Arguments, validity and form

Two people disagree about whether a conclusion really follows from what was said, and nothing in the words themselves settles it.

That is the problem this subject solves. Not whether a claim is true, which is usually a question for chemistry or history or the accounts, but whether accepting some claims commits you to another one. The answer turns out to depend on the shape of the argument and on nothing else, and shapes can be written down, listed and checked. Everything in the fourteen lessons that follow is machinery for doing that check, and this lesson fixes what the check is meant to test.

What an argument is

An argument here is not a quarrel. It is a set of statements, the premises, offered in support of one further statement, the conclusion. A calm paragraph in a contract and a shouted claim can have the same argument in them.

A statement is a sentence that is true or false. "The Thames flows through London" is one. "Close the door", "what time is it" and "if only I had known" are not, because they make no claim that could be right or wrong. That is a real restriction, worth stating up front: the machinery only judges arguments whose parts are true or false.

English marks the parts with signposts. "Therefore", "so", "hence" and "it follows that" introduce conclusions; "because", "since" and "given that" introduce premises. The order on the page means nothing, so "Socrates is mortal, since all men are mortal and he is a man" is the same argument written backwards.

Two things that look like arguments are not. An explanation takes an agreed fact and says why it happened: "the bridge collapsed because the bearings seized" is not trying to convince anyone the bridge collapsed. And a confident chain of assertions with no support relation is a description. The test is whether one statement is being offered as a reason for another.

Validity is an impossibility claim

Here is the central definition, and everything else in the subject depends on getting it exactly right.

An argument is valid if and only if there is no possible situation in which all its premises are true and its conclusion is false.

Read what that does and does not say. It does not say the premises are true. It does not say the conclusion is true. It says the combination "premises all true, conclusion false" cannot happen, which is a claim about what is possible, not about what is the case. Validity is the guarantee that truth is transmitted: feed true premises into a valid argument and a true conclusion must come out. Feed false ones in and the guarantee is silent.

So testing validity means trying to imagine the bad combination. Take: "All whales are fish. All fish are mammals. Therefore all whales are mammals." Both premises are false and the conclusion is true, but that is not the question. The question is whether any situation could make both premises true while leaving the conclusion false, and none could: if every whale is among the fish and every fish is among the mammals, every whale is among the mammals. The argument is valid.

Now take: "If the company committed fraud, the quarterly figures would be unusually smooth. The figures are unusually smooth. Therefore the company committed fraud." Both premises might well be true, and so might the conclusion. But the bad combination is easy to picture: a genuinely stable business with a small product range produces smooth figures with no fraud anywhere. Premises true, conclusion false. The argument is invalid, and it stays invalid even in the cases where the company did commit fraud.

The four combinations

Since validity and truth are independent, it pays to see how they combine. Of the four pairings of premise truth with conclusion truth, three are open to a valid argument and one is not. True premises with a true conclusion is the ordinary good case. False premises with a false conclusion is fine too, since such an argument fails on its facts rather than its logic. False premises with a true conclusion is the whales case above. The single forbidden pairing is true premises with a false conclusion, and it is forbidden by the definition itself, which is what makes validity worth having.

Invalid arguments can show all four, including true premises with a true conclusion, which is the trap. "Paris is the capital of France, therefore water is a compound of hydrogen and oxygen" has a true premise and a true conclusion and no connection whatever between them. Checking that the premises and the conclusion are all true tells you nothing about whether the argument works.

Example. Build an argument that is valid, has at least one false premise, and has a true conclusion.

Start from a form known to transmit truth, then feed it a falsehood that happens to land somewhere true. "All metals conduct electricity. Graphite is a metal. Therefore graphite conducts electricity." The second premise is false, since graphite is a form of carbon, a non-metal. The conclusion is true, since graphite conducts well enough to be used for the brushes in electric motors. And the argument is valid: if everything metallic conducted and graphite were metallic, graphite would conduct. Validity was never a claim about the premises.

Now you. Build an argument that is invalid, has two true premises, and has a true conclusion.

Soundness

Validity is a low bar on its own, because it never asks whether the premises are true. The property that actually matters in argument is soundness: an argument is sound when it is valid and all its premises are in fact true. A sound argument has a true conclusion, guaranteed, and that is exactly what a proof is.

The division of labour is the point. Logic owns validity completely and can settle it by inspection of form. Nothing in logic can settle whether "all metals conduct electricity" is true; that is measurement. So an argument is attacked in one of two quite different ways, and confusing them wastes everybody's time. Either you deny a premise, which is a dispute about the world, or you deny that the conclusion follows, which is a dispute about form. "Your figures are wrong" and "your figures do not show that" are different objections and need different evidence.

This subject is entirely about the second kind, which is a narrow speciality: most bad arguments in the wild fail on their premises rather than their form. The last lesson comes back to that honestly.

Form is what does the work

Look again at the whales argument. Nothing about whales, fish or mammals was used. Replace those three words with any others and the guarantee survives:

All {A} are {B}. All {B} are {C}. Therefore all {A} are {C}.

