Introduction to Logic

Basic Definitions

Logic is the study of the criteria used in evaluating inferences or arguments.

An inference is a process of reasoning in which a new belief is formed on the basis of or in virtue of evidence or proof supposedly provided by other beliefs.

An argument is a collection of statements or propositions, some of which are intended to provide support or evidence in favor of one of the others.

A statement or proposition is something that can either be true or false. We usually think of a statement as a declarative sentence, or part of a sentence.

How many statements are there in the example below?

I have two brothers, and I have no sisters. (The answer is 3!)

The premises of an argument are those statements or propositions in it that are intended to provide the support or evidence.

The conclusion of an argument is that statement or proposition for which the premises are intended to provide support. (In short, it is the point the argument is trying to make.)

(Important note: premises are always intended to provide support or evidence for the conclusion, but they donn't always succeed! It's still an argument, and there are still premises and a conclusion, even if the premises don't really provide any support at all.)

Some Example Arguments

God is defined as the most perfect being. A perfect being must have every trait or property that it's better to have than not to have. It is better to exist than not to exist. Therefore, God exists.

Hillary Clinton must be a communist spy. She supports socialized health care, and everyone who supports socialized health care is a communist spy.

It has rained more than 15 inches per year in Amherst every year for the past 30 years. So you can safely bet it will rain more than 15 inches in Amherst this year.

Professor Chappell said that the ratio of female to male students in the class was exactly 3:1. This means that there are 112 female students in the class, because there are 148 students in the class total.

The Encyclopaedia Britannica has an article on symbiosis. It stands to reason that the Encyclopedia Americana has an article on symbiosis as well, since the two reference works tend to cover the same topics.

1 is a prime number. 3 is a prime number. 5 is a prime number. 7 is a prime number. Therefore, all odd integers between 0 and 8 are prime numbers.

Jason isn't an NRA member. Almost 90% of NRA members are republicans, and Jason isn't a republican.

Inductive Logic and Deductive Logic

This can be a tricky subject, because many people are taught the distinction wrongly in high school. Many people think deduction is reasoning from the general to the specific, and induction is reasoning from the specific to general. This is NOT how these words are actually used by most logicians, nor in this course.

The distinction actually has to do with how strong the author of an argument intends the evidence or support to be.

An argument is deductive if the author intends it to be so strong that it is impossible for the premises to be true and the conclusion to be false, or in other words, that the conclusion follows necessarily from the premises. A deductive argument attempts (successfully or unsuccessfully) to provide full proof of the conclusion.

An argument is inductive if the author intends it only to be so strong that it is improbable that the premises could be true and the conclusion false, or in other words, that the conclusion is likely if the premises are true. An inductive argument only attempts (successfully or unsuccessfully) to provide evidence for the likely truth of the conclusion, rather than outright proof.

From now on, I'm going to focus only on deductive logic.

Validity and Soundness

A deductive argument is valid if it has a form that would make it impossible for the premises to be true and the conclusion false, or if the conclusion follows necessarily from the premises.

To test whether an argument is valid, you should first imagine that the premises are true—whether or not they actually are—and then ask yourself, without appealing to any other knowledge you have, could you still imagine the conclusion being false? If you can, the argument is invalid. If you can't, then the argument is valid.

Note that validity does not have to do with the actual truth or falsity of the premises. It only has to do with what would follow from them if they were true. A valid argument can have false premises. For example:

All toasters are items made of gold.
All items made of gold are time-travel devices.
Therefore, all toasters are time-travel devices.

It may be hard to imagine these premises as true, but it is not hard to recognize that if they were true, the conclusion would also be true.

So there's more to an argument's being a good one than validity. To be a good argument, an argument must also have true premises. An argument with true premises is called factually correct.

A sound argument is an argument that is both valid and factually correct.

An invalid argument may have true or false premises, and a true or false conclusion. A valid argument may have false premises with either a true or a false conclusion. The only combination that is ruled out is a valid argument with true premises and a false conclusion. Sound arguments always have true conclusions.

Argument Form

The validity of a deductive argument is determined entirely by its form. Consider these arguments.

1)    All tigers are mammals.
No mammals are creatures with scales.
Therefore, no tigers are creatures with scales.

2)    All spider monkeys are elephants.
No elephants are animals.
Therefore, no spider monkeys are animals.

These arguments share the same form: All A are B, No B are C, Therefore, No A are C. All arguments with this form are valid. So the examples above are valid. (What is wrong with #2?) Now consider:

The Earth is round.
Therefore, the Earth is a basketball.

4)    All Jedis are one with the force.
Yoda is one with the force.
Therefore, Yoda is a Jedi.

