Statement Type Form Diagram of Statement Meaning Negated if Diagram of Negation
Universal Affirmative All A are B. Every member of A is also a member of B. There is at least one A not in B.
Universal Negation No A is B. No member of A is also a member of B. There is at least one A in B.
Particular Affirmative Some A are B. At least one member of A (possibly all) is also a member of B. No members of A are in B.
Particular Negation Some A are not B. At least one member of A (possibly all) is not a member of B. Every A is in B.

 

The diagram below summarizes the relationships between these types of statements and their respective negations.

 

 

Rules and Fallacies

 

In contemporary logic (Boolean interpretation) there are five rules to which categorical syllogisms must conform if they are to be valid.  If one of these rules is violated, a specific formal fallacy is committed and the syllogism is invalid.  If none of these rules is violated, the syllogism is valid. 

 

These rules provide a convenient cross-check for validity against the mood and figure method or the Venn diagram method for checking validity.

 

 

A.  RULES BASED ON THE CONCEPT OF DISTRIBUTION:

 

The first two rules are based on the concept of distribution.  REMEMBER: a term is distributed when it refers to every member of the class it denotes.  It is easy to determine when a term is distributed.  Simply note the type of proposition in which it is used (A, E, I, or O).  The terms are distributed as follows:

 

          Statement type     Term(s) distributed in this type of statement

                   A                                    Subject

                   E                                    Both Subject and Predicate

                   I                                     Neither

 O                                    Predicate

 

 

Rule 1:        The middle term must be distributed at least once.

                   Fallacy: Undistributed middle.

 

Rule 2:        If a term is distributed in the conclusion, then it must be distributed in the premise in which it occurs.

Fallacies: Illicit major, Illicit minor.

(If a term is distributed in the conclusion, it makes a claim about every member of the class it denotes.  If the term is not distributed in the premises, it makes an assertion about only part of the class.  Hence the conclusion would go beyond what is justified by the premises, and the syllogism would be invalid.)

 

            Rule 3:        Two negative premises are not allowed.

                             Fallacy:  Exclusive premises.

 

Rule 4:   A negative premise requires a negative conclusion, and a negative conclusion requires a negative premise.    

                   Fallacies:  Drawing an affirmative conclusion from a negative premise.  (It is not necessary for both premises to be negative for this fallacy to be committed.   If either premise is negative, you cannot draw an affirmative conclusion.)

                                                          Or

                   Drawing a negative conclusion from affirmative premises.  (If both premises are affirmative, you cannot draw a negative conclusion.)

 

 

Rule 5.        If both premises are universal, the conclusion cannot be particular.

                   Fallacy:  Existential fallacy.

 

A Few Valid Syllogism Forms 

  AAA, EAE, AII, EIO, AEE, AOO,  IAI, OAO,

 

 

Pres. Bush speech, Jan 22, 2002. Pres. thanks the military.

 

I know I'm like many moms -- many dads, and Laura is like many moms who yearn for peace.  We want nothing more than our children to be in a peaceful world.  But I understand that in order to defeat the evil ones, we must use the mighty U.S. military to put -- after we have put them on notice, to rout them out of their caves and to bring them to justice.  And that's exactly what our nation will do.  (Applause.)

Our military has performed brilliantly.  I gave them a task with clear objectives, and they're accomplishing those tasks and those objectives.  I said real clear to the world that -- real clearly to the world -- to old West Texan in me slipping out -- (laughter) -- clearly to the world, I said that either you are with us or you are against us when it comes to finding terror.  (Applause.)  I'm proud to report many, many, many nations have signed up to be with us.