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:
I see a few problems with your argument against the liar's paradox.
First, it seems to beg the question. That is, your argument, if I understand it correctly, merely states that if the liar's paradox is sound then the law of non-contradiction is false. But, you claim, the law is not false. So, the paradox is not sound. Although your argument is valid, it's not a legitimate way out since the paradox's very purpose is to question the law. To merely restate the law only amounts to fist pounding.
Now, you might respond by saying that this is not really your argument. Rather your argument is that the liar's sentence is meaningless. This brings me to the second problem. Your argument that the liar's sentence is meaningless seems to be that it violates the law of non-contradiction. However, meaningless sentences can't violate the law. Take your example of a meaningless statement "Polycarp butter up vacuum establishing." This statement doesn't and can't violate the law of non-contradiction because it is indeed meaningless. It is absolutely devoid of any content. It neither affirms nor denies anything. On the other hand, the liar's sentence isn't devoid of content. In fact, you even concede that liar's sentence has content when you break it down and explain it further. In fact, you even state "that is pretty much all the _content_ that is in the sentence." But, of course, if the sentence has content, it's not meaningless.
Furthermore, if your argument worked, it seems that it would make it such that every contradiction is meaningless instead of necessarily false.
What do you think?
To which I responded:
Your comments point out a legitimate point of lack of clarity in my reasoning, or at least the articulation of my reasoning. So let me try to address that.
One problem with a sentence like "This sentence is false" is that it is not putting forward and articulating all of the claims it is making. There is a concealed claim in the sentence, which is the claim that "this sentence is true." The sentence sounds like it is making only one claim--asserting the falseness of the sentence--but, by virtue of the fact that the sentence is intended as a propositional TRUTH claim (I don't know how to do italics in Facebook), it is making another claim as well--asserting the truth of the sentence.
Therefore, in order to understand the sentence fully, we must bring out that concealed truth claim. Here are two possible revised sentences that result from bringing out that concealed claim:
1. This sentence is both true and false.
2. This sentence is true/false (where "true/false" is taken as one single state of affairs)
Both 1 and 2 are actually the same statement, but they are phrased in different ways verbally to the extent that our response to them might be different. With regard to #1, we could say that the sentence is false, on the grounds that no sentence can be both true and false. By articulating the concealed claim, we have avoided the paradoxical implications of saying the sentence is false. The paradox arose because of verbal confusion arising from an inappropriately articulated sentence with a concealed claim. Once the sentence is restructured verbally to to express its actual claims more clearly and accurately, the verbal paradox disappears, thus dissolving the problem and the argument against the law of non-contradiction based on it.
With regard to #2, we could say that the sentence is meaningless, and thus neither true nor false. We could say this because "true/false" is a meaningless word with no content, along the lines of "non-red red." "True/false" communicates no meaningful state of affairs.
This is where the ambiguity in my articulation of my reasoning came in. I was not adequately distinguishing between 1 and 2, because in my mind they ultimately mean the same thing. The problem with contradictions is that they are ultimately meaningless things that have no content, but I kept switching back and forth between saying that "'A is non-A' is wrong" and saying that "'A/non-A' is meaningless. Both are true and really express the same point, but are different verbally responding to different verbal formulations of what they are responding to. I should have made that clearer. So the original statement, "This sentence is false," could validly be said to be both false and meaningless, depending on how we rephrase it once we un-conceal the concealed additional claim. In no case are we forced to give up bivalence or the law of non-contradiction, because the paradox was the result only of the verbally misleading form of the sentence.
Regarding your point that I seemed to be begging the question in that I seemed to be merely asserting and not proving that the law of non-contradiction (LNC) is true: Actually, that is not the case. I went further and examined the ontological foundation of the LNC. The LNC is really just an implication of the fact of difference, and this fact is one that is known from direct observation. For example, red is different from blue (speaking merely of the colors as perceived). How do I know that? By direct observation. It is immediately evident. The fact that red is different from blue means that where there is red, there is not blue. If exactly where there was red there was also blue, red and blue would have to be the same thing, not different. So difference inherently involves exclusion. And exclusion is nothing other than the LNC.
We can apply this to "truth" and "falsehood". Just as with red and blue, and can see by direct observation that the concept of truth is different from and thus excludes the concept of falsehood. If something is true, to that extent and in that same way, then, it cannot be false. Therefore, a true sentence cannot also be false in the same way, and vice versa.
