Antillogism Test

- An argument is valid if its antillogism satisfies the three steps in the table below.

Step 1 | There must be one and only one inequation.
Step 2 | There must be one less pair than the number of equations.
Step 3 | The unpaired terms in the equations must appear in the inequation exactly as they appear in the equations.

- An antillogism is arrived at by contradicting the conclusion. (SP = 0 becomes SP != 0) and (SP != 0 becomes SP = 0)
- An equation is ( = 0 ).
- An inequation is ( != 0 ).
- A pair is a term and its negation. (A and ~A), (B and ~B)...

Examples:

All logicians are realists.
Some dreamers are not realists.
Some dreamers are not logicians.

Old Form | Boolean Form | Antillogism
All L are R Some D are not R Some D are not L | L~R = 0 D~R != 0 D~L != 0 | L~R = 0 D~R != 0 D~L = 0

The test:

- Step 1: The antillogism has one and only one inequations. (D~R != 0)
- Step 2: There are two equations ( L~R = 0 and D~L = 0). There should be one less pair between the two equations. And there is exactly pair marked in red above.
- Step 3: The unpaired terms in the two equations (L~R = 0 and D~L = 0) appear in the inequations (D~R != 0) exactly as they are in the two equations.
- Since the three steps are satisfied in the antillogism, the argument is valid.

No nuclear powered submarines are commercial vessels.
All nuclear powered submarines are warships.
So no warships are commercial vessels.

Old Form | Boolean Form | Antillogism
No N are C All N are W No W are C | NC = 0 N~W = 0 WC = 0 | NC = 0 N~W = 0 WC != 0

The test:

- Step 1: The antillogism does have one and only one inequation. (WC != 0)
- Step 2: The should be one pair in the two equations (NC = 0 and N~W = 0). But the term they have in common (marked in red) is not negated in either equations so no pair is formed.
- The argument is invalid because it violates step two.