Sorites

Definition:

- A sorites is two or more categorical syllogisms where the intermediate conclusion(s) have been omitted.
- A sorites is in standard form when:
  - The predicate term of the conclusion is in the first premise.
  - Each term occurs exactly twice.
  - Each premise after the first has a term in common with the preceding one.

Example:

No persons entitled to the right of cross-examination are entitled to examine all the evidence before the trail.
All persons accused of crimes are entitled to the right of cross-examination.
Some innocent persons are persons accused of crimes.
So, some innocent persons are not entitled to examine all the evidence before the trail.

Old Form | Boolean Form | Antillogism
No R are E All P are R Some I are P Some I are not E | RE = 0 P~R = 0 IP != 0 I~E != 0 | RE = 0 P~R = 0 IP != 0 I~E = 0

The easiest way to test the validity of a sorites is to use the antillogism test.

- Step 1: There is one and only one inequation (IP != 0).
- Step 2: There are three equations. So there should be exactly 2 pairs found in the equations. These two pairs are to be found in the equations: (I~E = 0 and RE = 0), (RE = 0 and P~R = 0).
- Step 3: The remaining unpaired terms in the equations (I and P) appear in the inequation exactly as they are in the equations.
- All tree steps of the antillogism is satisfied. So this sorites is valid.

There is another way to test a sorites by deducing the omitted conclusions. This is they way you would do it with the Applet.

- Place the sorites in standard form
- Take the first two premises and treat them as an enthymeme and grind out a conclusion:

| Step 1 | Step 2 | Step 3
RE = 0 P~R = 0 | RE = 0 P~R = 0 != 0 | RE = 0 P~R = 0 != 0 | RE = 0 P~R = 0 PE!= 0

- Take the conclusion of this enthymeme, treat is as a premise combined with the third premise and grind out another conclusion. NOTE: Don't forget to contradict the conclusion of the enthymeme before you use it as a premise.
- Keep this up 'till you either derive the conclusion ( in which case it is valid) or it fails along the way (in which case it is invalid).

| Step 1 | Step 2 | Step 3
PE = 0 IP ! = 0 | PE = 0 IP != 0 = 0 | PE = 0 IP != 0 ~E= 0 | PE = 0 IP != 0 I~E= 0

- Contradicting the conclusion of this last enthymeme we get (I~E != 0) which is the conclusion of the sorites.