Sub-conclusions, etc.

We went over this in my 11:45 session, but it's an important enough point that everyone should see it. The issue is, how do you string together an argument with more than just three steps? Here's an example argument.

1. All apples are red.
2. Object x is an apple.
3. If object x is red, then it is sweet.

4. Object x is red.
5. Object x is sweet.

The above argument actually only has three premises-- that is, three independent statements that lead to the conclusion-- because statement 4 is a sub-conclusion. Premises 1 and 2 form a syllogism that gets us to statement 4. Then that statement is used with Premise 3 in a modus ponens argument to get to the final conclusion.

Oftentimes, it is clearer to present the argument as follows.

P1. All apples are red.
P2. Object x is an apple.
C1. Object x is red.
P3. If object x is red, then it is sweet.
C2. Object x is sweet.

In this presentation, the flow among all the statements is a little more obvious. But notice that the same statements are labelled premises-- regardless of where they appear within the argument. This is because the "premise", "sub-conclusion", and "conclusion" labels are attached according to the logical form of the argument.

When providing support for this argument, only the three premises need evidence. The sub-conclusion is arrived at through a valid argument, so its truth is, of course, guaranteed by the truth of its premises.