“Squares of Opposition”
A square of opposition is a visual display
of the logical relationships between the
four standard-form categorical sentences.
Logical Relationship
(between sentences).*
A logical relationship is a
relationship which is useful for
determining rules and methods for
correct reasoning.
Formal Logical Relationship*
A formal logical relationship between
sentences does not involve the subject
matter of the sentences.
S1: No dogs are cats.
S2: No cats are dogs.
Which are Formal Relationships ?
Which are Logical Relationships ?
(a) S1 is above S2.
(b) S1 is equal in length to S2.
(c) S1 cannot differ in truth value from S2.
(d) S1 and S2are not very informative.
Four Categorical Sentences. Hypothetical Viewpoint.
A: All S are P
E: No S are P
S
P
I: Some S are P
S
P
*
P
S
O: Some S are not P
S
P
*
Hypothetical Viewpoint*
To assume the hypothetical viewpoint is
to make no assumptions (add no premises)
about the existence or nonexistence of what
the terms refer to.
Some Assertions from the
Hypothetical Veiwpoint:
• Molecules at absolute zero are motionless.
• Contrary to popular opinion, Martians do not
live in Rothemal Hall.
• The perfect marriage is made in heaven.
• All bodies uninfluenced by external forces
maintain constant velocity.
Four Categorical Sentences. Hypothetical Viewpoint.
A: All S are P
E: No S are P
S
P
I: Some S are P
S
P
*
S
P
O: Some S are not P
S
P
*
Four Categorical Sentences: Hypothetical
S
P
I: Some S are P
S
P
*
S
P
O: Some S are not P
S
P
*
Square of Opposition: Hypothetical
A: All S are P
E: No S are P
Contradictory
I: Some S are P
*
O: Some S are not P
*
Contradictory Sentences*
Contradictory sentences are sentences which
have opposite truth values in all circumstances.
In every situation, one must be true and the other
false.
Existential Viewpoint*
The existential viewpoint is taken when it is
assumed that the thing(s) mentioned by the
subject term S in the square of opposition
actually exist(s).
Four Categorical Sentences: Existential
A: All S are P
S
*
E: No S are P
P
I: Some S are P
S
P
*
S
P
*
O: Some S are not P
S
P
*
Contrary Sentences*
A: All S are P
*
E: No S are P
*
Contrary sentences are sentences which cannot
both be true. Though they could both be false.
Example: All dogs are collies. No dogs are collies.
Given the existential assumption (viewpoint),
A and E are contrary sentences.
Subcontrary Sentences*
I: Some S are P
S
*
O: Some S are not P
P
S
*
P
Subcontrary sentences are sentences which
cannot both be false, though they could both
be true.
Example: Some dogs are collies. Some dogs are not collies.
Given the existential assumption (viewpoint),
the I and O sentences are subcontraries.
Implication (Entailment)*
Sentence S1 implies (entails) sentence S2 if
(and only if) S2 is true whenever S1 is true.
S1 cannot true without S2.also being true.
Implication from the Existential Viewpoint
A: All S are P
S
*
E: No S are P
P
S
*
T
T
I: Some S are P
S
P
*
P
O: Some S are not P
S
P
*
Implication:
A implies I, given the existential viewpoint.
E implies O, given the existential viewpoint.
No implication relationships hold in the
hypothetical viewpoint.
Square of Opposition: Existential
A: All S are P
E: No S are P
Contrary
T
Implies
T
Contradictory
I: Some S are P
*
Implies
O: Some S are not P
Subcontrary
*