Liar Paradox

Form and Reality II

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This is a followup to this earlier post, but will use a number of other threads to get a fuller understanding of the matter. Rather than presenting this in the form of a single essay, I will present it as a number of distinct theses, many of which have already been argued or suggested in various forms elsewhere on the blog.

(1) Everything that exists or can exist has or could have some relationship with the mind: relationship is in fact intrinsic to the nature of existence.

This was argued here, with related remarks in several recent posts. In a sense the claim is not only true but obviously so. You are the one who says or can say “this exists,” and you could not say or understand it unless the thing had or could have some relationship with your mind.

Perhaps this seems a bit unfair to reality, as though the limits of reality were being set by the limits of the thinker. What if there were a limited being that could only think of some things, but other things could exist that it could not think about? It is easy to see that in this situation the limited being does not have the concept of “everything,” and so can neither affirm nor deny (1). It is not that it would affirm it but be mistaken. It would simply never think of it.

Someone could insist: I myself am limited. It might be that there are better thinkers in the world that can think about things I could never conceive of. But again, if you have concept of “everything,” then you just thought of those things: they are the things that those thinkers would think about. So you just thought about them too, and brought them into relationship with yourself.

Thus, anyone who actually has the idea of “everything,” and thinks about the matter clearly, will agree with (1).

(2) Nothing can be true which could not in principle (in some sense of “in principle”) in some way be said to be true.

Thesis (1) can be taken as saying that anything that can be, can also be understood, at least in some way; and thesis (2) can be taken as saying that anything that can be understood, can also be said, at least in some way.

Since language is conventional, this does not need much of an argument. If I think that something exists, and I don’t have a name for it, I can make up a name. If I think that one thing is another thing, but don’t have words for these things, I can make up words for them. Even if I am not quite sure what I am thinking, I can say, “I have a thought in my mind but don’t quite have the words for it,” and in some way I have already put it into words.

One particular objection to the thesis might be made from self-reference paradoxes. The player in the Liar Game cannot correctly say whether the third statement is true or false, even though it is in fact true or false. But note two things: first, he cannot do this while he is playing, but once the game is over, he can explicitly and correctly say whether it was true or false. Second, even while playing, he can say, “the third statement has a truth value,” and in this way he speaks of its truth in a generic way. This is in part why I added the hedges to (2), “at least in some way”, and “in principle.”

(3) Things do not have hidden essences. That is, they may have essences, but those essences can be explained in words.

This follows in a straightforward way from (1) and (2). The essence of a thing is just “what it is,” or perhaps, “what it most truly is.” The question “what is this thing?” is formed with words, and it is evident that anyone who answers the question, will answer the question by using words.

Now someone might object that the essence of a thing might be hidden because perhaps in some cases the question does not have an answer. But then it would not be true that it has an essence but is hidden: rather, it would be false that it has an essence. Similarly, if the question “where is this thing,” does not have any answer, it does not mean the thing is in a hidden place, but that the thing is not in a place at all.

Another objection might be that an essence might be hidden because the answer to the question exists, but cannot be known. A discussion of this would depend on what is meant by “can be known” and “cannot be known” in this context. That is, if the objector is merely saying that we do not know such things infallibly, including the answer to the question, “what is this?”, then I agree, but would add that (3) does not speak to this point one way or another. But if it is meant that “cannot be known” means that there is something there, the “thing in itself,” which in no way can be known or expressed in words, this would be the Kantian error. This is indeed contrary to (3), and implicitly to (1) or (2) or both, but it is also false.

People might also think that the essence cannot be known because they notice that the question “what is this?” can have many legitimate answers, and suppose that one of these, and only one, must be really and truly true, but think that we have no way to find out which one it is. While there are certainly cases where an apparent answer to the question is not a true answer, the main response here is that if both answers are true, both answers are true: there does not need to be a deeper but hidden level where one is true and the other false. There may however be a deeper level which speaks to other matters and possibly explains both answers. Thus I said in the post linked above that the discussion was not limited to “how many,” but would apply in some way to every question about the being of things.

(4) Reductionism, as it is commonly understood, is false.

I have argued this in various places, but more recently and in particular here and here. It is not just one-sided to say for example that the universe and everything in it is just a multitude of particles. It is false, because it takes one of several truths, and says that one is “really” true and that the other is “really” false.

(5) Anti-reductionism, as it is commonly understood, is false.

This follows from the same arguments. Anti-reductionism, as for example the sort advocated by Alexander Pruss, takes the opposite side of the above argument, saying that certain things are “really” one and in no way many. And this is also false.

