# IQ Test Labs

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### Syllogistic arguments

When evaluating the validity of syllogistic arguments it helps to gain confidence working with Euler diagrams.

Practice drawing accurate diagrams that represent the relationships between the statement types and which help to solve syllogisms visually.

• Positive statements: (a) Some Sulptors are Magicians, (b) All Programmers are Maximalists.
• Negative statements: (a) No Plumbers are Minimalists, (b) Some Sailors are not Medalists.
• Particular statements: (a) Some Sailors are not Medalists, (b) Some Painters are Mentalists.
• Universal statements: (a) All Programmers are Maximalists, (b) No Swimmers are Motorists.

### 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.