Question Database: Philosophical Logic
Paradoxes of material implication
A LaTex version of the following questions is available. The .tex file makes use of the beamer packages.
Consider the argument form: ~P Therefore: P ⊃ Q
1. Is this argument form valid according to the truth-table for ⊃ ?
A. The argument form is valid according to the truth-table.
B. The argument form is invalid according to the truth-table.
Answer: A
Topic:
paradoxes of material implication
Course Level:
First year philosophical logic
Notes:
Consider the argument form: ~P Therefore: P ⊃ Q
2. Which of the following arguments is an instance of the form?
I. Not all cats are mammals. Therefore, if all cats are mammals, pigs can fly.
II. If all mammals are cats, some mammals are dogs, since not all mammals are cats.
III. If all mammals are cats, then no dog is a mammal. But since dogs are mammals, not all mammals are cats.
A. I only
B. I and II
C. I and III
D. I, II and III
Answer: B
Topic:
paradoxes of material implication
Course Level:
First year philosophical logic
Notes:
Consider the argument form: ~P Therefore: P ⊃ Q
3. Which of these arguments could provide a counter-example to the validity of the argument form?
I. Not all cats are mammals. Therefore, if all cats are mammals, pigs can fly.
II. If all mammals are cats, some mammals are dogs, since not all mammals are cats.
III. If all mammals are cats, then no dog is a mammal. But since dogs are mammals, not all mammals are cats.
A. I only
B. I and II
C. II only
D. I, II and III
Answer: C
Topic:
paradoxes of material implication
Course Level:
First year philosophical logic
Notes:
I has a false premise and III is not of the right form.
Consider the argument form: ~P Therefore: P ⊃ Q
4. Could there be an instance of the above form of argument in which:
The premise is true.
The conclusion is false because it has a true antecedent and a false consequent.
A. Yes, there could be such an instance.
B. No, it is not possible for there to be such an instance.
C. It is impossible to tell.
Answer: B
Topic:
paradoxes of material implication
Course Level:
First year philosophical logic
Notes:
P would have to be true and false. Assuming this is not possible, the answer is B.
Consider the following argument:
P1. If John is in Paris, he is in France.
P2. If John is in London, he is in England.
Therefore:
C. Either John is in England if he is in Paris, or John is in France if he is in London.
1. Opinion poll: Do you think the argument is valid?
A. Yes, the argument is valid.
B. No, the argument is invalid.
Answer:
Topic:
paradoxes of material implication
Course Level:
First year philosophical logic
Notes:
The point of this question is to simply to collect student intuitions about the example, before showing that the argument is valid according to the truth-table for material implication.
2. Which of the following best represents the form of this argument?
A. (P ⊃ Q), (R ⊃ S) Therefore: (S ⊃ P) ∨ (Q ⊃ R)
B. (P ⊃ Q), (P ⊃ S) Therefore: (P ⊃ S) ∨ (P ⊃ Q)
C. (P ⊃ Q), (R ⊃ S) Therefore: (P ⊃ S) ∨ (R ⊃ Q)
D. (P ⊃ Q) ∨ (R ⊃ S) Therefore: (P ⊃ S) & (R ⊃ Q)
Answer: C
Topic:
paradoxes of material implication
Course Level:
First year philosophical logic
Notes:
3. Is the argument form valid or invalid, according to the truth-table for ⊃ ?
(P ⊃ Q), (R ⊃ S) Therefore: (P ⊃ S) ∨ (R ⊃ Q)
Hint: check using trees rather than truth-tables.
A. The argument form is valid according to the truth-tables.
B. The argument form is invalid according to the truth-tables.
Answer: A
Topic:
paradoxes of material implication
Course Level:
First year philosophical logic
Notes:
Out of interest, this argument form is still valid if either of the premises are removed.
