Law of Excluded Middle (LoEM) states that “There is no Third”. What does this mean? How is it motivated? Is it valid?
What are the First and the Second? Apparently they are P and ~P, but in which order?
What do these two represent?
— P represents the naming of P.
— ~P represents a simple operation on P that reaches something different from P.
The naming of P is a reference to P. The simplest operation on P is to access the content of P. So which of the following two comes first?
— A reference to P = P
— The content of P = ~P
Obviously, the content of P comes before a reference to P. As a result, the order of the First and the Second is as follows:
1) The First: ~P
2) The Second: P
The naming comes after the content, because the name is the way to access that content. The notation ~P is like the “pointer” notation *p that should be familiar to C++ programmers.
What is the Third? Let’s call it Q. Q is neither ~P nor P.
1) ~P (the content of P)
2) P (the name for accessing ~P)
3) Q (a new name)
Q denotes a place beyond ~P and P. In other words, Q is unreachable through P (only the content ~P is reachable through P); Q is a new variable.
So what is the content of Q? What is ~Q (something different from Q that’s directly reachable through Q)?
The content of Q is the pair (~P,P) = ~Q. There was nothing else before the naming of Q, so these two must constitute the content of Q.
Therefore, the Third exists. It names the coupling of the First and the Second.
What is the motivation to exclude the third?
LoEM holds that the Third must be excluded, and what would replace it?
In LoEM, a ranking of the First and the Second replaces the Third: “Either P is true, or ~P is true.” This implies two alternative dismissals.
If ~P is true, then P can be dismissed. If P is true, then ~P can be dismissed. But there is an asymmetry between these two dismissals.
1) To dismiss the content ~P is to keep the name P while changing its content ~P.
2) But to dismiss the name P is to replace it with another name (say) R, so that ~P will now be called ~R.
Let’s exemplify the First and the Second by a metaphor: “Earth is like a ball” where the content “Earth” is characterized by the name “ball”. The negation ~P makes sense because obviously the Earth is not a ball. Nonetheless, the content ~P (Earth) is characterized and made accessible through the name P (ball).
1) To dismiss the content “Earth” is to keep the ballness of the ball while excluding the content of the metaphor because of the simple fact that the Earth is not a ball. The Earth cannot be characterized by the likeness of a ball. The Earth-in-itself is quite beyond the likeness of a ball. The ballness of a ball can characterize another content, such as a football, but not the Earth.
2) But to dismiss the characterization “ball” is to replace it with another one (say) “Earth is like a geoid” that is supposedly better and more exact than the previous one. This second dismissal hierarchizes the two characterizations “geoid > ball” with respect to “Earth”.
The first dismissal makes “Earth” a logical basis for a hierarchization by invoking the Earth-in-itself, like the Kantian Thing-in-itself.
The second dismissal performs the hierarchization to prefer the geoidness of a geoid to the ballness of a ball based on the logical basis “Earthness of the Earth”.
As such, the exclusion of the third is motivated by a will to hierarchize names and characterizations.
In the example above, the exclusion of the third amounts to the forbidding of the metaphor Q = “Earth is like a ball”.
Let’s now consider the negation of the metaphor, ~Q = “Earth is not like a ball” as the content of the metaphor Q, just like we did with P and ~P.
In the new situation, there is a New First and a New Second, both of which emerge from the Third:
1) New First: ~Q = “Earth is not like a ball”
2) New Second: Q = “Earth is like a ball”
As you see here, the claim “There is no Third” is invalid. The Third exists and moreover it can be extracted to obtain a new set of a First and a Second.
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