ArmandLTan Lecture Notes
For class discussions only
SillimanUniversity
CATEGORICAL STATEMENTS
Categorical logic, often referred to as traditional or classical logic, is a deductive system of reasoning developed by the ancient Greek philosopher Aristotle (bc. 384-322).For obvious reason, it is also called Aristotelian logic.
s25. INTERPRETATIONS OF CATEGORICAL STATEMENTS
A categorical statement is a statement that asserts or denies something without condition or qualification. Itis thought to contain four components: quantifier, subject term, copula, and predicate term. The quantifier is expressed by thewords, `all,'`no,' and `some.' The copula is expressed by such words as `are' and `are not.' The classic example of a categorical statement is the frequently quoted premiseof Aristotelian logic: `Allmen are mortal.' In this statement, `men' is the subject term and `mortal' is the predicate term.
In classical logic, a categorical statement is usually understood as an assertion about classes or categories. These classes are designated by the subject term and the predicate term of any given categorical statement. A part of any class is called its member. Members of the subject class mayeitherbe totally or partially included in or excluded from the predicate class.Based on this interpretation of class inclusion and exclusion, four possible types of categorical statements may be stated:
1. those that assert total inclusion of the subject class in the predicate class; 2. those that assert total exclusion of the subject class from the predicate class; 3. those that assert partial inclusion of the subject class in the predicate class; 4. those that assert partial exclusion of the subject class from the predicate class. Having the same subject term and predicate term, these various types are illustrated by the following statements:
(1). All scholars are scientists.
(2). No scholars are scientists.
(3). Some scholars are scientists.
(4). Some scholars are not scientists.
Where S and P represent respectively the subject term and the predicate term, these various types of categorical statements are expressed in abbreviated forms as follows:
1. All S are P.
2. No S are P.
3. Some S are P.
4. Some S are not P.
Page 2
25a. The Quantity and Quality of Statements
Every categorical statement has a quantity. The quantity of the statement is either universal or particular.
Every categorical statement also has a quality.Since a categorical statement either affirms or denies, its quality is either affirmative or negative.
The letter names for the four types of categorical statements are: A and I for affirmative statements and E and O for negative statements. Accordingly, these letter names were derived from the Latin words `AffIrmo' (I affirm) and `nEgO' (I deny).
Affirmative All S are P A
Universal <
Negative No S are P E
Affirmative Some S are P I
Particular <
Negative Some S are not P O
25b. Distribution
The term `distribution' refers to the total or partial inclusion or exclusion of the terms of the statement. A term is said to be distributed if such term is totally included in or excluded from the other term. Thus both the A and E statements distribute their subject termssince A asserts total inclusion of S in P and E asserts total exclusion of S from P.
On the other hand, a term is said to be undistributed ifsuch term is partially included in or excluded from the other term. Thus both the I and O statements do not distribute their subject terms since the I asserts partial inclusion of Sin P and the O asserts partial exclusion of S from P.
As for the predicate terms, both the A and I statements do not distribute their predicate terms since the A asserts partial inclusion of P in S as well as the I. However both the E and O distribute their predicate terms since the E asserts total exclusion of P from S; so does the O.
A: S (dt) -- P (un)
E: S (dt) -- P (dt)
I: S (un) -- P (un)
O: S (un) -- P (dt)
Page 3
s26. THE SQUARE OF OPPOSITION
Contrary
superaltern (A) (E) superaltern
\ /
\ /
sub \ / sub
al \ / al
ter Contra\d/iction ter
na / \ na
tion / \ tion
/ \
/ \
subaltern subaltern
(I) (O)
Subcontrary
To complete our picture of the Square of Opposition, the following tables may be used as a guide for validating immediate inferences or elementary argument forms such as those based on one or a single premise.
A : True : False:
-----:------:------:
E : 0 : UN :
I : 1 : UN :
O : 0 : 1 :
-----:------:------:
E : True : False :
------:------:------:
A : 0 : UN :
I : 0 : 1 :
O : 1 : UN :
------:------:------:
S27. CONVERSION OF STATEMENTS
27a. Conversion. A statement is said to be the converse (the converted statement) of any given statement (called the convertend) if it has the same form and terms but expressed in reverse order. Accordingly, both E and I have equivalent converses for they have identical truth values when their terms are transposed. For instance,
the converse of `no idealists are realists' is `no realists are idealists' and the converseof `some teachers are women' is `some women are teachers.'
Page 4
This means to say that as a type of immediate inference, conversion, which proceeds by reversing the terms of the statement, is valid for E and I.
