Logic and Inference

This is a rather deep topic, which has been studied since ancient times. In simple terms, if a fact is known, logic and inference tell us what other facts are also known because of it. This is not, of course, a simple process. In my previous post, I discussed theorems, which are essentially statements of these facts. Some theorems are extremely complicated. Logic and inference are what we generally use in proofs to connect known facts with those we want to prove.

As an example, let us take the statement "All my friends are taller than I am". Then, let us take the statement "Ed is my friend". Inference tells us that "Ed is taller than I am". If both the first two statements (premises) are true, then so is the third statement (conclusion). If, however, at least one of the premises is false, then the conclusion need not be true. For example, it might not be the case that all my friends are taller than I am. Then Ed may or may not be taller than I am. The creation of this conclusion from the premises above is a valid rule of inference, known as Universal Instantiation. This rule takes a characteristic of a set and applies it to a specific element in the set.

There are various other rules of inference, most of them dealing with taking two premises that are somehow related and extracting from them a conclusion which is related to each of them. The following is a short list of some common rules, with examples. I will use the letters p, q and r as statements, ~ as the negation symbol "NOT", ^ as the conjunction symbol "AND", * as the disjunction symbol "OR" and => (with other similar arrows) as the implication symbols.

Modus ponens (affirming the antecedent). Premises: p, p => q. Conclusion: q.
Here, one of the statements implies the other. Whenever p is true, it makes q true as well. We also have that p is true. Therefore, q must be true as well.
Example: If there is cake, I will eat it. There is cake. So, I will eat it. Here, p is "there is cake" and q is "I will eat the cake".

Modus tollens (denying the precedent). Premises: p => q, ~q. Conclusion: ~p.
This, in a sense, is the reversal of the previous rule. Again, p implies q. If p were true, q would be true. But q is false. So p cannot be true either.
Example: If there is cake, I will eat it. I will not eat cake. Therefore, there is no cake. (Otherwise, I'd be eating it).

Reductio ad absurdum (reduction to the absurd). Premises: p => q, p => ~q. Conclusion: ~p.
This one is a bit of a tricky one at first. In essence, if some statement being true makes another statement both true and false, then the first statement must be false.
Example: If I have a first-born, it is a boy. If I have a first-born, it is a girl. So, I do not have a first-born. (If I did, it would have to be both a boy and a girl).

Disjunctive syllogism. Premises: p * q, ~p. Conclusion: q.
Here we are given that at least one of our premises must be true. We are also given that one of them is definitely false. Then, we know that the other premise has to be the true one.
Example: My wall is white, or the door is open. The door is closed. So, my wall is white.
Note that here, the word "or" does not mean that only one of the two statements is true. In logic, that would be known as an "exclusive or". What we have instead in this case is an inclusive or, where it is possible for both statements to be true and at least one of them is true.

Hypothetical syllogism. Premises: p => q, q => r. Conclusion: p => r
This can be seen as a sequence of implication. When p is true, it causes q to be true. When q is true, it causes r to be true. So, if p is true, it will also cause r to be true. Also, using the same reversals as before, if r is false, it causes q to be false, which in turn causes p to be false.
Example: If it rains, I will bring my umbrella. If I bring my umbrella, I will lose it. so, if it rains, I will lose my umbrella.

There are many more rules of inference, some of which include more (and sometimes fewer) premises. In all of them, the single characteristic is the following: the conclusion follows directly from the premises. All that is at work are the rules of logic and inference. These rules are important; as I mentioned earlier, they give us guidelines in finding new facts, from previously known ones. This is not limited to Mathematics. Science, Law, Medicine all rely on the application of logic to their specific fields. A good understanding of logic is almost indispensable in most academic pursuits.

Of course, logic is often misused. There are many fallacies and errors which can be inadvertently committed by the misapplication of logical rules. I will be going over some of the more common errors of logic in my next post.