From the square of opposition to the logical square

Ayn Rand defined logic as “the art of non-contradictory identification” (p.74, her italics). Logic is the metaphysical art of consciousness.

Let us look at the logic we all know. I am trying to connect Aristotle’s square of oppositions (see Figure 1) to the law of identity through Rand’s axioms. The square of opposition deals with propositions, but identity is viewed as only a part of a propositional statement. For example, in “all men are mortal” you have two identities: “men” and “mortal.” This proposition can be expressed in the following form: All S is P, where S is “men” and P is “mortal.” “All” is a logical operator meaning “Every.” In propositional logic, to say that “Some S is P” does not presuppose that “Every S is P.” However, if you know that “Every S is P” is true, then it is necessarily that “Some S is P” is also true. In other words, it is an implication in the form: IF (Every S is P), THEN (Some S is P), where the inverse without negation is not valid. I would like to convert this into the ontological form. One way is to take “is P” to mean “is an existent” (P==existent), and we can write it as the pure action of existing, that is, simply the verb “is,” i.e., exists (in some place and/or in some time). Then we have: “Every S exists” and “Some S exists.” Without substituting anything for S, we can infer that the sum of all S that exists is (metaphysical) Existence and some part of that sum is (ontological) Identity. Then we have, IF (Existence), THEN (Identity). Since we know that existential cause is an action that follows the law of identity, we can say that “Existence is Identity.” In the square of oppositions, this converts to “A is I” (see below).[1] Both A and I are true individually and as a subaltern.

square of opposition

Figure 1: the square of opposition (source:

As we have seen in the previous post, Existence is finite but exists in an everlasting context – Nonexistence. Existence may be continuous or discontinuous, but it cannot necessarily be continuous. When looking at Existence in an eternal context, it becomes unbounded, i.e., potentially infinite but never actually infinite.

Digressing for a bit, I want to place these ideas in context before exploring the right side of the square. Hegel was inspired by Kant’s flawed dialectic–hence the great failure of Hegel. It’s a metaphysical contradiction to claim that existence can cease to exist. Instead, it is not the negation that occurs (it cannot), but the negation that leads to the becoming of an existent. In other words, instead of Hegelian thesis-antithesis-synthesis, it should be antithesis-thesis-synthesis because a thesis cannot exist without its context (i.e., its antithesis), and synthesis is neither antithesis nor thesis. Kant accepted both views, which contradict each other. He should have only accepted the latter view, in which antithesis always precedes but the same antithesis never follows the same thesis. Applying the incorrect view to reality, we get contradictions that take lives to bring them forth and keep on maintaining them. “A is A” is always actual, but actualizing a potential A is done through Non-A as its implicit potentiality, and A thus becomes an explicit actuality. In other words, an entity gets re-identified by its own action or cause. So, in the Aristotelian sense, non-A is becoming its own being A diachronically (across spacetime). From outside when A is A, it looks as if A jumped to a new A by abandoning its old self (non-A) with which it identified through the process of becoming.

As in Lakovian metaphors, non-A is source domain and A is target domain while we write this dialectic metaphorically: A is non-A. However, A (i.e., new or current concept) is not literally non-A (i.e., old or known concept). A implies non-A. (A non-A). If A, then non-A. A is A only if non-A is A. Or simply, A only if non-A. It is interesting to note that some metaphors are conceptual and directly relate to the way we experience the world, the way we think about it and express it in words. This new understanding of metaphors shows that conceptual metaphors are conventional and mostly unconscious and that they are physical, i.e., reflected while connecting neurobiological and cognitive levels. However, since source domain is generally physical and maps onto conceptual target domain to explain it, we should be looking bottom-up, not top-down (see CRH), like Lakoff.

To keep the proper direction, our proposition is as follows: Only if nothing is everything, is everything something. Here is our reasoning.

Using the square of opposition, we convert the above statement to:

(1) Only if (No S is P) is (All S is P), (All S is P) is (Some S is P).

(2) Only if contrary(E is A), subaltern(A is I).

(3) If A is I, then E is A.

In axiomatic form:

(4) If Existence is Identity, then Nonexistence is Existence.

From subaltern of (2): If Existence is TRUE, then Identity is TRUE.

Existence is TRUE, therefore Identity is TRUE.

(5) (Existence is Identity) is TRUE

From (1) and (4): Only if Nonexistence is Existence

From contrary of (2): Existence is TRUE, therefore Nonexistence is FALSE

(6) However, Only if {Nonexistence (FALSE) is Existence (INDET.)} is TRUE

OR: Only if {Nonexistence (INDET.) is Existence (TRUE)} is FALSE

From the necessary premise (5), (6) is the necessary conclusion, and Existence cannot prove or disprove (Nonexistence is Existence).

Therefore, (Nonexistence is Existence) is also an axiom.

By Godel’s completeness theorem: (5) is consistent, (4) is complete.

(5) or (4) is a weak disjunction. If (5) is TRUE, then (4) can be chosen TRUE or FALSE, but only if (4) is FALSE, (5) must be TRUE.

Existence is Identity, Only if Nonexistence is Existence

From the above axiomatic proposition, we take Nonexistence to be FALSE, and therefore Existence is INDETERMINATE (TRUE or FALSE). From this point on, we have Existence (TRUE) is Identity (TRUE). Nonexistence is absolute nothing, Existence is absolute everything (or the sum of everything), Existence in (Existence is Identity) is absolute everything AND something specific (a mix of Existence and Identity through the verb “is”), and Identity is something specific. Notice how Existence in (Nonexistence is Existence) and Existence in (Existence is Identity) share the same definition and the same value. However, the first is pure everything and the last can be every thing (in other words, fragmented). The FALSE value of the first makes it absolutely whole, whereas the second can descend into Identity. Thus, Nonexistence does not mix (and is not the same) with either Existence or Identity (in Existence is Identity), and the final proposition is valid.

We add the right-side subaltern (E is O) as the FALSE Nonexistence (No S exists) becoming INDET. Non-identity (Some S is not existent, or Some S doesn’t exist) and, in the subcontrary (O is I), TRUE Non-identity (Some S doesn’t exist) becoming INDET. Identity (Some S exists). At the end, Identity is TRUE when it has actualized. Thus we have merged the metaphysical laws governing the Model with the 34 of its epistemological identities or non-identities.

We can represent the outlined Law of Becoming visually in The Logical Square below.

logical square

This post concludes the explanation of the fundamentals of the new philosophy beyond Objectivism.

[1] Note that “I” can be substituted for “A,” but “A” cannot be substituted for “I.” This only means that “A is I” but not the reverse.


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