The Simplest Options Possible in the Theory of Identity in Concise Statement

0. In the course of my continued reading and rereading Réginald Garrigou-Lagrange, O.P., Reality. A Synthesis of Thomistic Thought,* it has occurred to me that the simplest options possible in the theory of identity are susceptible of very concise statement. In the present post I will identify those most simple options and give them concise expression. In, then, Sections 1 through 6, I will identify and provide concise statements of the several most simple options in the theory of identity. Then, in Sections 7 through 13, I will review and provide concise statements of the next most simple options in the theory of identity. I will close with some comments in Sections 14 and 15.

The Several Most Simple Options Possible in the Theory of Identity

1. Thus, first, there is the thesis or principle of the reflexivity of identity, expressed by means of the universal affirmative proposition:

For any existent x, x is identical with x.

In a more ordinary English:

Any existent is identical with itself.

It is to Parmenides, of course, that we owe the first, if perhaps not quite perfect, expression of the principle of the reflexivity of identity, a thesis utterly basic to not only Parmenidean thought, but also that of Aristotelianism and the rest of classical rationalism.

2. Now, as I enjoy putting it, for every great universal affirmative truth there are two equally great and opposite falsehoods, the one the contradictory and the other the contrary of the truth. The great falsehood contradicting the principle of the reflexivity of identity is the thesis denying the universality of the reflexivity of identity:

It is not the case that, for any existent x, x is identical with x.

This is, of course, equivalent to the particular negative proposition:

For some existent x, x is not identical with x.

3. The great falsehood standing as the contrary of the, as I hold it, true principle of the reflexivity of identity is the thesis of the irreflexivity of identity, expressed by means of the universal negative proposition:

For any existent x, x is not identical with x.

I take it that this universal negative thesis is the one towards which the thought of Heraclitus was trending, though his express statements seemed more in accordance with the more complex thesis:

For any existent x, it is the case both that x is identical with x and that x is not identical with x.

Whatever may have been the position of the historical Heraclitus, let me identify the thesis of the irreflexivity of identity as that of radical Heracliteanism, at least for the ahistorical purposes of the present post.

4. Just as the thesis contradicting the universal affirmative principle of the reflexivity of identity, given in Section 1, is the particular negative thesis denying it given in Section 2, so the thesis contradicting the universal negative principle of the irreflexivity of identity given in Section 3 is the particular affirmative thesis:

For some existent x, x is identical with x.

5. a. It is further the case that, as can be demonstrated using only the resources of elementary logic, the two universal propositions, the affirmative and the negative, imply the corresponding particular propositions, affirmative and negative. So, the universal affirmative proposition given in Section 1:

For any existent x, x is identical with x.

implies the particular affirmative proposition given in Section 4:

For some existent x, x is identical with x.

That is, if the universal affirmative proposition is true, the particular affirmative proposition is also true. The Parmenidean ontology, the Aristotelian ontology, and those of the rest of classical rationalism hold both of these two propositions to be true.

5. b. So too, the universal negative proposition given in Section 3:

For any existent x, x is not identical with x.

implies the particular negative proposition given in Section 2:

For some existent x, x is not identical with x.

That is, were the universal negative proposition true, the particular negative proposition would also be true. The ontology of radical Heraclitianism would hold both of these latter two propositions to be true (if and as truth itself is intelligible in the context of philosophical irrationalism).

6. a. The universal affirmative proposition expressing the thesis of the reflexivity of identity and the universal negative proposition expressing the thesis of the irreflexivity of identity stand as contraries the one to the other. That is, as contraries, they cannot both be true, though they can both be false. In the case of the two universal theses immediately at hand, the affirmative proposition is true and the negative false.

6. b. The particular affirmative proposition expressing the thesis that some existent is self-identical and the particular negative proposition expressing the thesis that some existent is not self-identical stand as sub-contraries the one to the other. In the case of the two particular theses immediately at hand, the affirmative proposition is true and the negative false.

The Several Next Most Simple Options Possible in the Theory of Identity

7. The next most simple options in the theory of identity differ from the ones just reviewed in that they feature the two variables, “x” and “y,” instead of the one variable, “x.” Thus, first, the universal affirmative proposition, expressing the thesis of radical monism:

For any existent x and any existent y, x is identical with y.

or, in a more ordinary English (despite the extraordinary claim it expresses):

Any existent is identical with any existent.

