Introduction

The following essay has two aims. First, to elucidate Irad Kimhi's concept of the "dual counterpart" that he presents in Thinking and Being. Second, to employ Kimhi's notion of the "dual counterpart" in an elucidation of Hegel's notion of the inverted world from his Phenomenology of Spirit. Before I do this, I am going to begin by describing a well-known children's game known as "Opposite Day" in English, and "Verkehrte Welt" in German. My description of this game will help build the general intuition that I then attempt to work out more rigorously in the discussions of Kimhi and Hegel.

Opposite Day

My children sometimes play a game that I and my wife also played when we were little that is called "Opposite Day" in English and "Verkehrte Welt" in German. The people playing agree that from now on, anything they say will mean the opposite. For example, my son says to my daughter "You are poop", and everyone knows that by that he means that she is wonderful. "I want to stay inside" means "I want to go outside." "I hate playing being a doggie" means that I love that game, etc. Part of the delight of this game is that the expressive options are limited. The game works on the basis of logical contraries, so that part of the challenge is to think of things that narrow one down to a specific contrary. For example, it is clear in this game that "hate" implies "love". But it is not clear what contrary one intends when one says "the sky is red". Perhaps one means that the sky is blue, or black, but rather than implying a single contrary attribute, "the sky is red" only seems to mean "the sky is not red."

There is a way to change this game that makes it significantly less fun, but infinitely expand its expressive options. Namely, one could agree that one means not the contrary of what one says, but rather the contradictory. So, for example, "The sky is red" means "The sky is not red", and "The sky is not red" means "The sky is red". The implication is that in this inverted world of contradictories one can say absolutely everything one could otherwise say in the normal world, without any loss of information. It is simply a matter of applying or removing the negation from any thing one says. Nevertheless, such an inverted world seems pointless. For it only works because we have secretly agreed that the positive predication means the negative predication and vice versa. In public we are employing the inverted formulations. But privately, everyone is "undoing" them to get at the sense of the utterance. Thus, this inverted world turns on the notion of a transparent masquerade - we cloak our utterances in negation, but understand our utterances by removing that cloak. This suggests that, in fact, the public form of the utterances aside, we have not left the normal world at all, but merely added a superfluous layer of encoding on our otherwise transparent utterances.

Consider, by way of contrast, what it would mean, if we applied the inversion not to our public utterances but to the very form of our understanding. In this scenario we would set up the following rules:

• When I say "p", I actually mean to say "¬p"
• When I say "¬p", I actually mean to say "p"

This brings us into an infinite loop of saying "p" and saying "¬p". I say "p", but realize, by the first rule, that what I actually mean to say by that is "¬p". But then, the second rule comes to bear: having now said "¬p", I realize that I actually mean to say "p". And so on and so forth. In this inverted world, the inversion is not a transparent layer of convention in how we choose to communicate. Rather, the inversion goes right to the heart of what I mean. In such an inverted world, the result is that I mean nothing at all. I cannot settle on whether I mean "p" or "¬p". Having said one thing, I immediately contradict myself, and contradict myself again and again. In this inverted world we therefore don't mean anything at all.

The contrast between these two inverted worlds presents a puzzle. In the one, I can communicate what I mean, but only because I, in truth, have not inverted anything beside the arbitrary signs by which I communicate. The first inverted world is not an inverted world at all. However, in the second, when I try to invert what it is I mean, I find that I mean nothing at all. Another way of saying this is that there is no coherent inverted world. A truly inverted world is not a world at all, it consists of no claims at all. We can say nothing of it.

