Section 5: Practice Exercises

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If you want additional practice, you can construct truth tables for any of the sentences and arguments in the exercises for the previous chapter.

*Part A Determine whether each sentence is a tautology, a contradiction, or a contingent sentence. Justify your answer with a complete or partial truth table where appropriate.

1. \(A\) → \(A\)
2. ¬ \(B\) &\(B\)
3. \(C\) →¬\(C\)
4. ¬\(D\)∨\(D\)
5. (\(A\) ↔ \(B\)) ↔¬(\(A\) ↔¬\(B\))
6. (\(A\)&\(B\))∨(\(B\)&\(A\))
7. (\(A\) → \(B\))∨(\(B\) → \(A\))
8. ¬[\(A\) → (\(B\) → \(A\))]
9. (\(A\)&\(B\)) → (\(B\)∨\(A\))
10. \(A\) ↔ [\(A\) → (\(B\) &¬\(B\))]
11. ¬(\(A\)∨\(B\)) ↔ (¬\(A\)&¬\(B\)
12. ¬(\(A\)&\(B\)) ↔ \(A\)
13. [(\(A\)&\(B\))&¬(\(A\)&\(B\))]&\(C\)
14. \(A\) → (\(B\)∨\(C\)
15. [(\(A\)&\(B\))&\(C\)] → \(B\)
16. (\(A\)&¬\(A\)) → (\(B\)∨\(C\))
17. ¬[(\(C\) ∨\(A\))∨\(B\)]
18. (\(B\) &D) ↔ [\(A\) ↔ (\(A\)∨\(C\))]

* Part B Determine whether each pair of sentences is logically equivalent. Justify your answer with a complete or partial truth table where appropriate.

1. \(A\), ¬\(A\)
2. \(A\), \(A\)∨\(A\)
3. \(A\) → \(A\), \(A\) ↔ \(A\)
4. \(A\)∨¬\(B\), \(A\) → \(B\)
5. \(A\)&¬\(A\), ¬\(B\) ↔ \(B\)
6. ¬(\(A\)&\(B\)), ¬\(A\)∨¬\(B\)
7. ¬(\(A\) → \(B\)), ¬\(A\) →¬\(B\)
8. (\(A\) → \(B\)), (¬\(B\) →¬\(A\))
9. [(\(A\)∨\(B\))∨\(C\)], [\(A\)∨(\(B\)∨\(C\))]
10. [(\(A\)∨\(B\))&\(C\)], [\(A\)∨(\(B\) &\(C\))]

* Part C Determine whether each set of sentences is consistent or inconsistent. Justify your answer with a complete or partial truth table where appropriate.

1. \(A\) → \(A\), ¬\(A\) →¬\(A\), \(A\)&\(A\), \(A\)∨\(A\)
2. \(A\)&\(B\), \(C\) →¬\(B\), \(C\)
3. \(A\)∨\(B\), \(A\) → \(C\), \(B\) → \(C\)
4. \(A\) → \(B\), \(B\) → \(C\), \(A\), ¬\(C\)
5. \(B\) &(\(C\) ∨\(A\)), \(A\) → \(B\), ¬(\(B\)∨\(C\))
6. \(A\)∨\(B\), \(B\)∨\(C\), \(C\) →¬\(A\)
7. \(A\) ↔ (\(B\)∨\(C\)), \(C\) →¬\(A\), \(A\) →¬\(B\)
8. \(A\), \(B\), \(C\), ¬\(D\), ¬\(E\), \(F\)

* Part D Determine whether each argument is valid or invalid. Justify your answer with a complete or partial truth table where appropriate.

