# Logic Text Chapter 1 Solutions

## Chapter 1: Propositions and Arguments

### Basic

Question {1.1}

1. Proposition

2. Question

3. Proposition (though of course, it’s false)

4. Question

5. Exclamation

6. Wish

7. Command or request

8. Proposition (which states that I have a wish, so you might say it expresses a wish too. It differs from the example in part 6 in that it can be true or false, and 6 cannot be.)

9. Proposition (though false, again)

10. Proposition (it can be true or false — though it might also express a wish or a command or something else too.)

Question {1.2}

All of them are valid except affirming the consequent.

For affirming the consequent, consider the instance

• If I’m doing logic, I’m happy.
• I’m happy.
• Therefore, I’m doing logic.

That’s not valid, as you can have the premises true, while the conclusion is false. For example, suppose I’m also happy when having dinner with friends in a good cafe (and not doing logic). Then in this situation you have the premises true but the conclusion false. As a result, this instance of the argument is invalid, and so, the argument form is invalid.

However, this form has valid instances. Here’s one

• If I’m doing logic I’m doing logic.
• I’m doing logic.
• Therefore, I’m doing logic.

This argument is stupid — the first premise is not needed and the conclusion is a restatement of the second premise. However, it’s valid according our definition of validity. There’s no way that the premises could be true and the conclusion false.

Question {1.3}

Form 1 is the form of all arguments with two premises both of which include “and”, and with any conclusion. Arguments 1 and 2 have this form, but argument 3 does not, as the second premise of argument 3 is not of the form r and s.

All of the arguments have form 2, since they have two premises and one conclusion, and form 2 is the form of any argument with this general shape.

Only argument 1 has form 3. Argument 2 does not have form 3 as there is no statement q shared by both premises in this argument. Argument 3 does not have form 3 as the second premise of this argument is not of the form q and r. Argument 1 has form 3 as we can take p to be “Greg lectures PHIL137”, q to be “Caroline lectures PHIL137” and r to be “Caroline Lectures PHIL132” (see errata).

Question {1.4}

I think that the first four express propositions, and that the fifth does only if you give it a context in which the non-English words have some kind of meaning. But many disagree with me on this.

Some say that (1) has a “category mistake”. It makes no sense to think of the sum of 2 and the Pacific Ocean.

Some think that (2) has a “reference failure” as the phrase “The present King of France” doesn’t refer to anyone or anything, so he cannot be bald (and he cannot fail to be bald). See Chapter 12 for more on this topic.

Some think that (3) cannot express a proposition, for if it’s true it is false, and if it’s false it’s true.

Some think that (4) cannot express a proposition, for there is nothing to decide whether it’s true or false.

Question {1.5}

Not every invalid argument form has valid instances. I think that the form:

• p or not p
• Therefore, q and not q

has no valid instances. Each instance of this argument will have a true premise and a false conclusion, and therefore, be invalid.

Question {1.6}

This is a very interesting question. To answer it, you need to have a clear understanding of what counts as the form of an argument. Some arguments are clearly valid, and have no valid form as far as we have seen so far. The argument

• All footballers are bipeds.
• Socrates is a footballer.
• So, Socrates is a biped.

is pretty obviously valid, but the only form it has — for the level of analysis we’ve looked at so far is:

• p
• q
• So, r

since none of the premises are conditionals, biconditionals, conjunctions, disjunctions or negations. However, it has some kind of form. (Exactly what, we’ll see in Chapter 8.)