The Categorical Proposition in Logic

 

 © 2011 by Jensen dG. Mañebog    

IN LOGIC, the statement that relates two classes or “categories” is called a categorical proposition. The classes in question are denoted respectively by the subject term and the predicate term. In effect, this type of proposition gives a direct assertion of agreement or disagreement between the two terms. The proposition asserts that either all or part of the class denoted by the subject term is included in or excluded from the classes denoted by the predicate term. Here are some examples of categorical statement.

 

1. All dogs are mammals.

2. No acids are bases.                                                                                 

3. Some philosophers are mathematicians.

4. Some Americans are not cheaters.

                                               

         The first example asserts that the whole class of dogs are included in the class of mammals; the second declares that the entire class of acids are excluded from the class of bases; the third states that a part of philosophers are included in the class of mathematicians; and the last one claims that a part of the class of Americans are excluded from the class of cheaters.

 

The “quantity” of a categorical statement is either universal orparticular, depending on whether the statement makes a claim about all members or just some members of the class denoted by the subject term. The quantity of the subject term determines the quantity of the whole proposition. Since the propositions which take the form “All S are P” and “No S are P” obviously assert something about every member of class S, they are considereduniversal.

            e.g. All balls are round objects; No fish are rational being.

 

Statements that take the form “Some S are P” and “Some S are not P” logically assert something about “at least one but not all” members of class S and hence, are particular.

e.g. Somepoliticians are females; Some government officials are not corrupt.

 

In summary, the quantity of a categorical statement can be determined through the quantifier of the subject term: “All” and “No” imply universal quantity, while “some”, particular quantity.

 

The “quality” of a categorical statement is either affirmative or negative, depending on whether the statement affirms or denies class membership. Propositions which have the form “All S are P” and “Some S are P” are affirmative because in these forms, the subject asserts the predicate.

            e.g. All rocks are solid objects; Some animals are carnivores.

 

On the other hand, the predicate is denied or negated by the subject in the statements which have the form “No S are P” and “Some S are not P.” Hence, they are negative.

e.g. No women are priests (=All women are not priests); Some animals are not aquatic creatures.

 

Evidently, the copula indicates the quality of a proposition. If the copula is affirmative (am, is, are), the entire proposition is affirmative, regardless of whether or not the terms are negative. Similarly, negative copula (am not, is not, are not) necessarily makes the whole proposition negative.

 

Combining the quality and quantity of propositions results to four structures of statements known as the four standard forms of categorical propositions:

 

1. Universal Affirmative where the whole of the subject class is included in the predicate class.

 

2. Universal Negativewhere the whole of the subject class is excluded in the predicate class.

 

3. Particular Affirmativewhere a part of the subject class is included in the predicate class.

 

4. Particular Negativewhere a part of the subject class is excluded in the predicate class.

 

Since the early Middle Ages, the four types of categorical statements have been designated by letters:

 

A for Universal Affirmative (All S are P)

E for Universal Negative (No S are P)

I for Particular Affirmative (Some S are P)

O for Particular Negative (Some S are not P)

 

The letters A and I come from the first two vowels of the Latin word “affirmo”, which means “I affirm” and are thus assigned to two affirmative propositions. Letters E and O come from the two vowels of the Latin word “nego” which means “I deny” and are so designated to two negative propositions.

          At this point, it should be made clear that a categorical statement, in standard form, contains four elements:

 

1. Quantifier – the word thatindicates the range of individuals or items referred to in the subject term (All for A proposition; No for E proposition; Some for I and O propositions).

 

2. Subject term– that which designates the idea about which the pronouncement is made. (e.g. the word “men” in All men are mortals.)

 

3. Predicate term– that which designates the idea which is affirmed or denied of the subject. (e.g. the word “mammals” in All dogs are mammals.)

 

4. Copula – is the linking verb is or is not (am, am not, are, are no,) expressing the agreement or disagreement between the subject term and the predicate term.

 

As our purpose in Logic is to study the mode in which the mind represents the real order, the question of present, past or future is purely accidental. The time-determination does not affect the mental representation as such. Therefore, the copula should always be in the present tense of the verb “to be” as it must express the present act of the mind.

          Notice that even propositions which refer to some past or future event can be reduced to present tense without altering the meaning. For instance, The Republicans did not win the last election can be translated as The Republican party is not the party which won the last election.

 

Notice that the four standard forms of categorical propositions follow the same structure: Quantifier-Subject term-Copula-Predicate term (Q-S-C-P). This arrangement is the logical structure of a sentence which is in the indicative mood, stated in the present tense, and the (logical) predicate of which is separated from the copula.  Having this form, sentences can be used in logical arguments.

