In my
previous post, I mentioned the philosophical position of
dialetheism--which holds that not all contradictions are impossible. Some contradictions can be true and meaningful.
Of course, one way to establish the truth of this proposition (which wouldn't, of course, mean that the proposition is not also false--sorry, I couldn't resist) would be to provide examples of situations where a contradiction must be true. That is precisely the strategy of an argument called the Liar paradox.
Here's how the Liar paradox works: Any propositional statement--that is, any statement which makes propositional truth claims--must be either true or false. This is the principle of
bivalence. For example, take the statement, "There is an apple on the table." This statement must be either true or false. Either there is an apple on the table or there is not. There is no possible third option.
The paradox then tries to put forward statements that contain inherent contradictions in them. It is called the Liar paradox because one famous propositional statement associated with the argument goes like this: "I am lying." Let's analyze this. It is a propositional statement, so it must be either true or false. Either I am lying or I am not. Let's say the statement is true. I am lying. Well then, what am I lying about? I am lying in saying that I am lying. So it is true that I am lying when I claim that I am lying, which means that I am
not actually lying but rather telling the truth. But if I am telling the truth, then I cannot be lying, which means that my statement "I am lying" is actually false, which means that it is not true that I am lying, which means that I am actually telling the truth, which means that I am lying, which means that I am telling the truth, etc. The statement requires that I am both lying and telling the truth about the very same thing.
Here is another example: "This statement is false." Well, this statement must either be true or false. If it is true, then it must be correct in stating that it is false, which means it can't be true. But if it is false, then it must be the case that it is incorrect in stating that it is false, which means it must be true. So the statement must be both true and false. Both are required. You get the picture.
Since all statements must be either true or false, all of these kinds of paradoxical statements must be either true or false as well. We must assign either "true" or "false" to all of them. But if we assign "true" to any of them, we are required also by logic to assign "false" to them as well, and vice versa. Therefore, we are under an inescapable necessity of saying that these statements are both true and false. Thus, contradictions are an unavoidably real part of reality. The law of non-contradiction, then, which holds that no contradictions can be a real part of reality, must be false.
Now, before we examine this claim made by the Liar paradox argument, I want to say something about how one ought to respond when one encounters an argument like this, even before one knows how to respond to it. In my previous post, I showed how the universality of logic in all of reality is immediately evident from a clear and consistent examination of our basic observations about reality. That logic is a universal aspect of reality is just as evident as 2+2=4, or "something exists." It is impossible that we could be wrong about that, once we understand clearly what we are saying. Therefore, when we are confronted with an argument that is trying to refute what is certainly evident, we should not respond by simply throwing out everything we knew before and embracing it. If the universality of logic is immediately evident and thus certainly true, then we know that something is wrong with the Liar paradox argument. We may not know what in particular is wrong with it, but we know that something must be wrong with it. So what do we do? We continue to hold on to what we already know, and we examine the new argument like crazy to find out where it is going wrong. If we find in doing so that there was something wrong with our previous reasoning, so be it. We change our minds. But, in this case, the evidence that logic is universal is so clear and strong that we can be absolutely confident that this won't happen. That doesn't mean we do not need to examine and find the particular flaws of the Liar paradox argument; it just means that as we do so, we can be confident of success. And we certainly shouldn't cower in fear of asserting the universality of logic--and thus the truth of the law of non-contradiction--in the meantime.
Now, let's examine the Liar paradox. It does seem to have clinched its case. All propositional statements must be either true or false. There certainly are statements that contain inherent contradictions, so that if they are true they must be false and vice versa. It would seem, then, that we are required by clear observation and logic itself to conclude that contradictions are possible and that the law of non-contradiction is thus not true.
But things are not always as they first seem. The problem with Liar paradox statements is that they are playing a verbal trick on those who try to analyze and respond to them.
