Scope fallacy

A scope fallacy is a logical fallacy that occurs when the scope of a logical operator (e.g., "not", "some", or "all") is vague, and allows misinterpretation and incorrect conclusions.

The fallacy is a fallacy of ambiguity, and an informal fallacy.

Alternate names

 * quantifier shift (specifically, this applies to the ambiguous scope of a quantifier; see below)
 * modal scope fallacy (specifically, this applies to the ambiguous scope of a modal word; see below)
 * illicit quantifier shift

Form

 * Every X has the relation R to some Y.
 * Therefore, some Y has the inverse of relation R to every X.

Alternately:
 * For every X, there is a Y, such that Z.
 * Therefore, there is a Y, such that for every X, Z.

More formally, $$\forall x \exists y\, R(x, y)$$ does not imply $$\exists y \forall x\, R(x, y)$$, though the reverse implication is valid. (The use of the ∀ and ∃ quantifiers in the formal statement of the fallacy is why this case is called a "quantifier shift".)

Explanation
Logical operators like "not" have a particular "scope": they affect some portion of the proposition in which they occur. Thus, the term "not" can affect some portion of the proposition or all of the proposition. Understanding what is affected is key to understanding the argument.

Glitter?

 * All that glitters is not gold.

This could mean two things:


 * 1) Everything that glitters is a non-gold substance. (This is a narrow interpretation, concluding that the word "not" only negates "is gold".)
 * 2) Not everything that glitters is gold. (This is a broad interpretation, concluding that the word "not" negates the rest of the sentence.)

Normally, whatever occurs in the second phrase is considered to be "in the scope of" the first phrase. Thus, "is not gold" falls under the scope or influence of "all that glitters."

Consider that the first interpretation would lead to the following fallacious argument:

Yet this conclusion is not necessarily true, making this fallacious logic.

Love?
Compare:
 * 1) For every person, there exists one true soulmate.
 * 2) There is one true soulmate for every person.

The first statement argues that every individual has another individual whom they can love. The second statement probably is intended to mean the same as the first, yet it could be interpreted as saying that there exists one individual who is the true soulmate of every other individual.

The second statement could be written into the "logical" argument:
 * Everyone loves someone.
 * Therefore, there is (a very lucky) someone whom everyone loves.

Is there someone that everyone loves?

God?
A common argument for God:

The fallacy occurs in the first two sentences, with the scope of the idea of "does not exist." The first sentence asserts that "All contingent beings do not exist at some point in time."

The second sentence assumes that the first sentence's scope is so broad that, at some point in time, no contingent being existed. But this is not the necessary interpretation of the premise — it could be that some contingent beings have always been around, but that they keep changing.

Importance
This is one of the reasons philosophical arguments are represented by symbols instead of words. When used properly, the symbols can't create ambiguities. For example, everything constrained by a symbol for "not" or "all" will be contained in parenthesis, as in math, avoiding this fallacy.