 A rule of inference is a way we can derive a true statement from other statements we know to be true. They have Latin names. The names aren't used outside of formal logic, so we won't worry about them, except perhaps as interesting pieces of trivia. One way to look at the rules of inference is to imagine you're playing a game where you can bet on whether a simple statement is true or false. And you know that a particular compound statement is true, and so the question you've got to ask yourself is, do you feel lucky? Or, more specifically, what will you bet on? So for example, let's find a rule of inference based on the conjunction a and b. So to begin with, let's set up our truth table for this conjunction. So each row in the truth table corresponds to a certain assignment of the truth values of a and of b, and that in turn gives us a truth value for the conjunction a and b. So if we know that a and b is true, what can we conclude? Well, first of all, we know we're not dealing with any of the cases where a and b is false. And so here's the question. You have $20. What will you put it on? So remember that in order for our conjunction to be true, both statements have to be true. And so we know a is true, and if we put our $20 on a is true, we win. We could have also put it on b as true because we also know that, and this leads to the rule of inference. If a conjunction is true, its component statements are true. So for example, let's assume the following is true. 36 is a perfect square that is divisible by 9. What can we conclude from this? So this corresponds to the conjunction. 36 is a perfect square, and 36 is divisible by 9. Now, we're assuming this is true. So if a conjunction is true, its component statements are true. And so we can conclude 36 is a perfect square. Now, since the conjunction a and b is the same as the conjunction b and a, or is it? You can show that these are logically equivalent. And so we can also conclude that 36 is divisible by 9. How about the disjunction a or b? As before, we could set up a truth table. Every row of the truth table corresponds to an assignment of the truth values of a and of b. And there's a corresponding truth value of the disjunction a or b. And we know that a or b is true, so we're one of these three cases, but we don't know which one. And so the question is, what are you willing to put $20 on? Let's think about this a little. Now, the disjunction a or b could be true in any of three cases. Both a and b are true. A is true, but b is false. And b is true, but a is false. And the problem is we know that at least one of a or b has to be true, but we don't know which one, and we should hesitate to put down any money at all. So if we know the disjunction is true, then we know not very much. And so it's important to keep in mind knowing that a disjunction is true isn't very useful. How about the conditional? Suppose you know that both a then b is true and also that a is true. Find the corresponding rule of inference. Well, again, let's set up our truth table. What we know is that a is true, and if a then b is also true. And the only time that occurs is in this line of the truth table. And that's because if a is true, the only way for the conditional to be true is if b is true. And since we know we have to be in this line of the truth table, that means we should be happy to put $20 and say that b must also be true. And this gives us another useful rule of inference. If the antecedent of a conditional is true, the consequent is also true. This particular rule of inference is known as modus ponens. That's one of those Latin names that are occasionally used for the rules of inference. So suppose all primes greater than two are odd. Let n be a prime greater than two. What can you say about n? So first of all, suppose is mathematical speak for assume this is true. So first we'll rewrite our premise as a conditional. If n is a prime greater than two, then it is odd. Now let is also mathematical speak for assume this is true. So we're assuming that n is a prime greater than two. And that's the antecedent of our conditional. And remember, if the antecedent of a conditional is true, the consequent is also true. And so we might say the following, since n is a prime greater than two, we can conclude that n is odd. One more important rule of inference comes from the following. Suppose we know that b is false and that if a then b is true, let's find the corresponding rule of inference. And we've set up truth tables for everything else, so this time we should set up a truth table. So we know that b is false and the conditional if a then b is true. And that means we have to be in this line of the truth table. If b is false, the only way that the conditional can be true is if a is false. And so we know that a is false and I'm willing to put $20 on that claim. And this leads to another important rule of inference. If a conditional is true but the consequent is false, then the antecedent must also be false. And the Latin name for this is modus tallens. So we claim every odd perfect square is one more than a multiple of four. What can you say about 51? We claim is another way to say the following statement is true. So let's rewrite our statement as a conditional. If an odd number is a perfect square, then it is one more than a multiple of four. And we note that 51 is not one more than a multiple of four. So remember, if a conditional is true but the consequent is false, then the antecedent must also be false. And that means our antecedent must be false. That it is not an odd number that is a perfect square. But there's a bonus. Odd number that is a perfect square is a conjunction of two statements. The number is odd. The number is a perfect square. Since this conjunction is false, remember that for a conjunction to be false, one of the component statements must be false. But we know that 51 is odd. And since 51 is odd, the first statement is true. And this means the second statement must be false. And so we also know that 51 is not a perfect square.