What makes an argument logically valid?

What makes an argument logically valid?

Can an argument be logically well built even when one of its claims is false? At first that sounds contradictory, but perhaps we are mixing up two different questions: whether the starting claims are true and whether the conclusion follows from them.

Consider the form: all members of group A are in group B; this object is in group A; therefore this object is in group B. If the two premises were true, could the conclusion still be false? No. The structure closes off that possibility.

That is the heart of deductive validity: there is no possible case in which all the premises are true and the conclusion is false. Validity therefore concerns the connection between claims, not whether the claims happen to match reality.

Notice what we never did in checking it. We never asked whether any A exists, or what A and B are. Put "cats" and "creatures that bark" in their places and the argument stays valid while becoming absurd. So validity belongs to the shape, and the shape survives any substitution of content — which gives us, if we look at it from the other end, a method.

Suppose you suspect an argument is bad. How would you show it? Not by pointing out that its conclusion is false; a valid argument can have a false conclusion whenever a premise is false. You show it by finding a counterexample — another argument of the same shape whose premises are plainly true and whose conclusion is plainly false. If the shape lets that happen even once, the shape guarantees nothing. Try it on this: if it rained, the pavement is wet; the pavement is wet; therefore it rained. Feels reasonable. Now imagine a sprinkler. Both premises hold, the conclusion fails, and the form is dead. That form has a name — affirming the consequent — and it has a twin that fails to the same sprinkler: if it rained the pavement is wet; it did not rain; therefore the pavement is not wet. Denying the antecedent. The two forms that do hold sit right beside them: from "if it rained, the pavement is wet" you may go forward from the rain to the wetness, or backward from dry pavement to no rain, and nowhere else.

What should we call an argument that has both a valid structure and true premises? Logicians call it sound. Soundness gives us what validity alone cannot: a justified true conclusion. The distinction looks pedantic until you notice how much work it does. Two people arguing politics usually disagree about premises while accusing each other of illogic, and the two complaints have entirely different remedies — one is settled by evidence, the other by inspection of the form alone.

One consequence of the definition is worth sitting with, because it feels like cheating. If validity means "no possible case with all premises true and conclusion false", then an argument whose premises cannot all be true is valid whatever it concludes — there is no such case to be found. Contradict yourself in the premises and you may validly derive that the moon is cheese. Related: standard logic reads "if P then Q" as false in exactly one situation, P true and Q false, so any conditional with a false antecedent counts as true. "If Rome is in Norway, then I am the king" is a true statement in this system. Does that match what you mean by "if"? Probably not, and you are in good company — it is precisely this that motivated the relevance logicians to build systems in which premises must bear on the conclusion. Standard logic keeps the odd cases because dropping them costs more elsewhere.

A true conclusion does not prove that the reasoning was valid; someone can reach a true answer by a bad route. Likewise, a valid argument can have a false conclusion when at least one premise is false. To evaluate reasoning well, inspect both the structure and the premises.

The harder point is that real arguments do not arrive with their structure exposed. "Socrates is a man, so he is mortal" is not valid as it stands — nothing in it links men to mortality. It becomes valid the moment you supply the premise nobody bothered to say. Almost all everyday argument works this way, on premises left suppressed because they are obvious, or because stating them would be tedious, or because the arguer would rather not. And that creates a genuine difficulty: since you can rescue nearly any argument by adding a premise, "is this valid?" is rarely a question you can answer by looking. You first have to decide what the argument is, which means guessing at unstated commitments — and a reader who supplies a wild premise can make nonsense valid, while a reader who refuses to supply any can make reasonable argument look like a fallacy. Formalizing is an interpretive act before it is a technical one, and most disputes about whether an argument "is logical" are really disputes about which hidden premise the arguer was leaning on.

Validity is also relative to how much structure you have chosen to formalize. De Morgan's old example — all horses are animals, therefore all horse heads are animal heads — is unmistakably valid, but no syllogism captures it; the machinery of the older logic simply cannot see the shape it depends on. Failing to fit a particular system is not the same as failing to follow.

• How inductive strength differs from deductive validity, given that no induction is ever valid.
• What relevance logics change, and what they give up to get it.
• Why a proof system is judged by soundness and completeness, and how those differ from an argument's.

Every term the collection defines is gathered in the glossary.