Solutions to Chapter 2 The scope of logic

A. Which of the following arguments is valid? Which is invalid?

1. 1.

   • Socrates is a man.

   • All men are carrots.

   • ∴
     ​

     Socrates is a carrot. Valid

2. 2.

   • Abe Lincoln was either born in Illinois or he was once president.

   • Abe Lincoln was never president.

   • ∴
     ​

     Abe Lincoln was born in Illinois. Valid

3. 3.

   • If I pull the trigger, Abe Lincoln will die.

   • I do not pull the trigger.

   • ∴
     ​

     Abe Lincoln will not die. Invalid
     Abe Lincoln might die for some other reason: someone else might pull the trigger; he might die of old age.

4. 4.

   • Abe Lincoln was either from France or from Luxemborg.

   • Abe Lincoln was not from Luxemborg.

   • ∴
     ​

     Abe Lincoln was from France. Valid

5. 5.

   • If the world were to end today, then I would not need to get up tomorrow morning.

   • I will need to get up tomorrow morning.

   • ∴
     ​

     The world will not end today. Valid

6. 6.

   • Joe is now 19 years old.

   • Joe is now 87 years old.

   • ∴
     ​

     Bob is now 20 years old. Valid
     An argument is valid if and only if it is impossible for all the premises to be true and the conclusion false. It is impossible for all the premises to be true; so it is certainly impossible that the premises are all true and the conclusion is false.

B. Could there be:

1. 1.
   ​

   A valid argument that has one false premise and one true premise? Yes.
   Example: the first argument, above.

2. 2.
   ​

   A valid argument that has only false premises? Yes.
   Example: Socrates is a frog, all frogs are excellent pianists, therefore Socrates is an excellent pianist.

3. 3.
   ​

   A valid argument with only false premises and a false conclusion? Yes.
   The same example will suffice.

4. 4.
   ​

   An invalid argument that can be made valid by the addition of a new premise? Yes.
   Plenty of examples, but let me offer a more general observation. We can always make an invalid argument valid, by adding a contradiction into the premises. For an argument is valid if and only if it is impossible for all the premises to be true and the conclusion false. If the premises are contradictory, then it is impossible for them all to be true (and the conclusion false).

5. 5.
   ​

   A valid argument that can be made invalid by the addition of a new premise? No.
   An argument is valid if and only if it is impossible for all the premises to be true and the conclusion false. Adding another premise will only make it harder for the premises all to be true together.

In each case: if so, give an example; if not, explain why not.