Philosophy 120
Introduction to Logic

MidTerm Exam Prep
Theory Quiz

The first exam will include not only problems to solve (translating sentences, evaluating truth-tables, evaluating sentences in a world, filling in proof steps and justifications for steps, etc.) but also some questions designed to test your understanding of the basic logical concepts of validity, soundness, consequence, logical truth, tautology, etc.

Here are some practice problems about logical concepts. As you click on your answers, you will get instant feedback — you will not receive a score for the entire set of practice questions. At the end of the quiz, you will find a link to a separate page containing fuller explanations for each question.

1. If you negate a tautology, what kind of sentence do you get?

a. A tautology
b. A TT-contradiction
c. A sentence that is neither a TT-contradiction nor a tautology.
d. You can’t tell — it could be any of (a), (b), or (c).
2. If you negate a TT-contradiction, what kind of sentence do you get?
a. A tautology
b. A TT-contradiction
c. A sentence that is neither a TT-contradiction nor a tautology.
d. You can’t tell — it could be any of (a), (b), or (c).
Call a sentence TT-contingent if neither it nor its negation is a tautology (i.e., the sentence is neither a tautology nor a TT- contradiction).
3. If you negate a TT-contingent sentence, what kind of sentence do you get? Explain.
a. A tautology
b. A TT-contradiction
c. A TT-contingent sentence
d. You can’t tell — it could be any of (a), (b), or (c).
4. If you conjoin (i.e., form a conjunction out of) two tautologies, will the resulting sentence always be a tautology? Explain.
a. Yes, it will be a tautology.
b. No, it will be a TT-contradiction
c. No, it will be a TT-contingent sentence.
d. You can’t tell — it could be any of (a), (b), or (c).
5. If you conjoin two TT-contingent sentences, will the resulting sentence always be TT-contingent? Explain.
a. Yes, it will be TT-contingent.
b. No, it will be a TT-contradiction
c. No, it will be a tautology.
d. You can’t tell — it could be either (a) or (b).
6. If an argument has a false conclusion and all of its premises are true, can you tell whether it is valid? Can you tell whether it is sound? Explain.

