Completing truth-tables for complex sentences

Now we’ll see how to complete the truth table for our target sentence, working up from the truth values of the basic sentences. We start working through our truth table by looking for grammatical sentential clauses that are made by plugging only basic sentences into connectives. As we noted when we ran through the structure of our target sentence, there are two here. ‘(A & B) and ~C. We work out the truth values of these clauses for each way things could be with the relevant basic sentences, using the defining truth table for the relevant connective and writing the truth value of the clause under its connective.

For example on the first row ‘A’ and ‘B’ are both true, so ‘(A & B)’ is true, and we write T under the ampersand on the first row, like this. You’ll notice that we’ve used lowercase t and f for the basic sentence truth values and that we cross out truth values as they’re used up in working out truth values of progressively larger sentential clauses. These are optional extras that help us to keep track of where were up to in our working. You don’t need to copy this way of working but you might find that it helps. Both ‘A’ and ‘B’ are true on the second row, so we write T under the ampersand there too.

On the third row, however, ‘A’ is true and ‘B’ is false so ‘(A & B)’ is false so we write F in the ampersand column like that. Now we continue filling out the truth values of ‘(A & B)’ of the remaining rows with the results depending on the truth values of ‘A’ and ‘B’ on those rows.

Remember that there’s another clause made by plugging

only basic sentences into a connective: ~C. We now work out the truth values of that clause in each row using the truth table for tilde. Where ‘C’ is true, ‘~C’ is false. Where ‘C’ is false, ‘~C’ is true. That gives us this. Now we need to move up to the next level where sentential clauses we’ve dealt with already are plugged into connectives. For the complex sentence we’re looking at, this involves just the second ampersand. This has ‘(A & B)’ and ‘~C’ plugged into it. On the first row, ‘(A & B)’ is true, but ‘~C’ is false so the conjunction is false. On the second row, ‘(A & B)’ is again true but ‘~C’ is true.

So this time we have true.

Working down the remaining rows of the table we see that on each row at least one of the conjuncts is false.

So the conjunction is false on all of those rows. Now we’ve reached the highest level of structure for this sentence. The last connective involved is the second ampersand. This is the main connective. So, it’s under this that we find the truth values of the whole sentence in all of the different kinds of situation. We can see that there’s only one kind of situation in which ‘((A & B) &~C) is true and that’s where ‘A’ is true, ‘B’ is true and ‘C’ is false. The sentence is false in all other kinds of situation, so that’s how to work out the full truth table for a complex sentence.