In short, I am not begging the question when I assert that the LNC is true, because that assertion is based on direct observation of reality and evident truths about it. I know that the LNC is true the same way I know that Aardvarks are not boats--by direct observation of difference/exclusion.
Does that help a bit?
I'm not sure why you say there's a concealed claim in the liar's sentence. It's true that the falsity of the liar's sentence _entails_ its truth. But this certainly doesn't mean that it's asserting or concealing its truth. If it did you'd have to claim that anytime anyone says anything false they're asserting or concealing everything else that's true and false since every truth and falsehood is entailed by something false (i.e., every hypothetical statement that has a false antecedent is true). But that doesn't seem right. So it doesn't seem that the liar's sentence is saying it's true. It says that it's false and nothing more.
But let's suppose it does assert or conceal its truth. I don't see how this makes the sentence meaningless. You want to say it's meaningless because it says of itself that it's both true and false. But that doesn't make it meaningless. If anything, it makes it a contradiction and, thus, necessarily false. But that's the problem! Nothing false is meaningless and nothing meaningless is false. Take your red-blue example. Are you saying that the sentence "X is all red and all blue" is meaningless? That doesn't seem right. It's not meaningless, it's a contradiction. Contradictions aren't meaningless. They're necessarily false. As I stated before, it seems that your position would make all contradictions, narrow and broad, meaningless instead of necessarily false, which would wreak havoc on logic. I think this is just too big of a consequence to pay for a solution to the paradox.
But let's suppose the liar's sentence is, as you argue, meaningless. We could then change the liar's sentence to say "This sentence is true/false, i.e., meaningless." Now, since, ex hypothesi, you've proven it is meaningless, the new sentence turns out to be true. But, of course, if it is then it's not! So ... you're back in the same boat!
1. What I mean by talking about a concealed claim is this: Every propositional truth claim is asserting the truth of what it is asserting. If I say "There is an apple on the table," my sentence could be rephrased thus: "There is an apple on the table and this statement is true." The added part is implicit in the original sentence, because it is a TRUTH claim. Normally, we don't need to bring it out in that way, but the verbally misleading nature of the liar paradox statements make it necessary.
To put it another way, all propositional claims are trying to portray a state of affairs. "There is an apple on the table" is trying to portray a situation where we have a a table, and there is an apple on it. We can imagine this in our minds. So it can help in analyzing a statement to ask what state of affairs it is trying to portray. The statement "This sentence is false" is trying to portray a situation where we have some sentence and it is both true and false. The statement only explicitly mentions its falsehood, but because it is a propositional truth claim the assertion of the truth of the sentence is implicit as well. The whole point of the statement is to say that what it is saying is true, but within the sentence is also the claim that that same statement is false. So the state of affairs conveyed by the sentence is that there is a sentence that is both true and false.
2. Now that we've figured out what the statement is actually trying to say/portray, we can ask, Is it true, or false, or meaningless? We have two verbal options here that are often both legitimate more or less depending on the verbal construction of the sentence we are talking about. Let's take the phrase "A is non-A." Is this statement false or meaningless? We can really say truly that it is both, depending on how we look at it. We can say it is false, because, in fact, A is NOT non-A, so it is false to say it is. On the other hand, why is it false to say that A is non-A. Why couldn't A be non-A? Because it is a matter of evident, observed fact that it isn't. A is simply different from non-A, and thus the two characteristics exclude each other, and thus the same single thing cannot be A and non-A in the same way and to the same extent, and we certainly cannot say that the two characteristics are identical to each other. How do I know that A is not non-A? by direct observation, the same way I know that cats are not dogs, blue is not red, etc. I observe evident differences.
Because A and non-A convey mutually exclusive, because different, characteristics, it is impossible to form a coherent idea of single entity with both characteristics--an A/non-A (or a blue/red, etc.). The very idea of an A/non-A is meaningless, because the word conveys no coherent and thus meaningful idea. The statement "A is non-A" is attempting to convey a state of affairs, and that state of affairs is one where we have an A/non-A. Since this state of affairs is incoherent and meaningless, the statement itself can be said to be meaningless, just as the sentence "I am a glompoti" can be said to be meaningless because "glompoti" has not meaning. There is a difference between "glompoti" and "A/non-A," however, and that is that the former word COULD possibly refer to something coherent since I know less about what it is trying to assert, whereas we know that the latter word COULD NOT refer to something coherent because we know what it is trying to assert and we know that it is an incoherent and thus meaningless idea.