Validity
a. It is valid.
b. It is invalid.
c. You can’t tell — it could be either valid or invalid.
Soundness
a. It is sound.
b. It is unsound.
c. You can’t tell — it could be either sound or unsound.
7. If an argument has at least one false premise and a true conclusion, can you tell whether it is valid? Can you tell whether it is sound? Explain.
Validity
a. It is valid.
b. It is invalid.
c. You can’t tell — it could be either valid or invalid.
Soundness
a. It is sound.
b. It is unsound.
c. You can’t tell — it could be either sound or unsound.
8. If a disjunction (“or” sentence) has a tautology as one of its disjuncts, can you tell whether the disjunction is a TT-contradiction, a tautology, or TT-contingent? Explain.
a. TT-contradiction
b. Tautology
c. TT-contingent
d. You can’t tell — it could be any of (a), (b), or (c).
9. If a conjunction (“and” sentence) has a tautology as one of its conjuncts, can you tell whether the conjunction is a contradiction, a tautology, or TT-contingent? Explain.
a. TT-contradiction
b. Tautology
c. TT-contingent
d. You can’t tell — it could be any of (a), (b), or (c).
10. If a disjunction has a TT-contradiction as one of its disjuncts, can you tell whether the disjunction is a TT-contradiction, a tautology, or TT-contingent? Explain.
a. TT-contradiction
b. Tautology
c. TT-contingent
d. You can’t tell — it could be any of (a), (b), or (c).
11. If a conjunction has a TT-contradiction as one of its conjuncts, can you tell whether the conjunction is a TT-contradiction, a tautology, or TT-contingent? Explain.
a. TT-contradiction
b. Tautology
c. TT-contingent
d. You can’t tell — it could be any of (a), (b), or (c).
12. If a conditional sentence has a TT-contradiction as its antecedent, can you tell whether the conditional is a TT-contradiction, a tautology, or TT-contingent? Explain.
a.TT-contradiction
b. Tautology
c. TT-contingent
d. You can’t tell — it could be any of (a), (b), or (c).
13. If a conditional sentence has a tautology as its antecedent, can you tell whether the conditional is a TT-contradiction, a tautology, or TT-contingent? Explain.
a. TT-contradiction
b. Tautology
c. TT-contingent
d. You can’t tell — it could be any of (a), (b), or (c).
14. If a conditional sentence has a TT-contradiction as its consequent, can you tell whether the conditional is a TT-contradiction, a tautology, or TT-contingent? Explain.
a. TT-contradiction
b. Tautology
c. TT-contingent
d. You can’t tell — it could be any of (a), (b), or (c).
15. If a conditional sentence has a tautology as its consequent, can you tell whether the conditional is a TT-contradiction, a tautology, or TT-contingent? Explain.
a. TT-contradiction
b. Tautology
c. TT-contingent
d. You can’t tell — it could be any of (a), (b), or (c).
16. Suppose a conditional sentence is a TT-contradiction. Can you tell what kind of sentence its antecedent is? Can you tell what kind of sentence its consequent is? Explain.
Antecedent
a. TT-contradiction
b. Tautology
c. TT-contingent
d. You can’t tell — it could be any of (a), (b), or (c).
Consequent
a. TT-contradiction
b. Tautology
c. TT-contingent
d. You can’t tell — it could be any of (a), (b), or (c).
17. If a set of sentences has a contradiction as one of its members, can you tell whether the set is inconsistent? Explain.
a. The set is consistent.
b. The set is inconsistent.
c. You can’t tell — it could be either consistent or inconsistent.
18. If a set of sentences is inconsistent, can you tell whether the set has a contradiction as one of its members? Explain.
a. Yes, you can tell; one of the sentences in the set is a contradiction.
b. Yes, you can be sure that the set does not contain any sentence that is a contradiction.
c. You can’t tell — there may or may not be a contradiction in the set.
19. If an argument’s premise set is inconsistent, can you tell whether or not the argument is valid? Can you tell whether or not the argument is sound? Explain.
Validity
a. It is valid.
b. It is invalid.
c. You can’t tell — it could be either valid or invalid.
Soundness
a. It is sound.
b. It is unsound.
c. You can’t tell — it could be either sound or unsound.
20. [Hard] If a set of sentences is consistent, what (if anything) can you tell about the logical relationship between one of those sentences (or its negation) and the remaining sentences in the set? For example, suppose the set {P1, P2, P3} is consistent. What is the logical relationship between P1 and {P2, P3}? What is the logical relationship between ¬P1 and {P2, P3}?
a. P1 is a logical consequence of {P2, P3}.
b. P1 is not a logical consequence of {P2, P3}.
c. ¬P1 is a logical consequence of {P2, P3}.
d. ¬P1 is a not a logical consequence of {P2, P3}.
e. Nothing follows about the logical relationship between either P1 or ¬P1 and {P2, P3}.
21. Suppose X is a valid argument, and X+1 is another argument that is exactly like X (e.g., it has the same conclusion) except that X+1 has one additional premise. Is X+1 valid, or invalid, or is it impossible to tell?
a. It is valid.
b. It is invalid.
c. It is impossible to tell.

For each of questions 22-27, classify the FOL sentence correctly. (Review the diagram on p. 102 of LPL if you need to.)

22. The FOL sentence SameSize(a, b) → ¬Larger(b, a) is:
a. A tautology
b. Logically necessary but not a tautology
c. TW-necessary but not logically necessary
d. None of the above.
23. The FOL sentence (Cube(a) ∧ a = b) → Cube(a) is:
a. A tautology
b. Logically necessary but not a tautology
c. TW-necessary but not logically necessary
d. None of the above.
24. The FOL sentence (Cube(a) ∧ a = b) → Cube(b) is:
a. A tautology
b. Logically necessary but not a tautology
c. TW-necessary but not logically necessary
d. None of the above.
25. The FOL sentence (¬Cube(a) ∧ ¬Tet(a)) → Dodec(a) is:
a. A tautology
b. Logically necessary but not a tautology
c. TW-necessary but not logically necessary
d. None of the above.
26. The FOL sentence ¬(SameCol(b, c) ∧ SameRow(b, c)) is:
a. A tautology
b. Logically necessary but not a tautology
c. TW-necessary but not logically necessary
d. None of the above.
27. The FOL sentence Tet(a) ∧ ¬Tet(b) ∧ a = b is:
a. A tautology
b. Logically necessary but not a tautology
c. TW-necessary but not logically necessary
d. None of the above.

For more complete explanations of the answers to these questions, see the Answer Page for this theory quiz.

Copyright © 2004, S. Marc Cohen