Solutions to Exercises

I. Validity    

1. Valid. "All mammals have fur" is equivalent to "If something is a mammal, then it has fur." (Satisfy yourself). We can now see that this argument is an example of modus ponens, so it is valid.
     
2. Valid. We can test for validity by assuming that all the premises are true and then seeing whether the conclusion must be true (this is a "direct test.")  So assume premise 1 is true.  The first premise of the argument is a disjunction.  As stated in the definition of disjunction, a disjunction is true unless both disjuncts are false, that is, at least one disjunct must be true.  The second premise we assume is true and it says that first disjunct of premise 1 is false.  So given the truth of the first two premises and the definition of disjunction, it must be the case that the second disjunct of the first premise is true.  But second disjunct of the first premise is identical to the conclusion.  So if all the premises are true, it must be the case that the conclusion is true.  Therefore, the argument is valid.
     
3. Invalid. We can also test for validity indirectly by trying to construct a counterexample. If we can, then the argument is invalid; if we can't, then it is valid. A counterexample to an argument is a case in which all the premises are true and the conclusion is false. So, begin by assigning truth values to r and p that make the conclusion false. From the definition of a conditional, we see that the only case in which the conclusion (r implies p) is false is that in which r is true and p is false. So, if r is true and p is false, can both premises still be true? Yes. Let q be false. In this case, premise one is true (satisfy yourself) and premise two is true (satisfy yourself). We can construct a counterexample (p: false, q: false, r: true), so the argument is invalid.
**Exercise: Can we construct a counterexample in which q is true?
     
4. Valid. If the premises are taken to be true, then the conclusion must be true.  In this exercise, the only premise that factors into the truth value of the conclusion is premise 2.  If we assume the premises to be true, including assuming that premise two is true (remember that a conjunction is only true if both conjuncts are true), then the conclusion must be true because it asserts the same thing that the second conjunct of premise two asserts.  Therefore, the argument must be valid.
     
5. Valid. We can test for validity by substituting statement variables into the argument.  If we do the substitution, we get:
If p, then q.
Not q.
Therefore, not p.
     Upon inspection, we find that this is one of the common patterns of valid inference discussed above—modus tollens.  Take both premises to be true. In this case, the first premise is a conditional with a false consequent. From the definition of a conditional, we find that the only case in which such a conditional is true is when the antecedent is false. Therefore, p must be false. If p is false, then the conclusion (not p) must be true. So the argument is valid.
     
6. Invalid. We can construct a counterexample of the following form:
"Jacob has a lion tattoo" is true.
"Max a pottery wheel" is true.
"Jacob has a lion tattoo and Max has a clay pot" is false.
     
7. Valid. If the premise is true, then the conclusion must be true because the conclusion is a disjunction.  So, as long as one of the disjuncts in the conclusion is true, then the entire conclusion is true.  The argument can be translated into statement variables as follows:
p.
Therefore, p or q.
     If the premise is true, then the truth of p necessitates the truth of the conclusion.
     
II. Soundness  

1. Sound. The argument is valid and all the premises are true.  The argument is valid because, when we take all of the premises to be true, the conclusion must be true.  As we've seen, we can rewrite the first premise as "If something is a fish, then it can swim and has gills." We can then see that this is an instance of a common pattern of valid inference, modus ponens. So the argument is valid. Also, both premises regarding fish are true, so the argument meets the definition of soundness.
     
2. Unsound. The argument is valid; however, since not all the premises are true, the argument is unsound.  The argument is valid because if we take the premises to be true, then the conclusion must be true.  You can be sure of this because example 2 is of the very same form as a common pattern of valid inference we told you about: modus ponens. As for soundness, Martha Stewart is not the President of the United States.

3. Unsound. The argument is invalid and therefore unsound. By the definition of disjunction, the first premise is true unless both disjuncts are false. So consider the case:
Elisa dyes her hair blond. TRUE
Jessica dyes her hair green. TRUE
In this case, both premises are true and the conclusion is false. Therefore, the argument is invalid.

IX. Rotating Validity Exercises