Going back to our liar sentence, "This sentence is false," we can likewise say of the sentence both that it is false and that it is meaningless. It is meaningless because it is trying to portray a state of affairs in which there is a "true/false," which is an incoherent and meaningless concept, since the characteristics of truth and falsehood are observably and evidently different from each other and thus exclude each other and thus cannot coherently be the same or be to the same extent characteristics of the same entity. But we can also say that the statement is false because it is not the case that that sentence is both true and false, or true/false, because no sentence can be true/false, because true/false can't be at all ever. So, again, we can say both that the sentence is false and that it is meaningless. Both are valid things to say about it and express true components of what it is.
The paradox that caused the problems in the beginning was caused by that paradox being embedded in the verbal form of the sentence. We can avoid the confusion simply by rephrasing the sentence to make explicit the concealed truth claim and thus see better what we are dealing with rather than running into it when we don't expect it. Once the confusing verbal form is entangled, the statement is seen to be nothing more than the assertion of a contradiction which has no more magical power to make contradictions true than any other straightforward assertion of a contradiction.
Now, you seem concerned to avoid the conclusion that all contradictions are meaningless. But it is true that, in a sense, all contradictions are meaningless, because all contradictions are incoherent and can refer to no actual idea. That is precisely why they can also be said to be wrong. "A is non-A" can be said to be false precisely because the state of affairs it tries to convey cannot actually be because it is incoherent. I think we're getting tripped up by semantics at this point. You are correct in pointing out that "A is non-A" is in a sense more meaningful than "Ordinary go apple washing nor," in that the former we have a better idea what the former sentence is trying to convey whereas we have no idea what the latter sentence is trying to convey. That is why it is appropriate to say that the former sentence is false but it would not be appropriate to say that about the latter sentence. However, since an A/non-A is supposed to convey a real idea but fails to so so because it is incoherent, it is appropriate to say that in a sense "A is non-A" is also meaningless. When we understand clearly in what sense contradictions can be said to be meaningless and in what sense they can be said to be false, we can see that no havoc is going to be wreaked on logic by acknowledging both appropriately.
"But let's suppose the liar's sentence is, as you argue, meaningless. We could then change the liar's sentence to say "This sentence is true/false, i.e., meaningless." Now, since, ex hypothesi, you've proven it is meaningless, the new sentence turns out to be true. But, of course, if it is then it's not! So ... you're back in the same boat!"
I don't think this is accurate. The statement "This sentence is false" can be said in some sense to be meaningless, in that it is trying to portray a state of affairs as real which actually is incoherent and thus cannot be a real idea or a real thing. But the statement itself is not TRYING to say that it itself is meaningless. I think that "This sentence is meaningless" is a different sentence with different content.
In this case, though, the sentence does not push a paradox because we can just say that it is false without any trouble. The sentence is not meaningless. It is therefore wrong to say that it is. It is not meaningless to call something meaningless. "Meaningless" is a perfectly coherent word with real meaning, unlike true/false or A/non-A.
In conclusion for now, let me go back with all of this in mind and restate my solution to the liar paradox problem:
We have a statement like "This sentence is false." The statement sounds like an ordinary propositional claim but it really is a verbal trick. The statement makes explicit one part of its claims but conceals another part. It is actually an assertion that itself is both true and false. It is saying, "This sentence is false and it is true." By concealing the claim of truth within it, it forces the responder verbally to affirm its own claimed paradox: If we say that the sentence is true, we are agreeing with it that it is false, so it is both true and false. If we say that the sentence is false, we are agreeing with it and thus saying it is true, because it is asserting that it is true that it is false, and thus we are saying it is both true and false. The trick is that the verbal form of the statement forces us to deal only with one part of the whole claim, the explicit part, while not noticing that there is another implicit part of the claim. When we respond with either yea or nay to the explicit part, we then find that we have responded with the opposite to the implicit part. Since the explicit part and the implicit part are contradictory, we are unwittingly tricked into affirming a contradiction.
