This article was originally posted on 7 January 2009 at www.thinkingmatters.org.nz
An Introduction to Logic
I often tell people that the most useful and fundamental tool in the apologist’s tool-kit is logic. However, it’s depressing how often I see basic logical mistakes being made. Even more depressing is when an opponent or detractor of an argument actually thinks he or she is making a good point, when a basic mistake is obvious. To help with avoiding this, I’m going to lay out a few ground rules for debating: what logic is, and how to use it.
Logic is the study of the rules for right thinking. This is often not properly understood. People tend to use “logic” as a popular synonym for “common sense,” but it is actually a highly technical sub-discipline of philosophy, akin to mathematics. It’s a discipline that itself has many sub-disciplines, and like mathematics, it’s very easy to understand the basics and to use them correctly. For now, we’ll just be surveying basic deductive logic, by asking the question: what makes a good argument?
For our purposes, I’m taking a good argument to be one which is persuasive to a broad range of people. It is one which would persuade a reasonable person, and ideally even an unreasonable person. There are four criteria which make a good argument.
I. It must be formally valid
Formal validity means that the conclusion of the argument follows necessarily from its premises. Such an argument can be structured into a syllogism, which is a way of writing it with its various premises, and a conclusion, as in the examples below. If an argument breaks one of the rules of inference, it is formally invalid. We’ll look at the nine essential rules of inference, first using proper notation, and then using plain language with an example. Leibniz, the German philosopher who formulated many of these rules and gave them their notation, believed that all thought was constructed from these simple rules.
1. Modus ponens
Modus ponens is Latin. It means “the mode which affirms”. Knowing the English translation makes it very easy to follow:
- P -> Q
In plain English: if P, then Q; P, therefore Q. “P” and “Q” represent propositions, so it’s helpful to substitute in simple phrases for them, to get a better idea of what the rule is saying. For example, let P mean “it is raining”, and let Q mean “the ground is wet”. So:
- If it is raining, then the ground is wet.
- It is raining.
- Therefore, the ground is wet.
As you see, this is really a very simple and obvious rule—as you’ll find that all the fundamental rules of logic are.
2. Modus tollens
Like modus ponens, modus tollens is also Latin. It means “the mode which denies”. Again, this gives a helpful clue as to its form:
- P -> Q
- ¬ Q
- ¬ P
In plain English: P implies Q; not Q, therefore not P. Substituting in some propositions:
- If it tastes sweet, then I like it.
- It don’t like it.
- Therefore, it doesn’t taste sweet.
3. Hypothetical syllogism
- P -> Q
- Q -> R
- P -> R
If P implies Q, and Q implies R, then P implies R.
- If it is cheese, then it is delicious.
- If it is delicious, then I want to eat it.
- Therefore, if it is cheese, then I want to eat it.
- P & Q
If P, and if Q, then P and Q.
- I have chocolate.
- I have cheese.
- Therefore, I have chocolate and cheese.
- P & Q
If P and Q, then Q. This isn’t a trick rule; you shouldn’t read it to be excluding P by stating the conclusion Q. It’s just saying that if two things are true, then one of those things is true. Of course, the other is also true, so we can equally say, if P and Q, then P. Both are valid simplifications of the same premise.
- I like chocolate and I like cheese.
- Therefore, I like cheese.
- I like chocolate and I like cheese.
- Therefore, I like chocolate.
- P -> Q
- P -> ( P & Q )
If P implies Q, then P implies both P and Q.
- If it is raining, then the grass is wet.
- If it is raining, then it is raining and the grass is wet.
This is also known as the disjunction introduction, since it introduces disjunctions (“and/or” statements).
- P v Q
P, therefore P or Q. In other words, given some proposition P, either P is true, or some unrelated proposition Q is true. The truth of Q doesn’t exclude the truth of P, though typically it is assumed.
- The sun rises in the east.
- Therefore, the sun rises in the east, or I have a secret identity as a superhero.
8. Constructive Dilemma
- ( P -> Q ) & ( R -> S )
- P v R
- Q v S
If P implies Q, and R implies S, P or Q is true; so R or S is true.
- If Yvette comes along on the trip, then Jim will be happy; and if Jim goes on the trip without Yvette, then he will be lonely.
- Either Yvette comes along on the trip or Jim goes on the trip without Yvette.
- Therefore, either Jim will be happy or he will be lonely.
9. Disjunctive syllogism
This is basically an “either/or” statement:
- P v Q
- ¬ P
P or Q; not P, therefore Q. Of course, the converse is also true: P or Q; not Q, therefore P.