In the case of A, there is no equivalent or true converse.For the converse of `all priests are persons who believe in God' (which is true) is `all persons who believe in God are priests'(which is false). Accordingly, though A has no exact or true converse, we can have what logicians call `conversion by limitation' (or per accidens) since the particular converse has somehow diminished the universal quantity of the convertend. In other words, the converse will retain the truth value of the convertend, though expressed in its particular form. Thus `all Filipinos areAsians' is just as trueas`someAsiansare Filipinos.'
Equally for the O, there is generally no true converse. For the true statement, `some human beings are not priests' will have as its converse the false statement, `some priests are not human beings.'
In inference by conversion, it is important tonote that statements whose terms mean the same by definitionmay have equivalent converses. Such terms as bachelor/unmarried man, famous/well known, neat/clean and the like can be switched inan A statement.But this should be an exception rather than the rule. The following table summarizes our discussion of conversion:
Convertend : Converse
------:------
A : I: Some P are S (by limitation)
E : E: No P are S
I : I: Some P are S
O : Not in general equivalent
27b. Obversion. With obversion the procedure involves adding the word `non' to the predicate term and changing the quality of the statement. For example, the A statement, `all mathematicians are intelligent' has its obverse the E statement, no mathematicians are non-intelligent.' This shows that only the quality but not the quantity of the statement is changed. In our example, both statements are universal, but the givenstatement is affirmative and the obverse is negative. Unlike that of conversion, the subject and the predicate terms of the original statement (called the obvertend) remain in their original position. Like the converse, the obverse must have the same truth value asthe obvertend.The following table summarizes our discussion of obversion.
Page 5
Obvertend | Obverse
------
A | E: No S are non-P
E | A: All S are non-P
I | O: Some S are not non-P
O | I: Some S are non-P
27c. Contraposition. Conversion by contraposition can be formed in two ways: The first is to switch the subject and the predicate terms (as in conversion) and then add the word `non' to both terms. But the problem with this procedure is that it applies only to both A and O statements.
The second way of forming a contrapositive is to obvert the converse of the obverse of any given statement. In otherwords, given a statement (called the premise), the procedure is 1. to obvertthe premise; 2. convert the obverse of the premise and again 3. obvert the converse of the obverse of the premise.The following example illustrates this procedure:
All A are B (premise)
1. No A are non-B (obverse of the premise)
2. No non-B are A (converse of the obverse)
3. All non-B are non A (obverse of the converse)
So the contrapositive of `all A are B' is `all non-B are non-A.' As can be seen, the subject term and the predicate term are switched and the word `non' is added to both terms.
Accordingly, only the A and the O statements have logically equivalent contrapositives, which means that they have identical truth values as their premises.The contrapositive of E can only be formedby limitation.
The following table summarizes our discussion of contraposition:
Premise | Contrapositive
------|------
A | A: All non-P are non-S
E | O: Some non-P are not non-S (by
limitation)
I | none
O | O: Some non-P are not non-S
s28. BOOLEAN MODIFICATION AND VENN DIAGRAM
Modern logic claims that universal statements, whether affirmativeor negative, do not necessarily imply the existence of members of their subject classes. They are said to have no `existential import.’ This idea is based onthe interpretation of the so-called
Page 6
`non-existential universal' following the view of the British mathematician and logician, George Boole (1815-1864).
The difference in terms of existentialpresupposition or assertion between these two categorical statements
(1) all nurses are sensitive helpers
and
(2) all dragons are romantic animals
is obvious. The interpretation of traditional logic concerning a universal statementsuch as (1) as having existentialimport couldhardly be (justifiably) applied to (2). From themodern (Boolean) perspective, however, (1)is interpreted to meanthat the class of non-sensitive nurses has no members or is emptyof members.The same applies to (2), that is, that the class of non-romantic dragons is empty of members. Of course there are no non-romantic dragons since dragons do not exist in the first place.
Perhaps no logician has better clarified the Boolean interpretation of non-existential universals than his follower, John Venn (1834-1923), who in 1880 devised a famous diagram that now bear his name.
In summary, the Venn diagrammatic representations of the four categorical propositions or statements of Aristotelian logic are:
S P S P
All S are P No S are P
S P S P
X X
Some S are P Some S are not P
Due tothese alternativeinterpretationsof categorical statements, certain modifications in the traditional Squareof Opposition have been made. If both A and E are claimed to be non-existentialstatements,then
Page 7
they are not contraries, for they assert no contrary relation or position, and thus can both be true. On the other hand, though both I and O are claimedto be existential statements, they can both be false if their subject classes are deemed empty of members. Hence, they assert no subcontrary relation or opposition.