(6) Form makes a thing to be what it is, and makes it to be one thing.

This is largely a question of definition. It is what is meant by form in this context.

Someone might object that perhaps there is nothing that makes a thing what it is, or there is nothing that makes it one thing. But if it is what it is of itself, or if it is one of itself, then by this definition it is its own form, and we do not necessarily have an issue with that.

Again, someone might say that the definition conflates two potentially distinct things. Perhaps one thing makes a thing what it is, and another thing makes it one thing. But this is not possible because of the convertibility of being and unity: to be a thing at all, is to be one thing.

(7) Form is what is in common between the mind and the thing it understands, and is the reason the mind understands at all.

This is very distinctly not a question of definition. This needs to be proved from (6), along with what we know about understanding.

It is not so strange to think that you would need to have something in common with a thing in order to understand it. Thus Aristotle presents the words of Empedocles:

For ’tis by Earth we see Earth, by Water Water,

By Ether Ether divine, by Fire destructive Fire,

By Love Love, and Hate by cruel Hate.

On the other hand, there is also obviously something wrong with this. I don’t need to be a tree in order to see or think about a tree, and it is not terribly obvious that there is even anything in common between us. In fact, one of Hilary Lawson’s arguments for his anti-realist position is that there frequently seems to be nothing in common between causes and effects, and that therefore there may be (or certainly will be) nothing in common between our minds and reality, and thus we cannot ultimately know anything. Thus he says in Chapter 2 of his book on closure:

For a system of closure to provide a means of intervention in openness and thus to function as a closure machine, it requires a means of converting the flux of openness into an array of particularities. This initial layer of closure will be identified as ‘preliminary closure’. As with closure generally, preliminary closure consists in the realisation of particularity as a consequence of holding that which is different as the same. This is achieved through the realisation of material in response to openness. The most minimal example of a system of closure consists of a single preliminary closure. Such a system requires two discrete states, or at least states that can be held as if they were discrete. It is not difficult to provide mechanical examples of such systems which allow for a single preliminary closure. A mousetrap for example, can be regarded as having two discrete states: it is either set, it is ready, or it has sprung, it has gone off. Many different causes may have led to it being in one state or another: it may have been sprung by a mouse, but it could also have been knocked by someone or something, or someone could have deliberately set it off. In the context of the mechanism all of these variations are of no consequence, it is either set or it has sprung. The diversity of the immediate environment is thereby reduced to single state and its absence: it is either set or it is not set. Any mechanical arrangement that enables a system to alternate between two or more discrete states is thereby capable of providing the basis for preliminary closure. For example, a bell or a gate could function as the basis for preliminary closure. The bell can either ring or not ring, the gate can be closed or not closed. The bell may ring as the result of the wind, or a person or animal shaking it, but the cause of the response is in the context of system of no consequence. The bell either rings or it doesn’t. Similarly, the gate may be in one state or another because it has been deliberately moved, or because something or someone has dislodged it accidentally, but these variations are not relevant in the context of the state of system, which in this case is the position of the gate. In either case the cause of the bell ringing or the gate closing is infinitely varied, but in the context of the system the variety of inputs is not accessible to the system and thus of no consequence.

A useful way to think about Lawson is that he is in some way a disciple of Heraclitus. Thus closure is “holding that which is different as the same,” but in reality nothing is ever the same because everything is in flux. In the context of this passage, the mousetrap is either set or sprung, and so it divides the world into two states, the “set” state and the “sprung” state. But the universes with the set mousetrap have nothing in common with one another besides the set mousetrap, and the universes with the sprung mousetrap have nothing in common with one another besides the sprung mousetrap.

We can see how this could lead to the conclusion that knowledge is impossible. Sight divides parts of the world up with various colors. Leaves are green, the sky is blue, the keyboard I am using is black. But if I look at two different green things, or two different blue things, they may have nothing in common besides the fact that they affected my sight in a similar way. The sky and a blue couch are blue for very different reasons. We discussed this particular point elsewhere, but the general concern would be that we have no reason to think there is anything in common between our mind and the world, and some reason to think there must be something in common in order for us to understand anything.

Fortunately, the solution can be found right in the examples which supposedly suggest that there is nothing in common between the mind and the world. Consider the mousetrap. Do the universes with the set mousetrap have something in common? Yes, they have the set mousetrap in common. But Lawson does not deny this. His concern is that they have nothing else in common. But they do have something else in common: they have the same relationship to the mousetrap, different from the relationship that the universes with the sprung mousetrap have to their mousetrap. What about the mousetrap itself? Do those universes have something in common with the mousetrap? If we consider the relationship between the mousetrap and the universe as a kind of single thing with two ends, then they do, although they share in it from different ends, just as a father and son have a relationship in common (in this particular sense.) The same things will be true in the case of sensible qualities. “Blue” may divide up surface reflectance properties in a somewhat arbitrary way, but it does divide them into things that have something in common, namely their relationship with the sense of sight.

