What does a reason actually support?

From reasons to a conclusion

Back at the garden, the gardener’s case fits in one line: “Residents pay for its upkeep; therefore residents should enter first.” We can accept the point about payment and still question the priority. The word “therefore” announces a connection, but it can’t create one.

The claims offered as reasons are the premises, and the claim they are meant to support is the conclusion. These are roles a claim plays within an argument, so a conclusion reached in one step can become a premise in the next. [1]

Here is one way to reconstruct the gardener’s reasoning:

People who pay for a shared resource should have priority in using it. Residents pay for this garden. Therefore residents should have priority here.

The first sentence supplies a reason the gardener never stated. Because it is our reconstruction, we should ask whether the gardener would accept it. It may claim more than they intended; perhaps they meant only that paying is one reason for priority among several.

When could the premises be true and the conclusion false?

Now take a different argument, also invented:

Every key marked G opens the garden gate. This key is marked G. Therefore this key opens the garden gate.

These premises leave no room for the conclusion to be false. Now swap the second premise and the conclusion:

Every key marked G opens the garden gate. This key opens the garden gate. Therefore this key is marked G.

Suppose the key in question is an unmarked master key. Every G key still opens the gate, and so does this one, yet it is not marked G: both premises are true and the conclusion is false. We have found a counterexample to the inference, a case showing that the step from these reasons to this conclusion can fail.

An argument is deductively valid when its premises cannot all be true while its conclusion is false. It is sound when it is valid and its premises are true. Keep the two standards apart: validity concerns only the connection. [2]

So the first key argument is valid before anyone has checked a single G key. If a broken G key turned up, the argument’s first premise would be false, but the argument would still be valid: had the premises been true, the conclusion would have had to follow. It would not be sound, because a premise is false.

An invalid argument, by contrast, can still arrive at a true conclusion; its fault is that the premises do not guarantee that conclusion, even when it happens to be true.

Aristotle on what follows necessarily

At the start of the Prior Analytics, Aristotle defines a syllogism as an argument in which something follows necessarily from what has been stated. His subject is the connection we have been testing, between what we begin with and what we can conclude. [3]

We will return to Aristotle in his own historical setting. The key examples are our own exercises; they do not reproduce his logical system as a whole. Nor does starting with his definition imply that he invented reasoning or described every form it can take.

Support can fall short of a guarantee

Suppose the gardener tests a large and varied sample from a new batch of keys, and every one of them works. That gives us some reason to expect an untested key from the batch to work too, though it could still be defective. How much confidence is warranted depends on how the sample was chosen, and on whether the untested key really comes from the same manufacturing run.

This is defeasible support: a reason that further information could weaken, even while the original observations remain true. Introductory logic often calls reasoning of this kind inductive, though philosophers disagree about exactly how to distinguish inductive from deductive arguments. For now the useful difference is this: some premises guarantee a conclusion, while others give us grounds for expecting it and leave room for error. The two categories need not cover every kind of reasoning. [2]

Now return to the residents’ priority. The premise we supplied, that paying for something gives a person the right to use it first, still has to be examined. Should it hold for a donated public garden? What might give others a claim to enter? Stating the premise openly shows us where the disagreement lies; whether the premise is true is what we then have to argue about.

Pause and try

Consider this argument: “All members may enter before noon. Noor may enter before noon. Therefore Noor is a member.” Describe a possible situation in which both premises are true and the conclusion is false. Then say exactly what your example shows.

Read an explanatory answer

Suppose volunteers may also enter before noon, and Noor is a volunteer who is not a member. Both premises are then true and the conclusion false, which shows that the inference is invalid. It does not tell us whether Noor is actually a member. The example establishes what could be the case, not what is.

Carry this forward

Keep two questions apart: Should we accept these premises? What do they actually support? The next lesson turns this habit on historical evidence. The counterexample workshop offers further practice, though nothing later depends on it.

Sources and limits

[1] Matthew McKeon, “Argument,” Internet Encyclopedia of Philosophy, introduction and §1, for the terminology and the problem of unstated premises. The reconstructed garden argument is an original example.

[2] Timothy Shanahan, “Deductive and Inductive Arguments,” Internet Encyclopedia of Philosophy, introduction, §1 and §10, for the definitions used here and for the disagreement about how to classify arguments. The key and membership cases were invented for the course.

[3] Aristotle, Prior Analytics, I.1 (book I, chapter 1), translated by A. J. Jenkinson: the paragraph defining a syllogism. The lesson makes no claim to have reviewed the whole treatise or the Greek text.