Squares of Opposition

1
Squares of Opposition

• A square of opposition is a visual displayof
  the logical relationships between thefour
  standard-form categorical sentences.

2
Logical Relationship(between sentences).

• A logical relationship is a relationship which
  is useful for determining rules and methods
  forcorrect reasoning.

3
Formal Logical Relationship

• A formal logical relationship between
  sentences does not involve the subject matter of
  the sentences.

4
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.

5
Four Categorical Sentences. Hypothetical
Viewpoint.

A All S are P E No
S are P
S
P
P
S
I Some S are P O Some S are not P
S
P
S
P


6
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.
7
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.

8

Four Categorical Sentences. Hypothetical
Viewpoint.
A All S are P E No S
are P
S
P
S
P
I Some S are P O Some S are not P
S
P
S
P


9
Four Categorical Sentences Hypothetical
S
P
S
P
I Some S are P O Some S are not P
S
P
S
P


10
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


11
Contradictory Sentences
Contradictory sentences are sentences whichhave
opposite truth values in all circumstances. In
every situation, one must be true and the
other false.
12
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).

13
Four Categorical Sentences Existential
A All S are P E No S are P
S
P
S
P


I Some S are P O Some S are not P
S
P
S
P


14
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.

15
Subcontrary Sentences
I Some S are P O Some S are not P
S
S
P
P

• Subcontrary sentences are sentences which
  cannot both be false, though they could bothbe
  true.
• Example Some dogs are collies. Some dogs
  are not collies.
• Given the existential assumption
  (viewpoint), the I and O sentences are
  subcontraries.

16
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.

17
Implication from the Existential Viewpoint
A All S are P E No S are P
S
P
S
P


T
T
I Some S are P O Some S are not P
S
P
S
P


18
Implication

• A implies I, given the existential viewpoint.
• E implies O, given the existential viewpoint.
• No implication relationships hold in the
  hypothetical viewpoint.

19
Square of Opposition Existential
A All S are P E No S are P
Contrary
T
T
Implies
Implies
Contradictory
I Some S are P O Some S are not P


Subcontrary