I think your claim is a little too sweeping and categoric, and I would also distinguish more carefully between equivocation, an incomplete demonstration, and an invalid inference, because there are three distinct things.
Equivocation occurs when a term changes meaning within an argument and that shift is improperly used to support a conclusion, but establishing the existence of something under a particular description, without thereby establishing every attribute it possesses, is not equivocation.
Think about Aquinas, who explicitly employs Aristotelian metaphysical principles in his First Way (Summa Theologiae, I, q. 2, a. 3).
Aquinas argues from motion, understood metaphysically as the actualization of potentiality, toward the existence of a First Unmoved Mover, but notice what he does next: he doesn’t claim that this argument, taken by itself, establishes the Trinity, the Incarnation, or every other truth revealed by Christianity.
Instead, he proceeds to investigate the attributes of this First Principle through further philosophical reasoning, addressing divine simplicity, perfection, goodness, infinity, immutability, eternity, unity, knowledge, and other properties.
And when he comes to the Trinity, Aquinas explicitly states that it cannot be demonstrated by natural reason (Summa Theologiae, I, q. 32, a. 1).
This distinction is not merely a peculiarity of Thomistic philosophy. It is also reflected in the dogmatic teaching of the First Vatican Council.
In Dei Filius, chapter II, the Council teaches that the one true God, the beginning and end of all things, can be known with certainty through the natural light of human reason from created things. *
Yet in chapter IV, the same Council distinguishes truths accessible to natural reason from supernatural mysteries that cannot be known without divine revelation. **
Indeed, the first canon on faith and reason explicitly condemns the proposition that all the dogmas of faith can be understood and demonstrated through natural reason alone.
In other words, Catholic theology explicitly distinguishes between knowing that God exists, knowing certain attributes of God through natural reason, and knowing the supernatural mysteries of His inner life through revelation.
There is no equivocation inherent in that distinction.
Now, suppose an argument successfully establishes the existence of a First Unmoved Mover, that conclusion doesn’t, by itself, establish that the First Unmoved Mover is the Holy Trinity, i agree entirely with this, but the fact that an argument doesn’t establish every attribute of a proposed being does not mean that it cannot establish something relevant to that being’s existence.
Christian classical theism maintains that God is the ultimate, uncaused source of all created reality: consequently, an argument establishing the existence of such a First Principle would establish something that Christian classical theism affirms about God.
It would not establish the whole of Christian theism, nor would it distinguish Christianity from every other metaphysical or religious worldview. But it could still contribute to a cumulative argument for Christian theism.
For example, if a conclusion establishes that a First Cause exists, and Christian theism holds that God is that First Cause, we have established a proposition consistent with and required by that conception of God, we have not thereby established every additional proposition Christian theology makes about Him.
To infer those additional propositions without further premises would indeed be unjustified, but an unjustified inferential leap is not automatically equivocation.
There is also another important distinction: Aquinas does not simply adopt Aristotle’s entire conception of the divine without qualification.
Aristotle’s own account of the Unmoved Mover, particularly its relationship to the world, differs in significant respects from the Christian understanding of God as Creator.
Aquinas takes elements of Aristotelian metaphysics, develops arguments from them, and integrates their conclusions into a broader philosophical and theological framework, but using an Aristotelian argument is not equivalent to endorsing every belief Aristotle held about its conclusion, and there is nothing logically illegitimate about correcting, developing, or extending an earlier philosophical argument, provided the relevant conclusions are properly justified.
So I would distinguish the following claims:
First: Aristotle’s argument, by itself, establishes the entire Christian doctrine of God, including the Trinity.
That is false.
Second: Aristotelian metaphysical arguments can establish, or contribute to establishing, certain truths about the existence and nature of the First Cause, which Christian theology identifies with the God revealed in Scripture.
That doesn’t necessarily involve equivocation at all.
Of course, the arguments must succeed on their own terms, and the identification of the First Cause with the Christian God must be justified rather than merely asserted, but that is an issue concerning the validity and completeness of the demonstration, not automatically an issue of equivocation.
So i would say that your use of the word “whenever” is therefore unjustified.
To establish equivocation in a particular argument, you need to identify the precise term whose meaning changes, explain the two meanings involved, and show how the argument relies on that shift to reach its conclusion, simply observing that the Christian God possesses attributes not established by Aristotle’s argument does none of those things.
A proof of some attributes of a being is not necessarily a proof of all its attributes, but neither is it an equivocation merely because it is incomplete.
And a difference between two descriptions does not, by itself, establish a difference in the reality being described.
*1. On the natural knowledge of God
First Vatican Council, Dei Filius, Canons, II — On Revelation, Canon 1 (Denzinger–Hünermann, DH 3026): “If anyone says that the one true God, our Creator and Lord, cannot be known with certainty through created things by the natural light of human reason, let him be anathema.”
**2. On the mysteries of faith and natural reason
First Vatican Council, Dei Filius, Canons, IV — On Faith and Reason, Canon 1 (Denzinger–Hünermann, DH 3041): “If anyone says that divine revelation contains no mysteries truly and properly so called, but that all dogmas of faith can be understood and demonstrated by adequately cultivated reason on the basis of natural principles, let him be anathema.”
