Supporting A Conditional Premise

How do you support a conditional premise? In general, for purposes of our class, this support can be thought of as connecting the A with the B in the conditional statement: If A, then B. It can be thought of as explaining how B follows from A. What is it about A that results in B? Why was A chosen as the criteria for B?

In our class this could take several forms. If there is a well-established scientific theory connecting A with B we can use it as support for the conditional statement. You could use the theory of gravity to support the statement: if you drop a rock, then it will fall toward the earth. Sometimes, when we are not familiar with the topic, it may be necessary to use expert opinion. Sometimes we may use statistical arguments. Other times may require that we use personal observations or information that is generally regarded as true by most people. In all of the cases we are trying to show that B follows from A.

For this class the support doesn't need to be a formal argument, but how would a more formal argument look? To support a conditional premise the conditional itself must be the conclusion to an argument. In general there are three stategies to support a conditional premise. These general strategies for proving a conditional statement "if A, then B" are:

1. Start the evidence with assuming A to be the case and show that from A you can derive B. It then follows "if A then B" from this evidence.
2. Show that A is sufficient for B to happen, usually causally sufficient is the way to go. It then follows "if A then B" from this evidence.
3. Show that B is necessary for A, that A could not exist or be true without B existing or being true. It then follows "if A then B" from this evidence.

I will now intoduce a new kind of conditional form that can be used to support a conditional premise using the first strategy.

P1: If A, then K.
P2: If K, then B.
C: If A, then B.

Here is an example:

P1: If air is 80% nitrogen, then air is mostly nitrogen.
P2: If air is mostly nitrogen, then there isn't as much oxygen as nitrogen in air.
C: If air is 80% nitrogen, then there isn't as much oxygen as nitrogen in air.

This may be a little simple, but you get the idea. Here is another argument where the premises and the conclusion are conditionals.

P1: If a bicycle tire is fully pumped up, then the tire pressure is high.
P2: If the tire pressure is high, then the tire will feel hard.
C: Therefore, if a bicycle tire is fully pumped up, then the tire will feel hard.

Now convince someone that this is a sound argument. To do this you need to support the premises.

Here is one way to support the first premise.

P1a: If a bicycle tire is fully pumped up, then there has been a lot of air placed into a small volume.
P1b: If there has been a lot of air placed in a small volume, then the tire pressure is high.
IC/NBC (P1): If a bicycle tire is fully pumped up, then the tire pressure is high.

Here is one way to support the second premise.

P2a: If the tire pressure is high, then the tire will resist inward forces.
P2b: If the tire resists inward forces, then the tire will feel hard.
IC/NBC (P2): If the tire pressure is high, then the tire will feel hard.

The first premise in the support for the second main premise (P2a) may itself need some support. One way to do that could be as follows:

P2a1: If the tire pressure is high, then it has a lot of air in a small volume.
P2a2: If a tire has a lot of air in a small volume, then the air is constantly pushing against the inside of the tire.
P2a3: If the air is constantly pushing against the inside of the tire, then the tire will resist inward forces.
IC/NBC (P2a): If the tire pressure is high, then the tire will resist inward forces.

Notice that this last example has three conditional premises that are linked together in this new form. I have found this approach to be very useful when giving support for conditional premises.