Working out your working out

When you first start making truth-tables for complex sentences, it can be a little puzzling seeing what order to do your workings. This video shows a helpful way to identify the order.

The procedure

Here’s a breakdown of the procedure described in the video:

1. Write out the complex sentence
2. Underline the basic sentences
3. Find and underline grammatical clauses made by plugging only basic sentences into connectives. (At each stage, any new underlining goes below any existing underlining. Remember to extend your underlining to any brackets which are parts of the clause. Making sure they’re there ensures you’ve found a grammatical wff.)
4. Find and underline grammatical clauses made by plugging only sentential clauses you’ve already underlined into connectives.

Now, repeat step (4) until your underlining extends the full length of the sentence. The last connective involved is the main connective.

When you work through the truth-table, working out truth-values of progressively larger clauses, you simply follow the order of your underlining. Start with the values of the basic sentences, then work through the levels. At each stage, write your results under the connective that the clauses you’ve already worked out are plugged into.

When you’ve finished, you’ll have the truth-values of the full complex sentence in the column under the main connective.

Here’s the underlining for the example in the video. Note that the tilde at the start of the sentence is the main connective.