The proper response, then, is to refuse to play the verbal game and instead to rephrase the sentence to be more clear so that a more accurate response can be made to the whole claim with both its parts. The state of affairs conveyed by the statement is that there is a sentence that is both true and false. Now that the whole claim is out there, we can ask, is this true or false? Well, the claim is basically that there can be a meaningful propositional claim that can be both true and false. But this claim is false, because, in reality, there can be no such true and false propositional claim, because true and false are different and thus mutually exclusive. So our clarified claim can be said to be false.
We can also say, in a sense, that the claim is meaningless, in that it tries to envision a state of affairs that cannot be envisioned because it is essentially incoherent. To make this clearer, think of another claim: "There are square circles." Is this claim true, false, or meaningless. It can be said to be false, because in fact there aren't any square circles. But if we think of what the phrase 'square circle" actually means, we find that it really conveys no coherent idea to the mind and thus can be said in that sense to be meaningless. Looking at it that way, we can say that the claim that there are square circles is meaningless in that no coherent idea is conveyed to the mind by the phrase "square circle." Both are accurate ways of describing the situation, depending on what you want to emphasize. Do you want to emphasize the lack of coherence and thus meaning in a contradictory phrase, or do you want to emphasize that what the phrase is trying to convey, because incoherent, cannot be, and thus the claim that it is is false? Both are true.
At any rate, you were quite right in pointing out to me the inadequacy of my response. I need now to go back in to the article and fix these problems. Perhaps I am still not dealing with the issue adequately, so feel free to keep raising concerns. This has been very helpful thus far!
Would you have any problem with my pasting this conversation into the blog post as part of an addendum, either anonymously or not (and which would you prefer)? I think it would be useful to use this as an illustration of how conversation/dialogue can improve an argument.
Let me point out one more area of concern. I think your argument may also confuse _utterances_ with _assertions_. I can utter a sentence without implying that it is either true or false. However, if I assert something, not only do I utter it, it also seems that I'm saying my utterance is true.
So, for instance, I can utter "You are sitting." But, in doing so, I am not claiming that it is either true or false. I'm merely saying the sentence. Now, it might turn out that my utterance is true or it might turn out that it's false. But regardless of how it turns out, I did not imply either one. On the other hand, if I assert that "You are sitting" then, not only do I utter the sentence, it does seem that I'm also saying that it's true.
You seem to think that the liar's paradox works with the _assertion_ of the liar's sentence. But it need not do so. It could be formulated by merely _uttering_ it. If so formulated, then the liar's sentence would neither imply that it is true nor false. It might turn out that the sentence is true or it might turn out that the sentence is false. But either way, if you merely utter the liar's sentence, you did not imply one way or the other.
So, even if your responses above successfully avoid my objections -- which I don't think they do -- there's still a way to avoid your objection.
Oh..., as for posting this conversation on your blog, I'm fine with that. But I would prefer that it be anonymous. Thanks!
Regardless of whether or not some particular utterer intends to assert his/her utterance as a true assertion, the assertive character of the utterance is inherent in the utterance itself. The statement that "There is an apple on the table" is a propositional truth claim. That is its essential form and meaning. Propositional truth claims inherently make claims/assertions about reality, even if they are uttered by people who are uttering them for some other reason (such as simply reporting what someone else said, or whatever). "There is an apple on the table" means to assert that there is an apple on the table. That is its whole point as a statement, and that assertive essence cannot be removed from it. Thus the claim of the truth of the statement is indeed inherent in every propositional statement.
Thanks for letting me post the conversation (anonymously)!
And that's the whole of the conversation (thus far).
UPDATE 2/15/13: My friend has posted a new response to my argument. Here it is:
One last thing: another way to avoid your objection would be to formulate the paradox without self-referential statements. For instance,
(N) The next sentence is false;
(P) The previous sentence is true.
We can now generate the paradox as follows:
(2) If (N) then (~P)
(3) So, (~P)
(4) If (~P) then (~N)
(5) So, (~N)
(6) If (~N) then (P)
(7) So, (P)
(8) So, [(N)&(~N)] & [(P)&(~P)]
So, even your solution worked, it'd be a hollow victory. You'd only have escaped goblins to be devoured by wolves!
