Want to keep learning?

This content is taken from the University of York's online course, Logic: The Language of Truth. Join the course to learn more.

Skip to 0 minutes and 0 seconds As we’ve said, when you first start making truth tables for complex sentences, it can be a little puzzling, seeing in what order you should do your workings. Here’s a helpful way to identify the order. First, write out your sentence. Here we’ll use this complex example. The first step is to underline the basic sentences, like this. The next step is to underline grammatical clauses made by plugging into connective only basic sentences. Which do you think these are in our example? Think for a moment before we go on. Be careful. Remember, you’re looking for grammatical clauses made by plugging only basic sentences into connectives. In fact, there’s only one in this example. It’s here. ‘Tilde B’.

Skip to 1 minute and 1 second We underline underneath the first set of underlining. When you underline a clause, underline all of it. So here you’ll underline tilde and B that make up ‘tilde B’. The rule for the next step is a little different. Here, we look for grammatical clauses that are made by plugging into connectives only clauses we’ve underlined already. Can you see any? Again, there’s only one in our example. It’s

Skip to 1 minute and 35 seconds here: Open bracket A ampersand tilde B close bracket. When you underline a clause which is surrounded by brackets, underline the brackets too. They’re part of the well-formed formula which makes up the clause and making sure they’re there will ensure you’ll have a grammatical clause. Now, we just apply this rule again and again to find new levels. At each stage look for grammatical clauses that are made by plugging only clauses we’ve underlined already into connectives and each time. we add a further layer of underlining. Can you see the next clause to get underlined? Careful.

Skip to 2 minutes and 12 seconds The next one is here: “It’s not the case that both A and not B”. Notice that the rule for tilde says that plugging a well-formed formula into tilde gives you a well-formed formula. Here we plug A and not B into tilde. Notice that if we try and plug ‘A and not B’ into this ampersand together with this ‘C’, we wouldn’t have a grammatical clause. We’d have a closing bracket after the ‘C’, here, but no opening bracket before ‘bracket A’ here. What about the next level? That’s this, the conjunction of ‘it’s not the case that both A and not B’ with ‘C’ plugging them into this ampersand. What about the next level? That’s this.

Skip to 3 minutes and 16 seconds Notice that we take the whole of that last clause that we underlined and plug it into this tilde at the beginning of the sentence. Notice that we’ve now underlined the whole sentence so this is the last connective to get involved and, if we produced a truth table for the sentence, it would be under this tilde that we found the truth values for the whole sentence. Now, when you work through a truth table working out truth values of progressively larger clauses, you simply follow the order of the underlining that we’ve established, at each stage writing your results under the connective that the clauses you’ve already worked out are plugged into.

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.

Underlining example

Share this video:

This video is from the free online course:

Logic: The Language of Truth

University of York