Lecture 7

Categorical Propositions

and

Immediate Inferences

 

Categorical Propositions:  Any proposition which relates two or more classes of objects

 

All categorical propositions are divided into two distinct logical parts which are associated with the terms which make up the object classes being compared

 

I. Parts of Categorical Propositions:

 

A. Subject Term - first category or class

 

B. Predicate Term - second category or class

 

Examples:

      

              Cats are members of the mammal family.

                   ­                                       ­

       Subject Term                 Predicate Term

 

 

              Asteroids are astronomical objects.

                       ­                                  ­

         Subject Term            Predicate Term

 

 

C. Copula - that which links the terms of a C. Proposition together

 

Examples:

 

              Cats are members of the mammal class.

                           ­

                   Copula

 

 

Asteroids are astronomical objects.

            ­

                             Copula

 

D. Quantifies - the words which specify the quantity of the subject class which is related to the predicate class

 

1. Universal:

 

a) 'All' - inclusive of a whole class

 

b) 'No' - exclusive of a whole class

 

2. Particular:  'Some' - inclusive of part of a class

 

 

 


Note:  We should keep in mind that the quantifiers ('all', 'no', and 'some') are implied in all categorical propositions even though they are not explicitly stated.  If these quantifiers are not explicitly used in the formulation of the proposition it is said to be in non-standard form.  The standard form for all categorical propositions is the following:

 

                     

All S are P.

                      No S are P.

                      Some S are P.

                      Some S are not P.

 

 


If our examples above were in standard form they would look like this:

 

All cats are members of the mammal family.

 

All asteroids are astronomical objects.

 

E. Quality - the kind of affirmation made by the proposition

 

1. Affirmative - asserts a quality to a class

 

2. Negative - denies a quality to a class

 

When we put the quality and quantity of categorical proposition together we see that there are four and only four possible arrangements:

 

Universal Affirmative                         Particular Affirmative

Universal Negative                           Particular Negative

 

 

 

 

Thus, there are only four types of standard form categorical proposition possible.  As a shortcut we may tag each with a letter as follows:

 

              A    Universal Affirmative  All S are P.

              E     Universal Negative      No S are P.

              I      Particular Affirmative  Some S are P.

              O    Particular Negative     Some S are not P.

 

F. Distribution - the quantity associated with either the subject or predicate term of a categorical proposition

 

Definition:  a term is 'distributed' if the proposition makes an assertion about every member of the class denoted by the term.

 

 

1. Universal Affirmative - All S are P.

 

                                

 

 

2. Universal Negative - No S are P.

                         

 

 


Note:  in light of this distribution we might be tempted to formulate the E proposition as "All S are not P."  But this formulation is ambiguous since it could mean one of two different things:  suppose we said "All pit bulls are not mean."  This could be interpreted as

 

       1. No pit bulls are mean, or

       2. Some pit bulls are not mean

 

Without a specific context it is impossible to determine which is the intended meaning.  Thus, even though both terms are distributed in the E claim, we always use the standard form "No s are P."

 

 


3. Particular Affirmative - Some S are P.

 

                                                            

 

 

4. Particular Negative - Some S are not P.

                         

 

II. Categorical Propositions and Immediate Inferences:

 

 

Aristotle first divided the proposition horizontally by quantity and vertically by quality.

                                

From this arrangement Aristotle derived four logical relations       

 

       1. From A to E and E to A

     2. From I to O and O to I

     3. From A to O and O to A, and from E to I and I to E

     4. From A to I and I to A, and E to O and O to E

 

 

A. Aristotle's Square of Opposition and Immediate Inference

                                

 

1. Contrary - at least one claim is false (they cannot both be true)

      

2. Subcontrary - at least one is true (they cannot both be false)

 

3. Contradiction they have opposite truth values

 

4. Subalternation - truth of the universal implies the truth of the particular, and falsity of the particular implies the falsity of the universal.

 

B. Modern Square of Opposition and Immediate Inference

 

1. Universal propositions (Boolean)

 

            All S are P. = No members of S are outside P.

            No S are P. = No members of S are inside P.

 

2. Particular propositions (Boolean)

 

    Some S are P. = Some S exists, and it is a P.

    Some S are not P. = Some S exists and it is not a P.

 

The Boolean interpretation of the Categorical Propositions leaves only one possible relationship for immediate inference:  Contradiction.

                                   

 

3. Existential Fallacy - occurs when the traditional square is used in conjunction with non-existent entities.

 

 

III. Categorical Propositions and Venn Diagrams:

 

The most useful device for understanding the nature of modern Categorical Propositions is the circle diagram of John Venn.  A Venn diagram is a model for some universe of discourse.

 

                    

 

Note that there are four regions in the diagram each corresponding to some element of a categorical proposition:

 

 

1. The region of S and only S (not P).

 

2. The region of S and P (both S and P).

 

3. The region of P and only P (not S).

 

4. The region of not S, and not P, and not S and P (i.e., what lies beyond both S and P (not S and not P).

 

 

             

 

UA/A All S is P. (None of S is outside of P.)

 

                            

 

 

UN/E No S is P. (None of S is inside of P.)

                            

 

                     

 

PA/I Some S is P. (At least one S is inside of P.)

                            

 

             

PN/O Some S is not P. (At least one S is not inside of P.)

                            

 

 

 

IV.  Translating Ordinary Sentences to Categorical Propositions:

 

 

A. The main rule to observe is that all standard form CPs must have a subject term, predicate term, quantifier, and copula in the following form:

 

Quantifier - Subject Term - Copula - Predicate Term

 

In the case of ambiguous sentences, isolate what you think is the primary meaning, then follow the guidelines below:

 

1. Where there is a Term without a noun:  supply the missing noun

 

2. Nonstandard Verbs:  always replace with 'are' and 'are not'

 

3. Singular Propositions:  translate as 'all' with a parameter

 

4. Adverbs and Pronouns: translate as follows -

 

a) spatial adverbs - 'places'

 

b) temporal adverbs - 'times'

 

c) pronouns - 'persons' or 'things'

 

5. Unexpressed Quantifiers:  choose the probable quantity

 

6. Nonstandard Quantifiers: translate as follows -

 

a) few - 'some'

 

b) any or anyone - 'all'

 

c) not every or not all - 'some ______ are not'

 

 


Note:  a statement beginning with 'few' must be translated as a compound I/O proposition.

 

 


7. Conditionals:  translate as follows -

 

a) if . . . then - 'all' or 'no'

 

b) where 'if' occurs in the middle - move 'if predicate' to beginning and translate 'all' or 'no'

 

c) transposition - negate both antecedent and consequent and switch their order, then translate 'all' or 'no'

 

8. Exclusive Propositions:  conditionalize then categorize

 

 

 

 


Note:  'only' and 'none but' at the beginning require term reversal

 

 


9. 'The only' - translate as 'all'

 

10. Exceptive Propositions - must be translated as a compound E/A proposition