Or consider the same thing with a picture. Does the picture have anything in common with the thing it represents? Since a picture is meant to actually look similar to the eye to the object pictured, it may have certain shapes in common, the straightness of certain lines, and so on. It may have some colors in common. This kind of literal commonness might have suggested to Empedocles that we should know “earth by earth,” but one difference is that a picture and the object look alike to the eye, but an idea is not something that the mind looks at, and which happens to look like a thing: rather the idea is what the mind uses in order to look at a thing at all.

Thus a better comparison would be between the the thing seen and the image in the eye or the activity of the visual cortex. It is easy enough to see by looking that the image in a person’s eye bears some resemblance to the thing seen, even the sort of resemblance that a picture has. In a vaguer way, something similar turns out to be true even in the visual cortex:

V1 has a very well-defined map of the spatial information in vision. For example, in humans, the upper bank of the calcarine sulcus responds strongly to the lower half of visual field (below the center), and the lower bank of the calcarine to the upper half of visual field. In concept, this retinotopic mapping is a transformation of the visual image from retina to V1. The correspondence between a given location in V1 and in the subjective visual field is very precise: even the blind spots are mapped into V1. In terms of evolution, this correspondence is very basic and found in most animals that possess a V1. In humans and animals with a fovea in the retina, a large portion of V1 is mapped to the small, central portion of visual field, a phenomenon known as cortical magnification. Perhaps for the purpose of accurate spatial encoding, neurons in V1 have the smallest receptive field size of any visual cortex microscopic regions.

However, as I said, this is in a much vaguer way. In particular, it is not so much an image which is in common, but certain spatial relationships. If we go back to the idea of the mousetrap, this is entirely unsurprising. Causes and effects will always have something in common, and always in this particular way, namely with a commonality of relationship, because causes and effects, as such, are defined by their relationship to each other.

How does all this bear on our thesis (7)? Consider the color blue, and the question, “what is it to be blue?” What is the essence of blue? We could answer this in at least two different ways:

  1. To be blue is to have certain reflectance properties.
  2. To be blue is to be the sort of thing that looks blue.

But in the way intended, these are one and the same thing. A thing looks blue if it has those properties, and it has those properties if it looks blue. Now someone might say that this is a direct refutation of our thesis, since the visual cortex presumably does not look blue or have those properties when you look at something blue. But this is like Lawson’s claim that the universe has nothing in common with the sprung mousetrap. It does have something in common, if you look at the relationship from the other end. The same thing happens when we consider the meaning of “certain reflectance properties,” and “the sort of thing that looks blue.” We are actually talking about the properties that make a thing look blue, so both definitions are relative to the sense of sight. And this means that sight has something relative in common with them, and the relation it has in common is the very one that defines the nature of blue. As this is what we mean by form (thesis 6), the form of blue must be present in the sense of sight in order to see something blue.

In fact, it followed directly from thesis (1) that the nature of blue would need to include something relative. And it followed from (2) and (3) that the very same nature would turn out to be present in our senses, thoughts, and words.

The same argument applies to the mind as to the senses. I will draw additional conclusions in a later post, and in particular, show the relevance of theses (4) and (5) to the rest.

Self Reference Paradox Summarized

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Hilary Lawson is right to connect the issue of the completeness and consistency of truth with paradoxes of self-reference.

As a kind of summary, consider this story:

It was a dark and stormy night,
and all the Cub Scouts where huddled around their campfire.
One scout looked up to the Scout Master and said:
“Tell us a story.”
And the story went like this:

It was a dark and stormy night,
and all the Cub Scouts where huddled around their campfire.
One scout looked up to the Scout Master and said:
“Tell us a story.”
And the story went like this:

It was a dark and stormy night,
and all the Cub Scouts where huddled around their campfire.
One scout looked up to the Scout Master and said:
“Tell us a story.”
And the story went like this:

It was a dark and stormy night,
and all the Cub Scouts where huddled around their campfire.
One scout looked up to the Scout Master and said:
“Tell us a story.”
And the story went like this:
etc.

In this form, the story obviously exists, but in its implied form, the story cannot be told, because for the story to be “told” is for it to be completed, and it is impossible for it be completed, since it will not be complete until it contains itself, and this cannot happen.