To accept such an interpretation would leave only the relationof contradiction valid for certain immediate inferences. Inferences by subalternation is considered invalid because, unlike their particular forms, universal statements arethought to assert no existential import. And so are the inferences based on conversion (and equally contraposition) by limitation.Togetherwiththe Venn diagrammatic representationof categoricalstatements,the following modified version of the oppositionalrelation between categorical statements represents themodern (Boolean) perspective (Figure 6) of theSquareof Opposition.
A: SP=0 E: SP=0
Figure 6
I: SP=0 O: SP=0
X X
As can be observed, only the contradictory relations hold betweenthe universal statements and their particular formsin the Square of Opposition. For clearly the non-existential universalsareequal to zero; whereas theexistential particulars are not equal to zero. On the whole, this is all that is left of the Aristotelian inferential relations between categorical propositions or statements.
s25 EXERCISES
Identify the subject term, predicate term, copula, and the quantifier in each of the following statements.
1. Some soldiers are men of honor.
2. No dogs with long ears are wild animals.
3. Some school administrators are persons with low IQ.
4. No mathematicians are men who have squared the circle.
5. Some women with no M.A. are university professors.
6. All PAL passengers are travelers who are always late.
7. Some sensitive writers are critical of the status quo.
8. All Marxist-oriented theologians are students who do not believe in the last days.
Page 8
s25a/b EXERCISES
State thequantity and the quality of the statements in s25 exercises, and then determine which terms are dt.
s26 EXERCISES
I. Determine the truth and falsity of the following assuming the given statement is true and assuming it is false.
1. (T) All theologians are optimists. (F)
Some theologians are optimists.
No theologians are optimists.
Some theologians are not optimists.
2. (T) Some ducks are wild animals. (F)
No ducks are wild animals.
All ducks are wild animals.
Some ducks are not wild animals.
*3. (T) No mathematicians are scientists. (F)
Some mathematicians are scientists.
Some mathematicians are not scientists.
All mathematicians are scientists.
4. (T) Some pastors are not men of good will. (F)
All pastors are men of good will.
Some pastors are men of goodwill.
No pastors are men of good will.
II. Determine thevalidity of the following arguments using the square of opposition.
1. It is false that all men are mortal. Hence some menare not mortal.
2. Some roses are red. So it is false that some roses are not red.
3. No lazy person are rich people. So some lazy person are rich people.
4. It is false that some birds are not animals. / all birds are animals.
5. Some evolutionists are monkeys. / some evolutionists are not monkeys.
6. It is false that no scholars are ignorant./ some scholars are not ignorant.
7. No square-circles are mathematical figures. / some square-circles are not mathematical figures.
8. No skeptics are persons who believed in God. /some skeptics are persons who believed in God.
Page9
s27 EXERCISES
I. State the converses, obverses, and the contra positives of the following statements.
1. All spies are liars. 2. No idealists are politicians.
3. Some flowers are not violets. 4. All fanatics are morons.
5. Some college teachers are sensitive persons.
II. In the light of our discussions of conversion, obversion, and contraposition, determine the validity of thefollowing arguments.
*1. All spies are liars.
/ all liars are spies.
2. No fanatics are idealists.
/ some idealist are fanatics.
3. Some philosophers are symbolic logicians.
/ some philosophers are not non-symbolic logicians.
4. Some birds are non-wild animals.
/ some non-wild animals are birds.
5. No neurotics are good writers.
/ all non-good writers are neurotics.
6. All children are considerate.
/ all non-considerate persons are non-children.
7. Some politicians are not liars.
/ some non-liars are politicians.
8. Some psychologists are neurotics.
/ some neurotic persons are psychologists.
9. No pessimists are realists.
/ all non-realists are non-pessimists.
10. All theologians are men who do not mean what they say. / all men who do not mean what they say are theologians.
s28 EXERCISES III. Draw a Venn diagram for the following:
*1. SP=0 2. PS=0 3. PS = 0 4. SP = 0
5. SP=0 6. PS=0 7. PS = 0 8. SP = 0
Review Questions
1. What is a categorical statement?
2. State the four types of categorical statements.
3. From where did we get the symbols AEIO?
4. What is the quantity and quality of statement?
5. Define the term distribution?
6. What is meant by the word `some'?
7. What is meant by the term `complement'?
8. What is the meaning of `existential import' of statements?
9. Explain the difference between contrary and subcontrary statements?
10.Are there no exact converse for the A and O statements? Y or y not?
11. What is an immediate inference? Cite some examples.
12. Why is it invalid to infer "some P is not S" from "some S is not P"?