A common mistake made by many people in drawing inferences from an argument is to confuse the the quantifiers “all X” and “some X.”

[The next part is technical so please feel free to skip it.]

Universal quantification, or all X, can be symbolized by ∀x. The upside down A means “for all x.” For example. ∀x P(x) = for all x, P(x) holds. Suppose x = unicorns. ∀x P(x) = for all unicorns, property P holds.

Universal quantification is very different from existential quantification, some X. Existential quantification can be symbolized by: ∃x. The reverse E means “there exists an x” or “there exists at least one x.” For example. ∃x P(x) = there exists an x, such that P(x). Suppose again x = unicorns. ∃x P(x) = there exists a unicorn (or there exists at least one unicorn), such that property P holds.

Compare sentences 1) and 2) below:

1) All unicorns have a white mane.

2) There is a unicorn, such that it has a white mane.

Suppose 1) were true. It does not follow from 1) that unicorns exists. So simply because 1) is true, does not mean that 2) is true. Suppose 2) were true. It does not follow from 2), that all unicorns have white manes. 2) claims there is at least one unicorn and that unicorn has a white mane.

So the truth of 1) and 2) are independent of one another. An effective communicator must not confuse 1) or 2) or assume they are the same.

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