Introduction to Logic: Worksheet 1



Introduction to Logic: Worksheet 1 14 Nov. 08

Cathy (Xiaofan) Chen Class: Thurs 11-12

1. Which of the following are best construed as arguments? If so construable, write out showing premises and conclusion: if not, say why not. Which are deductively valid? Explain your answers.

a) If God is omnipotent, he is able to prevent credit crunches. But if God is benevolent, and able to, he would prevent credit crunches. So God is not both omnipotent and benevolent.

This is an invalid argument; the sentence is constructed so that the first two are the premises, and the last one (following keyword “so”) is the conclusion. The premises state some things about God, but there is no premise stating whether there is currently a credit crunch. Therefore, it is possible for the premises to be both true and the conclusion to be false. This violates the definition of deductive validity, so this is an invalid argument. This argument can be written as follows:

If God is omnipotent, he is able to prevent credit crunches

If God is benevolent, and able to, he would prevent credit crunches

God is not both omnipotent and benevolent

b) Some mental events are physical events, as some mental events cause physical events, but only physical events cause physical events.

This is not an argument. Although “as..” seemed to suggest a cause-effect relationship, the connectives among the three parts of this sentence can all be read as “and” connectives. These are three random statements about mental and physical events, and do not make sense with each other.

c) The Cookie Monster eats all the cookies he likes; these cookies would have been eaten unless he didn't like them. So he must not like these cookies – they haven't been eaten.

The sentences construct three premises and one conclusion. This is a valid argument, as it is impossible for all the premises to be true and conclusion to be false.

The Cookie Monster eats all the cookies he likes

If cookie is not eaten, Cookie Monster does not like the cookie

Cookies have not been eaten

Cookie Monster does not like the cookies

2. Which of the following are true and which false? Explain your answers.

a) A valid argument may have a false conclusion.

True; an argument is valid as long as all the premises cannot be true while the conclusion is false. Provided that there is a false conclusion, an argument can be valid as long as both premises cannot be true at the same time (that they are not consistent).

b) An unsound argument may have all true premises.

True; a sound argument must have two facts: all true premises and that the argument is valid. Since we are not provided information on the validity of the arguments, an unsound argument may have all true premises as long as they are not also valid.

c) Every valid argument with a true premise is sound.

False; a sound argument requires that the argument to be valid, and also that all premises are true, not just one. Therefore, it is false that every valid argument with a true premise is sound.

3. Give an example of:

a) An invalid argument with at least one true premise and a false conclusion.

Everyone in this world loves tea with sugar

Joann is in this world

Joann does not love tea with sugar

b) A valid argument with all false premises and a true conclusion.

It is raining dogs and it is not raining dogs

It is raining cats and it is not raining cats

It is cats and dogs or it is not raining

4. Which of the following claims are true, and which false? Explain your answers.

a) No logically indeterminate sentences are logically equivalent.

False; logically indeterminacy means that a sentence is not either always true or always false (logically true or logically false). This means that a sentence may have truth value of “true” and “false” depending on circumstances. This means that some logically indeterminate sentences will have the same truth values, or that there are some combinations of logically indeterminate sentences that are logically equivalent.

b) If the premise of an argument is logically equivalent to its conclusion, the argument is valid.

True; an argument is valid if it is impossible for all the premises to be true and the conclusion to be false. Since one premise is logically equivalent to its conclusion, the conclusion is always true when all the premises are true. This would make the argument valid.

c) If the conclusion of an argument is logically false, the argument is invalid.

False; an argument is invalid if it is possible for the premises to be true and the conclusion to be false. Provided that the conclusion of an argument is always false (logically false), an argument can still be valid as long as the premises are inconsistent (impossible for them to be both true at the same time).

d) If at least one member of a set of sentences is logically false, the set is logically inconsistent.

True; in order for a set of sentence to be logically consistent, it must be possible for them to be all true at the same time. Logical falseness makes it so that it is impossible for them to be all true, therefore the set of sentences cannot be logically consistent. The set is logically inconsistent, because it is not logically consistent.

e) If a valid argument has a logically false conclusion, it is unsound.

True; a valid argument that has a logically false conclusion requires that its premises cannot be all true at the same time, since logical validity states so. Provided this, this argument is unsound because a sound argument must be valid and have true premises.

5. Answer the following questions:

a) Suppose all the premises of a valid argument are logically equivalent to logical truths. What can we conclude about the conclusion? Explain.

The conclusion must always be true (or logically true). Logical validity states that it must be impossible for all the premises to be true and conclusion to be false. If all the premises of an valid argument are logically equivalent to logical truths, this means all the premises are logically true, and that for the argument to uphold its validity, the conclusion must be logically true as well.

b) Suppose an argument is valid but has a logically false conclusion. What can we conclude about its premises? Explain.

The premises will be inconsistent. By definition, an argument that is valid cannot have all true premises and a false conclusion. Given that the conclusion is always false, or logically false, the premises of this argument, then, must never be all true.

c) Suppose the premises of a valid argument form a consistent set. What can we conclude about its conclusion? Explain.

The conclusion must not be logically false, and must be true when the premises are true. Premises that form a consistent set means that it is possible for the premises to be all true, but does not mean that they will always be true. This means, it is not necessary to have a conclusion that is always true to uphold the argument’s validity. The conclusion needs to be true when all the premises are true.

6. Symbolize each of the following sentences as a negation in SL, stating what symbolization you use for atomic sentences:

a) Elmo is not in Fraggle Rock.

E: Elmo is in Fraggle Rock

~E

b) Some Fraggle is not a Muppett.

F: Every Fraggle is a Muppett

~F

c) No Muppett is in Fraggle Rock.

M: Some Muppet are in Fraggle Rock

~M

7. Symbolise the following sentences in SL using the symbolization key given.

P: John is happy

Q: Yoko is sad.

R: Paul will sing

a) John is not happy but Yoko is not sad

Rephrase: John is not happy and Yoko is not sad.

~P & ~Q

b) If John is happy, either Yoko is sad or Paul will sing

P ⊃ (Q ∨ R)

c) Unless Paul will not sing, Yoko is not sad.

Rephrase: If Paul sings, Yoko is not sad.

R ⊃ ~Q

d) It is not the case that either Paul will sing or Yoko is not sad.

~(R ∨ ~Q)

e) Paul will not sing only if either John is happy or Yoko is sad

Rephrase: If Paul will not sing, then John is happy or Yoko is sad

~R ⊃ (P ∨ Q)

f) Either John is happy and Yoko is sad or John is not happy and Yoko is not sad.

(P & Q) ∨ (~P & ~Q)

g) Only if John is happy if Paul will sing is Yoko sad if John is not happy.

Only if (John is happy if Paul will sing) is (Yoko sad if John is not happy).

Rephrase: If Yoko is sad if John is not happy, then John is happy if Paul will sing.

(~P ⊃ Q) ⊃ (R ⊃ P)

8. Show the truth conditions for the exclusive ‘or’ by means of a truth table. How might the exclusive ‘or’ be expressed as a biconditional?

Express (P ∨ Q) & ~(P & Q) with ≡

Answer: ~ (P ≡ Q)

P |Q | |(P |∨ |Q) |& |~ |(P |& |Q) |~ |(P |≡ |Q) | |T |T | |T |T |T |F |F |T |T |T |F |T |T |T | |T |F | |T |T |F |T |T |T |F |F |T |T |F |F | |F |T | |F |T |T |T |T |F |F |T |T |F |F |T | |F |F | |F |F |F |F |T |F |F |F |F |F |T |F | |

Highlighted columns are the main connectives of both forms of expressing exclusive or.

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