1. \(A\) → \(A\), .˙. \(A\)
2. \(A\)∨[\(A\) → (\(A\) ↔ \(A\))], .˙. \(A\)
3. \(A\) → (\(A\)&¬\(A\)), .˙. ¬\(A\)
4. \(A\) ↔¬(\(B\) ↔ \(A\)), .˙. \(A\)
5. \(A\)∨(\(B\) → \(A\)), .˙. ¬\(A\) →¬\(B\)
6. \(A\) → \(B\), \(B\), .˙. \(A\)
7. \(A\)∨\(B\), \(B\)∨\(C\), ¬\(A\), .˙. \(B\)&\(C\)
8. \(A\)∨\(B\), \(B\)∨\(C\), ¬\(B\), .˙. \(A\)&\(C\)
9. (\(B\) &\(A\)) → \(C\), (\(C\) &\(A\)) → \(B\), .˙. (\(C\) &\(B\)) → \(A\)
10. \(A\) ↔ \(B\), \(B\) ↔ \(C\), .˙. \(A\) ↔ \(C\)

* Part E Answer each of the questions below and justify your answer.

1. Suppose that \(\mathcal{A}\) and \(\mathcal{B}\) are logically equivalent. What can you say about \(\mathcal{A}\)↔\(\mathcal{B}\)?
2. Suppose that (\(\mathcal{A}\)&\(\mathcal{B}\)) → \(\mathcal{C}\) is contingent. What can you say about the argument “\(\mathcal{A}\), \(\mathcal{B}\), .˙.
3. Suppose that{\(\mathcal{A}\),\(\mathcal{B}\),\(\mathcal{C}\)}is inconsistent. What can you say about (\(\mathcal{A}\)&\(\mathcal{B}\)&\(\mathcal{C}\))?
4. Suppose that \(\mathcal{A}\) is a contradiction. What can you say about the argument “\(\mathcal{A}\), \(\mathcal{B}\), .˙.\(\mathcal{C}\)”?
5. Suppose that \(\mathcal{C}\) is a tautology. What can you say about the argument “\(\mathcal{A}\), \(\mathcal{B}\), .˙.\(\mathcal{C}\)”?
6. Suppose that \(\mathcal{A}\) and \(\mathcal{B}\) are logically equivalent. What can you say about (\(\mathcal{A}\)∨\(\mathcal{B}\))?
7. Suppose that \(\mathcal{A}\) and \(\mathcal{B}\) are not logically equivalent. What can you say about (\(\mathcal{A}\)∨\(\mathcal{B}\))?

Part F We could leave the biconditional (↔) out of the language. If we did that, we could still write ‘\(A\) ↔ \(B\)’ so as to make sentences easier to read, but that would be shorthand for (\(A\) → \(B\))&(\(B\) → \(A\)). The resulting language would be formally equivalent to SL, since \(A\) ↔ \(B\) and (\(A\) → \(B\))&(\(B\) → \(A\)) are logically equivalent in SL. If we valued formal simplicity over expressive richness, we could replace more of the connectives with notational conventions and still have a language equivalent to SL.

There are a number of equivalent languages with only two connectives. It would be enough to have only negation and the material conditional. Show this by writing sentences that are logically equivalent to each of the following using only parentheses, sentence letters, negation (¬), and the material conditional (→).

*1. \(A\)∨\(B\)
*2. \(A\)&\(B\)
*3. \(A\) ↔ \(B\)

We could have a language that is equivalent to SL with only negation and disjunction as connectives. Show this: Using only parentheses, sentence letters, negation (¬), and disjunction (∨), write sentences that are logically equivalent to each of the following.

4. \(A\)&\(B\)
5. \(A\) → \(B\)
6. \(A\) ↔ \(B\)

The Sheffer stroke is a logical connective with the following characteristic truthtable:

\(\mathcal{A}\)

\(\mathcal{B}\)

\(\mathcal{A}\)|\(\mathcal{B}\)

7. Write a sentence using the connectives of SL that is logically equivalent to (\(A\)|\(B\)).

Every sentence written using a connective of SL can be rewritten as a logically equivalent sentence using one or more Sheffer strokes. Using only the Sheffer stroke, write sentences that are equivalent to each of the following.

8. ¬\(A\)
9. (\(A\)&\(B\)
10. (\(A\)∨\(B\))
11. (\(A\) → \(B\))
12. (\(A\) ↔ \(B\))