Which of these conversational maxims is broken in the following dialogue:
A. Quality. Be honest: do not assert what you know or beleive to be false.
B. Quantity. Be as informative as possible: do not assert less than you can.
C. Relevance. Make your responses as relevant as possible.
D. Neither.
American tourist in London: Excuse me, do you know the way to Piccadilly Circus?
Surly Londoner: Yes thanks.
Answer: B
Topic:
Gricean maxims
Course Level:
First year philosophical logic
Notes:
Which of these conversational maxims is broken in the following dialogue:
A. Quality. Be honest: do not assert what you know or beleive to be false.
B. Quantity. Be as informative as possible: do not assert less than you can.
C. Relevance. Make your responses as relevant as possible.
D. Neither.
American tourist in Sydney: Excuse me, can you tell me the way to the Opera House?
Surly Sydneysider: No mate, we knocked it down and built a McDonalds.
Answer: A
Topic:
Gricean maxims
Course Level:
First year philosophical logic
Notes:
Which of these conversational maxims is broken in the following dialogue:
A. Quality. Be honest: do not assert what you know or beleive to be false.
B. Quantity. Be as informative as possible: do not assert less than you can.
C. Relevance. Make your responses as relevant as possible.
D. Neither.
Concerned Parent: How is my daughter Sally doing in mathematics?
Sally's Maths Teacher: Well, Sally is always very well-behaved in class.
Answer:
Topic:
Gricean maxims
Course Level: C
First year philosophical logic
Notes:
Which of these conversational maxims is broken in the following dialogue:
A. Quality. Be honest: do not assert what you know or beleive to be false.
B. Quantity. Be as informative as possible: do not assert less than you can.
C. Relevance. Make your responses as relevant as possible.
D. Neither.
Student: Is Melbourne in Australia or Germany?
Teacher: Some people who live in Melbourne live in Australia.
Answer: B
Topic:
Gricean maxims
Course Level:
First year philosophical logic
Notes:
Which of these conversational maxims is broken in the following dialogue:
A. Quality. Be honest: do not assert what you know or beleive to be false.
B. Quantity. Be as informative as possible: do not assert less than you can.
C. Relevance. Make your responses as relevant as possible.
D. Neither.
Child to Father: Dad, are we having fish and chips for dinner again tonight?
Father who has just bought fish and chips home for dinner: We're having either fish and chips or pizza.
Answer: B
Topic:
Gricean maxims
Course Level:
First year philosophical logic
Notes:
Which of these conversational maxims is broken in the following dialogue:
A. Quality. Be honest: do not assert what you know or beleive to be false.
B. Quantity. Be as informative as possible: do not assert less than you can.
C. Relevance. Make your responses as relevant as possible.
D. Neither.
Police officer to suspect: Did you murder your wife Mr. Bones?
Mr. Bones (who has just murdered his wife): Yes I certainly did.
Answer: D
Topic:
Gricean maxims
Course Level:
First year philosophical logic
Notes:
Which of these conversational maxims is broken in the following dialogue:
A. Quality. Be honest: do not assert what you know or beleive to be false.
B. Quantity. Be as informative as possible: do not assert less than you can.
C. Relevance. Make your responses as relevant as possible.
D. Neither.
English tourist in Paris: Excuse me, do you speak English?
Witty Parisian: No I'm terribly sorry, I'm afraid I don't.
Answer: A
Topic:
Gricean maxims
Course Level:
First year philosophical logic
Notes:
Here is a definition of the concept of a conversational implicature:
An assertion A conversationally implies B iff the fact that the speaker has asserted A entails that B is true, assuming that the speaker is conforming to all the conversational maxims.
Example:
Child: Can I go out to play?
Mother: It's raining outside.
The mother's response conversationally implies:
That she knows or believes it to be raining outside, by the maxim of quality.
That the child may not be allowed out to play, by the maxim of relevance.
Student: Is Melbourne in Australia or Germany?
Teacher: Some people who live in Melbourne live in Australia.
