Quantifier-Shift Fallacy

Taxonomy: Logical Fallacy > Formal Fallacy > Fallacy of Quantificational Logic > Quantifier-Shift Fallacy < Scope Fallacy < Amphiboly < Ambiguity < Informal Fallacy < Logical Fallacy

Alias: Illicit Quantifier Shift

Form:

Every P bears the relation R to some Q.
Therefore, some Q bears the inverse of relation R to every P.

Similar Validating Form:

Some Q bears the relation R to every P.
Therefore, every P bears the inverse of relation R to some Q.

Example:

We find in nature things that are possible to be and not to be…. But it is impossible for these always to exist, for that which is possible not to be at some time is not. Therefore, if everything is possible not to be, then at one time there could have been nothing in existence.
Source: Thomas Aquinas, Summa Theologica, I, Q. 2, A. 3

Analysis

Counter-Example:

Everybody loves someone.
Therefore, there is somebody whom everyone loves. (Lucky person!)

Exposition:

The phrase "quantifier-shift" refers to the two quantifiers in the premiss and conclusion of arguments of this form, namely, "every" and "some". "Shift" refers to the fact that the difference between the premiss and conclusion of this form of argument consists in a shift in the order—or, technically, the scope—of the quantifiers. In the premiss, the universal quantifier, "every", is followed by the existential one, "some", whereas in the conclusion the order is reversed. This means that in the premiss the universal quantifier has widest scope, while in the conclusion the existential quantifier has wider scope.

The fallaciousness of this form of argument is most easily seen by examining some counter-examples that would fool no one. For instance, everyone has a mother—that is, for every person, there is some mother of that person. However, it is false that there is a mother of us all—that is, it is not true that some woman is the mother of everyone.

The converse inference is validating (see the Similar Validating Form, above); for instance, if there was truly someone who loved everyone, then it would follow that everyone was loved by someone—namely, the all-lover. But it does not follow from the fact that everyone is loved by someone that there is someone who loves everyone—that is, an all-lover. The fact that these two inferences differ only in the direction in which the quantifiers are shifted is probably one psychological reason why this fallacy is so easy to commit.

Exposure:

The Taxonomy―see above―for this fallacy has two branches, one tracing back to Logical Fallacy through Formal Fallacy, and the other passing through Informal Fallacy. This is not a contradiction, but an indication of the fact that this fallacy has both a formal and informal aspect. The formal aspect come from the fact that shifting quantifiers in this way is not validating in quantificational logic. The informal aspect enters when such a shift is expressed in a natural language, such as English, where ambiguity conceals the shift.

Sources:


Technical Appendix:

For those conversant with quantificational logic, the form of the fallacy is as follows:

∀x∃yRxy
∴∃y∀xRxy

Here we have clearly shifted the order of the two quantifiers. From the premiss that for every x there is a y such that x bears R to y it does not follow that there is a y such that every x bears R to y. For instance, ∀x∃y(x < y) is true for all integers, that is, for every integer there is a greater. However, it is false that ∃y∀x(x < y), which says that there is a greatest integer.

In contrast, the inverse inference is validating:

∃y∀xRxy
∴∀x∃yRxy

If there is a y such that every x bears R to y, then it follows that for every x there is some y such that x bears R to y. This is easy to see: assume that b is such that ∀xRxb and that z is an arbitrary member of our domain of quantification. It follows from our assumptions that Rzb, from which we can existentially generalize on b to get ∃yRzy. Finally, since z was an arbitrary member of the domain, we can univerally generalize on it to get ∀x∃yRxy.

Finally, though I have presented the fallacy in terms of a relational sentence in order to simplify the exposition, it is committed by any quantificational argument whose premiss begins with a universal quantifier followed by an existential one, and whose conclusion shifts the order of those quantifiers. Similarly, any quantificational argument with a premiss that begins with an existential quantifier followed by a universal one, and whose conclusion simply reverses the order of these quantifiers, is valid.

Analysis of the Example:

This passage is taken from the third of five "ways" that Aquinas offers to prove the existence of God. Aquinas goes on to argue that there must be a necessarily existing entity, namely "God", but the example is taken from an earlier stage in the argument. At this point, he argues that if everything were contingent―that is, "possible to be and not to be"―then there would have been a time when nothing existed. He then argues that this latter is impossible, therefore there would have to be a necessary being, that is, God. However, we are only interested in the earlier step of the argument from everything being contingent to there being a time when nothing existed. There are actually two arguments in this passage, but the relevant one is the following:

  1. "It is impossible for these [contingent things] always to exist."
  2. "Therefore, if everything is possible not to be [that is, contingent], then at one time there could have been nothing in existence."

Let's assume that everything is contingent, as Aquinas does in the conclusion of this argument―the argument is part of a reduction to absurdity of this assumption. Thus, our quantifiers will range over only contingent things, and the argument can be restated as follows:

  1. Everything fails to exist at some time.
  2. Therefore, at some time everything fails to exist.

It's now clear that the order of the quantifiers has been shifted in the fallacious way. Representing that an object x exists at time t by "Exists(x,t)", the argument can be symbolized:

∀x∃t¬Exists(x,t)
∴∃t∀x¬Exists(x,t)

As a result of the fallaciousness of this step in the argument, Aquinas' "third way" is invalid.