Consider a similar example. You sit in a room at a desk, and decide to draw a picture of the room. You draw the walls. Then you draw yourself and your desk. But then you realize, “there is also a picture in the room. I need to draw the picture.” You draw the picture itself as a tiny image within the image of your desktop, and add tiny details: the walls of the room, your desk and yourself.

Of course, you then realize that your artwork can never be complete, in exactly the same way that the story above cannot be complete.

There is essentially the same problem in these situations as in all the situations we have described which involve self-reference: the paradox of the liar, the liar game, the impossibility of detailed future prediction, the list of all true statements, Gödel’s theorem, and so on.

In two of the above posts, namely on future prediction and Gödel’s theorem, there are discussions of James Chastek’s attempts to use the issue of self-reference to prove that the human mind is not a “mechanism.” I noted in those places that such supposed proofs fail, and at this point it is easy to see that they will fail in general, if they depend on such reasoning. What is possible or impossible here has nothing to do with such things, and everything to do with self-reference. You cannot have a mirror and a camera so perfect that you can get an actually infinite series of images by taking a picture of the mirror with the camera, but there is nothing about such a situation that could not be captured by an image outside the situation, just as a man outside the room could draw everything in the room, including the picture and its details. This does not show that a man outside the room has a superior drawing ability compared with the man in the room. The ability of someone else to say whether the third statement in the liar game is true or false does not prove that the other person does not have a “merely human” mind (analogous to a mere mechanism), despite the fact that you yourself cannot say whether it is true or false.

There is a grain of truth in Chastek’s argument, however. It does follow that if someone says that reality as a whole is a formal system, and adds that we can know what that system is, their position would be absurd, since if we knew such a system we could indeed derive a specific arithmetical truth, namely one that we could state in detail, which would be unprovable from the system, namely from reality, but nonetheless proved to be true by us. And this is logically impossible, since we are a part of reality.

At this point one might be tempted to say, “At this point we have fully understood the situation. So all of these paradoxes and so on don’t prevent us from understanding reality perfectly, even if that was the original appearance.”

But this is similar to one of two things.

First, a man can stand outside the room and draw a picture of everything in it, including the picture, and say, “Behold. A picture of the room and everything in it.” Yes, as long as you are not in the room. But if the room is all of reality, you cannot get outside it, and so you cannot draw such a picture.

Second, the man in the room can draw the room, the desk and himself, and draw a smudge on the center of the picture of the desk, and say, “Behold. A smudged drawing of the room and everything in it, including the drawing.” But one only imagines a picture of the drawing underneath the smudge: there is actually no such drawing in the picture of the room, nor can there be.

In the same way, we can fully understand some local situation, from outside that situation, or we can have a smudged understanding of the whole situation, but there cannot be any detailed understanding of the whole situation underneath the smudge.

I noted that I disagreed with Lawson’s attempt to resolve the question of truth. I did not go into detail, and I will not, as the book is very long and an adequate discussion would be much longer than I am willing to attempt, at least at this time, but I will give some general remarks. He sees, correctly, that there are problems both with saying that “truth exists” and that “truth does not exist,” taken according to the usual concept of truth, but in the end his position amounts to saying that the denial of truth is truer than the affirmation of truth. This seems absurd, and it is, but not quite so much as appears, because he does recognize the incoherence and makes an attempt to get around it. The way of thinking is something like this: we need to avoid the concept of truth. But this means we also need to avoid the concept of asserting something, because if you assert something, you are saying that it is true. So he needs to say, “assertion does not exist,” but without asserting it. Consequently he comes up with the concept of “closure,” which is meant to replace the concept of asserting, and “asserts” things in the new sense. This sense is not intended to assert anything at all in the usual sense. In fact, he concludes that language does not refer to the world at all.

Apart from the evident absurdity, exacerbated by my own realist description of his position, we can see from the general account of self-reference why this is the wrong answer. The man in the room might start out wanting to draw a picture of the room and everything in it, and then come to realize that this project is impossible, at least for someone in his situation. But suppose he concludes: “After all, there is no such thing as a picture. I thought pictures were possible, but they are not. There are just marks on paper.” The conclusion is obviously wrong. The fact that pictures are things themselves does prevent pictures from being exhaustive pictures of themselves, but it does not prevent them from being pictures in general. And in the same way, the fact that we are part of reality prevents us from having an exhaustive understanding of reality, but it does not prevent us from understanding in general.

