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“Squares of Opposition”

“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).*.

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“Squares of Opposition”

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  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. A: All S are P E: No S are P Four Categorical Sentences. Hypothetical Viewpoint. 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 S2if (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 *

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