The aim of the present post is three-fold. I wish first to draw attention to the ontological theory of identity, i.e., of existents as identical with existents, and three quite basic principles of the theory. I then wish to then to draw attention to four arithmetical definitions that the introduction of non-identity into the ontology makes possible, statements, that is, of what it is to be one existent, what it is to be two existents, what it is to be three existents, and what it is to be four existents. There is little if anything in this post that is truly original, though the accompanying philosophical realism, according to which the ontology and arithmetic in question are theories of real existents, may startle some.

I am more inclined, though still only inclined, to believe that the results which will be presented in the next post are, though absolutely elementary, yet original; I say “inclined,” because I just don’t know if others have preceded me in arriving at them. That is, in the next post, I will spell out how one can prove, demonstrate, that one existent plus one existent are two existents, that two existents plus one existent are three existents, and that two existents plus two existents are four existents. It will follow that the philosophical realism of the ontology and arithmetic at hand is an apodicticism, according to which they are, not just theories, but demonstrative sciences. [Correction, December 10, 2020: The proof that two existents plus two existents are four existents was not published until my “Number and Reality 3: The Demonstration That 2 Plus 2 Are 4 Spelled Out,” posted on December 5, 2020.]

Ontology. There is, then, an ontology of identity. I will state the three basic principles of the theory in, first, a nearly standard English and then in a rendering typical of what you might find in a textbook of (so-called) “mathematical logic” (but I would rather identify as one of ontology); the nearly standard English rendering is for those who are not familiar with contemporary elementary logic and the rendering of “mathematical logic” is for those who are. I’ll also offer exemplifications of each of the three principles.

The Principle of the Reflexivity of Identity:

In the nearly standard English: For any existent x, x is identical with x.

In the rendering of “mathematical logic”: (x)(Ixx)

E.g.: Donald Trump is identical with Donald Trump.

The Principle of the Symmetry of Identity:

In the nearly standard English: For any existent x and any existent y, if x is identical with y, then y is identical with x.

In the rendering of “mathematical logic”: (x)(y)(Ixy —> Iyx)

E.g.: If Donald Trump is identical with the President of the United States, then the President of the United States is identical with Donald Trump.

The Principle of the Transitivity of Identity:

In the nearly standard English: For any existent x, any existent y, and any existent z, if x is identical with y and y is identical with z, then x is identical with z.

In the rendering of “mathematical logic”: (x)(y)(z)((Ixy & Iyz) —> Ixz)

E.g.: If Donald Trump is identical with the President of the United States and the President of the United States is identical with the Commander in Chief of the United States’ Army and Navy, etc., then Donald Trump is identical with the Commander in Chief of the United States’ Army and Navy, etc.

All extra-mental and extra-linguistic existents, including Donald Trump, are precisely that, extra-mental and extra-linguistic existents, and so we can note that the ontology of identity is a theory of extra-mental and extra-linguistic existents. To underline the point: the objects of the theory are not intra-mentally existent (existing in the mind) thoughts about the real or thoughts instead of the real; and they are not intra-linguistically existent (existing in language) words about the real or words instead of the real. We can also note, and not merely parenthetically, that the three principles just set forth are all absolutely true and true of absolutely everything.

(In the following expressing of the definitions in the “rendering of ‘mathematical logic’,” I found myself not knowing how to express, with the means available to WordPress users, the “if and only if” relation using the standard double arrow. So I had to make do with “iff.” I similarly found myself not knowing how to express the existential quantifier using the backwards capital letter “E” customary among logicians, and so had to make do with the capital letter “E” facing in the direction customary in the rest of the world.)

Ontological Arithmetic. Four Definitions. The three principles just placed before you are principles bearing upon identity. If we add non-identity to our ontology, such that one existent is not another, we will find ourselves having entered the realm of ontological arithmetic, via the following four arithmetical definitions.

The Definition of What It Is to Be One Existent.

In the nearly standard English: For any existent x, x is one existent if and only if it is not the case that there is an existent y and there is an existent z such that y is not identical with z and such that both y is identical with x and that z is identical with x.

In the rendering of “mathematical logic”: (x)(1x iff ~(Ey)(Ez)(~Iyz & (Iyx & Izx)))

[Note added, December 10, 2020: In the weeks that followed this post, I have grown less and less satisfied with this attempt at defining what it is to be one existent.]

The Definition of What It Is to Be Two Existents (A definition rather simpler than the previous).

In the nearly standard English: For any existent x and any existent y, x and y are two existents if and only if
x is not identical with y.

In the rendering of “mathematical logic”: (x)(y)(2xy iff ~Ixy)

The Definition of What It Is to Be Three Existents.

In the nearly standard English: For any existent x, any existent y and any existent z, x, y and z are three existents if and only if
x is not identical with y, x is not identical with z, and y is not identical with z.

In the rendering of “mathematical logic”: (x)(y)(z)(3xyz iff (~Ixy & ~Ixz & ~Iyz))

The Definition of What It Is to Be Four Existents.

In the nearly standard English: For any existent x, any existent y, any existent z, and any existent w, x, y, z, and w are four existents if and only if x is not identical with y, x is not identical with z, x is not identical with w, y is not identical with z, y is not identical with w, and z is not identical with w.

In the rendering of “mathematical logic”: (x)(y)(z)(w)(4xyzw iff (~Ixy & ~Ixz & ~Ixw & ~I yz & ~Iyw & ~Izw))

I could continue indefinitely, but I trust that that will not be necessary.

Summing Up. In the present post I have drawn attention to the ontological theory of identity, i.e., of existents as identical with existents, and to three quite basic principles of the theory. I have further drawn attention to four arithmetical definitions that the introduction of non-identity into the ontology makes possible, i.e., of what it is to be one existent, what it is to be two existents, what it is to be three existents, and what it is to be four existents.

With those definitions, I think it safe to say that I have one realistic foot in the door of the philosophy of mathematics.

In the next post, as I said above, I will spell out how one can demonstrate that one existent plus one existent are two existents, that two existents plus one existent are three existents, and that two existents plus two existents are four existents; I could continue indefinitely, but I trust that that will not be necessary. I will affirm that the philosophical realism of the ontology and arithmetic at hand is an apodicticism, according to which they are, not just theories of extra-mental and extra-linguistic existents, but demonstrative sciences thereof. With that done, I believe I will have two realistic feet in the door of the philosophy of mathematics.

Until next time.

Richard