There is one last temptation in addition to the two ways discussed above of saying that there can be an exhaustive drawing of the room and the picture. The room itself and everything in it, is itself an exhaustive representation of itself and everything in it, someone might say. Apart from being an abuse of the word “representation,” I think this is delusional, but this a story for another time.

Consistency and Reality

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Consistency and inconsistency, in their logical sense, are relationships between statements or between the parts of a statement. They are not properties of reality as such.

“Wait,” you will say. “If consistency is not a property of reality, then you are implying that reality is not consistent. So reality is inconsistent?”

Not at all. Consistency and inconsistency are contraries, not contradictories, and they are properties of statements. So reality as such is neither consistent nor inconsistent, in the same way that sounds are neither white nor black.

We can however speak of consistency with respect to reality in an extended sense, just as we can speak of truth with respect to reality in an extended sense, even though truth refers first to things that are said or thought. In this way we can say that a thing is true insofar as it is capable of being known, and similarly we might say that reality is consistent, insofar as it is capable of being known by consistent claims, and incapable of being known by inconsistent claims. And reality indeed seems consistent in this way: I might know the weather if I say “it is raining,” or if I say, “it is not raining,” depending on conditions, but to say “it is both raining and not raining in the same way” is not a way of knowing the weather.

Consider the last point more precisely. Why can’t we use such statements to understand the world? The statement about the weather is rather different from statements like, “The normal color of the sky is not blue but rather green.” We know what it would be like for this to be the case. For example, we know what we would expect if it were the case. It cannot be used to understand the world in fact, because these expectations fail. But if they did not, we could use it to understand the world. Now consider instead the statement, “The sky is both blue and not blue in exactly the same way.” There is now no way to describe the expectations we would have if this were the case. It is not that we understand the situation and know that it does not apply, as with the claim about the color of the sky: rather, the situation described cannot be understood. It is literally unintelligible.

This also explains why we should not think of consistency as a property of reality in a primary sense. If it were, it would be like the color blue as a property of the sky. The sky is in fact blue, but we know what it would be like for it to be otherwise. We cannot equally say, “reality is in fact consistent, but we know what it would be like for it to be inconsistent.” Instead, the supposedly inconsistent situation is a situation that cannot be understood in the first place. Reality is thus consistent not in the primary sense but in a secondary sense, namely that it is rightly understood by consistent things.

But this also implies that we cannot push the secondary consistency of reality too far, in several ways and for several reasons.

First, while inconsistency as such does not contribute to our understanding of the world, a concrete inconsistent set of claims can help us understand the world, and in many situations better than any particular consistent set of claims that we might currently come up with. This was discussed in a previous post on consistency.

Second, we might respond to the above by pointing out that it is always possible in principle to formulate a consistent explanation of things which would be better than the inconsistent one. We might not currently be able to arrive at the consistent explanation, but it must exist.

But even this needs to be understood in a somewhat limited way. Any consistent explanation of things will necessarily be incomplete, which means that more complete explanations, whether consistent or inconsistent, will be possible. Consider for example these recent remarks of James Chastek on Gödel’s theorem:

1.) Given any formal system, let proposition (P) be this formula is unprovable in the system

2.) If P is provable, a contradiction occurs.

3.) Therefore, P is known to be unprovable.

4.) If P is known to be unprovable it is known to be true.

5.) Therefore, P is (a) unprovable in a system and (b) known to be true.

In the article linked by Chastek, John Lucas argues that this is a proof that the human mind is not a “mechanism,” since we can know to be true something that the mechanism will not able to prove.

But consider what happens if we simply take the “formal system” to be you, and “this formula is unprovable in the system” to mean “you cannot prove this statement to be true.” Is it true, or not? And can you prove it?

If you say that it is true but that you cannot prove it, the question is how you know that it is true. If you know by the above reasoning, then you have a syllogistic proof that it is true, and so it is false that you cannot prove it, and so it is false.

If you say that it is false, then you cannot prove it, because false things cannot be proven, and so it is true.

It is evident here that you can give no consistent response that you can know to be true; “it is true but I cannot know it to be true,” may be consistent, but obviously if it is true, you cannot know it to be true, and if it is false, you cannot know it to be true. What is really proven by Gödel’s theorem is not that the mind is not a “mechanism,” whatever that might be, but that any consistent account of arithmetic must be incomplete. And if any consistent account of arithmetic alone is incomplete, much  more must any consistent explanation of reality as a whole be incomplete. And among more complete explanations, there will be some inconsistent ones as well as consistent ones. Thus you might well improve any particular inconsistent position by adopting a consistent one, but you might again improve any particular consistent position by adopting an inconsistent one which is more complete.