The teacher's response conversationally implies:
A. that Melbourne is in Germany, by the maxim of quantity.
B. that some people who live in Melbourne do not live in Australia, by the maxim of quantity.
C. that some people who live in Melbourne do not live in Australia, by the maxim of quality.
D. that everyone who lives in Melbourne lives in Australia, by the maxim of quality.
Answer: B
Topic:
conversational implicature
Course Level:
First year philosophical logic
Notes:
Child to Father: Dad, are we having fish and chips for dinner again tonight?
Father who has just bought fish and chips home for dinner: We're having either fish and chips or pizza.
The father's response conversationally implies
A. that they will be having pizza, rather than fish and chips by the maxims of quality and relevance.
B. that they will be having fish and chips, rather than pizza by the maxims of quality and relevance.
C. that he does not know whether they will be having fish and chips or pizza for dinner, by the maxim of quantity.
D. that he does not know whether they will be having fish and chips or pizza for dinner, by the maxim of quality.
Answer: C
Topic:
conversational implicature
Course Level:
First year philosophical logic
Notes:
How would the Gricean solution to the paradoxes of material implication apply to this example?
~P Therefore P ⊃ Q
A. If a speaker knows that P is false, then P ⊃ Q is false, but assertible, since no conversational maxim would be broken.
B. If a speaker knows that P is false, then P ⊃ Q is true, but unassertible, since the conversational maxim `do not assert what you know or believe to be false' would be broken.
C. If a speaker knows that P is false, then P ⊃ Q is true, but unassertible, since there is a more informative assertion the speaker could make.
D. If a speaker knows that P is false, then P ⊃ Q is both true and assertible.
Answer: D
Topic:
Gricean solution to paradoxes of implication
Course Level:
First year philosophical logic
Notes:
Alternatively:
How would the Gricean solution to the paradoxes of material implication apply to this example?
~P Therefore P ⊃ Q
For example:
Not all mammals are cats. Therefore, if all mammals are cats, some mammals are dogs.
A. The argument is valid. However, someone who knew that the premise is true would not assert the conclusion, because it would be more informative to assert the premise.
B. The argument is valid. However, since the conclusion is false, asserting it would violate the maxim of quality: `do not assert what you know or believe to be false'.
C. The argument is invalid. However, someone who knew that the premise is true would be able to assert the conclusion, since the conclusion is more informative than the premise.
D. The argument is invalid. It has a true premise and a false (and therefore unassertible) conclusion.
Answer: A
Topic:
Gricean solution to paradoxes of implication
Course Level:
First year philosophical logic
Notes:
How would the Gricean solution to the paradoxes of material implication apply to this example:
(P ⊃ Q), (R ⊃ S) Therefore: (P ⊃ S) ∨ (R ⊃ Q)
P1. If John is in Paris, he is in France.
P2. If John is in London, he is in England.
Therefore:
C. Either (If John is in Paris, he is in England) or (If John is in London, he is in France).
A. The argument is invalid. However, it appears to be valid because the conclusion is assertible in any situation in which both premises are assertible.
B. The argument is invalid, because although both premises are true, neither disjunct is true. The conclusion is therefore unassertible.
C. The argument is valid. However, although the conclusion is true, asserting it would violate a conversational maxim.
D. The argument is valid. However, since both disjuncts are false, the conclusion is unassertible.
Answer: C
Topic:
Gricean solution to paradoxes of implication
Course Level:
First year philosophical logic
Notes:
Which conversational maxim do you think is violated by the conclusion of this argument?
P1. If John is in Paris, he is in France.
P2. If John is in London, he is in England.
Therefore:
C. Either (If John is in Paris, he is in England) or (If John is in London, he is in France).
A. Quality. Be honest: do not assert what you know or believe to be false.
B. Quantity. Be as informative as possible: do not assert less than you can.
C. Relevance. Make your responses as relevant as possible.
D. None of the above.
Answer: D?
