Common syllogism cases
Case 1 - All A is B.
Immediate inferences between A and B.
Some A is B.
Some B is A.
All A is B means that all member os class A are in class B.
The following terminology can also be used for All A is B:
- Only Bs are As.
- If anything is A, then it's a B.
- If A, then B.
Case 2 - All A is B, and All B is C.
Immediate inferences between A and B.
Some A is B.
Some B is A.
Immediate inferences between B and C.
Some B is C.
Some C is B.
Valid conclusions between A and C.
All A is C.
Some A is C.
Some C is A.
Example:
All Middlemen are Programmers,
and all Sculptors are Middlemen;
thus all Sculptors are Programmers.
Case 3 - All A is B, and All C is B.
Immediate inferences between A and B.
Some A is B.
Some B is A.
Immediate inferences between C and B.
Some C is B.
Some B is C.
Valid conclusions between A and C.
No direct connection.
Case 4 - All B is A, and All B is C.
Immediate inferences between A and B.
Some B is A.
Some A is B.
Immediate inferences between B and C.
Some B is C.
Some C is B.
Valid conclusions between A and C.
Some C is A.
Some A is C.
Example:
All Mimes are Poets,
and all Squares are Mimes;
thus some Squares are Poets.
Case 5 - Some A is B.
Immediate inferences between A and B.
Some B is A.
At least one, but not all of A, is B.
Case 6 - Some A is B, and some B is C.
Immediate inferences between A and B.
Some A is B.
Some B is A.
Immediate inferences between B and C.
Some B is C.
Some C is B.
Valid conclusions between A and C.
No direct connection.
Case 7 - Some A is B, and All B is C.
Immediate inferences between A and B.
Some A is B.
Some B is A.
Immediate inferences between B and C.
Some B is C.
Some C is B.
Valid conclusions between A and C.
Some A is C.
Some C is A.
Example:
All Musicians are Personas,
and some Stoics are Musicians;
thus some Stoics are Personas.
Case 8 - All A is B, and Some B is C.
Immediate inferences between A and B.
Some A is B.
Some B is A.
Immediate inferences between B and C.
Some B is C.
Some C is B.
Valid conclusions between A and C.
Some A is C.
Some C is A.
Example:
All Musicians are Personas,
and some Stoics are Musicians;
thus some Stoics are Personas.
Case 9 - No A is B.
Immediate inferences between A and B.
No B is A.
Some A is not B.
Some B is not A.
In other words:
No member of A is a member of B.
If A, then not B.
Case 10 - All A is B, and No B is C.
Immediate inferences between A and B.
Some A is B.
Some B is A.
Immediate inferences between B and C.
Some B is not C.
Some C is not B.
No C is B.
Valid conclusions between A and C.
Some C is not A.
No C is A.
No A is C.
Some A is not C.
Example:
No Mavericks are Performers,
and all Swimmers are Mavericks;
thus no Swimmers are Performers.
Case 11 - All B is A, and No B is C.
Immediate inferences between A and B.
Some B is A.
Some A is B.
Immediate inferences between B and C.
No C is B.
Some B is not C.
Some C is not B.
Valid conclusions between A and C.
Some A is not C.
Example:
No Managers are Painters,
and all Managers are Socialists;
thus some Socialists are not Painters.
Case 12 - Some A is B, and No B is C.
Immediate inferences between A and B.
Some B is A.
Immediate inferences between B and C.
No C is B.
Some B is not C.
Some C is not B.
Valid conclusions between A and C.
Some A is not C.
Example:
No Motorists are Photographers,
and some Sheriffs are Motorists;
thus some Sheriffs are not Photographers.
Case 13 - No A is B, and Some B is C.
Immediate inferences between A and B.
No B is A.
Some A is not B.
Some B is not A.
Immediate inferences between B and C.
Some C is B.
Valid conclusions between A and C.
Some C is not A.
Example:
No Painters are Matchmakers,
and some Squares are Matchmakers;
thus some Squares are not Painters.
Case 14 - All A is B, and No C is B.
Immediate inferences between A and B.
Some A is B.
Some B is A.
Immediate inferences between B and C.
No B is C.
Some B is not C.
Some C is not B.
Valid conclusions between A and C.
No C is A.
Some C is not A.
No A is C.
Some A is not C.
Example:
No Philanthropists are Megalomaniacs,
and some Scientists are Megalomaniacs;
thus some Squares are not Painters.
Case 15 - No A is B, and All B is C.
Immediate inferences between A and B.
No B is A.
Some A is not B.
Some B is not A.
Immediate inferences between B and C.
Some B is C.
Some C is B.
Valid conclusions between A and C.
Some C is not A.
Example:
No Perpetrators are Mentalists,
and all Mentalists are Specialists;
thus some Specialists are not Perpetrators.
Case 16 - Some A is not B.
No immediate inferences.
'Some A is not B' can be depicted by any of the three figures..
In other words:
- At least one part of A is excluded from all of B.
- Some parts of A is excluded from B.
- All B is not A.
Case 17 - All A is B, and Some C is not B.
Immediate inferences between A and B.
Some A is B.
Some B is A.
Immediate inferences between C and B.
'Some C is not B,' in other words: All B is not C.
Valid conclusions between A and C.
Some C is not A.
Example:
All Performers are Mimes,
and some Singers are not Mimes;
thus some Singers are not Performers.
Case 18 - Some B is not A, and All B is C.
Immediate inferences between A and B.
'Some B is not A,' in other words: All A is not B.
Immediate inferences between B and C.
Some B is C.
Some C is B.
Valid conclusions between A and C.
Some C is not A.
Example:
Some Moldmakers are not Poets,
and all Moldmakers are Songwriters;
thus some Songwriters are not Poets.