Trilemmas Define Syllogistics, 2 of 11

 Various Trilemmas

 

 

The previous section’s “Superman” trilemma, and associated troika and triad, is a special case of the Equality Glitch:

 

A has property P;

B does not have property P;

A equals B

 

The associated troika is:

 

Moe: A has property P; B hasn’t property P; A doesn’t equal B

Larry: A has property P; B has property P; A equals B

Curly: A hasn’t property P; B hasn’t property P; A equals B

 

          The associated rule triad is:

 

From          A has property P;

And            B does not have property P;

Deduce      A does not equal B

 

From          A equals B;

And            A has property P;

Deduce      B has property P

 

From          B does not have property P;

And            A equals B;

Deduce      A does not have property P

 

Here is the Implication Glitch:

A is true;

A implies B;

B is false.

 

This trilemma is the voter’s-paradox supported by this troika:

Moe: A is true, A implies B, B is true.

Larry: A implies B, B is false, A is false.

Curly: B is false, A is true, A does not imply B.

 

Its rule triad is:

From: A is true; A implies B; Deduce: B is true.

From: A implies B; B is false; Deduce: A is false.

From: B is false; A is true; Deduce: A does not imply B.

 

The first is Modus Ponens, the second is Modus Tollens, and the third deserves its own name. I propose “Anti-implication”.

 

Related to Implication glitch is Equivalence Glitch:

A is true;

A is equivalent to B;

B is false.

 

Also related to Implication glitch is Modus Ponens Breakdown:

All A are B;

all B are C;

Some A are not C.

I also call this the  Barbarism Trilemma because it defies the classic Aristotelian syllogism, BARBARA. Its rule triad is:

From: All A are B; all B are C; Deduce all A are C.

From: All B are C; some A are not C; Deduce some A are not B.

From: some A are not C; all A are B; Deduce some B are not C.

 

         Its troika is:

Moe: All A are B; all B are C; all A are C.

Larry: All B are C; some A are not C; some A are not B.

Curly: some A are not C; all A are B; some B are not C.

 

 

Here is the Weak And Glitch:

A is true;

B is true;

A and B is not true

 

Its troika is:

          Moe: A is true, B is true, (A and B) is true

          Larry: B is true, (A and B) is not true: A is not true

          Curly: (A and B) is not true; A is true; B is not true

 

Its rule triad is:

          From A is true; B is true;  Deduce: (A and B) is true

          From B is true; (A and B) isn’t true; Deduce: A isn’t true

          From (A and B) isn’t true; A is true; Deduce: B isn’t true

 

Here is the Strong Or Glitch:

Not-A is true;

Not-B is true;

A or B is true

 

Its troika is:

          Moe: not-A is true; not-B is true; not-(A or B) is true

          Larry: not-B is true; (A or B) is true; A is true

          Curly: (A or B) is true; not-A is true; B is true

 

Its rule triad is:

          From not-A is true; not-B is true; Deduce: not-(A or B) is true

          From not-B is true; (A or B) is true; Deduce: A is true

          From (A or B) is true; not-A is true; Deduce: B is true

 

 

            In the Reductio Trilemma, reductio ad absurdum fails:

                        A implies B

                        A implies not-B

                        A

            It’s associated with this triad:

                        From:  A implies B; A implies not-B; Deduce: not-A

                        From:  A implies not-B; A; Deduce: A does not imply B

                        From:  A; A implies B; Deduce: A does not imply not-B

            It’s associated with this troika:

                        Moe:  not-A

                        Larry: A and not-B

                        Curly: A and B

 

            Here is the Reductio Trilemma’s dual; the Failed  Dilemma:

                        A implies B

                        Not-A implies B

               Not-B

           

            It’s associated with this triad:

                        From: A implies B; not-A implies B; Deduce: B

                        From: not-A implies B; not-B; Deduce: A does not imply B

                        From: not-B; A implies B; Deduce: not-A does not imply B

 

            It’s associated with this troika:

                        Moe:  B

                        Larry: not-B and A

                        Curly: not-B and not-A

 

 

 

Here is a Linear Loop Trilemma:

          A > B;

          B > C;

          C > A.

 

For instance:

A crook is better than a fool;

A fool is better than a wimp;

A wimp is better than a crook.

 

         This voter’s paradox is supported by this troika:

Moe:

A crook is better than a fool, and a fool is better than a wimp.

Larry:

A wimp is better than a crook, and a crook is better than a fool.

Curly:       

A fool is better than a wimp, and a wimp is better than a crook.

 

 

 

 

         

 

Here is the Constancy Trilemma:

 

          Some X has property P;

          Some Y does not have property P;

          Property P is constant.

 

          It is supported by this troika:

          Moe: P(x), not P(y), P is not constant.

          Larry: P is constant, P(x), P(y)

          Curly: not P(y), P is constant, not P(x)

 

          For instance:

          Some men are good;

          Some men are not good;

          All men are equally good.

         

It yields this triad:

          If some men are good, and some men are not good,

then not all men are equally good.

          If some men are not good, and all men are equally good,

then no men are good.

          If all men are equally good, and some men are good,

then all men are good.