Topic:
Gricean solution to paradoxes of implication
Course Level:
First year philosophical logic
Notes:
None of the answers seems particularly apt. It does not appear for example that the conclusion is unassertible because one of the disjuncts is assertible - neither disjunct seems assertible (or indeed, even true). Perhaps then, this question could be used to suggest that the Gricean solution may not always work, so a different kind of solution might be needed.
Strict Implication
The following questions make use of the following definition of a strict conditional:
P → Q is true at a world w, iff at every world where P is true, Q is also true.
Consider the following universe:
w1: ~P, Q, ~R
w2: P, ~Q, R
What is the truth value P ∨ Q at w1?
A. True.
B. False.
C. Not enough information to decide.
Answer: A
Topic:
strict implication
Course Level:
First year philosophical logic
Notes:
w1: ~P, Q, ~R
w2: P, ~Q, R
What is the truth value of ~(P & S) at w1?
A. True.
B. False.
C. Not enough information to decide.
Answer: A
Topic:
strict implication
Course Level:
First year philosophical logic
Notes:
w1: ~P, Q, ~R
w2: P, ~Q, R
What is the truth value of Q → R at w1?
A. True.
B. False.
C. Not enough information to decide.
Answer: B
Topic:
strict implication
Course Level:
First year philosophical logic
Notes:
w1: ~P, Q, ~R
w2: P, ~Q, R
What is the truth value of P → Q at w2?
A. True.
B. False.
C. Not enough information to decide.
Answer: B
Topic:
strict implication
Course Level:
First year philosophical logic
Notes:
w1: ~P, Q, ~R
w2: P, ~Q, R
What is the truth value of P → Q at w1?
A. True.
B. False.
C. Not enough information to decide.
Answer: B
Topic:
strict implication
Course Level:
First year philosophical logic
Notes:
w1: ~P, Q, ~R
w2: P, ~Q, R
What does this universe tell us about the formula:
(P → Q) ∨ (Q → R)
A. It is not logically valid, since it is false at w1 (and also at w2).
B. It is logically valid, because at every world in which the first disjunct is true, the second disjunct is also true.
C. It is logically valid, since it is true at all possible worlds.
D. It is not logically valid, since it is false at w1 (though it is true at w2).
Answer: A
Topic:
strict implication
Course Level:
First year philosophical logic
Notes:
Could there be a counter-model to the validity of this formula that consisted of just one world?
(P → Q) ∨ (Q → R)
A. Yes, there could be a one-world counter-example.
B. No, a one-world counter-model is impossible.
C. It is impossible to tell.
Answer: B
Topic:
strict implication
Course Level:
First year philosophical logic
Notes:
NOTE: To make the first disjunct false at a world, we need P true there and Q false there. But to make the second disjunct false, we need Q true there (and R false). If Q cannot be true and false at the same world, this is impossible.
Consider the following argument form:
(P ⊃ Q), (R ⊃ S) Therefore: (P ⊃ S) ∨ (R ⊃ Q)
We will attempt to construct a universe of possible worlds where the premise is true and the conclusion false.
1. Could there be a single world at which the premises of this argument are true and the conclusion false?
A. Yes, there could be a one-world counter-example.
B. No, a one-world counter-model is impossible.
C. It is impossible to tell.
Answer: B
Topic:
strict implication
Course Level:
First year philosophical logic
Notes:
Note: For the conclusion to be false, we need both disjuncts false. So we need P true and S false, and R true and Q false. But a world in which P is true and Q is false invalidates the first premise. And a world where R is true and Q is false invalidates the second premises.