If we examine more closely the statement, "This statement is false," we find in it both an explicit claim and a concealed claim. The explicit claim is, of course, that the statement is false. The implicit claim is that the statement is true. Every propositional truth claim explicitly or implicitly affirms the truth of itself. This is inherent in the fact that it is intended as a claim of
truth. We can always, if we want to, bring out the implicit claim of truth. For example, we can take our claim, "There is an apple on the table," and rephrase it to make explicit its implicit claim of truth: "There is an apple on the table and this statement is true." Our verbal addition here has actually added nothing that was not already there before. It was already implied. We can do the same sort of rephrasing with our paradoxical statement: "This statement is false and this statement is true." Again, we have added no content to the original statement but merely brought out explicitly what was before implicit in it.
Normally, we don't need to make explicit the implicit claim in a statement that it is a true statement. It would be unnecessarily redundant to do so. In this case, however, there is a trick embedded in leaving the implicit claim implicit. The trick is this: We are given a statement, "This statement is false," and asked to decide whether it is true or false. Our natural reaction is to try to evaluate the statement as it is given, which causes us to respond only to the explicit claim and not notice the implicit claim essentially bound up with it. The explicit claim by itself does not sound like a contradiction, because the contradiction is only apparent when the implicit claim is made explicit. Therefore, we are tricked into responding to the claim as if it is a straightforward, non-contradictory claim. When we try to straightforwardly assign a truth value to it, declaring it either true or false, because the statement contains an inherent contradiction we find that whatever we say we have inevitably affirmed also the opposite of what we have said, thus affirming a contradiction.
This verbal trick is broken when we make explicit the implicit claim: "This statement is false and this statement is true," or "This statement is both true and false." Instead of falling for trying to respond to one of the claims in the statement without noticing the other claim, we can now respond to the entire statement including both its claims.
So, now that the statement we are responding to has been clarified so that we can respond to it more accurately, what is our response to it? Is the statement true or false? One valid response we can make is to say that it is false. "This statement is both true and false" is a false statement--not false
and true, but simply false. We can now say this without affirming a contradiction. It is not the case that the statement is both true and false, because it is the case, alternatively, that the statement is simply false and not true. Affirming that the statement as a whole is simply false does not require me to affirm that it is also true. We fell into that trap when we conceived the claim to be simply that "This statement is false." In that case, if we affirmed it to be false, we were entirely agreeing with it and thus equally affirming it to be true. But once we understand the claim to be not simply that "This statement is false" but that "This statement is
both true
and false," we can affirm the entire statement to be simply false without agreeing with it and thus affirming it to be also true. Making the sentence more clear removes the need to affirm a contradiction in responding to it, thus dissolving the Liar paradox argument.
But let's add a couple of further comments to complete our analysis. For one thing, if we are going to say that "This statement is both true and false" is in fact false, on what grounds can we make this claim? We can make this claim on the grounds that no meaningful propositional statement can be both true and false. But how do we know this? We know this because the words "true" and "false" refer to different ideas, and difference inherently involves exclusion, as I argued in my previous post. To bring this out more clearly once again, let's look at the concepts of
red and
blue. They are different
concepts. Imagine something all-red in your mind. Now, imagine
something all-blue. Now, try to imagine something that is all-red and
partly blue. You can't, can you? (Unless you cheated in some way.)
Why not? Because red and blue, being different, exclude each other. To
the extent and in the place that you have one, you don't have the
other. This is an observationally-evident component of
red and
blue.
It is just as observationally-evident in the case of truth and falsehood. Truth is
different from falsehood, so to the extent and in the place that you have
one, you don't have the other, by necessity.
