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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

Answers


    1. A ⋅ B
    2. (A v C) ⊃ D /∴ A ⋅ D
    3. A Simplification 1
    4. A v C Addition 3
    5. D Modus ponens 2, 4
    6. A ⋅ D Conjunction 3, 5

    1. A
    2. B /∴ (A v C) ⋅ B
    3. A v C Addition 1
    4. (A v C) ⋅ B Conjunction 2, 3

    1. D ⊃ E
    2. D ⋅ F /∴ E
    3. D Simplification 2
    4. E Modus ponens 1, 3

    1. J ⊃ K
    2. J /∴ K v L
    3. K Modus ponens 1, 2
    4. K v L Addition 3

    1. A v B
    2. ~A ⋅ ~C /∴ B
    3. ~A Simplification 2
    4. B Disjunctive syllogism 1, 3

    1. A ⊃ B
    2. ~B ⋅ ~C /∴ ~A
    3. ~B Simplification 2
    4. ~A Modus tollens, 1, 3

    1. D ⊃ E
    2. (E ⊃ F) ⋅ (F⊃ D) /∴ D ⊃ F
    3. E ⊃ F Simplification 2
    4. D ⊃ F Hypothetical syllogism 1, 3

    1. (T ⊃ U) ⋅ (T ⊃ V)
    2. T /∴ U v V
    3. T v U Addition 2
    4. U v V Constructive dilemma 1, 2, 3

    1. (E ⋅ F) v (G ⊃ H)
    2. I ⊃ G
    3. ~(E ⋅ F) /∴ I ⊃ H
    4. G ⊃ H Disjunctive syllogism 1, 3
    5. I ⊃ H Hypothetical syllogism 2, 4

    1. M ⊃ N
    2. O ⊃ P
    3. N ⊃ P
    4. (N ⊃ P) ⊃ (M v O) /∴N v P
    5. M v O Modus ponens 3, 4
    6. N v P Constructive dilemma 1, 2, 5

    1. A v (B ⊃ A)
    2. ~A ⋅ C /∴ ~B
    3. ~A Simplification 2
    4. B ⊃ A Disjunctive syllogism 1, 3
    5. ~B Modus tollens 3, 4

    1. (D v E) ⊃ (F ⋅ G)
    2. D /∴ F
    3. D v E Addition 2
    4. F ⋅ G Modus ponens 1, 3
    5. F Simplification 4

    1. T ⊃ U
    2. V v ~U
    3. ~V ⋅ ~W /∴ ~T
    4. ~V Simplification 3
    5. ~U Disjunctive syllogism 2, 4
    6. ~T Modus tollens 1, 5

    1. (A v B) ⊃ ~C
    2. C v D
    3. A /∴ D
    4. A v B Addition 3
    5. ~C Modus ponens 1, 4
    6. D Disjunctive syllogism 2, 5

    1. L v (M ⊃ N)
    2. ~L ⊃ (N ⊃ O)
    3. ~L /∴ M ⊃ O
    4. N ⊃ O Modus ponens 2, 3
    5. M ⊃ N Disjunctive syllogism 1, 3
    6. M ⊃ O Hypothetical syllogism 4, 5

    1. A ⊃ B
    2. A v (C ⋅ D)
    3. ~B ⋅ ~E /∴ C
    4. ~B Simplification 3
    5. ~A Modus tollens 1, 4
    6. C ⋅ D Disjunctive syllogism 2, 5
    7. C Simplification 6

    1. (F ⊃ G) ⋅ (H ⊃ I)
    2. J ⊃ K
    3. (F v J) ⋅ (H v L) /∴ G v K
    4. F ⊃ G Simplification 1
    5. F v J Simplification 3
    6. G v K Constructive dilemma 2, 4, 5

    1. (E v F) ⊃ (G ⋅ H)
    2. (G v H) ⊃ I
    3. E /∴ I
    4. E v F Addition 3
    5. G ⋅ H Modus ponens 1, 4
    6. G Simplification 5
    7. G v H Addition 6
    8. I Modus ponens 2, 7

    1. (N v O) ⊃ P
    2. (P v Q) ⊃ R
    3. Q v N
    4. ~Q /∴ R
    5. N Disjunctive syllogism 3, 4
    6. N v O Addition 5
    7. P Modus ponens 1, 6
    8. P v Q Addition 7
    9. R Modus ponens 2, 8

    1. J ⊃ K
    2. K v L
    3. (L ⋅ ~J) ⊃ (M ⋅ ~J)
    4. ~K /∴ M
    5. L Disjunctive syllogism 2, 4
    6. ~J Modus tollens 1, 4
    7. L ⋅ ~J Conjunction 5, 6
    8. M ⋅ ~J Modus ponens 3, 7
    9. M Simplification 8
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