Validity is a property of this form, and every argument with this shape, on any subject, in any language, is valid. That is the discovery the subject is built on, and it was Aristotle's, in the Prior Analytics of around 350 BCE. His decisive move was the one just made: writing letters where the terms go, so that an argument can be studied without knowing what it is about.

Contrast the invalid argument. Its form is: if {P} then {Q}; {Q}; therefore {P}. Every argument of that shape is invalid, including the ones with true conclusions, because the shape carries the fault. This is why logic can be taught at all: there are not infinitely many arguments to memorise, only a stock of forms and a method for the forms nobody has catalogued.

Refutation by parallel form

The most useful practical consequence: if validity depends only on form, then two arguments with the same form stand or fall together. So to show that an argument is invalid, exhibit another argument of the same form whose premises are plainly true and whose conclusion is plainly false. That parallel argument cannot be valid, so neither can the original, and the person you are arguing with does not have to grant anything about the original subject matter.

Suppose someone argues: "Every economy that raised interest rates in 2022 saw inflation fall by 2024. Inflation in Japan fell by 2024. So Japan raised interest rates in 2022." Rather than argue about Japanese monetary policy, hand back the parallel: "Everyone with flu has a temperature. This patient has a temperature. So this patient has flu." The premise is true and the conclusion does not follow, since a hundred other things raise a temperature. Same form, obviously broken, argument over. The method has one requirement: the parallel must genuinely share the form rather than merely feel similar, and if the original says "some" where the parallel says "all", the refutation fails and will be seen to fail.

Example. Refute by parallel form: "If a country has a large trade deficit, its currency will weaken. The pound weakened. So the United Kingdom has a large trade deficit."

The form is: if {P} then {Q}; {Q}; therefore {P}. Substitute something where the second step is transparently bad. "If it snowed last night, the roof is wet. The roof is wet. So it snowed last night." The roof is wet because it rained, or because someone hosed it down. True premises, false conclusion, same shape, so the original is invalid. Notice that the United Kingdom does in fact run a large trade deficit. The conclusion is true and still does not follow, which is exactly the distinction this lesson is drilling.

Now you. Refute by parallel form: "No one who trains properly gets injured. Ferrer was injured. So Ferrer did not train properly."

Forms worth knowing by name

Four propositional forms account for most everyday reasoning, and they come in two valid pairs and two invalid impostors that differ from them by one word.

Modus ponens is: if {P} then {Q}; {P}; therefore {Q}. Valid, and the workhorse of every calculation and every application of a rule. Modus tollens is: if {P} then {Q}; not {Q}; therefore not {P}. Valid, and the engine of testing: a hypothesis predicts something, the something fails to appear, the hypothesis goes.

The impostors change one word each. Affirming the consequent is: if {P} then {Q}; {Q}; therefore {P}. Invalid, and the most common fault in reasoning about evidence, being the fraud argument, the flu argument and the trade deficit argument all at once. Denying the antecedent is: if {P} then {Q}; not {P}; therefore not {Q}. Also invalid. "If you inherit the gene you will get the disease. She did not inherit it. So she will not get it." Other causes remain.

Two more, both valid. Disjunctive syllogism: {P} or {Q}; not {P}; therefore {Q}. And hypothetical syllogism: if {P} then {Q}; if {Q} then {R}; therefore if {P} then {R}, which is how long chains of reasoning are stitched together.

Example. Name the form and say whether it is valid: "If the alloy contains nickel, the sample is magnetic. The sample is not magnetic. Therefore the alloy contains no nickel."

The second premise denies the consequent, so this is modus tollens, and it is valid. Any situation making both premises true has a non-magnetic sample, and a nickel-bearing alloy would have been magnetic, so there is no nickel. Whether it is sound is a separate question, and here the first premise is doubtful, since austenitic stainless steels contain about 8 per cent nickel and are essentially non-magnetic. The argument is valid and unsound, and only the second half of that verdict is metallurgy.

Now you. Name the form and say whether it is valid: "If the server were down, the dashboard would be blank. The server is not down. Therefore the dashboard is not blank."

Where the guarantee stops

Everything so far has demanded an absolute guarantee, and most real reasoning cannot supply one. "The last four hundred swans I saw were white, so the next one will be white" has premises that make the conclusion likely without forcing it, which is why Australia was a surprise. Such arguments are inductive, and they are judged on a different scale, strength rather than validity, one that belongs to probability and statistics rather than to logic. Deductive validity buys certainty and pays for it by never telling you anything the premises did not already contain.

Two edge cases fall out of the definition and are worth meeting now rather than in the middle of a proof. If the premises contradict each other, no situation makes them all true, so the forbidden combination cannot arise and the argument is valid whatever the conclusion says. And if the conclusion cannot be false, the same holds for any premises at all. Both feel wrong and both follow from the definition, which is a sign that it is doing work rather than restating intuition.

The tools of this lesson are already useful, and they do not scale. Nothing here says what to do with an argument whose form nobody has named, and the parallel-form method needs a fresh flash of invention every time. To go further, forms must be written in a language precise enough to compute with, which means fixing a small set of connectives whose meaning never varies. That is the next lesson.