Presented To :Sir Munir Afzal

Presented By :Hadiqa Zehra

CATEGORICAL PROPOSITIONS

Symbolism and Diagrams for Categorical Propositions

Categorical Propositions 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.

Symbolism and Diagrams for Categorical Propositions 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.

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.

A Proposition

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”

E Proposition

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.

I Proposition

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.

O Proposition

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.

Modern square of opposition

The relationship between these propositions contradict each other in several ways, as can be illustrated here.

This diagram shows that :1. Contradictories2. Contrary3. Sub-Contrary4. Sub-Alternation

Contradictories

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.

Example:if “all rubies are red stones” (A) is true, then “some rubies are not red stones” (O) must be false.

Similarlyif “no mammals are aquatic” (E) is false, then “some mammals are aquatic” (I) must be true.

Contrary

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.

Sub Contrary

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.

Example:“some nations are democratic” and “some nations are not democratic”. Again I and O propositions are sub contrary, but not contrary or contradictory

Sub Alternation

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).

Boolean Venn diagrams of the four types of categorical propositions

All S are P. No S are P.

Some S are P. Some S are not P.

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.

THANK YOU FOR YOUR ATTENTION!