Our inability to truly leave the normal world and inhabit the inverted world of contradictories points to a puzzling asymmetry between positive and negative utterances. On the one hand, it seems we can reverse the role of positive and negative utterances with zero loss of information, i.e. without any functional impediment to our ability to effectively make the meaning of our utterances known. On the other hand, when we consider that this reversal only works because we can, in secret, understand the apparently negative utterances for the positive utterances that they truly are, then we understand the stubborn persistence of the respective roles of positive and negative utterances. Meaning what we say prevents us from reversing the polarity of meaning p to meaning ¬p. It is as if propositions, a matter of meaning things, has something akin to the right-hand rule in physics. Based on the zero loss of information between the world and the inverted contradictory world, we might think that the two utterances "The sky is red" and "The sky is not red" are entirely symmetrical. But similar to our befuddlement that it is a physical law of the universe (the right hand rule) that a magnetic field has a direction, we discover that meaning p and meaning ¬p seems to be asymmetrical. The analysis that follows of Kimhi's concept of the "dual counterpart" and Hegel's "inverted world" is the attempt to draw out the significance of this asymmetry.

Kimhi's Dual Counterpart

In Thinking and Being, Kimhi argues that thinking is a "two-way capacity", which is to say that a simple positive predication like "The sky is red" forms a unity with its contradiction "The sky is not red", and that within the unity of the simple predication p and ¬p, the positive form is the dominant form of the two. There is no shortcut to understanding what Kimhi means by all of this, but I find that Kimhi's notion of the "dual counterpart" offers a reasonable entrypoint into understanding his claims. It is, we shall see, a version of the "opposite day" game.

Kimhi introduces the "dual counterpart" in order to show the deficiency of what he calls "c-factualism". "C-factualism" is a theory about how we make simple predicative claims about what is the case, such as when saying "The cat is on the mat". According to the c-factualist, the claim "The cat is on the mat" can be described in terms of a picture, or diagram, of a cat on top of a mat: C. The problem with such a view is that it is not clear whether the diagram (C) is being used to claim that the cat is on the mat, or to claim that the cat is not on the mat. This is why Kimhi quotes Spinoza:

​So they look on ideas as dumb pictures on a tablet, and misled by this preconception they fail to see that an idea, insofar as it is an idea, involves affirmation or negation.

The diagram C is dumb in the sense that it does not contain its own affirmation or negation. It is certainly telling us something about the cat and the mat, but it cannot speak as to whether it is bringing to our attention the fact that the cat is on the mat, or that the cat is not on the mat.

We can recognize here that this is a variation on the opposite day game. In this game, I say "The cat is on the mat", and it can mean either the cat is on the mat, or the cat is not on the mat. The game is to figure out which one of those options I mean when I say "The cat is on the mat". Perhaps I mean "The cat is on the mat", or I mean "The cat is not on the mat". In any case, my utterance "The cat is on the mat" is the equivalent of displaying a card on which stands: C, and leaving it up to a guessing game whether I intend, by raising this card, to assert that the cat is on the mat, or that the cat is not on the mat. We could call this game "Guess my Intention". Another variation of this guessing game, which is functionally the same game, is what we could "Truth and Lies". Rather than furnishing a card with a diagram on it, I can say either "The cat is on the mat" or "The cat is not on the mat", and the others have to guess if I am intending with my claim to tell the truth or to lie.

Kimhi explains what is happening in this game like this:

Since a proposition can be ­either true or false, being true cannot simply be a ­matter of depicting a state of affairs—of being an isomorphic image of a state of affairs.(Kimhi 102)

The card C which "depicts a state of affairs" is the "isomorphic image" of two states of affairs: 1) The cat being on the mat, and 2) the cat not being on the mat. If I try to communicate that the cat is on the mat by furnishing the card C, there is still the question what I mean by this: "being true cannot simply be a matter of depicting a state of affairs". The card itself is neither true nor false, it is not a proposition. In other words, "since a proposition can be either true or false", the card C cannot be a proposition because it is neither true nor false - by showing the card, it is not clear which of the two claims I am making: 1) affirming that the cat is on the mat or 2) denying that the cat is on the mat.

To describe c-factualism as such a guessing game is to say that c-factualism believes that it is possible for us to inhabit an inverted world in the true sense of the inverted. For according to c-factualism, I can furnish "propositions" in the form of "spatial models" or "images" that can both affirm and deny what they say. The crucial point, however, is that these "images", the card C, make no claim at all. As such, it is not, by itself, a depiction of the world. C-factualism cannot represent any world to begin with, and so there is no possible world for it to invert, and thus the inverted world it claims to lead us into, in which our "proposition" can deny what it claims to "propose", is an illusion.

