well as the ordinary form of Deductive Reasoning, is strictly _mediate_, that is, made through the medium of a third thing, or "two things being compared through their relation to a third."