The Triptychs of Publius

I. The Necessary Existence of Plurality

1. To deny plurality is to distinguish the denial of plurality from its affirmation.
2. Every distinction presupposes plurality: at minimum, a distinction between what is affirmed and what is denied.
3. Therefore, the denial of plurality presupposes plurality.
4. Consequently, plurality cannot be coherently denied.

Conclusion I: Plurality necessarily exists, at least in the structure of any intelligible denial.

II. The Necessary Existence of Unity

5. Every plurality is necessarily a plurality of units. The many presuppose the one.
6. This dependence is asymmetrical: unity does not, merely as unity, presuppose plurality, whereas plurality necessarily presupposes unity.
7. By the principle of Implicit Prior Reference, if something actually exists, then whatever is necessary for its existence must also actually exist.
8. Plurality actually exists (Conclusion I), and unity is a necessary condition of plurality.
9. Therefore, unity actually exists.

Conclusion II: Unity necessarily exists as a constitutive condition of plurality.

III. The Combined Argument

The denial of plurality presupposes plurality; plurality, in turn, presupposes unity. Therefore, the denial of plurality ultimately presupposes unity.

In symbols:

Denial of P ⇒ P
P ⇒ U
∴ Denial of P ⇒ U

Where P denotes actual plurality and U denotes actual unity.

Conclusion III: Unity cannot be coherently denied, since any determinate denial itself presupposes distinction, plurality, and therefore unity.

IV. Ontological Clarification

The argument requires only a minimal conception of existence: something exists insofar as it actually obtains or is instantiated.

A distinction between imaginary objects need not establish the existence of those objects. Nevertheless, the act of distinguishing actually occurs. Even the denial that thought exists is itself an occurrence of thought.

Thus, the argument does not require the mind-independent existence of the objects distinguished. The actuality of distinction itself suffices.

V. The Priority of Unity

Plurality cannot exist without units, whereas unity, considered simply as the determinacy of one thing, does not itself require plurality.

Therefore, unity is constitutively prior to plurality, even though plurality is the starting point of the demonstration.

The many presuppose the one; the one does not, merely by being one, presuppose the many.