A System of Logic, Ratiocinative and Inductive. John Stuart Mill

All A is B, All A is B, Some A is B, Some A is B, therefore therefore All A is C. No A is C. Some A is C. Some A is not C.

      Or, if more significant symbols are preferred:

      To prove an affirmative, the argument must admit of being stated in this form:

      All animals are mortal;

       All men/Some men/Socrates are animals;

       therefore

       All men/Some men/Socrates are mortal.

      To prove a negative, the argument must be capable of being expressed in this form:

      No one who is capable of self-control is necessarily vicious;

      No one who is capable of self-control is necessarily vicious;

       All negroes/Some negroes/Mr. A's negro are capable of self-control;

       therefore

       No negroes are/Some negroes are not/Mr. A's negro is not necessarily vicious.

      Though all ratiocination admits of being thrown into one or the other of these forms, and sometimes gains considerably by the transformation, both in clearness and in the obviousness of its consequence; there are, no doubt, cases in which the argument falls more naturally into one of the other three figures, and in which its conclusiveness is more apparent at the first glance in those figures, than when reduced to the first. Thus, if the proposition were that pagans may be virtuous, and the evidence to prove it were the example of Aristides; a syllogism in the third figure,

      Aristides was virtuous,

       Aristides was a pagan,

       therefore

       Some pagan was virtuous,

      would be a more natural mode of stating the argument, and would carry conviction more instantly home, than the same ratiocination strained into the first figure, thus—

      Aristides was virtuous,

       Some pagan was Aristides,

       therefore

       Some pagan was virtuous.

      A German philosopher, Lambert, whose Neues Organon (published in the year 1764) contains among other things one of the most elaborate and complete expositions which had ever been made of the syllogistic doctrine, has expressly examined what sort of arguments fall most naturally and suitably into each of the four figures; and his investigation is characterized by [pg 131] great ingenuity and clearness of thought.51 The argument, however, is one and the same, in whichever figure it is expressed; since, as we have already seen, the premises of a syllogism in the second, third, or fourth figure, and those of the syllogism in the first figure to which it may be reduced, are the same premises in every thing except language, or, at least, as much of them as contributes to the proof of the conclusion is the same. We are therefore at liberty, in conformity with the general opinion of logicians, to consider the two elementary forms of the first figure as the universal types of all correct ratiocination; the one, when the conclusion to be proved is affirmative, the other, when it is negative; even though certain arguments may have a tendency to clothe themselves in the forms of the second, third, and fourth figures; which, however, can not possibly happen with the only class of arguments which are of first-rate scientific importance, those in which the conclusion is a universal affirmative, such conclusions being susceptible of proof in the first figure alone.52

      [pg 132]

      § 2. On examining, then, these two general formulæ, we find that in both of them, one premise, the major, is a universal proposition; and according [pg 133] as this is affirmative or negative, the conclusion is so too. All ratiocination, therefore, starts from a general proposition, principle, or assumption: a proposition in which a predicate is affirmed or denied of an entire class; that is, in which some attribute, or the negation of some attribute, is asserted of an indefinite number of objects distinguished by a common characteristic, and designated, in consequence, by a common name.

      The other premise is always affirmative, and asserts that something (which may be either an individual, a class, or part of a class) belongs to, or is included in, the class respecting which something was affirmed or denied in the major premise. It follows that the attribute affirmed or denied of the entire class may (if that affirmation or denial was correct) be affirmed or denied of the object or objects alleged to be included in the class: and this is precisely the assertion made in the conclusion.

      Whether or not the foregoing is an adequate account of the constituent parts of the syllogism, will be presently considered; but as far as it goes it is a true account. It has accordingly been generalized, and erected into a logical maxim, on which all ratiocination is said to be founded, insomuch that to reason, and to apply the maxim, are supposed to be one and the same thing. The maxim is, That whatever can be affirmed (or denied) of a class, may be affirmed (or denied) of every thing included in the class. This axiom, supposed to be the basis of the syllogistic theory, is termed by logicians the dictum de omni et nullo.

      This maxim, however, when considered as a principle of reasoning, appears suited to a system of metaphysics once indeed generally received, but which for the last two centuries has been considered as finally abandoned, though there have not been wanting in our own day attempts at its revival. So long as what are termed Universals were regarded as a peculiar kind of substances, having an objective existence distinct from the individual objects classed under them, the dictum de omni conveyed an important meaning; because it expressed the intercommunity of nature, which it was necessary on that theory that we should suppose to exist between those general substances and the particular substances which were subordinated to them. That every thing predicable of the universal was predicable of the various individuals contained under it, was then no identical proposition, but a statement of what was conceived as a fundamental law of the universe. The assertion that the entire nature and properties of the substantia secunda formed part of the nature and properties of each of the individual substances called by the same name; that the properties of Man, for example, were properties of all men; was a proposition of real significance when man did not mean all men, but something inherent in men, and vastly superior to them in dignity. Now, however, when it is known that a class, a universal, a genus or species, is not an entity per se, but neither more nor less than the individual substances themselves which are placed in the class, and that there is nothing real in the matter except those objects, a common name given to them, and common attributes indicated by the name; what, I should be glad to know, do we learn by being told, that whatever can be affirmed of a class, may be affirmed of every object contained in the class? The class is nothing but the objects contained in it: and the dictum de omni merely amounts to the identical proposition, that whatever is true of certain objects, is true of each of those objects. If all ratiocination were no more than the application of this maxim to particular cases, the syllogism would indeed be, what it has so often been declared to [pg 134] be, solemn trifling. The dictum de omni is on a par with another truth, which in its time was also reckoned of great importance, “Whatever is, is.” To give any real meaning to the dictum de omni, we must consider it not as an axiom, but as a definition; we must look upon it as intended to explain, in a circuitous and paraphrastic manner, the meaning of the word class.

      An error which seemed finally refuted and dislodged from thought, often needs only put on a new suit of phrases, to be welcomed back to its old quarters, and allowed to repose unquestioned for another cycle of ages. Modern philosophers have not been sparing in their contempt for the scholastic dogma that genera and species are a peculiar kind of substances, which general substances being the only permanent things, while the individual substances comprehended under them are in a perpetual flux, knowledge, which necessarily imports stability, can only have relation to those general