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4.18

# Truth-table shortcuts

There are five basic sentences in our representation of the argument. So a full truth-table would have thirty-two rows. That would be quite a hefty truth-table!

But there is a way to test the argument for formal validity without having to fill out all of the rows. Can you see how you might do this?

There are three hints below, but see if you can work out some shortcuts yourself before you look at them. (If you feel you need hints, take them one at a time.)

If you feel like you’ve cracked how to do a conclusive test for validity for this argument without completing the working for every single row — or if you just feel like taking on the challenge of working all thirty-two rows — have a go before looking at the next step. (IMPORTANT NOTE: If you are going to construct all or part of the truth-table, put your basic sentence columns in this order: S, W, A, O, L. We’ll give you a truth-table with the sentences in this order, so you can check your working, and that will have the basic-sentence columns in this order.)

• Hint 1: If the argument is invalid, there will be a counterexample row — a row on which all of the premises are true and the conclusion false. If there’s no such row, then the argument is valid. This should help you cut down the number of rows you need to work on. (There are a number of shortcuts that fall out from this.)

• Hint 2: Note that in this argument, a basic sentence, ‘S’, is one of the premises.

• Hint 3: For a row to be a counterexample, the conclusion must be false on that row.

How did you get on with working out how to test for validity without having to do all of the working on all of the rows of the truth-table?

The key point is that we’re looking for counterexamples. There are a number of ways that rows can get ruled out quickly as potential counterexamples.

For instance, if the conclusion is true on a row, it can’t be a counterexample, so we don’t need to work out truth-values of premises on those rows.

Another point is that a row can only be a counterexample if all of the premises are true on it. So, once we’ve found that one premise is false on a particular row, we don’t need to do any more working on that row. In the case of the argument we’re looking at, a basic sentence, ‘S’, is one of the premises, so we don’t need to look at any of the rows on which ‘S’ is false. That halves the number of rows we need to look at to sixteen in a stroke!

OK. Now, if you haven’t already done it, have a go at testing the argument for formally validity using a truth-table. You can use the short-cuts we’ve noted or, if you’d like, just do the whole thing.

(Note: Remember to put your basic sentence columns in the order S, W, A, O, L, so you can check your table against ours.)