Sec. 3.  A Monosyllogism is a syllogism considered as standing alone or without relation to other arguments.  But, of course, a disputant may be asking to prove the premises of any syllogism; in which case other syllogisms may be advanced for that purpose.  When the conclusion of one syllogism is used to prove another, we have a chain-argument which, stated at full length, is a Polysyllogism.  In any Polysyllogism, again, a syllogism whose conclusion is used as the premise of another, is called in relation to that other a Prosyllogism; whilst a syllogism one of whose premises is the conclusion of another syllogism, is in relation to that other an Episyllogism.  Two modes of abbreviating a Polysyllogism, are usually discussed, the Epicheirema and the Sorites.

Sec. 4.  An Epicheirema is a syllogism for one or both of whose premises a reason is added; as—­

      All men are mortal, for they are animals;
      Socrates is a man, for rational bipeds are men: 
    .’.  Socrates is mortal.

The Epicheirema is called Single or Double, says Hamilton, according as an “adscititious proposition” attaches to one or both of the premises.  The above example is of the double kind.  The Single Epicheirema is said to be of the First Order, if the adscititious proposition attach to the major premise; if to the minor, of the Second Order. (Hamilton’s Logic:  Lecture xix.)

An Epicheirema, then, is an abbreviated chain of reasoning, or Polysyllogism, comprising an Episyllogism with one or two enthymematic Prosyllogisms.  The major premise in the above case, All men are mortal, for they are animals, is an Enthymeme of the First Order, suppressing its own major premise, and may be restored thus: 

      All animals are mortal;
      All men are animals: 
    .’.  All men are mortal.

The minor premise, Socrates is a man, for rational bipeds are men, is an Enthymeme of the Second Order, suppressing its own minor premise, and may be restored thus: 

      All rational bipeds are men;
      Socrates is a rational biped: 
    .’.  Socrates is a man.

Sec. 5.  The Sorites is a Polysyllogism in which the Conclusions, and even some of the Premises, are suppressed until the arguments end.  If the chain of arguments were freed of its enthymematic character, the suppressed conclusions would appear as premises of Episyllogisms.

Two varieties of Sorites are recognised, the Aristotelian (so called, though not treated of by Aristotle), and the Goclenian (named after its discoverer, Goclenius of Marburg, who flourished about 1600 A.D.).  In order to compare these two forms of argument, it will be convenient to place side by side Hamilton’s classical examples of them.