1.
Symbolism and Diagrams
for Categorical
Propositions
2.
Categorical proposition is the base for the
Classical Logic. They are called
categorical propositions because they are
about categories or classes.
Such propositions affirm or deny that
some class S is included in some other
class p, completely or partially.
3.
Boolean interpretation of categorical
propositions depends heavily on the notation
of an empty class, it is convenient to have a
symbol to represent it.
The symbol zero 0, is used for this purpose.
The term S has no members, we write an equal
sign between S and 0.
4.
Thus the equation S = 0 says that there are no
S’s, or that S has no members.
To deny that S is empty S does have members.
We symbolize that denial by drawing a
slanting line through the equals sign.
The inequality S ≠ 0 says that there are S’s, by
denying that S is empty.
5.
To symbolize A proposition.
The A proposition, “All S is P”, says that all
members of class S are also member of the class
P.
That is, there are no members of the class S
that that are not members of P or “No S is non-
P”
6.
To symbolize E propositions.
The E propositions, “No S is P”, says that no
members of the class S are the members of class
P.
This can be rephrased by saying that the
product of the two classes is empty which is
symbolized by the equation SP = 0.
7.
To symbolize I proposition.
The I proposition “Some S is P”, says that at
least one member of S is also a member of P.
This means that the product of the classes S
and P is not empty.
It is symbolized by the inequality SP ≠ 0.
8.
To symbolize O proposition.
The O proposition, “Some S is not P”, obverts
to the logically equivalent to I propositions,
“Some S is non-P”.
It is symbolized by the inequality SP ≠ 0.
9.
The relationship between these propositions contradict
each other in several ways, as can be illustrated here.
10.
This diagram shows that :
1. Contradictories
2. Contrary
3. Sub-Contrary
4. Sub-Alternation
11.
Contradictory means they have opposite truth
values.
A and O propositions are contradictory as are E
and I proposition.
They are opposite of each other in both Quantity
and Quality therefore, have opposite truth values.
When any Categorical statement is true, its partner
across the diagonal is false.
When false its contradictory must be true.
12.
Example:
if “all rubies are red stones” (A) is true, then
“some rubies are not red stones” (O) must be
false.
Similarly
if “no mammals are aquatic” (E) is false, then
“some mammals are aquatic” (I) must be true.
13.
A and E propositions are contrary.
Propositions are contrary when they cannot both
be true.
Example:
An A proposition e.g. “all giraffes have long
necks” cannot be true at the same time as the
corresponding E proposition: “no giraffes have
long necks”.
They are opposite in Quality only (both are
universals).
however that corresponding A and E proposition
while contrary are not contradictory.
14.
I and O propositions are Sub Contrary.
Propositions are Sub Contrary when it is
impossible for both to false.
Example:
“some lunches are free” is false, “some lunches
are not free” must be true.
However that is possible for corresponding I and
O both to be true.
15.
Example:
“some nations are democratic” and “some
nations are not democratic”.
Again I and O propositions are sub contrary,
but not contrary or contradictory
16.
Sub alternation are same in Quality but
different in Quantity.
From A to I ( is A is true then, the I is true).
From E to O ( if E is true then, O is true).
Now for Falsity
From I to A ( if the I is false, then A is false).
From O to E (if O is false then, E is false).
17.
All S are P. No S are P.
Some S are P. Some S are not P.
18.
The Venn diagrams constitute an iconic
representation of the standard form categorical
propositions, in which spatial inclusions and
exclusions correspond to the non-spatial
inclusions and exclusions of classes.
They provide an exceptionally clear method of
notation.
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