# Existential assumption

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An existential assumption (or existential fallacy) occurs whenever the conclusion of a syllogism requires that a class has at least one member, but one or more of the premises do not.

The fallacy is a syllogical fallacy and a formal fallacy.

## Form

The syllogistic arguments AAI-1, EAO-1, AEO-2, EAO-2, AAI-3, IAI-3, AII-3, EAO-3, AAI-4, AEO-4 and EAO-4 all assume that one of the terms contains at least one element.

In predicate notation:

• P. ${\displaystyle \forall x\in S:P(x)}$.
• C. Therefore, ${\displaystyle \exists x\in S:P(x)}$.

## Explanation

Universal statements, unlike the particulars, do not instantiate anything. For example, the statement, "All trespassers will be shot" (or rather, in standard form, "All trespassers are people who will be shot") does not say whether the trespassers exist or not. People will vow to shoot at any and all trespassers precisely because the statement does not guarantee the existence or non-existence of them. But when the trespassers do exist, they will be shot.

Particular statements (e.g. "Some dogs are fantastic pets.") do instantiate the subject term ("dogs"). In categorical logic, the precise definition of the word some is often taken to be at least one.

The fallacy occurs when there is a mismatch between the necessity of existence for the terms that are mentioned in both the premises and the conclusion. For example:

P1: All animals are dangerous beings.
P2: All unicorns are animals.
C: Therefore, (necessarily,) some unicorns are dangerous beings. (Valid provided that the subject term, unicorns, exists)

This argument is not sound because the premises don't imply that conclusion is true, because the subject, a unicorn, is not guaranteed to exist as the conclusion claims it does (the word "some" means, again, at least one.) An informal version of the argument would read:

P1: Grandma read from the Scroll, "The legend says all Animalia are very, very dangerous in their nature…
P2: "Also, that all unicorns in the Aragon Forest will be Animalia!"
C1: I interrupted her, asking, "So, there is a dangerous unicorn out there in the Aragon Forest?"

The grandma can say either yes or no. Maybe the last sighting of a unicorn was 1,000 years ago, or unicorns are abundant and active and killing humans right now downtown. As you can see, the premises cannot prove the conclusion out of the premises. The key thing to understand is that in a categorical syllogism, the word some means there is at least one instance of something. So, it is a declaration of the existence of something. The original argument declared that unicorns existed, and specified a particular type of unicorns and finally concluded that these particular unicorns were dangerous. The latter two parts were guaranteed by the premises; the first part, the existence of unicorns, was assumed. In the informal phrasing of the argument, the existential question in the conclusion line emphasized the fact that that was assumed.