Logic for beginners: what makes an argument valid

I tutor logic to first-year students, and every year the same thing happens in the second week. Someone hands in an exercise in which they have marked an argument "invalid" because its conclusion is false, or "valid" because they agree with it. They are not being careless. They are using "valid" the way everyone uses it in ordinary speech, to mean roughly "a fair point". In logic it means something narrower and much more useful. This guide explains what, and how to test for it.

What an argument is

In logic an argument is not a quarrel. It is a set of statements, called premises, offered in support of another statement, the conclusion. "It's raining, so the match will be cancelled" is an argument with one premise and one conclusion. Most real arguments have premises left unstated, which we will come back to.

Logic asks one question about an argument: does the conclusion follow from the premises? Not "is the conclusion true?" and not "are the premises true?" Just: if the premises were true, would the conclusion have to be?

Validity

An argument is valid when it is impossible for all its premises to be true and its conclusion false. That is the whole definition, and it is worth reading twice.

Here is a valid argument:

1. All whales are mammals.
2. All mammals breathe air.
3. So all whales breathe air.

And here is another, equally valid:

1. All cats are made of glass.
2. Felix is a cat.
3. So Felix is made of glass.

The second argument has a false premise and a false conclusion, and it is still valid, because if all cats were made of glass and Felix were a cat, Felix would have to be made of glass. Validity is about the connection between premises and conclusion, not about whether any of them is true.

Soundness

That is why logicians use a second word. An argument is sound when it is valid and all its premises are true. A sound argument guarantees a true conclusion. The whale argument is sound. The Felix argument is valid but unsound.

This one distinction clears up a remarkable amount of confused debate. When you disagree with an argument, there are exactly two kinds of objection you can make: "that doesn't follow" (an attack on validity) or "that isn't true" (an attack on a premise). Most arguments that go in circles do so because people make one kind of objection while believing they are making the other.

Validity is about form

The reason the Felix argument is valid is not anything about cats or glass. It is the argument's shape: all A are B; x is an A; so x is B. Any argument with that shape is valid, whatever you put in for A, B and x. You can even use nonsense words: all wugs are blickets; Tom is a wug; so Tom is a blicket. You know the conclusion follows without knowing what a wug is.

That insight goes back to Aristotle, whose syllogistic in the Prior Analytics was the first formal system of logic in Europe. It catalogued the valid patterns built from statements like "all S are P", "no S are P" and "some S are P". The Stoic Chrysippus developed a logic of whole statements joined by "if", "and", "or", much of which is lost and was only appreciated again in the modern era. In the nineteenth century George Boole showed that such reasoning could be treated algebraically, and in 1879 Gottlob Frege introduced the quantifiers that let modern logic handle arguments about relations ("every student read some book") that Aristotle's system could not.

Four patterns worth memorising

Two valid forms:

• Modus ponens. If P then Q. P. So Q. (If it rained, the pitch is wet. It rained. So the pitch is wet.)
• Modus tollens. If P then Q. Not Q. So not P. (If it rained, the pitch is wet. The pitch is dry. So it didn't rain.)

And two invalid look-alikes, which are among the most common errors in everyday reasoning:

• Affirming the consequent. If P then Q. Q. So P. (If it rained, the pitch is wet. The pitch is wet. So it rained.) But the groundsman may have used a hose.
• Denying the antecedent. If P then Q. Not P. So not Q. (If it rained, the pitch is wet. It didn't rain. So the pitch is dry.) Again, the hose.

The counterexample test

The best practical test for validity is to look for a counterexample: a situation, however strange, in which every premise is true and the conclusion false. If you can describe one, the argument is invalid. The hose is a counterexample to both invalid forms above.

A useful trick is to keep the shape and swap in content where the answer is obvious. "If someone is a Londoner, they are British. Anna is British. So Anna is a Londoner." Anna might live in Glasgow. The shape is affirming the consequent, and now you can see why it fails.

Another tradition, another purpose

It is worth knowing that formal inference was developed independently in India. The Nyāya school, whose founding text is attributed to Gautama (Akṣapāda), set out a five-membered inference. Thesis: the hill has fire. Reason: because it has smoke. Example: wherever there is smoke there is fire, as in a kitchen. Application: the hill has smoke, as the kitchen does. Conclusion: so the hill has fire.

Western readers long thought the extra steps were padding. They are not. The Nyāya inference was built for debate, where you must persuade an opponent, and the example grounds the general rule in a case both sides accept. Some scholars read the example as rhetorical, others as supplying the evidence for the general rule. Either way it reminds us that validity is not the only virtue an argument can have. Being convincing to someone who doubts you is another.

What validity cannot do for you

Two warnings, and then you are ready.

First, a valid argument can be worthless. The Felix argument is valid and tells you nothing. Validity is a guarantee only as good as the premises you put in.

Second, most good reasoning in real life is not valid in the logician's sense. "Every swan I have seen is white, so the next one will be" is not valid: the next one could be black, as Europeans eventually found out in Australia. It is an inductive argument, which makes its conclusion probable rather than certain. We rely on such arguments constantly, in medicine, science and daily life. Calling them "invalid" is technically correct and practically misleading; they are simply measured by a different standard, strength rather than validity.

So my advice to anyone starting out is this. When you meet an argument you dislike, ask first whether it is valid. If it is, your disagreement must be with a premise, and you should be able to say which. If it is not, find the counterexample. And when an argument seems obviously right to someone and obviously wrong to you, look for the premise neither of you has said out loud. It is usually there.

If you want to go further, Graham Priest's Logic: A Very Short Introduction is the best short book, and P. D. Magnus's free textbook forall x will teach you to do the formal work yourself.