These arguments also have the same form. All A are F. X is F. Therefore, X is an A. All arguments with this form are invalid. #4 may seem like a good argument because all the premises and the conclusion are true (at least in fiction), but note that the conclusion isn't made true by the premises. It could be possible for the premises to be true and the conclusion false. #4 is invalid, and all invalid arguments are unsound. #4 is not a good argument.

Evaluating Arguments Logically

Logic is very important in philosophy, because so much of what philosophers do involves putting forth arguments, and assessing those of others. There are two steps in evaluating an argument. First, ignore for the moment whether or not the premises actually are true, and ask yourself whether or not, if you imagine that they are, the conclusion follows from them.  If it does, that is, if the process of reasoning is a good one, then the argument is valid. Next, ask yourself whether or not the premises are true.  If all of them are true (regardless of the conclusion), then the argument is factually correct. These are the only two ways of evaluating an argument that are important for the purposes of this class. If the argument is both valid and factually correct, then the conclusion must be true.

1) If God existed, then J. Lo and Puffy would still be dating. But J. Lo and Puffy are not dating anymore. Therefore, God does not exist.

2) If Minnie Driver has agreed to go on date with Kevin, then God exists. Minnie Driver has not agreed to go on a date with Kevin. Therefore, God does not exist.

3) Either materialism is false or all emotions are physical processes in the brain. If all emotions are physical processes in the brain, then it is possible to know what love feels like simply by studying brain chemistry. It is not possible to know what love feels like simply by studying brain chemistry. Therefore, materialism is false.

It is worth noting that an argument may still have a true conclusion even if it is invalid or factually incorrect (or both). Consider:

Christina Aguilera loves all of Eminem's lyrics. The Eifel tower is made of cheese. Therefore, Atlanta is the capital of Georgia.

The real upshot of this, however, is that just because you believe a certain conclusion, you do not logically have to accept the soundness of every argument for that conclusion.

Some Logical Pitfalls

Begging the Question

An argument begs the question when it makes use of a premise that no one who didn't already accept the conclusion would believe. Simply put, an argument begs the question when it reasons in a circle or presupposes the truth of the very thing it's trying to prove.

Example: God exists, because it says that God exists in the Bible, and everything in the Bible is the true word of God.

The Appeal to Ignorance Fallacy

This fallacy is committed when something is concluded to be true simply because it hasn't been proven to be false, or is concluded to be false just because it hasn't been proven to be true. Reasoning in such a way is invalid. Something can be true even if no one has succeeded in showing it to be true.

Example: No one has even proven that there is life after death. Therefore, there is no life after death.

The Wishful Thinking Fallacy

This fallacy is committed when someone concludes that something must be true in virtue of what he or she wants to be true (or doesn't want to be false) instead of what the evidence suggests. Unfortunately, just because there are better consequences to something's being true rather than false does not provide evidence that it is true.

Example: The idea of life in a universe without God would be frightening and depressing, and very difficult to accept. Therefore, God must exist.

This fallacy is committed when an argument or position is rejected not in virtue of its logical merits, but rather in virtue of the character, personality, background or motivation of the person giving the argument or holding the position. However, a position can be true, and an argument can be sound, no matter how deplorable the person is. Who holds a belief has nothing to do with whether or not it's true.

Example: Former president Clinton has argued in favor of increasing restrictions on the sale of guns. But President Clinton is a lecherous, adulterous, untrustworthy, draft-dodging old pervert, so his views must surely be misguided.

What I call the "who's to say" fallacy is an instance of ad hominem reasoning. E.g.: "Descartes has argued that all persons consist of two distinct substances: a material body and an immaterial mind. But who is Descartes to say what is true of all persons?"

Opinion and Fact

An opinion is something that someone believes to be true.

A fact is something that is true.

Sometimes people disagree about what the fact of the matter is with regard to a certain question. In those cases, there are many opinions, but only one fact. Those people whose opinion agrees with the facts are correct, those who have other opinions are incorrect.

Some points to remember:
1. Just because there is disagreement about what the facts are DOES NOT mean that there is no fact of the matter.
2. Something can be a fact even if no one, or even if very few people, believe it. (Copernicus's opinion that the Earth revolved around the sun was a fact, even before he managed to prove it and even before anyone else believed it.)
3. Just because people disagree about something does NOT mean that the thing in question is "true for for some people" but not "true for others".
4. Most things we talk about in philosophy are not in any way relative or subjective, in the same way that tastes is music or food are relative.  If God exists, God exists for everyone everywhere, and those people who believe that there is no God are mistaken. This may not sound nice, but it's the only coherent thing to say.
5. If you believe something is true, you cannot coherently NOT believe that people who believe that it is not true are mistaken.
6. It is not an objection to something to call it "just an opinion" or "just a theory".

Example: Sartre believes that human free will is incompatible with the existence of God. But that's just his opinion about the relationship between God and free will, not everyone's.

Is this an objection to Sartre?