Quantifier-Shift Fallacy

Alias: Illicit Quantifier Shift1

Taxonomy: Logical Fallacy > Formal Fallacy > Fallacy of Quantificational Logic > Quantifier-Shift Fallacy2 < Scope Fallacy < Amphiboly < Ambiguity < Informal Fallacy < Logical Fallacy3

Form Example
Every P bears the relation R to some Q.4
∴5 Some Q bears the inverse of relation R to every P. 		Every event is caused by some event.
∴ Some event caused every event.
In the example, both P and Q are the class of events, and the relation R is the "caused by" relation, the inverse of which, in the conclusion, is "caused". What would an event that caused every event be? The Big Bang maybe?
Similar Validating Form Example
Some Q bears the relation R to every P.
∴ Every P bears the inverse of relation R to some Q.		 Some event caused every event.
∴ Every event is caused by some event.
This argument is clearly valid, for if a single event caused every event then all events would have a cause, namely, that single event.
Real-Life Example Counter-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."6
See: Analysis of the Real-Life Example. 		Everybody loves someone.
∴ There is somebody whom everyone loves.7

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 widest 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. See, also, the Counter-Example above.

The converse inference is validating (see the Similar Validating Form, above); for instance, if there were 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.

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.

Technical Analysis of the Real-Life Example:

This passage is taken from the third of five "ways" that Aquinas offers to prove the existence of God8. 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.9

Notes:

  1. Robert Audi (General Editor), The Cambridge Dictionary of Philosophy (Second Edition, 1999), p. 317.
  2. A. R. Lacey, Dictionary of Philosophy (Third Revised Edition) (Barnes & Noble, 1996). P. T. Geach mentions this name for the fallacy in a way that suggests he created it, see: "History of a fallacy", in Logic Matters (1980), p. 1.
  3. The Taxonomy 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 comes 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 scope ambiguity conceals the shift: see the Scope superfallacy, above, for more on scope ambiguity.
  4. In these forms, P and Q stand for classes―or one-place predicates in quantificational logic―while R is, as indicated, a relation―or a two-place predicate.
  5. This symbol is read "therefore" and indicates that the conclusion of an argument follows it.
  6. Thomas Aquinas, Summa Theologica, I, Q. 2, A. 3. I found this example in Howard Pospesel's Introduction to Logic: Predicate Logic (1976), p. 159; see also his discussion of the fallacy on pp. 151-152.
  7. Lucky person!
  8. If even one of these ways worked, the others wouldn't be needed, so why five?
  9. Taken together with some additional premisses, Aquinas could have shown that everything in the universe will cease to exist at some time. The additional assumptions are:
    • There is a finite number of individual things in the universe.
    • If an individual thing ceases to exist then it cannot come back into existence.
    • New individual things are not being created.

    Given these three assumption, over the course of time every individual thing will cease to exist, and no new things will be created to replace those lost. Eventually, the universe will be empty, but that time is obviously some time in the future, not now, and it's not impossible that everything might cease to exist at some time in the future.

    Each of these three assumptions is necessary to make the proof work since, if there is an infinite number of individual things in the universe, then a finite number can cease to exist at any point in time, leaving an infinite number in existence. Alternatively, if individual things that have ceased to exist can come back into existence, or if new things are created to replace those lost, then the universe may never be emptied out.

Acknowledgment: Thanks to Gabriel Frohaug for pointing out a broken link.

Revised: 3/19/2023