Enthymemes

Definition:

- A categorical syllogism that has either the conclusion or a premise missing. The antillogism method makes finding the missing part of a valid argument quite straight forward.

Example:

All men who pursue the truth are wise.
...
So all philosophers are wise.

- This enthymeme is missing the minor premise.

Old Form | Boolean Form | Antillogism | Step 1 | Step 2 | Step 3
All M are W ... All P are W | M~W = 0 ... P~W = 0 | M~W = 0 ... P~W != 0 | M~W = 0 = 0 P~W != 0 | M~W = 0 ~M= 0 P~W != 0 | M~W = 0 P~M= 0 P~W != 0

- The following steps set forth in the above table will find the missing minor premise.
  - Represent the enthymeme in the Old form.
  - Convert it to Boolean form.
  - contradict the conclusion to get the antillogism.
  - Step 1:
    - There must be one and only one inequation (!= 0) in the antillogism. Since, in this case, the conclusion is an inequation, the missing minor premise has to be an equation (= 0).
  - Step 2:
    - There must be a pair, a term and its negation, in the two equations. (The paired term is always the term not found in the inequation.) So in this case the paired term has to be M. Since it is positive in the one equation, fill in a negative M in the other equation.
  - Step 3:
    - The unpaired terms in the equations must appear in the inequation exactly as they're found in the equations. (Another way of stating this rule: the terms in the inequation must be split between the two equations.) In this case ~W appears in one equation so P must appear in the other.
  - Hint: When filling in a missing premise or conclusion:
    - If one of the terms to fill in is negated, place it second as in the above example.
    - If what has been filled in is an inequation and the second term is negated, translate it back to an O type categorical proposition.
    - If,on the other hand, the second term is negated and it is a equation, translate it back to an A type categorical proposition.
    - If neither term is negated the order of the terms is interchangeable.
    - If it is an equation translate it back to an E type categorical proposition.
    - And for an inequation translate it back to an I type proposition.
  - Note: In the case of a missing conclusion, do not forget to contradict the Boolean form in the antillogism before translating it back to a categorical proposition.

Example:

- The missing premise for the above enthymeme turned out to be: (P~M = 0).
- Not being a missing conclusion, it does not have to be contradicted.
- It should be translated back to an A type proposition because the second term is negated and it is an equation.
- An A type proposition has the form:
  - All ... are ...
- Subject terms are never changed when converting to Boolean form. So:
  - All P are ...
- Predicate terms of A and O type propositions are negated when converting to Boolean form. So in this case the predicate in the Old form had to be:
  - All P are M
- Filling with subject and predicate terms that P and M represent:
  - All philosophers are men who pursue the truth.