Introduction to Logic

Answer Page

1. A tautology is true on every row of its truth-table, so when you negate a tautology, the resulting sentence is false on every row of its table. That is, the negation of a tautology is a TT-contradiction.

2. A TT-contradiction is false in every row of its truth-table, so when you negate a TT-contradiction, the resulting sentence is true on every row of its table. That is, the negation of a TT-contradiction is a tautology.

3. A TT-contingent sentence comes out true on at least one row of its truth-table and false on at least one row. So the negation of such a sentence will be false on at least one row and true on at least one row, i.e., it will be TT-contingent. So the negation of a contingent sentence is contingent.

4. If you conjoin two tautologies, the result will be a tautology. A tautology comes out true on every row of its truth-table. So a conjunction of tautologies will have true conjuncts on every row. So the conjunction itself will be true on every row, i.e., a tautology.

5. The conjunction of two TT-contingent sentences need not be TT-contingent. It might be a TT-contradiction. The reason is that each TT-contingent sentence comes out true on some row(s), but there may not be a single row on which both come out true together. For example: ‘Cube(a)’ is TT-contingent and ‘¬Cube(a)’ is also TT-contingent. But the conjunction ‘Cube(a) ‚ ¬Cube(a)’ is not TT-contingent — it is a TT-contradiction.

6. An argument with a false conclusion and all true premises is invalid. In the case of such an argument, it is not only possible for the premises to be true and the conclusion false, it is actually the case. Such an argument is also unsound, of course, since an argument cannot be sound unless it is also valid.

7. You cannot tell whether an argument is valid if all you know is that it has at least one false premise and a true conclusion. But you can tell that it is unsound, for a sound argument is not only valid, but all of its premises are true.

8. If a disjunction has a tautology as one of its disjuncts, the disjunction itself is also a tautology. For a disjunction comes out true whenever at least one disjunct does, and if one disjunct is a tautology, then one disjunct always comes out true. So the disjunction always comes out true, i.e., is a tautology.

9. You cannot tell anything about a conjunction if all you know is that it has a tautology as one of its conjuncts. Everything will depend on the other conjunct. If it’s a tautology, so is the conjunction; if it’s a contradiction, so is the conjunction; if it’s contingent, so is the conjunction.

10. You cannot tell anything about a disjunction if all you know is that it has a TT-contradiction as one of its disjuncts. Everything will depend on the other disjunct. If it’s a tautology, so is the disjunction; if it’s a TT-contradiction, so is the disjunction; if it’s TT-contingent, so is the disjunction.

11. A conjunction that has a TT-contradiction as one of its conjuncts is itself a TT-contradiction. A conjunction comes out false whenever either of its conjuncts does. Since one conjunct is a TT-contradiction, it always comes out false. So the conjunction always comes out false, i.e., is a TT-contradiction.

12. A conditional sentence with a TT-contradiction as its antecedent is a tautology. That’s because a conditional comes out true on every row in which its antecedent is false. But if the antecedent is a TT-contradiction, it’s false on every row. So the conditional is true on every row, i.e., is a tautology.

13. You cannot tell anything about a conditional if all you know is that it has a tautology as its antecedent. Everything will depend on the consequent. If it’s a tautology, so is the conditional; if it’s a TT-contradiction, so is the conditional; if it’s contingent, so is the conditional.

14. You cannot tell anything about a conditional if all you know is that it has a TT-contradiction as its consequent. Everything will depend on the antecedent. If it’s a tautology, the conditional is a TT-contradiction; if it’s a TT-contradiction, the conditional is a tautology; if it’s TT-contingent, so is the conditional.

15. A conditional sentence with a tautology as its consequent is a tautology. That’s because a conditional comes out true on every row in which its consequent is true. But if the consequent is a tautology, it’s true on every row. So the conditional is true on every row, i.e., is a tautology.

16. If a conditional sentence is a TT-contradiction, it comes out false on every row of its truth-table. Since a conditional is false only in the cases in which its antecedent is true and its consequent is false, a conditional that is a TT-contradiction will have an antecedent that is true on every row (i.e., a tautology) and a consequent that is false on every row (i.e., a TT-contradiction).

17. A set of sentences with a contradiction as one of its members is inconsistent. If one sentence in the set is a contradiction, it always comes out false. In that case, there is no way in which all of the members of the set can come out true.

