Abstract: The
most important properties of standard form categorical propositions are explained and
illustrated.
I. Categorical propositions and classes. |
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A. The long range goal is to give a theory of
deduction, i.e., to explain the relationship between the premisses and conclusion
of a valid argument and provide techniques for the appraisal of deductive arguments.
Hence, we will be distinguishing between valid and invalid arguments. |
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1. A deductive argument is defined as one
whose premisses are claimed to provide conclusive evidence for the truth of its
conclusion. |
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2. A valid deductive argument is one in
which it is impossible for the premisses to be true without the conclusion being true
also. |
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B. Our study of deduction, for the present, will
be about arguments stated in categorical propositions, e.g., |
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No honest people are persons who embroider the
truth.
Some politicians are persons who embroider the truth.
Some politicians are not honest people. |
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1. A categorical proposition is defined as
any proposition that can be interpreted as asserting a relation of inclusion or exclusion,
complete or partial, between two classes. |
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2. A class is defined as a collection of
all objects which have some specified characteristic in common. This is no more
complicated than observing that the class of "lightbulbs" all have the common
characteristic of "being a lightbulb." |
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Thus, we can have four class relations in the
various kinds of categorical propositions: |
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Utilizing the classes, "people" and
"good beings": |
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a. complete inclusion>>>"All people
are good beings." |
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b. complete exclusion>>>"No people
are good beings." |
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c. partial inclusion>>>"Some people
are good beings." |
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d. partial exclusion>>>"Some people
are not good beings." |
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We can also describe these four kind of
statements respectively as |
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a. universal affirmative |
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b. universal negative |
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c. particular affirmative |
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d. particular negative |
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3. Often, it is convenient to look at the general
form of the statements given above. To these forms, special names are given: A, E,
I, and O. |
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A: All S is P. |
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E: No S is P. |
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I: Some S is P. |
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O: Some S is not P. |
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...where S and P stand for the
logical subject and the logical predicate of the statement respectively. |
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4. A mnemonic device for the four kinds of
statements is to remember Affirmo and Nego. |
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5. Note, here, the logical subject differs from
the grammatical subject of a statement. |
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For example, in the statement, "All (unfledged
floithoisters) are (things apt to become unflaggled)," the
logical subject is everything between the "all" and the "are," and the
logical predicate is everything after the "are." |
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6. Also note that the word "some" is
taken to mean "at least one." This meaning differs somewhat from ordinary
language. |
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7. A model statement, then, can be represented as |
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Quantifier
[subject term] copula [predicate term]. |
| II. Analysis of the Categorical Proposition:
Quality, Quantity, and Distribution |
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A. The quantity of a categorical
proposition is determined by whether or not it refers to all members of its subject class
(i.e., universal or particular). The question "How many?"
is asking for quantity. |
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B. The quality of a categorical
proposition is determined by whether the asserted class relation is one of exclusion or
inclusion (i.e., affirmative or negative). |
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C. Indicators of "how much" are called
quantity indicators (quantifiers) and specifically are "all,"
"no," and "some." |
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D. Indicators of affirmative and negative are
quality indicators (qualifiers) and specifically are "are," "are
not," "is," "is not," and "no," |
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Note that "no" is both a quantifier and
a qualifier. |
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E. Memorize the
following table: |
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| Name |
Form |
Quantity |
Quality |
Distribution |
| Subject |
Predicate |
| A |
All S is P |
universal |
affirmative |
distributed |
undistributed |
| E |
No S is P |
universal |
negative |
distributed |
distributed |
| I |
Some S is P |
particular |
affirmative |
undistributed |
undistributed |
| O |
Some S is not P |
particular |
negative |
undistributed |
distributed |
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F. Distribution of a term. |
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1. A distributed term is a term of a
categorical proposition that is used with reference to every member of a class. If the
term is not being used to refer to each and every member of the class, it is said to be undistributed. |
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2. Consider the following propositions: |
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A: All birds are winged creatures. |
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E: No birds are wingless creatures. |
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I: Some birds are black things. |
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O: Some birds are not black things. |
| Read the above statements and see how the
following chart represents distribution. |
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Subject |
Predicate |
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A: refers to all
birds |
does not refer to every member,
e.g., bats, flying fish. |
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E: refers to all birds
by indicating that they are not part of the predicate class |
refers to all wingless
creatures by indicating that they are not part of the subject class |
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I: refers only to some
birds |
refers only to some black
things, viz., those which are birds |
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O: refers only to some
birds, not all of them |
refers to all members of the
class! Viz., not one of them is in the class referred to by "some birds" |
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3. For the predicate of the O proposition,
consider the following analogy. If we know that there is a book not in a bookcase, then we
know something about each and every shelf in that bookcase-- the book is not on that
shelf. |
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4. There are three ways to remember the
distribution status of subject and predicate for standard form categorical propositions: |
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a. Memorize it. |
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b. Figure it out from an example (as was done
above). |
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c. Remember the following rule: |
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The quantity of a standard form
categorical proposition determines the distribution of the subject (such that if the
quantity is universal, the subject is distributed and if the quantity is particular, the
subject is undistributed), and ... |
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the quality of a standard form categorical
proposition determines the distribution status of the predicate (such that if the quality
is affirmative, the predicate is undistributed, and if the quality is negative, the
predicate is distributed). |