In the
statement "This statement is both true and false," we have a single claim which tries to make one
what is really two. It tries to portray one "thing"--one statement--as
having two different and exclusionary characteristics--truth and
falsehood--to the same extent and in the same place. It is like trying
to make
red and
non-red--or some specific sort of
non-red like
blue--the
same thing when they clearly are not. I can say the words, "non-red
red," but these words have no actual content. They refer to no actual
meaningful idea. I know what
red is. I know what
blue is. But I have no idea whatsoever what
non-red red or
blue red
is. These words are as meaningless as "ghotrealopony." Phrases like
"red is non-red" or "A is non-A"or "truth is falsehood" are simply meaningless combinations of
words that are attempting to deny evident observed difference-ness in
things. And since this difference-ness is really there in those things,
and cannot be separated from their essences, therefore these phrases
commit the fallacy mentioned in the previous post--the fallacy of
forgetting the essential meanings of the words being used and so
slipping from meaning into meaningless gibberish. Therefore, since it is manifestly not the case that truth and falsehood are the same thing, we can say of the statement "This statement is both true and false" that it is definitely false. No statement can be both true and false, for the same reason no object can be, in the same sense, both all-red and all-blue--because the ideas are different and thus exclude each other.
One more thing is worth noting here, before we conclude: It is fully appropriate to say of the claim "This statement is both true and false" that it is false. But it would also be appropriate to say of the claim that it is neither true nor false because it is really meaningless. It is meaningless in the sense that the state of affairs it is trying to portray--a state of affairs where we have one statement that is both true and false--is incoherent and therefore cannot be a meaningful idea. A "true/false statement" is like a "square circle." Both seem superficially to be conveying some particular idea, but upon closer examination it turns out that both are conveying no idea at all because they are incoherent. So is the claim "There are square circles" false or meaningless? Really, the answer is "both," depending on how you want to look at it and what you want to emphasize. If you want to emphasize the fact that, although "square circle" is incoherent, it is an attempt to describe some specific thing, and thus we can say that there are no square circles in reality, you can say that the claim "There are square circles" is false because there aren't. If, on the other hand, you want to emphasize that, although "square circle" is trying to describe some specific thing, it is really incoherent and thus describes nothing, and thus that one cannot properly deny the existence of something whose essence is meaningless gibberish, you can say that the the claim that "There are square circles" is meaningless because "square circle" is a meaningless phrase. Both are correct and point out important truths about the phrase "square circle." The same sorts of things can be said of the claim that "This sentence is both false and true." It is, in different ways, both false and meaningless. But in neither case are we forced to affirm a contradiction.
So, in the end, it turns out that the sorts of paradoxical statements provided in the Liar paradox argument do not at all provide any
examples of necessary contradictions, and thus this argument against the law of non-contradiction doesn't successfully refute it. We ought not to be surprised.
ADDENDUM 2/15/13: The above version of this post was altered from a previous version. I've posted part of the previous version below, along with the conversation with a friend which helped me to revise it to what it is now. If you are interested in how dialogue can help thought progress, or you just find yourself with nothing better to do, feel free to have a look at it.
The second version doesn't really disagree with the first version or correct anything of substance in it. It merely states a few points more clearly and explicitly and points out some things the first post did not. Both versions are, I think, useful in attacking the Liar paradox.
My original post was the same as my current post up until the sentence, "But things are not always as they seem." At this point the two versions diverge. Here is the original post after that point:
But things are not always as they first seem. There is a fatal flaw
in the Liar paradox argument. Can you spot what it is? Before you read
on, try to figure it out. I'll wait . . .
. . .
.
. . OK, did you figure it out? Let's see if you've got the same answer
I am going to suggest. The problem with the Liar paradox argument is
that the paradoxical propositional statements are not true propositional
statements. They look, superficially, like true propositional
statements. But on further investigation it turns out that they are
nothing but meaningless gibberish.
The statement,
"There is an apple on the table," has definite meaning and content. It
is a meaningful statement. However, the phrase, "This sentence is
false," is not a meaningful statement. Why not? To help us to see
this, let's break up the sentence further to see clearly what it is
saying: "This sentence, which, by making it, I am asserting to be true,
I also assert to be false." Of course, the contradiction is in the
part that says that "something true is also false." Or we can express
this component of the sentence by saying, "A is non-A." Really, that is
pretty much all the content that is in the sentence. But let's look at
this phrase, "A is non-A." A and non-A are clearly different things.