Now we are in a position to understand Kimhi's comments on the dual counterpart:

Consider therefore the dual counterpart of a state of affairs, which differs from the original state of affairs only insofar as the obtaining (or non-­ obtaining) of the counterpart is the same as the non-­obtaining (or obtaining) of the original. (Kimhi 103)

Kimhi is inviting us consider the possibility that we might be living in an inverted world, the dual counterpart. Either everything is as I say it is, or everything is precisely not as I say it is, but we don't know which of these two scenarios we are in fact living in. Now suppose I claim "The sky is red". Considering that I don't know which of the two worlds I am living in, I do not know if what I claim is true or false, and there is nothing in that claim itself that can help me. Kimhi writes:

There cannot be anything in a ­simple affirmation that determines ­whether the state of affairs it depicts is the original or the counterpart. (Kimhi 103)

There is nothing in the "simple affirmation" The sky is red that determines whether the sky is actually red or not actually red. In other words, the claim The sky is red could "depict" the "state of affairs" of a red sky, or it could "depict" the "state of affairs of a blue sky. In the first case, I am living in the normal world, the "original". In the second case, I am living in the inverted world, the "dual counterpart". However, I have no way to decide, based on the claim "The sky is red" which of the two worlds I am living in. Hence, the claim "The sky is red" and the denial "The sky is not red" say nothing at all. Kimhi writes:

There cannot be anything in a ­simple affirmation that determines ­whether the state of affairs it depicts is the original or the counterpart. Hence, it is meaningless to describe affirmation or denial as true or false. (Kimhi 103)

In this state of doubt about which world I am living in, I can say nothing at all. If I consider the possibility that I live in the inverted world, no claim of the world "The sky is red" or "The sky is not red" is going to help me determine which world I am living in, and what the actual claim is that is being made. Therefore, unsure of whether I live in the origianl or the counterpart, "it is meaningless to describe affirmation ["The sky is red"] and denial ["The sky is not red"] as true or false." Saying "The sky is red" is saying something that might be true, and might be false, depending on the world I am living in. As such, it says nothing at all. And this is true of any simple predicative claim I make concerning the world. They are all null and void, they all say nothing at all, all for the same reason. Thus, the possibility of living in an inverted world collapses. It is a world in which no proposition, no claim is possible, and hence it is not any world at all.

To review: contemplating even the mere possibility of an inverted world demonstrates that such a world is impossible. Thus, it is by virtue of the fact that the claim "The sky is red" means anything at all that I know that the inverted world is an incoherent impossibility. I cannot even so much as consider such a world. The uninvertable relationship between affirmation and denial, truth and falsity is a condition for having a world at all.

It might appear that this argument ignores an obvious way out of the uncertainty that leads to world collapse. Why can't I simply look up at the sky and see what color it is? That will tell me if I am living in the inverted world or not. If, having claimed that the sky is red, I look up to the sky and see redness, then I know I am living in the normal world, the "original". And if I look up to the sky and see blue, then I know that I am living in the inverted world, the "counterpart". But this generates a reductio. For, for me to see that the sky is red is for me to make the claim that the sky is red. Everything I could possibly observe about the world, however direct, must, in order to be an observation at all, be expressible in the form of a proposition. And so, when I attempt to look up at the sky to see if it is red, I look up at the sky, and see that it is, indeed, red, and exclaim "The sky is red! I live in the original". Alas, I find myself back where I started: unable to form any coherent claim at all. For perhaps my "direct" observation that the sky is red is the non-obtaining of the sky being red. Considering the possibility that I live in a truly inverted world, I find that, in fact, there is no such world that I can ever inhabit.

By the argument of reductio ad absurdum, I conclude that I live in the only possible world there is, in which claims mean what they do, and the world is as I say it is. I conclude that the polarity of truth and lies cannot be inverted.

Hegel's Inverted World

[To be continued...]