The above has some relation to our discussion of the Liar Paradox. Someone might be tempted to give the same response to “tonk” and to “true”:

The problem with “tonk” is that it is defined in such a way as to have inconsistent implications. So the right answer is to abolish it. Just do not use that word. In the same way, “true” is defined in such a way that it has inconsistent implications. So the right answer is to abolish it. Just do not use that word.

We can in fact avoid drawing inconsistent conclusions using this method. The problem with the method is obvious, however. The word “tonk” does not actually exist, so there is no problem with abolishing it. It never contributed to our understanding of the world in the first place. But the word “true” does exist, and it contributes to our understanding of the world. To abolish it, then, would remove some inconsistency, but it would also remove part of our understanding of the world. We would be adopting a less complete but more consistent understanding of things.

Hilary Lawson discusses this response in Closure: A Story of Everything:

Russell and Tarski’s solution to self-referential paradox succeeds only by arbitrarily outlawing the paradox and thus provides no solution at all.

Some have claimed to have a formal, logical, solution to the paradoxes of self-reference. Since if these were successful the problems associated with the contemporary predicament and the Great Project could be solved forthwith, it is important to briefly examine them before proceeding further. The argument I shall put forward aims to demonstrate that these theories offer no satisfactory solution to the problem, and that they only appear to do so by obscuring the fact that they have defined their terms in such a way that the paradox is not so much avoided as outlawed.

The problems of self-reference that we have identified are analogous to the ancient liar paradox. The ancient liar paradox stated that ‘All Cretans are liars’ but was itself uttered by a Cretan thus making its meaning undecidable. A modern equivalent of this ancient paradox would be ‘This sentence is not true’, and the more general claim that we have already encountered: ‘there is no truth’. In each case the application of the claim to itself results in paradox.

The supposed solutions, Lawson says, are like the one suggested above: “Just do not use that word.” Thus he remarks on Tarski’s proposal:

Adopting Tarski’s hierarchy of languages one can formulate sentences that have the appearance of being self-referential. For example, a Tarskian version of ‘This sentence is not true’ would be:

(I) The sentence (I) is not true-in-L.

So Tarski’s argument runs, this sentence is both a true sentence of the language meta-L, and false in the language L, because it refers to itself and is therefore, according to the rules of Tarski’s logic and the hierarchy of languages, not properly formed. The hierarchy of languages apparently therefore enables self-referential sentences but avoids paradox.

More careful inspection however shows the manoeuvre to be engaged in a sleight of hand for the sentence as constructed only appears to be self-referential. It is a true sentence of the meta-language that makes an assertion of a sentence in L, but these are two different sentences – although they have superficially the same form. What makes them different is that the meaning of the predicate ‘is not true’ is different in each case. In the meta-language it applies the meta-language predicate ‘true’ to the object language, while in the object language it is not a predicate at all. As a consequence the sentence is not self-referential. Another way of expressing this point would be to consider the sentence in the meta-language. The sentence purports to be a true sentence in the meta-language, and applies the predicate ‘is not true’ to a sentence in L, not to a sentence in meta-L. Yet what is this sentence in L? It cannot be the same sentence for this is expressed in meta-L. The evasion becomes more apparent if we revise the example so that the sentence is more explicitly self-referential:

(I) The sentence (I) is not true-in-this-language.

Tarski’s proposal that no language is allowed to contain its own truth-predicate is precisely designed to make this example impossible. The hierarchy of languages succeeds therefore only by providing an account of truth which makes genuine self-reference impossible. It can hardly be regarded therefore as a solution to the paradox of self-reference, since if all that was required to solve the paradox was to ban it, this could have been done at the outset.

Someone might be tempted to conclude that we should say that reality is inconsistent after all. Since any consistent account of reality is incomplete, it must be that the complete account of reality is inconsistent: and so someone who understood reality completely, would do so by means of an inconsistent theory. And just as we said that reality is consistent, in a secondary sense, insofar as it is understood by consistent things, so in that situation, one would say that reality is inconsistent, in a secondary sense, because it is understood by inconsistent things.

The problem with this is that it falsely assumes that a complete and intelligible account of reality is possible. This is not possible largely for the same reasons that there cannot be a list of all true statements. And although we might understand things through an account which is in fact inconsistent, the inconsistency itself contributes nothing to our understanding, because the inconsistency is in itself unintelligible, just as we said about the statement that the sky is both blue and not blue in the same way.

We might ask whether we can at least give a consistent account superior to an account which includes the inconsistencies resulting from the use of “truth.” This might very well be possible, but it appears to me that no one has actually done so. This is actually one of Lawson’s intentions with his book, but I would assert that his project fails overall, despite potentially making some real contributions. The reader is nonetheless welcome to investigate for themselves.

