Logical reasoning proceeds from given assumptions to some sought conclusion. The essence of assumptions is that they are hypothetical so that it is not determined if they hold, and the point with the steps of reasoning is that they produce correct conclusions whenever the assumptions are correct. These steps are two-fold: In one direction, we analyse the assumptions into their simpler parts, in another direction, we look at the conditions from which the sought for conclusion can be synthesized. The aim is to make these ends meet. Some examples lead us to a small collection of basic steps and it turns out that all logical arguments based on the connectives can be reproduced as combinations of the basic steps.

Steps in proofs

Consider our bather in Cap Breton. The argument was: We have assumptions of the forms A ⊃ B and ¬ B. Now a is added to these assumptions, and a contradiction follows. The argument can be presented as a succession of steps each one of which is in itself hard to doubt. We write the steps one after another together with a justification at right:

Example argument 2.1. Proof of a contradiction from A ⊃ B, ¬ B, and A.

1. A ⊃ B by assumption

2. ¬B by assumption

3. A assumed with the aim of proving a contradiction

4. B from 1 and 3

5. B & ¬ B from 4 and 2