18. If a set of sentences is inconsistent, you cannot tell whether the set has a contradiction as one of its members; it may, but it need not. Considered individually, each member of the set may come out true under at least one assignment (i.e., be either contingent or a tautology), but there may be no assignment under which all of the sentences in the set come out true together. Example: the set {A, B, ¬A v ¬B}. Each sentence by itself is contingent, but the set itself is inconsistent: there is no assignment under which the 3 sentences all come out true.

19. If an argument’s premise set is inconsistent, the argument is valid! This result is very unintuitive, so please consider it carefully. If the premise set is inconsistent, there is no assignment of truth-values under which all the premises come out true; that is, it’s impossible for them all to come out true. But if that is so, then it’s impossible for the premises all to come out true and the conclusion false. (If it’s impossible for me to fly, it’s impossible for me to fly and chew gum!) But that is precisely our definition of validity. Of course, such an argument is unsound, because for an argument to be sound it must not only be valid, but all of it premises must be true. And if the premise set is inconsistent, the premises cannot all be true.

20. Since the set {P_{1}, P_{2}, P_{3}} is consistent,
it is possible for the individual sentences all to be true together. That means
that it is possible for ¬P_{1} to be false while both P_{2}
and P_{3} are true. But if ¬P_{1} were a logical consequence
of {P_{2}, P_{3}}, it would come out true whenever they come
out true. Since ¬P_{1} does not come out true whenever they are true,
¬P_{1} is not a logical consequence of {P_{2}, P_{3}}.

21. X+1 is a valid argument. Since X is valid, it is impossible for all of the premises of X to come out true while its conclusion is false. But the only difference between X and X+1 is that X+1 has one additional premise. And it doesn’t matter what that premise is; even if it’s true, it still cannot make it possible for all of the premises of X+1 to come out true while its conclusion is false. For we already know that it’s impossible for all the premises of X to come out true while its conclusion is false. The addition of one (irrelevant) premise will not convert a valid argument into an invalid one.

22. ‘SameSize(a, b) → ¬Larger(b, a)’ is logically necessary, but it is not a tautology. To see why it is logically necessary, we have to appeal to the meanings of the predicates ‘SameSize’ and ‘LargerThan’. This sentence comes out true with respect to any world, not just a Tarski world. (It is not just blocks on a chess board that obey the law that when things are the same size, one of them is not larger than the other.)

23. ‘(Cube(a) ∧ a = b) → Cube(a)’ is a tautology. Notice that the form of the sentence is (P ∧ Q) → P. The left conjunct of the antecedent is the same sentence as the consequent. A truth table for this sentence comes out true on every row.

24. ‘(Cube(a) ∧ a = b) → Cube(b)’ is logically necessary but not a tautology. It is obviously necessary that if a is a cube and a = b, b is also a cube. If a and b are the same object, then a is a cube if and only if b is a cube. The reason this sentence is not a tautology is that we cannot establish its necessity just by means of a truth table; we have to appeal to the meaning of the identity predicate. In fact, a truth table for this sentence [whose truth-functional form is (P ∧ Q) → R)] shows that there is a row on which it comes out false. This row does not correspond to a “real” possibility, which is why the sentence is logically necessary.

25. ‘(¬Cube(a) ∧ ¬Tet(a)) → Dodec(a)’ is TW-necessary but not logically necessary. Every block in a Tarski world is either a cube or a tetrahedron or a dodecahedron. So, in every Tarski world if a block a is neither a cube nor a tetrahedron, it will be a dodecahdron. But this is not in general true in worlds outside of TW, where there are objects of different shapes than just the three in TW.

26. ‘¬(SameCol(b, c) ∧ SameRow(b, c))’ is not necessary in any sense. Did you think it was necessary? Perhaps you thought that there isn’t room in a single square for two blocks — at any rate it is not possible in a Tarski world for two blocks to occupy the same square — and so you concluded that this sentence was either logically necessary or at least TW-necessary. If this is what you thought, your mistake was in supposing that b and c had to be two different blocks. Just because there are two names doesn't mean there are two blocks. Remember, a single object can have more than one name.

27. ‘Tet(a) ∧ ¬Tet(b) ∧ a = b’ is necessarily **false**. The correct answer is
therefore “none of the above,” since our sentence is not a tautology,
not necessarily true, and not true in every Tarski world. Just because the truth
**value** of a sentence is a matter of necessity does not mean the sentence
is necessarily **true**.

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Copyright © 2004, S. Marc Cohen