The List of All True Statements

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Earlier we noted that it is not possible to have a correct and an extremely detailed prediction of the future, if the prediction is known to people.

In a similar way, there are often restrictions on the possibility of bringing together all true statements of a certain form. This is frequently independent of whether people know of them or not.

For example, consider the library of Babel. A great deal is contained in it, including every sentence in this blog post, as you can discover by testing the site’s search function. Suppose someone were to go through these texts and make a list of all the true ones, avoiding meaningless, indeterminate, and false ones. In time, he would come upon this page, which has the sentence, “this sentence is not contained in your list of true sentences.” Should he include this sentence, or not?

If he does not include the sentence, he will have missed a true sentence. And if he does include it, he will have included a false one. So it is impossible in principle for his project to succeed: he cannot make a list of all the true sentences in the library.

This is related to the Paradox of the Liar, but it is not the same thing. Whether or not a sentence is contained in someone’s list is a perfectly objective fact. It is either there, or not, and there is no inconsistency in reality on this account. But the list maker still cannot succeed. In practice this is a case of the Liar Game.

 

 

Liar Game

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While this is the name of a certain story, it is also the name I am giving to the game I am about to propose. The rules are that I propose a certain number of statements, and the player has to categorize them as true or false. The player wins if all of them are correctly categorized, and fails if he does not categorize them all, or if he mistakenly categorizes a true statement as false, or a false statement as true. It is against the rules for him to place a statement in both categories.

The statements I propose are the following:

  1. 2+2=4.
  2. 2+2=5.
  3. The player will categorize this as false.

It can easily be seen that the player is guaranteed to lose the game. If the player does not categorize the third statement, then it is false, and he has failed to categorize them all. On the other hand, if he categorizes it as true, it is false, and if he categorizes it as false, it is true. In any case either he fails to categorize it, or he categorizes it incorrectly.

It is evident that this is related to the paradox of the Liar, but there is a significant difference. The original liar statement is paradoxical, in the sense that applying the ordinary rules of logic results in a contradiction regardless of whether one considers the statement to be true or false.

This is not the case here. There is nothing paradoxical about the statement, in this sense. Given an actual player and an actual instance of playing the game, the statement will plainly be true or false in an objective sense, and without any contradiction being implied. It is just that the player cannot possibly categorize it correctly, since its truth is correlated with the player categorizing it as false.

 

The Liar

entirelyuseless Logic, Metaphysics , , ,

The paradox of the Liar is a logical problem that results from a sentence that implies that the very sentence itself is false, or at least that it is not true. Consider the following statement:

(1) Statement (1) is not true.

Is statement (1) true or not? We might reason about it as follows.

(2) If statement (1) is true, then statement (1) is not true, since this is what it says.

(3) But this is absurd, since statement (1) would then be both true and not true.

(4) Therefore (1) is not true.

(5) But this is just what (1) says. So (1) is true.

And so on. There does not appear any way to avoid the conclusion that (1) is both true and not true, which is a contradiction. Nor is it helpful to say that it is neither true nor not true, since the same contradiction will follow: if something fails to be not true, it is surely true.

Any statement whatever will follow from a contradiction, so if one accepts this contradiction, one will be forced to accept that every statement is both true and false.

A. N. Prior discusses the idea of an analytically valid inference:

It is sometimes alleged that there are inferences whose validity arises solely from the meanings of certain expressions occurring in them. The precise technicalities employed are not important, but let us say that such inferences, if any such there be, are analytically valid.

One sort of inference which is sometimes said to be in this sense analytically valid is the passage from a conjunction to either of its conjuncts, e.g., the inference ‘Grass is green and the sky is blue, therefore grass is green’. The validity of this inference is said to arise solely from the meaning of the word ‘and’. For if we are asked what is the meaning of the word ‘and’, at least in the purely conjunctive sense (as opposed to, e.g., its colloquial use to mean ‘and then’), the answer is said to be completely given by saying that (i) from any pair of statements P and Q we can infer the statement formed by joining P to Q by ‘and’ (which statement we hereafter describe as ‘the statement P-and-Q’), that (ii) from any conjunctive statement P-and-Q we can infer P, and (iii) from P-and-Q we can always infer Q. Anyone who has learnt to perform these inferences knows the meaning of ‘and’, for there is simply nothing more to knowing the meaning of ‘and’ than being able to perform these inferences.

