some thoughts on the so-called “liar's paradox” and a riddle.
Consider the following sentence:
The second sentence of this post is false. The previous sentence is said to engender a paradox because if the sentence is true, i.e. that it is true that the second sentence is false, then it also entails that the sentence cannot be true, for the sentence says of itself that it is false. Conversely, if the sentence is false, i.e. if the second sentence is false in saying that it is false, then the sentence is expressing a truth; but this is impossible because the sentence says that it is false. Nay, paradox is not avoided on either reading.
It has been said that we can avoid the paradox by showing that the second sentence cannot have a truth value in the first place because it fails to intend anything other than it self, i.e. that it does not refer to anything beyond itself. Consider the following sentence:
My bookshelf is false.
We think that such a sentence is non-sense because bookshelves are not the sort of things that can possibly be true or false and this is because bookshelves lack the power of intentionality, they do not ever refer to things. Sentences, on the other hand, do refer to things in virtue of their propositional content. Consider the following sentence:
My bookshelf is shorter than the ceiling.
This sentence refers to something beyond itself and therefore has the ability to be true or false. So what of the second sentence? Isn’t it a sentence and thereby eligible for truthvaluehood? True, the second sentence is a sentence, but because it refers to itself alone and not beyond itself it loses the power to intend, i.e. in failing to refer beyond itself it becomes an object like my bookshelf, and therefore it cannot have a truth value. And so the paradox is solved by showing that purely self-referential sentences cannot be true and false, and therefore are not paradox-eligible.
So what of the famous Cretan who said “all Cretans are liars”? Admittedly, such a Cretan would be expressing a paradox so long as we understand him to mean “all” in an unqualified sense. For if all Cretans are liars and this particular Cretan says “all Cretans are liars” then it means not all Cretans are liars, for this particular Cretan is telling the truth, in which case it would be false that all Cretans are liars, in which case his sentence would be false. The only way out of this paradox, it seems to me, is to say that if it is true that all Cretans are liars, then no Cretan should be able to say so.
And now I’ll end with a riddle. You’re on you way down a mountain and you come to a fork in the road in which there is two paths to take. You know that one path will bring you to your destruction and the other to safety, but you don’t know which. Lo and behold there are twin brothers living in a cabin at the fork in the road and you know that one always tells the truth and the other always lies, and you don't know who is who. What single question could you ask them both that will let you know for certain which road leads where?
The second sentence of this post is false. The previous sentence is said to engender a paradox because if the sentence is true, i.e. that it is true that the second sentence is false, then it also entails that the sentence cannot be true, for the sentence says of itself that it is false. Conversely, if the sentence is false, i.e. if the second sentence is false in saying that it is false, then the sentence is expressing a truth; but this is impossible because the sentence says that it is false. Nay, paradox is not avoided on either reading.
It has been said that we can avoid the paradox by showing that the second sentence cannot have a truth value in the first place because it fails to intend anything other than it self, i.e. that it does not refer to anything beyond itself. Consider the following sentence:
My bookshelf is false.
We think that such a sentence is non-sense because bookshelves are not the sort of things that can possibly be true or false and this is because bookshelves lack the power of intentionality, they do not ever refer to things. Sentences, on the other hand, do refer to things in virtue of their propositional content. Consider the following sentence:
My bookshelf is shorter than the ceiling.
This sentence refers to something beyond itself and therefore has the ability to be true or false. So what of the second sentence? Isn’t it a sentence and thereby eligible for truthvaluehood? True, the second sentence is a sentence, but because it refers to itself alone and not beyond itself it loses the power to intend, i.e. in failing to refer beyond itself it becomes an object like my bookshelf, and therefore it cannot have a truth value. And so the paradox is solved by showing that purely self-referential sentences cannot be true and false, and therefore are not paradox-eligible.
So what of the famous Cretan who said “all Cretans are liars”? Admittedly, such a Cretan would be expressing a paradox so long as we understand him to mean “all” in an unqualified sense. For if all Cretans are liars and this particular Cretan says “all Cretans are liars” then it means not all Cretans are liars, for this particular Cretan is telling the truth, in which case it would be false that all Cretans are liars, in which case his sentence would be false. The only way out of this paradox, it seems to me, is to say that if it is true that all Cretans are liars, then no Cretan should be able to say so.
And now I’ll end with a riddle. You’re on you way down a mountain and you come to a fork in the road in which there is two paths to take. You know that one path will bring you to your destruction and the other to safety, but you don’t know which. Lo and behold there are twin brothers living in a cabin at the fork in the road and you know that one always tells the truth and the other always lies, and you don't know who is who. What single question could you ask them both that will let you know for certain which road leads where?
Labels: liar's paradox
posted by Derek at 12:31 PM
2 Comments:
Derek, I'm glad to see you posting! I am always intrigued by your perspective on things like paradoxes. May I attempt to interact with this one?
*I think you mean "intension," not "intention."
*I would ask either individual, "Which path would your brother recommend I take?"
Your solution to the liars paradox raises several questions.
I am not convinced that every X that lacks any intension outside itself also lacks truth value. After some contentions and other notes on your post, I will offer a counterexample.
The first step in your argument for this claim is that bookshelves cannot have truth values, which is something I am prepared to grant. Then you explain this phenomenon by the fact that the reason why bookshelves cannot have truth values is because they do not intend. I am also prepared to grant this.
