Opposition of Propositions and Venn Diagrams

Opposition of Propositions and Venn Diagrams


Traditional logicians classified propositions into two types, namely, singular and general.

When the subject of a proposition represents one single individual, it is a singular proposition.

When the subject of a proposition represents a group of individuals, it is a general proposition.

General propositions are further classified into Universal and Particular.

When the subject tells something about the whole group represented by it, the proposition is known to be universal.

When the subject tells something about some members of the group represented by it, the proposition is known to be particular.

The propositions are also classified using another criteria of quality and this makes them affirmative or negative.

So, both the singular as well as general propositions are either affirmative or negative.

As a result, we have four types of general propositions, as the general propositions have both the quality as well as quantity.

The four types of general propositions are:

A, E, I, & O.

The quantity and quality of these are as follows:

A = Universal affirmative
E = Universal negative
I = Particular affirmative
O = Particular negative

Relation of opposition between these propositions is as follows:

When two universal propositions differ in quality, they are known as CONTRARY.

When two particular propositions differ in quality, they are known as SUB-CONTRARY.

When two propositions with same quality, differ in quantity, they are known as SUB-ALTERN.

When two propositions differ both in quality and quantity, they are known as CONTRADICTORY.

The relation of truth values between these opposite propositions is as follows:

If a universal proposition is true, its contrary is false, its sub-altern is true and its contradictory is false.

If a universal proposition is false, its contrary is uncertain, its sub-altern is uncertain and its contradictory is true.

If a particular proposition is true, its sub-contrary is uncertain, its sub-altern is uncertain and its contradictory is false.

If a particular proposition is false, its sub-contrary is true, its sub-altern is false and its contradictory is true.

The general propositions represent the relation of two groups indicated by the subject and predicate, and so, they can be represented symbolically using the venn diagram method used in mathematics.

To do this, we use two intersecting circles.

The circle on the left represents the subject, and the one on the right, represents the predicate.

To represent "A" proposition, we shade the part of the circle of subject, that is outside that of predicate. This shows that the set of subject outside the predicate is empty.

To represent "E" proposition, we shade the part of the circle of subject, that is inside predicate. This shows that the set of subject inside the predicate is empty.

To represent "I" proposition, we put a cross in the part of the circle of subject, that is inside that of predicate. This shows that the set of subject inside the predicate is not empty.

To represent "O" proposition, we put a cross in the part of the circle of subject, that is outside that of predicate. This shows that the set of subject outside the predicate is not empty.

Pl check the images for venn diagrams and opposition of proposition.

When these propositions are symbolized, we change them in a specific format so that we can show the class membership of the subject and the predicate terms. This method is known as the method of Quantification and the symbols used to indicate the quantity of the subject are known as quantifiers.

Let us see how this is done:

Singular propositions:

Affirmative:

Ramu is a boy.

is symbolized as:
Br

Negative:

Sita is not a boy.

is symbolized as:
~Bs

General propositions:

These are of four kinds as we have seen earlier. They are symbolized as follows:

"A" proposition:

Subject-less:

Everything perishes.

will be written as:
Given any x, x is Perishable.

This is symbolized as follows:
(x)(Px)

With subject:

All S is P.

will be written as:
Given any x, if x is S, then x is P

This is symbolized as follows:
(x)(Sx>Px)

[Since the implication sign cannot be put due to font limits of the portal, so, a similar sign is put here]

"E" proposition:

Subject-less:

Nothing is Permanent.

will be written as:
Given any x, x is not Permanent.

This is symbolized as follows:
(x)(~Px)

With subject:

No S is P.

will be written as:
Given any x, if x is S, then x is not P

This is symbolized as follows:
(x)(Sx>~Px)

[Since the implication sign cannot be put due to font limits of the portal, so, a similar sign is put here]

"I" proposition:

Subject-less:

Lions exist

will be written as:
There is an x, such that, x is a Lion.

This is symbolized as follows:
(Ex)(Lx)

[Since the Existential quantifier sign that is actually reverse as a mirror image as actual E, cannot be put due to font limits of the portal, so, E is put here]

With subject:

Some S is P.

will be written as:
There is an x, such that, x is S and x is P.

This is symbolized as follows:
(Ex)(Sx.Px)

[Since the Existential quantifier sign that is actually reverse as a mirror image as actual E, cannot be put due to font limits of the portal, so, E is put here]

"O" proposition:

Subject-less:

Ghosts do not exist

will be written as:
There is an x, such that, x is not a Ghost.

This is symbolized as follows:
(Ex)(~Gx)

[Since the Existential quantifier sign that is actually reverse as a mirror image as actual E, cannot be put due to font limits of the portal, so, E is put here]

With subject:

Some S is not P.

will be written as:
There is an x, such that, x is S and x is not P.

This is symbolized as follows:
(Ex)(Sx.~Px)

[Since the Existential quantifier sign that is actually reverse as a mirror image as actual E, cannot be put due to font limits of the portal, so, E is put here]

When we symbolize, the first letter of the subject term is taken as a capital letter and small x is written after it to indicate the singular variable that is quantified in the beginning.

This is how we symbolize the general propositions in traditional classification.