A doubt might be raised as to whether it is really the case that, for any pair of statements P and Q, there is always a statement R such that given P and given Q we can infer R, and given R we can infer P and can also infer Q. But on the view we are considering such a doubt is quite misplaced, once we have introduced a word, say the word ‘and’, precisely in order to form a statement R with these properties from any pair of statements P and Q. The doubt reflects the old superstitious view that an expression must have some independently determined meaning before we can discover whether inferences involving it are valid or invalid. With analytically valid inferences this just isn’t so.

I hope the conception of an analytically valid inference is now at least as clear to my readers as it is to myself. If not, further illumination is obtainable from Professor Popper’s paper on’ Logic without Assumptions’ in Proceedings of the Aristotelian Society for 1946-7, and from Professor Kneale’s contribution to Contemporary British Philosophy, Volume III. I have also been much helped in my understanding of the notion by some lectures of Mr. Strawson’s and some notes of Mr. Hare’s.

He proceeds to draw some conclusions from this:

I want now to draw attention to a point not generally noticed, namely that in this sense of ‘analytically valid’ any statement whatever may be inferred, in an analytically valid way, from any other. ‘2 and 2 are 5’, for instance, from ‘2 and 2 are 4 ‘. It is done in two steps, thus:

2 and 2 are 4.

Therefore, 2 and 2 are 4 tonk 2 and 2 are 5.

Therefore, 2 and 2 are 5.

There may well be readers who have not previously encountered this conjunction ‘tonk’, it being a comparatively recent addition to the language; but it is the simplest matter in the world to explain what it means. Its meaning is completely given by the rules that (i) from any statement P we can infer any statement formed by joining P to any statement Q by ‘tonk’ (which compound statement we hereafter describe as’ the statement P-tonk-Q ‘), and that (ii) from any ‘contonktive’ statement P-tonk-Q we can infer the contained statement Q.

A doubt might be raised as to whether it is really the case that, for any pair of statements P and Q, there is always a statement R such that given P we can infer R, and given R we can infer Q. But this doubt is of course quite misplaced, now that we have introduced the word ‘tonk’ precisely in order to form a statement R with these properties from any pair of statements P and Q.

As a matter of simple history, there have been logicians of some eminence who have seriously doubted whether sentences of the form ‘P and Q’ express single propositions (and so, e.g., have negations). Aristotle himself, in De Soph. Elench. 176 a 1 ff., denies that ‘Are Callias and Themistocles musical?’ is a single question; and J. S. Mill says of ‘Caesar is dead and Brutus is alive’ that ‘we might as well call a street a complex house, as these two propositions a complex proposition’ (System of Logic I, iv. 3). So it is not to be wondered at if the form ‘P tonk Q’ is greeted at first with similar scepticism. But more enlightened views will surely prevail at last, especially when men consider the extreme convenience of the new form, which promises to banish falsche Spitzfindigkeit from Logic for ever.

His point is quite clear. Given the way the word “tonk” is defined, one cannot avoid drawing all possible conclusions. But this means the word “tonk”, defined in this way, is quite unacceptable in the first place.

If we define the word “true” by saying that “P is true” is a statement such that it necessarily follows from P, and such that P necessarily follows from “P is true,” and we consider this an acceptable definition, then the rules of logic will force us to accept all possible conclusions.

Like the definition of the word “tonk”, therefore, this definition of the word “true” is unacceptable, and in the same sense, namely that if the definition is accepted, all possible conclusions follow.

This explains why all solutions to the Liar paradox seem to fail, in the sense that in the end either they admit a contradiction, or they insist that we change the meaning of our language, as for example by talking about levels of truth and so on. For despite the consequences, the word “true” does basically have the meaning stated. The only real difference in comparison with the word “tonk” is that the latter word would never be used in any real language, because the consequences are obvious. In the case of “true,” the consequences are subtle and only follow in special circumstances, namely the kind that are found in the case of the Liar paradox, and so the word could be incorporated into human language, and basically with this meaning, before the implications were noticed.

Note that this is quite different from saying that the word “true” has an inconsistent meaning. The problem is even deeper than that. We could define the word “zrackled” to mean “white and not white in the same respect,” and the meaning would be inconsistent. The only consequence would be that nothing is zrackled, and no contradiction would follow. But if we said that “true” has an inconsistent meaning, and consequently that nothing is true, it would follow from the meaning stated that “nothing is true” is true, and consequently that “nothing is true” is not true. The problem is that we are attempting to define the word, at least in part, by certain rules of usage, and those rules themselves force a contradiction, and ultimately force one to draw all possible conclusions, as with the word “tonk.”