Then you entertain the natural objection to the argument so far, which is that sentences can intend (in contradistinction to things like bookshelves), and so escape the conditions that you'd presented that disqualify a thing from the capacity to have a truth value.
But then, out of the blue, you somehow seem to think that you've established that when a thing (in our case a sentence) refers to itself alone, it loses the power to intend (como se dise "non sequitur?")!
You subsequently claim that the paradox is solved by showing this. Suffer me to point out that if that's true, you have yet to solved the paradox.
You've begged the question, "so why, exactly, can't a sentence meaningfully refer to itself?"
You also interacted with the case of the infamous Cretan who claims that all Cretans are liars. I think your answer is intuitive enough given your (yet unsupported) beliefs about the inability for self-referential statements to hold truth values.
But what interests me is that you operate as though your rules for sentences apply to people, such that a sentence that a human uses to refer, not to the sentence itself, but to the man himself, still lacks truth value!
What about the infamous Galilean who says "I am the way, the truth, and the life?"
When all is said and done, I am interested in the direction you are heading - that perhaps there is something about the nature of the Liar's Paradox that prevents it from being handled according to the ways we usually handle sentences.
What do you think? Have I gone wrong (misinterpreting you somewhere or reasoning incorrectly), or is there a way you can strengthen your excursus on the good ole Liar's Paradox?
Louis! Gosh I miss you! Thanks for the wonderful comments.
Concerning my choice of diction: I chose “intention” as opposed to “intension” because I think sentences (or more properly- propositions) are truth-value eligible because they exemplify intentionality- the relational property of referring or being about something besides themselves. The metaphysical relation of intentionality is to be distinguished from both its more common usage in the philosophy of action and ethics (e.g. when A didn’t intend to hurt B) and the logical relation of intension.
Okay now onto the meaty stuff. You object:
“…Then you entertain the natural objection to the argument so far, which is that sentences can intend (in contradistinction to things like bookshelves), and so escape the conditions that you'd presented that disqualify a thing from the capacity to have a truth value.
But then, out of the blue, you somehow seem to think that you've established that when a thing (in our case a sentence) refers to itself alone, it loses the power to intend (como se dise "non sequitur?")!
You subsequently claim that the paradox is solved by showing this. Suffer me to point out that if that's true, you have yet to solved the paradox.
You've begged the question, "so why, exactly, can't a sentence meaningfully refer to itself?"
It seems my conclusions were too quick, for I left out some details. I think sentences exhibit intentionality in virtue of their propositional content, and sentences have propositional content in virtue of being thought of by a mind. Consider again the following sentence:
This sentence is false.
The previous sentence only refers to itself because when a human reads it they are bringing before their mind the self-same sentence they are saying, which is made possible by yet another intention which is referring to the first intention, which is in turn making the original sentence non-intentional. By why of a clarifying (I hope) example, consider my phenomenological reasoning.
Say I begin to say “I am thinking about this sentence.” For me to meaningfully say the previous sentence I had to form at least two intentions. The first began with “I am thinking,” which was an intention that intended nothing specific, and then my mind began adopting a second intention when I said “about” and then came to finality with “this sentence.” The crucial point here is that as soon as I went from one intention to the next, the former intention becomes a non-intentional object; the second intention was directed towards the first, and even though the first was intentional, it no longer is when I’m using a new intention to refer to it. The first intention, since I’m no longer “having it,” is just another object in the world, like my bookshelf.
So let’s go back to the Liar’s Paradox:
The sentence following the previous colon is false. To say such a sentence meaningfully you must form two intentions: the first began with “the sentence”, since when you say “the sentence” you are looking to designate which sentence; once you continue to say “…following the previous colon is false”, you narrow down which sentence you are intending and it just so happens that the sentence which meets the description and context is the self-same sentence you are saying. Once you complete the sentence, you are now referring to the sentence as a single object, and since the sentence is an object you’re referring to, it’s rendered a non-intentional object at that point, and non-intentional objects cannot have truth values.
This is to show that it is minds that intend, and never sentences, although minds use semantics as vehicles of reference. But vehicles are not the objects they carry. Since sentences don’t refer, there can be no self-referential sentences.
Okay. I just realized I have two separate objections, and here goes the second. Pretend (Or continue to believe!) that everything I just said is crap, and sentences do in fact intend. Even so, sentences that do not predicate cannot be true or false. Consider the following:
My bookshelf.
This watch.
That banana.
This sentence.
None of these sentences predicate their direct object with a property or relation, and since they aren’t saying anything about their objects, they cannot be said to be true or false. The second sentence in this paragraph is false. The previous sentence does not predicate, and therefore cannot be true or false. And paradoxes cannot be generated without having a truth-value.
-I do not think my objection to the “the …. sentence is false” paradox works against the Cretan paradox, and this is because the Cretan, unlike sentences, can index himself. I concede that the Cretan Liar’s Paradox is one.
- When Jesus said I “… am the Truth” he didn’t mean he was the relation which holds between a correct belief I have and the world which that belief represents. He makes it possible that I am and have my being and have my correct beliefs, but he is not literally a piece in that relata.
-Did you heard of the riddle before? How long did it take you to figure out?
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