# Mathematical Logic and Reasoning

Often when people discuss critical thinking or other aspects of how we should or should not think, they talk about using logic. This can be a problem, since the word "logic" can mean substantially different things in different situations.

## Mathematical logic

At one extreme, logic is a branch of mathematics. This kind of logic is closely related to the study of logic as a branch of philosophy. It can often differ quite dramatically from some of the reasoning that people will sometimes call "logic".

Mathematical logic makes use of operations that use "true" and "false" instead of numbers. There are two common types of mathematical logic: propositional calculus and first order predicate calculus.

Propositional calculus is quite simple, at least to mathematicians, and corresponds to the logic that is used to define computer circuitry (ones and zeros on the computer correspond to true and false in the logic). Some of the common operations are AND, OR, NOT, and IMPLIES. An example would be the statement "(NOT A) AND (NOT B) IMPLIES NOT (A OR B)".

Predicate Calculus is a bit more complex and is normally the foundation that mathematicians use in proving theorems. It makes use of predicates, which are statements that are true or false depending upon values that are supplied to them. A predicate might be American(X), which would be true if person X was an American, or odd(X) which would be true if X represented an odd number and false otherwise. An example of a statement in predicate calculus is "For all x such that even(x), integer( x/2 )", which would mean that for all numbers that are even numbers, that number divided by two would be an integer.

Because the rules for mathematics are exact, and the rules for logic are properly designed, any valid mathematical deduction is guaranteed to be correct. If the deduction is not valid, then we must have made a mistake in following the rules. In that case the deduction is not truly logical. If we make a mistake in a mathematical deduction, we may be wrong but think our conclusion is logical. This can be a serious problem for complex mathematical proofs, but is rarely a problem for most of us because we almost never actually do mathematical logic. It is useful to know, however, that properly applied mathematical logic is guaranteed to be correct.

The Pulitzer Prize winning book "Godel, Escher, Bach" by Douglas Hofstadter discusses the nature of mathematical logic at considerable length for those who have a strong interest in it.

## Philosophical Logic

Philosophical logic is supposed to follow the same rules as mathematical, but I like to make a distinction because philosophical logic typically uses predicates that are modeled on real world experiences rather than mathematical abstractions, and so there are errors that can occur in an apparently valid philosophical deduction that can't happen in a mathematical proof. One problem is that while mathematical terms are completely unambiguous, real world terms (words) are not. If the meaning of a term changes even very slightly in the course of a logical deduction, that deduction may turn out to be false.

Consider this short argument:

Thomas Jefferson was a Christian.
Christians believe that Jesus was the Messiah.
Therefore Thomas Jefferson believed that Jesus was the messiah.

Historical sources indicate that Thomas Jefferson believed that Jesus was a great moral philosopher and Jefferson considered himself a Christian because he was a follower of Christ, but he didn't believe Christ was the Messiah. So the first statement is true if the term "Christian" is taken to mean a follower of Christ. The second statement is true if the definition of Christian is the one common today - a person who believes Christ is the Messiah and other Christian doctrines in addition to agreeing with the teachings of Jesus.

Similar problems can occur when, rather than having several distinct meanings, words are simply vague. Here is an example:

Cynthia is a great actor.
Great actors get paid millions of dollars to make a movie.
Cynthia gets paid millions of dollars to make a movie.

The words "great actor" might mean one of the best actors in town or one of the best in the world. The word "great" isn't very specific about this. Statement two is true if we are talking about the greatest actors in the world, but that is not necessarily the case for Cynthia.

## Everyday logic and reasoning

I sometimes hear people refer to an argument as logical when it doesn't seem logical to me, or people complain that someone made a bad choice because they were being too logical and not using their common sense. Other people might say we must learn to think logically in order to be critical thinkers. As a person who has studied mathematical logic, statements like these bother me because they could not really be about mathematical logic. Since mathematical logic is never false, it could not be the cause of a bad decision. However mathematical logic is fairly tedious and only applies when the premises are absolutely true, so it is very difficult to apply to everyday problems. Typically people making referring to logic in everyday situations are not really thinking of pure mathematical logic or even rigorous philosophical logic. The everyday reasoning they refer to might be thought of as "casual" logic. It still differs from intuition because it involves some sort of step by step verbal reasoning.

A lot of people may form their opinion of logical thinking from the characters Mr. Spock and Data from the Star Trek television series. If we pay attention, these characters are really not perfectly logical. What passes as logic is often just reasoning that ignores emotional influences. I recall situations where doing something according to a standard rule is considered "logical" by these characters, but if they were truly logical, they would recognize that the rule may not always be correct.

Practical reasoning is always limited by the fact that we don't have complete information about what is going on. If I want to buy a shirt, I might choose between going to a local store where I would estimate I would pay \$50 or going a longer distance to an outlet store where I think it would cost about \$35. If I figure it is worth \$20 to me to save the travel time and expense, I might choose to go to the local store. My neighbor, in the same situation, might choose the outlet store instead for a couple of reasons. One could be that his experience with the prices at the two stores is different from mine. Another consideration is that he might have more free time or enjoy the longer drive more than I would. Even if we were both perfectly logical, we might make different decisions in the same situations. The difference could be because of preferences (he likes driving more than I do) or experience (his estimates based on experience are different from mine).