(P ⊃ Q), (R ⊃ S) Therefore: (P ⊃ S) ∨ (R ⊃ Q)
We will start off the construction of our universe as follows:
w1: P, ~S
w2: R, ~Q
2. What is the truth-value of P → S at w1 and w2?
A. True at w1 and true at w2.
B. False at w1 and false at w2.
C. True at w1 and false at w2.
D. False at w1 and true at w2.
Answer: B
Topic:
strict implication
Course Level:
First year philosophical logic
Notes:
Counter-model for the argument form:
(P ⊃ Q), (R ⊃ S) Therefore: (P ⊃ S) ∨ (R ⊃ Q)
w1: P, ~S
w2: R, ~Q
3. What is the truth-value of R → Q at w1 and w2?
A. True at w1 and true at w2.
B. False at w1 and false at w2.
C. True at w1 and false at w2.
D. False at w1 and true at w2.
Answer: B
Topic:
strict implication
Course Level:
First year philosophical logic
Notes:
Counter-model for the argument form:
(P ⊃ Q), (R ⊃ S) Therefore: (P ⊃ S) ∨ (R ⊃ Q)
w1: P, ~S
w2: R, ~Q
4. What is the truth value of (P → S) ∨ (R → Q) at w1 and w2?
A. True at w1 and true at w2.
B. False at w1 and false at w2.
C. True at w1 and false at w2.
D. False at w1 and true at w2.
Answer: B
Topic:
strict implication
Course Level:
First year philosophical logic
Notes:
Counter-model for the argument form:
(P ⊃ Q), (R ⊃ S) Therefore: (P ⊃ S) ∨ (R ⊃ Q)
w1: P, ~S
w2: R, ~Q
5. In order to get a counter-example to the given argument form, what should the truth value of Q be at w1?
A. Q must be true at w1.
B. Q must be false at w1.
C. It doesn't matter what the value of Q is at w1.
Answer: A
Topic:
strict implication
Course Level:
First year philosophical logic
Notes:
(P ⊃ Q), (R ⊃ S) Therefore: (P ⊃ S) ∨ (R ⊃ Q)
Now the universe looks like this:
w1: P, ~S, Q
w2: R, ~Q
6. In order to get a counter-example to the above argument form, what should the truth value of R be at w1?
A. R must be true at w1.
B. R must be false at w1.
C. It doesn't matter what the value of R is at w1.
Answer: B
Topic:
strict implication
Course Level:
First year philosophical logic
Notes:
(P ⊃ Q), (R ⊃ S) Therefore: (P ⊃ S) ∨ (R ⊃ Q)
Now the universe looks like this:
w1: P, ~S, Q, ~R
w2: R, ~Q
7. In order to get a counter-example to the above argument form, what should the truth value of S be at w2?
A. S must be true at w2.
B. S must be false at w2.
C. It doesn't matter what the value of R is at w2.
Answer: A
Topic:
strict implication
Course Level:
First year philosophical logic
Notes:
Now the universe looks like this:
w1: P, ~S, Q, ~R
w2: R, ~Q, S
(P ⊃ Q), (R ⊃ S) Therefore: (P ⊃ S) ∨ (R ⊃ Q)
8. In order to get a counter-example to the above argument form, what should the truth value of P be at w2?
A. P must be true at w2.
B. P must be false at w2.
C. It doesn't matter what the value of P is at w2.
Answer: B
Topic:
strict implication
Course Level:
First year philosophical logic
Notes:
(P ⊃ Q), (R ⊃ S) Therefore: (P ⊃ S) ∨ (R ⊃ Q)
Now the universe looks like this:
w1: P, ~S, Q, ~R
w2: R, ~Q, S, ~P
9. What is the truth value of P → Q and R → S at w1 and w2?
A. P → Q is true at w1, false at w2. R → S is true at w2, false at w1.
B. P → Q is false at w1, true at w2. R → S is false at w2, true at w1.
C. P → Q is true at w1 and w2. R → S is true at w2 and w1.
D. P → Q is false at w1 and w2. R → S is false at w2 and w1.
Answer: C
Topic:
strict implication
Course Level:
First year philosophical logic
Notes:
10. Which of the following statements concerning the possible worlds semantics for → do you think are correct?
A. If A → B is true at some world, it is true at every world.
B. If A → B is false at some world, it is false at every world.
C. Both of the above.
D. A, but not B.
Answer: C
Topic:
strict implication
Course Level:
First year philosophical logic