They are words with different content. As we noted in my previous post,
difference is really nothing other than exclusion. Since A and non-A
are different ideas, to the extent you have one, you don't have the
other. Let's use a different, more specific example to bring this out
more clearly. Take the concepts of red and blue. They
are different concepts. Imagine something all-red in your mind. Now,
imagine something all-blue. Now, try to imagine something that is
all-red and partly blue. You can't, can you? (Unless you cheated in
some way.) Why not? Because red and blue, being different, exclude
each other. To the extent and in the place that you have one, you don't
have the other. This is an observationally-evident component of red and blue.
It is just as observationally-evident in the case of A and non-A. A is
different from non-A, so to the extent and in the place that you have
one, you don't have the other, by necessity.
In the
statement "A is non-A," we have a single claim which tries to make one
what is really two. It tries to portray one "thing"--one statement--as
having two different and exclusionary characteristics--truth and
falsehood--to the same extent and in the same place. It is like trying
to make red and non-red--or some specific sort of non-red like blue--the
same thing when they clearly are not. I can say the words, "non-red
red," but these words have no actual content. They refer to no actual
meaningful idea. I know what red is. I know what blue is. But I have no idea whatsoever what non-red red or blue red
is. These words are as meaningless as "ghotrealopony." Phrases like
"red is non-red" or "A is non-A" are simply meaningless combinations of
words that are attempting to deny evident observed difference-ness in
things. And since this difference-ness is really there in those things,
and cannot be separated from their essences, therefore these phrases
commit the fallacy mentioned in the previous post--the fallacy of
forgetting the essential meanings of the words being used and so
slipping from meaning into meaningless gibberish.
So
here, in short, is the answer to the Liar paradox argument, and thus a
defense of the law of non-contradiction from it: Statements like "This
sentence is false" do not have to be either true or false because
although they look like true propositional statements, in fact they are
not. They are actually nothing more than a meaningless string of words,
like "Polycarp butter up vacuum establishing." Is "Polycarp butter up
vacuum establishing" true or false? It is neither, not because the
principle of bivalence is false but because it is not a propositional
statement at all and thus the principle of bivalence doesn't apply to
it. Statements like "This sentence is false" fall into exactly the same
category. So these sorts of paradoxical statements do not at all
provide any examples of necessary contradictions. We ought not to be
surprised.
Now, here is the conversation that led me to change it:
My friend responded to my original argument thus:
UPDATE 6/2/23: A few years back, a student of mine named Jack Kiley pointed out in class an even simpler solution to the Liar's Paradox type of argument that had not occurred to me clearly before. Take, again, the sentence, "This sentence is false." The problem of the need to try to figure out if the statement is true or false is avoided simply by noting that that statement has no actual content. The sentence has the form of a propositional statement, but it actually affirms no proposition about reality. It seems to be claiming that something is false, but what is the thing that is claimed to be false? There is no actual what. It may seem, at first glance, that there is a what - the sentence itself. The sentence is an actual thing, and it is claimed to be false. But the claim is not that the sentence does not exist. That would be a meaningful - and false - claim. The claim is that some propositional statement - which, to be a propositional statement, must be a statement about some state of affairs in reality - is false. That is, the sentence is designed to be claiming that some alleged state of affairs in reality is not so. But what state of affairs in reality is actually being claimed to not be so? There isn't any actual state of affairs described in the sentence. The mere words "this sentence" don't describe a state of affairs in reality, and so they cannot be false any more than the word "banana" can be false. Are bananas true or false? This is a meaningless question, because the word "banana" does not describe a state of affairs that can be either true or false. (If we asked whether or not bananas exist, that would be a meaningful question, but it's not the same question.) So a simple answer to the Liar's Paradox argument with reference to this form of it (and this answer will also apply to all forms of it) is that the paradox arises from the illusion that the sentence "This sentence is false" is an actual propositional truth claim when, in reality, it is not. We try to treat it as one, but we are simply confused by the form of the sentence which suggests it is something it is not. Thanks, Jack, for providing yet more clarification on why the Liar's Paradox argument fails to prove the actual existence of contradictory conditions of reality!
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