            Many ordinary discourse statements do not display their logical form. To meet the needs of Logic, we have the right to change their wordings as long as the original meaning of the judgment remains the same. Thus, sentences have to be translated in a manner to conform to the Q-S-C-P form that they may be used in logical processes and analysis. In many occasions, a statement may appear clumsy or unusual when converted to logical form, but we are not concerned here with beautiful prose but with the substance of the thought expressed.

 

The following steps are helpful in transforming ordinary discourse statements into standard form:

1. Identify the two classes in the statement. Make sure to have two nouns or noun phrases which categorize kinds of things. Where a class term is not mentioned, use a parameter or category that captures the kind of things referred to (people, objects, places, etc.).

2. Identify which class is included in or excluded from the other. Use the former as the subject term, the latter as the predicate term.

3. Use the quantifier “All” if all of the subject class is included in the predicate term; “No” if all of the subject class is excluded from the predicate term; “some” if only part is included in or excluded from the predicate class.

4. If the subject term is totally or partly included in the predicate term, then the statement is affirmative; if the subject term is totally or partly excluded from the predicate term, then the statement is negative.

5. After determining the proper quantifier, copula, and two proper class terms, write the correct A, E, I, or O statement following the Q-S-C-P order.

 

The following are some of the guidelines which could further help in properly reducing ordinary statements to the standard categorical form:

1. Translate general statement as universal statement, unless it points to a “particular” usage.

          The statement Dogs bark should therefore take the quantifier “All”. But the proposition Books are expensive should take the quantifier “Some”.

 

2. Add the missing complement to an adjective or to a describing phrase to show that they refer to classes.

 

3. Quantifiers that indicate universality or particularity should be replaced by “all” or “no” or “some”, correspondingly.

“All” should be used in place of every, any, everybody, always, anything, everything, whoever, wherever, whatever, etc.

A prayerful student is always successful.  →  All   prayerful students aresuccessful individuals.

 

“No” should be used in place of no one, nobody, never, nothing, none, etc. 

Ex:  Nobody plays. →No persons are players.

       Nothing is permanent.→  No things are permanent things.

      My students never come late. → No students of mine are latecomers.

 

“Some” should be used in place of many, several, a few, certain, most, twenty-four, 90%, majority, minority, etc.

        24 firemen are brave.→Some firemen are brave people.

       Minority of the directors likes Bong Bread. →Some directors are personswho like Bong Bread.

 

4. Exclusive statements should be translated into universal statements by dropping the word-indicators of exclusivity ('None but, 'Only’, and 'None except^’) and reversing the order of the original statement.

      Only toddlers are pre-schoolers.→All pre-schoolers are toddlers.

      None except declarative sentences are statements.  →  All statementsare declarative sentences.

 

5.   Exceptive statements, that is, those that begin with ‘All except’ and‘all but’, may be translated intoE or an A proposition. In most cases however, E proposition is preferred.

All but Marxists are capitalists. →(E )  No Marxists are capitalists. / (A) All who are not Marxists are capitalists.

 

6. Statements that begin with “Not all” should be translated as an O statement.

 

7. “...Cannot be both..” statements must be reduced into an E proposition.

 

8. Basic conditional (‘If …..then’) statements must be reduced into universal propositions.

a) If A is B, then A is C →All B are C.

b) If A is B, then A is not C. →No B are C.

e.g. If it is a whale, then it is not a reptile.  →  No whales are reptiles.

 

There are kinds of sentences which are better left unchanged. For instance, to reduce singular statements like“Andrew is a priest” as “All persons who are like Andrew are priests” (as some authors suggest) is to have a problematic translation. Same is the case with statements that begin with article “a” or “an” like “A dog is sleeping under the table.”

            In using these sentences in logical processes, some guidelines have to be considered nonetheless. Sentences that begin with the article “a” or “an” are usually construed as universal, unless the context ascertains particularity. Thus, “A horse is a mammal” is considered as a universal proposition (A) while “A horse is sick” is treated as particular (I).

          Singular statements should also be treated as universal statements. Thus, “Sigmund Freud is a neurologist” is considered as an A proposition while “Carl Jung is not a mathematician”, an E statement.

 

Note: For logical reasons, discussion on Square of Opposition is deliberately placed under ‘Inference and Reasoning’. Furthermore, Distribution of Terms is discussed under the topic ‘Categorical Syllogism’ because knowledge about it is essentially needed in determining the validity of syllogistic arguments.

 

Jensen dG. Mañebog, the contributor, is a Debate and Philosophy professor in a university in Quezon City, Philippines.

 

Jensen dG. Mañebog. “The Categorical Proposition in Logic” @ www.OurHappySchool.com

 

1. Give 5 original examples for each standard form of categorical proposition (in a yellow paper or Microsoft Word file (.doc.) to be submitted to your professor’s e-mail address).

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