9/6/2026
The fallacy, in English
In Aristotelian philosophy there is the notion of Prime Mover (or Unmoved Mover). The idea here is most all of the things that are moved are moved by something else. Then, if these movement chains (or causal chains) were traced back in time, and the chains could not go back forever, there might be, or had to be, a Prime Mover, a first mover. At this point in the reasoning there is an issue, and it is an issue between 'a' and 'the'. That there is, or might be, a Prime Mover does not mean that there is only one of them i.e. the Prime Mover. There could be two of them (or more).
The same slip shows up in ordinary claims like "the tallest student in the class" — which presupposes there's exactly one, and breaks the moment there's a tie.
The diagram
Universe = {a, b, c}
P (PrimeMover) = {a,b} (two of them)
Try it
Formula (b), `∃x(PrimeMover(x)&∀y(PrimeMover(y)→y=x)) ` — 'there is exactly one Prime Mover' — is false. Endorse it and see how you get on. Then you can change the diagram so that there is exactly one and Endorse Formula (d), `∃x(PrimeMover(x)&∀y(PrimeMover(y)→y=x)) `
Existence and Uniqueness
In the wider intellectual world, statements similar to 'There is exactly one Prime Mover' are common. A mathematicians might undertake to prove a theorem like 'a particular set has exactly one largest element'. This is often done in two stages, as a task of two goals: proving that there is one, and proving that there are no more than one. That is, you prove that exactly one thing has a property by proving that something has the property and, separately, all things which also have that property are equal to the first one. That is, you want to prove something like ∃xF(x) ('existence') and ∀x∀y(F(x)&F(y)→x=y) ('uniqueness').
Once there is existence and uniqueness, use of the word 'the' can be in order (e.g. the largest element).
Failure of Uniqueness
Our earlier discussion of the possibility of two or more Prime Movers points to the possibility of a failure of uniqueness.
Failure of Existence
But use of the word 'the' can fail due to failures of existence. Modern philosophy often discusses this in terms of what it calls 'definite descriptions'.
The issue of definitive descriptions has its origin with Bertrand Russell. In his 1905 paper 'On Denoting', he discusses the sentence 'The present King of France is not bald' (where there was no present King of France so the 'the' word did not pick anything out). Russell, essentially, argued that sentence should be analysed as a pure quantificational expression which reads 'There is a King of France, and all other Kings of France are identical to that one, and he is not bald.' And this sentence is false. And so too is 'There is a King of France, and all other Kings of France are identical to that one, and he is bald.' Both sentences are false because there is no present King of France (a failure of existence). An accessible introduction to definite descriptions is Brian Holtz, Definite descriptions in one easy lesson [link good August 2026].
Generalization
The techniques on display here can be generalized to different numbers of items. Suppose you suspected that there were two Prime Movers and you wished to say 'the two Prime Movers', you could do this with a formula like Formula (e)
∃x∃y(PrimeMover(x)&PrimeMover(y)&~(x=y)&∀w(PrimeMover(w)→(w=x)∨(w=y)))
This says 'there are two Prime Movers and any other Prime Mover is identical to one of those two.'