Shorter truth tables can help us find if an argument is valid, or a set of propositions are consistent, or if two propositions are equivalent. However, when completing a shorter truth table, we must sometimes guess a truth value for a variable. This occurs when there are no “forced” truth values — that is, when there exists more than one way to complete the current truth value for every remaining proposition.
Here are two rules to keep in mind when you must guess a truth value:
- If guessing allows you to complete the shorter truth table without contradiction, then stop; your question is answered. Either you have shown the argument is invalid, or the given propositions are consistent, or the two propositions are not equivalent.
- If the guess leads to an unavoidable contradiction, then you must guess the opposite truth value for that variable and continue, because the contradiction just might be showing that your guess was wrong.
Take a look at this post for a flowchart for guessing with validity.