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Relationships in Truth Statements

Completion requirements

Read these sections to learn more about relationships among truth statements and using and constructing logical proofs.

These sections review materially equivalent propositions and three other relationships among statements: tautological, contradictory, and contingent relationships. They also review the eight valid forms of inference: modus ponens, modus tollens, hypothetical syllogism, simplification, conjunction, disjunctive syllogism, addition, and constructive dilemma. They show how to construct proofs, including strategies for working forward or backward, depending on which is easier according to your premises. Finally, they summarize everything you have learned about sentential and propositional logic.

Complete the exercises as you study, then check your answers against the key.

Exercise

Construct proofs for the following valid arguments. The first fifteen proofs can be complete in three or less additional lines. The next five proofs will be a bit longer. It is important to note that there is always more than one way to construct a proof. If your proof differs from the answer key, that doesn't mean it is wrong. #1


    1. A ⋅ B
    2. (A v C) ⊃ D /∴ A ⋅ D
  2.  
    1. A
    2. B /∴ (A v C) ⋅ B

    1. D ⊃ E
    2. D ⋅ F /∴ E

    1. J ⊃ K
    2.  J /∴ K v L

    1. A v B
    2. ~A ⋅ ~C /∴ B

    1. A ⊃ B
    2. ~B ⋅ ~C /∴ ~A

    1. D ⊃ E
    2. (E ⊃ F) ⋅ (F⊃ D) /∴D ⊃ F

    1. (T ⊃ U) ⋅ (T ⊃ V)
    2. T /∴ U v V

    1. (E ⋅ F) v (G ⊃ H)
    2. I ⊃ G
    3. ~(E ⋅ F) /∴ I ⊃ H

    1. M ⊃ N
    2. O ⊃ P
    3. N ⊃ P
    4. (N ⊃ P) ⊃ (M v O) /∴N v P

    1. A v (B ⊃ A)
    2. ~A ⋅ C /∴ ~B

    1. (D v E) ⊃ (F ⋅ G)
    2. D /∴ F

    1. T ⊃ U
    2. V v ~U
    3. ~V ⋅ ~W /∴ ~T

    1. (A v B) ⊃ ~C
    2. C v D
    3. A /∴ D

    1. L v (M ⊃ N)
    2. ~L ⊃ (N ⊃ O)
    3. ~L /∴ M ⊃ O

    1. A ⊃ B
    2. A v (C ⋅ D)
    3. ~B ⋅ ~E /∴ C

    1. (F ⊃ G) ⋅ (H ⊃ I)
    2. J ⊃ K
    3. (F v J) ⋅ (H v L) /∴ G v K

    1. (E v F) ⊃ (G ⋅ H)
    2. (G v H) ⊃ I
    3. E /∴ I

    1. (N v O) ⊃ P
    2. (P v Q) ⊃ R
    3. Q v N
    4. ~Q /∴ R

    1. J ⊃ K
    2. K v L
    3. (L ⋅ ~J) ⊃ (M ⋅ ~J)
    4. ~K    /∴ M

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