Aquinas and a Quantification Fallacy?

Joseph Schmid and Daniel Linford claim the Unmoved First Mover argument commits a quantifier shift fallacy, but their explanation seems erratic.

I’m following the Method of Engagement outlined when I began this publication. 1) Present the topic. 2) State the atheist (in this case agnostic) view. 3) State the theist view. 4) Analyze and compare. 5) Ask for feedback.

1. Present the topic.

In Chapter 2 “Aquinas’s First Way” of their book Existential Inertia and Classical Theistic Proofs, Joseph Schmid and Daniel Linford claim that the First Way argument for God’s existence is invalid because it commits a quantifier shift fallacy. This is a type of quantification fallacy in which the number of items in a proposition is shifted erroneously to another statement or conclusion. The example they give is one of counselors assigned to students. It would be false to conclude that there is one single counselor assisting all students from the statement that each student has a counselor. In the same way, Schmid and Linford say we “cannot conclude that there is a first cause of all chains of changes from the fact that each such chain has a fir…