- I am allowed either two plain biscuits or one chocolate biscuit.
- I choose to eat the one chocolate.
- Therefore, I don’t choose to eat the two plain.
II. It must be informally valid
The second of the criteria of a good argument is informal validity. Being formally valid is not enough; it must also not commit any informal fallacy. There are many kinds of informal fallacies—far too many to list here. However, I’ll cover the eight of the most common. When an argument commits any of these fallacies, it is informally invalid.
1. The strawman
This is misrepresenting your opponent’s position in some way—either by caricaturing it, or assuming that he holds to the most vulnerable possible variant of it—and then arguing against that misrepresentation as if it were his actual position. For example:
Person A: “Life got here by creation.”
Person B: “That’s impossible because the earth could not possibly have been created in six 24-hour days.”
2. Begging the question
Arguing in a circle and providing no reason for accepting a premise in your argument, other than the conclusion of the argument itself.
Person A: “I know that God exists because the Bible says He does.”
Person B: “How do you know the Bible is telling the truth?”
Person A: “Because the Bible was written by God.”
3. The genetic fallacy
Arguing for or against some belief based on the origin of that belief.
Because of our fear and ignorance of nature we invented God. Therefore he does not exist.This fellow often produces shoddy research, so his latest paper should be dismissed.
4. Argument from ignorance
Arguing that a belief is false because there is insufficient evidence for it.
No one can disprove the existence of God. Therefore, God exists. There’s no evidence that the Red Sea was ever parted. Therefore, the account in Exodus is a myth. (Notice, though, that an argument saying that there is evidence that the Red Sea was not ever parted would not be fallacious.)
Using a word or category in such a way that it has more than one meaning.
Margarine is better than nothing. Nothing is better than butter. Therefore, margarine is better than butter.
Formulating a premise in such a way as its meaning is ambiguous.
No food is better than our food.
Inferring that a whole has a certain property because all of its parts have that property.
Every part of an infinite past can be traversed. Therefore, an infinite past can be traversed. Every tile on the floor was cheap, therefore the tiled floor is cheap.
8. Ad hominem.
Attacking the person and not the argument (Latin: “at the man”).
What he says can’t be correct because he’s a religious nut. Calvinism is awful because John Calvin burned Servetus at the stake. (Notice how this example also constitutes the genetic fallacy.)
Is this a good argument?
- Either I am crazy, or I am dead.
- I am not crazy.
- Therefore, I am dead.
This disjunctive syllogism is both formally and informally valid, making it logically valid. But obviously this not a good argument. On reflection premise one is rather spurious. Why should anyone accept the dilemma presented there? There is also cause to doubt premise two. Therefore, logical validity is not all that is required to make a good, convincing argument. A good argument requires something else.
III. It must have true premises
The third criterion is that a good argument must have true premises. When an argument is formally and informally valid and has true premises, it is sound. But if it is invalid, or it has false premise, it is unsound. When an argument is sound the conclusion follows necessarily and inescapably from its premises. It doesn’t matter if you don’t like it. If you disagree, then you are wrong. When an argument is sound the conclusion is true. Of course, if we don’t know that some premise is true, then we can’t know if the argument is sound. But not knowing the truth of a premise does not make the argument unsound.
You’ll remember that we defined a good argument as one which will persuade a broad range of people, convince a reasonable man and hopefully even an unreasonable man. When an argument is sound, does this mean it is a good argument? This brings us to our final criterion.
IV. Premises more probable than their contradictories
When constructing an argument it would be a tall order indeed if we had to prove the truth of every premise. If that were the case we would be lost in utter scepticism, for we would have to prove the premises of the arguments for the premises of the argument, and the premises of those arguments backwards ad infinitum.
To save us from scepticism a good argument must have premises which are at least more probable than their contradictories. In other words, a premise should be more plausibly true than false. We need not know for certain a premise is true; we need only know that the alternative is less likely.
Obviously there are many differing opinions on many differing subjects, and we live in a very divergent and contrary world. The most powerful arguments will therefore be those structured on premises which are widely accepted and are the hardest to deny. The strength of an argument will depend on the strength of its premises.
Now, plausibility is a person-dependent notion, so in cases of disagreement we need to dig a little deeper to see what reasons we have for accepting or rejecting a premise. When we do, we may find that we have made the mistake—but we also may find that our opponent’s rejection is based on misinformation, ignorance of evidence, or a fallacious objection. Thus we may be able to persuade the other by giving better evidence, information, or gently correcting their error.