Here was my response to this:
Once we've figured out the trick in this line of argumentation, I don't think it will be difficult to disarm other versions of it. We just need to do the same thing we did the first time: Force clarity to unclear statements and watch what happens.
Let's do that in this case: What is N affirming? N is affirming that P is false. What is P affirming, and thus what does N think is false? P is affirming that N is true. So when N affirms that P is false, this is just a roundabout way of saying that N affirms that N is false. P functions merely as a mirror to N. So really what we have here is the statement from N that "N is false." This is the same self-referential denial we saw in the claim that 'This sentence is false." N, being itself a propositional claim, is saying, "I am false."
The answer, once again, is that when a propositional statement affirms anything, it is also affirming that what it is affirming is true. So, in this case, what we have is a propositional statement affirming (explicitly) that it is false and (implicitly) that it is true. N affirms that it is both true and false? Is this affirmation true or false? I cannot be looked on as true, because true/false cannot be. It can be looked on as false, because true/false cannot be. But no paradox s involved in saying it is false, because it is not agreeing with the statement in its denial of it. The statement affirms that true/false can be, and we are saying, contrary to that, that true/false can't be. Or, we could say that the statement is meaningless and thus neither true nor false, because true/false is meaningless incoherence. Once the basic claim is properly clarified, we see there is no need to affirm a contradiction to respond to it. And so the argument fails as an attempt to refute the LNC.
And, of course, we could follow P instead of N in the argument and arrive at the same conclusion--that P is saying ultimately that P is true and false. Basically, what we have here is simply a bunch of claims that, insofar as they have content, reduce to these: 'N is false and true" and "P is false and true."
And they are both false and meaningless.
Lack of clarity is the hiding place of both goblins and wolves! Avoid the one, and you avoid the others.
UPDATE 3/4/13: My friend has posted another attempt to save the Liar paradox. Here it is:
Hi Mark, I thought I'd throw another curve ball at you. Here's a very interesting version of the liar's paradox from John Pollock:
Suppose there is a special room where the Pope is supposed to go when he speaks _ex cathedra_ and no one else is ever supposed to assert a proposition in that room. However, I sneak into the room and assertively utter "At least one false proposition has been, is being, or will be asserted in this room." All other propositions asserted in that room are true, so if I succeed in asserting a proposition, it is true if [and only if] it is false. The only way to avoid a contradiction is to deny that I assert a proposition, but that seems intuitively unconvincing ("The Liar Strikes Back." _The Journal of Philosophy_. 74.10 (Oct. 1977): 605).
As Pollock notes, your solution just won't work here. It seems to me that in order to avoid the paradox you'd have to argue that such a world is logically impossible. But, of course, on the face of it, it certainly doesn't seem to be. What do you think?
And here is my response:
This example is no different from the previous examples. I find all these examples interesting, because they all find different semantic ways to get back to the same basic idea of a self-refuting statement.
As usual, it helps to remember that a propositional statement is always trying to paint a picture for us of some state of affairs that is asserted to be true. We must ask what that state of affairs is and examine it carefully. In this case, the state of affairs asserted by the proposition you uttered in the pope's room is that of a room in which at least one false statement is made (sometime). The only statements ever uttered in that room are the pope's infallible statements and the one statement from you. So we must ask, Which statement are you talking about when you say there sometime will be (or has been, etc.) a false statement uttered in the room? If you are talking about one of the pope's statements, then we can simply declare your statement false with no contradiction involved.
If you are referring to your own statement, then your statement is self-refuting. We can, again, restate your statement in this way: "My statement is false and my statement is true." Now we are back to where we were in previous examples, and my solution applies.
Either way, there is no need to run into an unsolvable contradiction. The apparent contradiction arises only because of hidden implications not noticed in the scenario and in your statement in it. Once we've looked at the issue with more specificity, we can see that the paradox arises because we know the pope can't be wrong and so it must be you who are wrong, and so your statement is self-refuting. But once we've spelled out all that your statement is saying explicitly--"This statement of mine is false and this statement of mine is true"--we can evaluate it without contradiction. We can say either that it is false--on the grounds that no statement can be both true or false because contradictions are impossible--or we can say that it is meaningless--on the grounds that true/false as a single idea is incoherent and thus meaningless. In neither case do we need to affirm the truth of true/false, and thus the contradiction is avoided.