some A is not B 
no A is B 
x is not A 
x is not y 
no P is B some C is B ∴ some C is not P 
a is F a is G ∴ some F is G 
some A is not B 
no A is B 
x is not A 
x is not y 

no P* is B* some C is B ∴ some C* is not P 
starred once and one righthand star 
no P* is B* some C is not B* ∴ some C* is P* 
starred twice and three righthand stars 
a is F a is G ∴ some F* is G* 
letters can be starred any number of times 
all M is P all S is M ∴ all S is P 
all M is P all S is M some S is not P 

(valid) all M* is P all S* is M ∴ all S is P* 
(inconsistent) all M is P all S is M some S is not P 
(valid) all M* is P all S* is M ∴ all S is P* 
(original) all M is P all S is M some S is not P 
(modified) all M is P all S is M a is S a is not P 
(valid) all M* is P all S* is M ∴ all S is P* 
(original) all M is P all S is M some S is not P 
(modified) all M is P all S is M a is S a is not P  ∴ a is M {from lines 2 and 3} ∴ a is P {from "a is M" and line 1} – and this contradicts line 4 
(valid) no A* is B* ∴ no B is A 
(inconsistent) no A is B some B is A 
(modified) no A is B x is B x is A 
∴ x is not A {from lines 1 and 2} – and this contradicts line 3 
(valid) ∴ all A is A* 
(inconsistent) some A is not A 
(modified) x is A x is not A 
not needed, since lines 1 and 2 contradict 
(valid) some A is B all B* is C no C* is D* ∴ some A* is not D 
(inconsistent) some A is B all B is C no C is D all A is D 
(modified) x is A x is B all B is C no C is D all A is D 
∴ x is C {from 2 and 3} ∴ x is not D {from "x is C" and 4} ∴ x is not A {from "x is not D" and 5} – and this contradicts line 1 
"a" and "b" in any statement. 
(valid) a is b ∴ b* is a* 
(inconsistent) a is b b is not a 
∴ a is not a {from line 2 using line 1} – and this is a selfcontradiction 
(valid) ∴ a* is a* 
(inconsistent) a is not a 
not needed, since the line is a selfcontradiction 
(valid) x is A y is x y is B all B is C ∴ some A is C 
(inconsistent) x is A y is x y is B all B is C no A is C 
∴ x is B {from line 3 using line 2} ∴ x is C {from "x is B" and 4} ∴ x is not A {from "x is C" and 5} – and this contradicts line 1 
by Harry J. Gensler, last modified on 21 December 2009