## Subsection5.1.2Review – Sound Arguments

Recall that, in our discussion of Boolean logic proofs, we said that an argument (proof) is:

• valid provided that every one of its steps can be justified by a sound inference rule.

• sound provided that it is valid and that its premises are true (in whatever world we are reasoning about).

We’ll consider these same properties of predicate logic proofs.

Note as before the somewhat unfortunate (but conventional) use of the word “sound” to mean one thing when applied to inference rules (i.e., the property that truth is preserved by the reasoning process) and another thing when applied to entire arguments (i.e., that truth is both introduced by the premises and preserved by the argument).

Consider the following argument:

Lucy is a cat.

All cats are mammals.

Therefore: Lucy is a mammal.

This argument is valid. (We’ll soon describe the logical inference rules that will let us construct this proof.) It’s also sound, since both of the premises are true.

Consider the following argument:

Lucy is a cat.

All cats live on Mars.

Therefore: Lucy lives on Mars.

This argument is also valid. (We can prove it using the same inference rules we used above.) But it isn’t sound, since it has a premise that isn’t true.

Consider the following argument:

Lucy lives on Mars.

All cats live on Mars.

Therefore: Lucy is a cat.

This argument isn’t valid. Our second premise is a quantified version of CAT  MARS. We know MARS (in the case of Lucy). But, since Converse isn’t a valid inference rule, we can’t go from that to CAT (in the case of Lucy).

As in our discussion of Boolean logic, our focus here will be on the construction of valid proofs. Choosing premises (axioms) is another issue, best left to experts in whatever problem domain we want to consider.

### ExercisesExercises

#### Exercise Group.

Imagine a world that contains:

##### 1.

(Part 1) Consider the following argument:

Smokey is a bear.

Smokey has a tail.

Therefore all bears have tails.

Which of the following is true of this argument:

1. It is sound.

2. It is valid but not sound.

3. Its premises are true but its reasoning is invalid.

4. It is total junk.

Solution.
Explanation: The premises are true. And so is the conclusion. But the conclusion doesn’t follow from the premises. The fact that Smokey has a tail doesn’t mean that all bears do.
##### 2.

(Part 2) Consider the following argument: Bambi is a bear.

All bears have tails.

Therefore Bambi has a tail.

Which of the following is true of this argument:

1. It is sound.

2. It is valid but not sound.

3. Its premises are true but its reasoning is invalid.

4. It is total junk.

Solution.
Explanation: It is valid. The reasoning is correct. But it isn’t sound because one of its premises (that Bambi is a bear) is incorrect.Explanation: It is valid. The reasoning is correct. But it isn’t sound because one of its premises (that Bambi is a bear) is incorrect.
##### 3.

(Part 3) Consider the following argument: Smokey is a bear.

All bears have tails.

Therefore Smokey has a tail.

Which of the following is true of this argument:

1. It is sound.

2. It is valid but not sound.

3. Its premises are true but its reasoning is invalid.

4. It is total junk.

Solution.
Explanation: It is sound. The reasoning is correct and the premises are true.
##### 4.

(Part 4) Consider the following argument: Bambi is a bear.

All bears are brown.

Therefore Bambi isn’t black.

Which of the following is true of this argument:

1. It is sound.

2. It is valid but not sound.

3. Its premises are true but its reasoning is invalid.

4. It is total junk.