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premises are A and I; then they contain only one distributed term, the subject of A, and this is required for the middle term by rule 3. Hence the minor term cannot be distributed without breaking rule 4, so that the conclusion must be the proposition I. The premises AO would contain two distributed terms, the subject of A and the predicate of 0; but if we were to draw from them the conclusion E, the major and minor terms would require to be distributed, so that the middle term would remain undistributed against rule 3. The reader can easily prove the other cases such as El by calculating the number of distributed terms in a similar manner: it will always be found that there are insufficient terms distributed in the premises to allow of a universal conclusion.
IRREGULAR AND COMPOUND SYLLOGISMS.
It may seem surprising that arguments which are met with in books or conversation are seldom or never thrown into the form of regular syllogisms. Even if a complete syllogism be sometimes met with, it is generally employed in mere affectation of logical precision. In former centuries it was, indeed, the practice for all students at the Universities to take part in public disputations, during which elaborate syllogistic arguments were put forward by one side and confuted by precise syllogisms on the other side. This practice has not been very long discontinued at the University of Oxford, and is said to be still maintained in some continental Universities; but except in such school disputations it must be allowed that perfectly formal syllogisms are seldom employed.
In truth, however, it is not syllogistic arguments which are wanting; wherever any one of the conjunctions, therefore, because, for, since, hence, inasmuch as, consequently occurs, it is certain that an inference is being drawn, and this will very probably be done by a true syllogism. It is merely the complete statement of the premises and conclusion, which is usually neglected because the reader is generally aware of one or other of the premises, or he can readily divine what is assumed; and it is tedious and even offensive to state at full length what the reader is already aware of. Thus, if I say "atmospheric air must have weight because it is a material substance," I certainly employ a syllogism; but I think it quite needless to state the premise, of which I clearly assume the truth, that “whatever is a material substance has weight.” The conclusion of the syllogism is the first proposition, viz. "atmospheric air has weight.” The middle term is “material substance," which does not occur in the conclusion; the minor is "atmospheric air," and the major, “having weight." The complete syllogism is evidently:
All material substances have weight,
Therefore atmospheric air has weight.
A syllogisrn when incompletely stated is usually called an enthymeme, and this name is often supposed to be li derived from two Greek words (év, in, and Ovuós, mind), so as to signify that some knowledge is held by the mind and is supplied in the form of a tacit, that is a silent or understood premise. Most commonly this will be the major premise, and then the enthymeme may be said to be of the First Order. Less commonly the minor premise is unexpressed, and the enthymeme is of the Second
Order. of this nature is the following argument: “Comets must be subject to the law of gravitation; for this is true of all bodies which move in elliptic orbits." It is so clearly implied that comets move in elliptic orbits, that it would be tedious to state this as the minor premise in a complete syllogism of the mood Barbara, thus : All bodies moving in elliptic orbits are subject to
the law of gravitation ; Comets move in elliptic orbits; Therefore comets are subject to the law of gravitation.
It may happen occasionally that the conclusion of a syllogism is left unexpressed, and the enthymeme may then be said to belong to the Third Order. This occurs in the case of epigrams or other witty sayings, of which the very wit often consists in making an unexpressed truth apparent. Sir W. Hamilton gives as an instance of this kind of enthymeme the celebrated epigram written by Porson the English scholar upon a contemporary German scholar:
“The Germans in Greek
And Hermann's a German." It is evident that while pretending to make an exception of Hermann, the writer ingeniously insinuates that since he is a German he has not a correct knowledge of Greek. The wonderful speech of Antony over the body of Cæsar, in Shakspeare's greatest historical play, contains a series of syllogistic arguments of which the conclusions are suggested only.
Even a single proposition may have a syllogistic force if it clearly suggest to the mind a second premise which
thus enables a conclusion to be drawn. The expression
Therefore we are not men who have no rights.
Syllogisms may be variously joined and combined together, and it is convenient to have special names for the several parts of a complex argument. Thus a syllogism which proves or furnishes a reason for one of the premises of another syllogism is called a Prosyllogism; and a syllogism which contains as a premise the conclusion of another syllogism is called an Episyllogism. Take the example :
All B's are A's,
But all D's are C's;
All D's are A's.
This evidently contains two syllogisms in the mood Barbara, the first of which is a Prosyllogism with respect to the second, while the second is an Episyllogism with respect to the first.
The peculiar name Epicheirema is given to a syllogism when either premise is proved or supported by a reason implying the existence of an imperfectly expressed prosyllogism; thus the form,
All B’s are A's, for they are P's,
Therefore all C's are A's, is a double Epicheirema, containing reasons for both premises. The reader will readily decompose it into three complete syllogisms of the mood Barbara.
A more interesting form of reasoning is found in the chain of syllogisms commonly called the Sorites, from the Greek word owpós, meaning heap. It is usually stated in
All A's are B's,
All D's are E's ; Therefore all A's are E's. The chain can be carried on to any length provided it is perfectly consecutive, so that each term except the first and last occurs twice, once as subject and once as predicate. It hardly needs to be pointed out that the sorites really contains a series of syllogisms imperfectly expressed; thus
First Syllogism. Second Syllogism. Last Syllogism. B's are C's, C's are D's, D's are E's, A's are B's; A's are C's; A's are D's; .. A's are C's. .. A's are D's. .. A's are E's. Each syllogism furnishes a premise to the succeeding one, of which it is therefore the prosyllogism, and any syllogism may equally be considered the episyllogism of that which precedes.
In the above sorites all the premises were universal and affirmative, but a sorites may contain one particular premise provided it be the first, and one negative premise provided it be the last. The reader may easily assure himself by trial, that if any premise except the first were