Whatever may have been the position of the historical Parmenides, let me identify the thesis of radical monism to be that of radical Parmenideanism, at least for the ahistorical purposes of the present post.

8. While I, along with Aristotelianism and the rest of classical rationalism, absolutely agree with radical Parmenideanism in holding the thesis of the reflexivity of identity to be true, we find the thesis of radical monism to be false. Rather, we accept the truth of the thesis contradicting it:

It is not the case that, for any existent x and any existent y, x is not identical with y.

or, equivalently, a particular negative thesis:

For some existent x and some existent y, x is not identical with y.

9. The contrary of the universal affirmative thesis that is the thesis of radical monism is the universal negative:

For any existent x and any existent y, x is not identical with y.

or, in a more ordinary English (despite the extraordinary claim it expresses):

No existent is identical with any existent.

I will identify this thesis too as that of the irreflexivity of identity, for, as can be demonstrated using only the resources of elementary logic, the proposition given concise expression in Section 3 above implies the one given expression here.

10. The thesis standing in contradiction to that of the irreflexivity of identity, as the latter has been most recently formulated, is:

It is not the case that, for any existent x and any existent y, x is not identical with y.

or, equivalently, as a particular affirmative proposition:

For some existent x and some existent y, x is identical with y.

11. a. Let us recall that, as can be demonstrated using only the resources of elementary logic, two universal propositions, the affirmative and the negative, imply the corresponding particular propositions, affirmative and negative. So, the universal affirmative proposition given in Section 7:

For any existent x and any existent y, x is identical with y.

implies the particular affirmative proposition given in Section 10:

For some existent x and some existent y, x is identical with y.

It is perhaps initially surprising that the false universal affirmative thesis here implies the true particular affirmative thesis. But such is the way of implication, for, as any textbook in elementary logic will make clear, if a proposition implying another is false, the implication is true; a true implication is one in which it is not the case that the implying proposition is true and the implied false.

11. b. Just as the false universal affirmative thesis given in Section 7 implies the true particular affirmative thesis given in Section 10, so the false universal negative thesis given in Section 9:

For any existent x and any existent y, x is not identical with y.

implies the true particular negative proposition given in Section 8:

For some existent x and some existent y, x is not identical with y.

12. a. It was noted above, in Section 6, that while universal affirmative propositions and the corresponding universal negative propositions cannot, as contraries, both be true, they can both be false. That is the case here, for both the thesis of radical Parmenideanism and that of the irreflexivity of identity of radical Heraclitianism are false.

12. b. On the other hand, particular affirmative propositions and the corresponding particular negative propositions cannot, as sub-contraries, both be false; they can however, both be true. That is the case here. If we conjoin the particular affirmative proposition reviewed in Section 10 with the particular negative proposition reviewed in Section 8, we have the following proposition:

For some existent x and some existent y, x is identical with y

and

for some existent x and some existent y, x is not identical with y.

As both conjuncts are true, the conjunction itself is true.

13. The moderate pluralism of Aristotelianism and the rest of classical rationalism stands as the middle position between the extremes of the radical Parmenideanism and the radical Heraclitianism presented above. That is, on the one hand, with radical Parmenideanism and against radical Heraclitianism, it upholds the thesis of the reflexivity of identity reviewed in Section 1. And, on the one hand, against radical Parmenideanism and with radical Heraclitianism, it upholds the particular negative thesis contradicting that of radical monism. The thesis of moderate pluralism relevant here is then the conjunction:

For any existent x, x is identical with x

and

for some existent x and some existent y, x is not identical with y.

14. I will leave it up to the reader to explore the several options which one can generate with propositions including the three variables, “x,” “y,” and “z.”

15. You may well have noticed that I have relied on the logic of categorical propositions typically exhibited by means of the “square of opposition” which can be found in the textbooks of traditional logic. I would have made use of the square of opposition here had I known how to “draw” it with the resources made available through WordPress.

Until next time.

* Réginald Garrigou-Lagrange, O.P., Reality. A Synthesis of Thomistic Thought, translated by Patrick Cummins, O.S.B. (Ex Fontibus, 2006-2012)

About Rchard E. Hennessey

See above, "About the Author/Editor."
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