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Components of Categorical propositions

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Matthew Flummer

on 18 September 2014

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Transcript of Components of Categorical propositions

Components of Categorical propositions
Proposition = Statement
Categorical Proposition: A proposition that relates two classes, or categories

We'll refer to these classes with two terms:
Subject terms and predicate terms.
Subject terms: the terms that come right after the quantifier (quantifier = all, no, or some)

Predicate terms: the terms that come right after the copula (copula = are, are not)
Standard-form categorical propositions:

All S are P
No S are P
Some S are P
Some S are not P
All puppies are cute.
Subject term
Predicate term
Quality: affirmative or negative

Quantity: either universal or particular

The four kinds of categorical propositions have been designated by letter names: A, E, I, O.

A proposition: Universal affirmative (All S are P).
E proposition: Universal negative (No S are P).
I proposition: Particular affirmative (Some S are P).
O proposition: Particular negative (Some S are not P).
Distribution is an attribute of the subject term and the predicate term.

A term is distributed if the proposition makes an assertion about every member of the class.
If it doesn't, then it is undistributed.
Universal propositions distribute subject terms.

Negative propositions distribute predicate terms.
No S are P (E proposition)
All S are P
Some S are P (I proposition)
Some S are not P (O proposition)
(A proposition)
A and E propositions (universal propositions) can be interpreted in two different ways. When used within arguments, the way we interpret the proposition might affect the validity of the argument.
Consider the following:

1. All lions are carnivores.

2. All fairies are magical.
Lions exist, fairies don't. We say that 1 has "existential import" and 2 doesn't.
In the history of logic, there have been two approaches; Aristotle's and Boole's.

Aristotelian standpoint: Universal propositions about existing things have existential import

Boolean Standpoint: No Universal propositions have existential import.
All eagles are raptors.

No dogs are cats.

All vampires are blood suckers.

No dragons are nice.
Both the Boolean and the Aristotelian standpoints recognize that particular propositions (I and O) make positive assertions about existence.

Some fish are not saltwater fish.
Some hobbits are heroes.
Boolean standpoint:

All S are P. = No members of S are outside P.
No S are P. = No members of S are inside P.
Some S are P. = At least one S exists, and that S is a P.
Some S are not P. = At least one S exists, and that S is not a P.
Venn Diagram: An arrangement of overlapping circles in which each circle represents the class denoted by a term in a categorical proposition.
Kinds of marks in a Venn Diagram:

Shading an area means that it is empty.
Placing an "X" in an area means that there is at least one thing in that area.
If there is no marking, then nothing is known about the area.

Because A and O propositions are contradictory, if we know the truth value for some A proposition, then we know that the O proposition must be the opposite truth value.

A proposition: All puppies are cute. - True
O proposition: Some puppies are not cute. - ?
Likewise with I and E propositions.

I proposition: Some snakes are poisonous. - T
E proposition: No Snakes are poisonous. - ?
We can use the modern square of opposition to test for validity.

1. First assume the premise is true.
2. Then use the square to compute the truth value of the conclusion.
3. If the square indicates that the conclusion is true, then the argument is valid. If not, then it is invalid.
P. Some trade spies are not masters at bribery.
C. Therefore, it is false that all trade spies are masters at bribery.

(This is an example of an immediate inference, i.e., an argument that has only one premise.)
P. It is false that all meteor showers are common spectacles.
C. Therefore, no meteor showers are common spectacles.
Using Venn diagrams to test for validity:

To diagram a false statement, simply diagram the opposite of what the statement says.
P. Some trade spies are not masters at bribery.
C. Therefore, it is false that all trade spies are masters at bribery.

1. It is false that all meteor showers are common spectacles.
2. Therefore, no meteor showers are common spectacles.

P. All hobbits are short creatures.
C. Therefore, some hobbits are short creatures.

They don't match so the argument is invalid. It commits the existential fallacy. From the Boolean standpoint, this fallacy is committed whenever an argument is invalid merely because the premise lacks existential support.
Class average: Average for those who've done all the homework:
84 90
Some hobbits are heroes.
No blow-outs are enjoyable Super Bowls.
Identify the quantifier, subject term, copula, and predicate term
Identify the letter name, quantity, and quality.
In class assignment #2:

Exercise 4.1 - #8
Exercise 4.2 parts II, III, & IV - #3

p. 210
Modern Square of Opposition - p. 211
In class assignment:
Exercise 4.3
I. 4, 8
II. 10, 15
III. 4, 7, 10, 15
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