_An epicheirema is a syllogism in which one or both of the premises is an enthymeme._ To put it in another way: An epicheirema is a syllogism in which one or both of the premises is supported by a reason. When one premise is an enthymeme the syllogism is termed a _single_ epicheirema; whereas when both premises are enthymemes it becomes a _double_ epicheirema. _Single epicheirema._ All men are mortal, _because all men die_, Socrates was a man, ∴ Socrates was mortal. _Double epicheirema._ All men are mortal, _because all men die_, Socrates was a man, _because he was a rational animal_, ∴ Socrates was mortal. It is obvious that supporting each premise with a reason lends strength to the argument. This justifies the use of the epicheirema. =3. POLYSYLLOGISM.= _A polysyllogism is a series of syllogisms in which the conclusion of a preceding syllogism becomes a premise of a succeeding one._ The syllogism in the series whose conclusion becomes a premise of the succeeding syllogism is termed a _prosyllogism_; while the syllogism which uses as one of its premises the conclusion of the preceding syllogism is called an _episyllogism_. _Illustrations._ { A quadruped is an animal, } { A dog is a quadruped, } _Prosyllogism_ _Polysyllogism_ {∴ A dog is an animal. } } { Fido is a dog, } _Episyllogism_ {∴ Fido is an animal. } { All who libel an associate } { are unprofessional, } { This teacher has libelled } _Prosyllogism_ { her associate, } _Polysyllogism_ {∴ This teacher is } } { unprofessional. } } { All who are unprofessional } _Episyllogism_ { should be disciplined, } {∴ This teacher should be } { disciplined. } =4. SORITES.= _A sorites is a series of syllogisms in which all of the conclusions are omitted except the last one._ Just as the epicheirema is a combination of enthymemes of the _first_ and _second orders_, so the sorites is a combination of enthymemes of the _third order_. If each conclusion were written, the sorites would take the form of prosyllogisms and episyllogisms. Two forms of the sorites are recognized by logicians. These are the _progressive_ or Aristotelian, and the _regressive_ or Goclenian. _Illustrations._ _Progressive_ _Symbolized._ _Put in Word Form._ All A is B Thomas Arnold was a teacher, All B is C A teacher is a man, All C is D A man is a biped, All D is E A biped is an animal, Hence all A is E Hence Thomas Arnold was an animal. _Regressive_