Existential fallacy

The textbook case

All unicorns are horned horses. All horned horses are mammals. So some unicorns are mammals.

On the modern reading, both premises are true, because there are no unicorns to break them. The conclusion says there is at least one unicorn that is a mammal, which is false. Nothing in the premises says unicorns exist, so nothing in them can deliver a conclusion that needs one.

A policy taken as a record

An auditor asks whether the manager sign-off on large refunds has ever actually been used. The office manager answers, “Yes. Every refund over $500 is signed off by a manager, so some refunds over $500 have been signed off.” The store has never issued a refund over $500.

The policy can be perfectly true and followed: it rules out an unsigned refund over $500, and there has been none. But a rule says what happens if a case comes up, not that one ever has. The auditor asked about the record, and the answer supplies only the rule.

“All tests passed”

A build report reads “All tests passed.” A developer tells the team, “Good, so the new feature has passed its tests.” The feature’s test file was never added, so no tests for it ran.

“All tests passed” is true when zero tests ran: there’s no failing test to make it false. The developer treated it as saying that some tests ran and passed. This version is easy to miss because the report looks like evidence of checking, when an empty result produces the same words.

The same two premises validly give “All P are R”, which, like them, doesn’t claim any P’s exist. The trouble starts only when the conclusion switches to “some”. Hurley’s textbook states the rule for syllogisms directly: if both premises are universal (“all” or “no”), the conclusion can’t be particular (“some”).

The one-step version is the same mistake: “All P are Q, so some P are Q”, or “No P are Q, so some P are not Q”.

Two readings of “all”. A statement has existential import if it implies that the things it is about exist. The two readings differ over whether “all” statements have it:

• The traditional reading (Hurley calls it the Aristotelian standpoint) treats “All P are Q” as a claim about P’s that exist. On this reading “All P are Q, so some P are Q” is a standard valid inference, and so is the syllogism “Every C is B; every C is A; so some A is B” (known as Darapti). Hurley calls such arguments conditionally valid: valid on condition that the P’s exist. If they don’t, as with unicorns, the argument commits the existential fallacy even on this reading.
• The modern reading (Hurley’s Boolean standpoint) treats “All P are Q” as saying only “there is no P that isn’t Q”, and “Some P are Q” as saying “there is at least one P that is Q”. Only the “some” statement claims existence. On this reading, every step from universal premises alone to a “some” conclusion is invalid, whatever the terms are.

Historians of logic reconstruct the older doctrine a little differently from the textbook version. The Stanford Encyclopedia’s entry on the square of opposition describes traditional logicians as holding that affirmative statements (“all”, “some”) carry existential import and negative ones do not, so on that account “No P are Q” and “Some P are not Q” were both true when there are no P’s.

• When a premise says the things exist. Add “there is at least one P” and the argument becomes valid. A “some” premise does the same job, because “some” claims existence.
• When their existence is already known. If everyone involved knows there are P’s (the argument is about the apples in a bag on the table, say), the missing premise is true and can fairly be taken as understood. This is Hurley’s “conditionally valid” case with the condition met.
• When reading what a speaker meant. In conversation, “all my cats sleep indoors” does suggest the speaker has cats, and taking them to mean that is fair. The error is carrying that assumption over to rules, definitions, guarantees and policies, which are true whether or not any case has ever come up.

The test: apart from the “all” or “no” statement, do we know there is at least one?

A “some” premise

All the ferns at the garden center are shade plants. Some of the plants on the clearance rack are ferns. So some plants on the clearance rack are shade plants.

There’s an “all” premise and a “some” conclusion, which can look like the fallacy. But the second premise is a “some” statement, and it says at least one fern is on the rack. The existence the conclusion needs comes from that premise, so the argument is valid. That’s the first condition under “When it isn’t an error”.

A group that is stated to exist

Every one of the 40 guests who replied yes gets a name tag at the door. So some guests will get name tags.

“Every one of the 40 guests” is an “all” statement that also says how many guests there are. The premise establishes that the group has members, so “some” follows. Compare the refund example, where the rule was true but nothing showed a single refund over $500 had happened.

In everyday speech, “all” usually does come with the assumption that there are some. The Stanford Encyclopedia entry observes that most English speakers tend to understand “Every S is P” as requiring for its truth that there be some S’s. Someone who says “all the rooms are booked” has rooms. That habit makes the step to “some” feel automatic.

The habit fails for general rules. A rule, a guarantee or a definition (“every refund over $500 is signed off”, “all tests passed”, “any customer who asks gets a discount”) can be true with no cases at all, and it reads exactly like a report of cases. Traditional logic also taught the step from “all” to “some” as valid for centuries, which is why it still turns up in older logic material as a correct inference.

Ask how many there have actually been. If the answer rests only on the rule or the “all” statement, the rule hasn’t shown that there are any. Checking this doesn’t show the conclusion is false: there may well be some. It shows the “all” statement alone can’t establish it (see Valid vs. true).