Is Truth an Inconsistent Concept?
I want to briefly consider Kevin Scharp's view on Truth from his book "Replacing Truth". I am about halfway through it, so the views I am looking at here concern his diagnoses of the problem rather than his solution.
Briefly, the concept 'Truth' is inconsistent and must be replaced for scientific and philosophical purposes. The concept works quite well in most cases, but we run into problems for certain scientific/philosophical reasons. Scharp uses an analogy from science to describe his view: the concept 'mass' works quite well for most of our purposes. However, the concept was inconsistent and replaced with two other concepts: proper mass and relativistic mass.
So the first question is this: why should we think that the concept of Truth is inconsistent?
The answer to this question is the liar paradox, which shows that there is a deep issue with the concept.
First, we need to outline the constitutive principles of Truth. Scharp argues that Truth has two such principles:
(1) If P is true then P (T-out)
(2) If P then P is true (T-in, also known as semantic ascent).
The argument is this: any competent user of the concept True possesses these constitutive principles, i.e., has a warrant to believe they are both true. However, the liar paradox shows that they are inconsistent (so, in fact, they can't both be true, despite the fact that we are justified in believing them). So let's see how the Liar Paradox shows this.
Consider the following sentence (Li): (Li) is not true. (Sorry has to use Li instead of L because it renders as an emoji). This is called the liar sentence and can be formulated in any natural language that has a truth predicate.
So let's consider what follows.
1. Assume (Li) is true
2. Then '(Li) is not true' is true (substitution).
3. Then (Li) is not true (T-out)
4. Then (Li) is true and (Li) is not true (from 1 and 3. Contradiction)
5. Then (Li) is not true (reductio 1-4)
6. Then '(Li) is not true' is true (T-in)
7. Then (Li) is true (substitution)
8. Then (Li) is true and (Li) is not true (from 5 and 7. Contradiction).
And so we get an outright contradiction. The Liar paradox shows that truth's constitutive principles are inconsistent (this Scharp calls the Obvious Argument).
Now I believe one of the central issues Scharp outlines is that the science of Linguistics uses truth-conditional semantics for natural languages. Since liar sentences can be formed in natural languages, then the linguist must provide a semantics for these sentences (on the assumption they are meaningful). But we cannot give such a semantics for such sentences, despite their being meaningful. This is a reason we need an alternative to the concept.
Thoughts?
Briefly, the concept 'Truth' is inconsistent and must be replaced for scientific and philosophical purposes. The concept works quite well in most cases, but we run into problems for certain scientific/philosophical reasons. Scharp uses an analogy from science to describe his view: the concept 'mass' works quite well for most of our purposes. However, the concept was inconsistent and replaced with two other concepts: proper mass and relativistic mass.
So the first question is this: why should we think that the concept of Truth is inconsistent?
The answer to this question is the liar paradox, which shows that there is a deep issue with the concept.
First, we need to outline the constitutive principles of Truth. Scharp argues that Truth has two such principles:
(1) If P is true then P (T-out)
(2) If P then P is true (T-in, also known as semantic ascent).
The argument is this: any competent user of the concept True possesses these constitutive principles, i.e., has a warrant to believe they are both true. However, the liar paradox shows that they are inconsistent (so, in fact, they can't both be true, despite the fact that we are justified in believing them). So let's see how the Liar Paradox shows this.
Consider the following sentence (Li): (Li) is not true. (Sorry has to use Li instead of L because it renders as an emoji). This is called the liar sentence and can be formulated in any natural language that has a truth predicate.
So let's consider what follows.
1. Assume (Li) is true
2. Then '(Li) is not true' is true (substitution).
3. Then (Li) is not true (T-out)
4. Then (Li) is true and (Li) is not true (from 1 and 3. Contradiction)
5. Then (Li) is not true (reductio 1-4)
6. Then '(Li) is not true' is true (T-in)
7. Then (Li) is true (substitution)
8. Then (Li) is true and (Li) is not true (from 5 and 7. Contradiction).
And so we get an outright contradiction. The Liar paradox shows that truth's constitutive principles are inconsistent (this Scharp calls the Obvious Argument).
Now I believe one of the central issues Scharp outlines is that the science of Linguistics uses truth-conditional semantics for natural languages. Since liar sentences can be formed in natural languages, then the linguist must provide a semantics for these sentences (on the assumption they are meaningful). But we cannot give such a semantics for such sentences, despite their being meaningful. This is a reason we need an alternative to the concept.
Thoughts?
Comments (198)
Nope.
If I say, "This sentence is false." What do I mean? That's not even proper English. Its like saying, "This sentence is run" is true. The sentence doesn't convey any truth to begin with so we can't label it true or false. Do something like, "The existence of this sentence is false", and that makes more sense.
The problem isn't truth. Its applying "truth" to something that doesn't make any sense to begin with.
We're skipping a step though, aren't we? Even if we're not going to reach for a separate criterion of meaningfulness (not relying on truth conditions), we have to argue for the meaningfulness of gaps (won't take a truth value) and gluts (takes too many, like the Liar).
What is the mistake in this inference?
Yes, this is another response to the liar paradox! As you say, maybe the Liar Sentence is meaningless and if that is the case it simply doesn't have truth-conditions and so no harm is done.
Still, I do think the claim that (Li) is meaningless is very counterintuitive. This is not to say that the claim is false, of course, but only that there needs to be stronger arguments for this claim. I dispute the fact that it isn't proper English, as you say. In what sense? It seems grammatical, and its constituent terms are all meaningful. Intuitively, if I say "That sentence is false", referring to some other sentence P, then I know precisely what that means. If I say "This sentence contains five words" this is perfectly meaningful (and true). So the problem is not self-reference. So if the issue isn't grammar, the meaning of constituent terms or self-reference, why think the sentence isn't meaningful? It is not analogous to the other sentences you use. For example "This sentence is run" is, of course, not meaningful because it is not grammatical. (Li) is a well-formed sentence of English.
I can be convinced otherwise, but so far I don't see a reason to accept the counterintuitive position that (Li) is not meaningful.
You are right. I know that Scharp argues against third-value semantics to solve the Liar Paradox because it falls prey to revenge paradoxes, which are essentially just Liar Paradoxes for languages with three+ value semantics. This is not to say that a three-value+ semantics isn't appropriate for natural languages, only that the Liar Paradox can be re-crafted, so we're not out of the woods.
Thanks! I honestly don't know much about this point so I will check out this video.
Here are the lyrics to a song I loved as a teenager:
https://genius.com/Avril-lavigne-sk8er-boi-lyrics
The lyrics are loopy in the sense that she is singing about a situation ("a song she wrote about a skater boy") but the song she wrote is the song... eh I can't do this. It's just loopy! I too am interested in what we can gather from loopy logic
What makes it a paradox is, as you say, it appears analogous to many other sentences that are just fine, but when we try to assign it a truth value, something goes wrong; we have two ways of deciding whether the Liar is meaningful and they give different answers.
I suggested we need an argument for why we should consider it meaningful. (I was thinking of the analogous efforts in support of treating questions, commands and the like -- over on the 'gap' side' -- as meaningful, despite starting from a model clearly designed for simple indicative statements.)
You suggest we need an argument for why we shouldn't treat it as meaningful.
We have an extra layer here now. There are arguments (M) that the Liar is or isn't meaningful; there are also arguments (A) that the Liar should or shouldn't be assumed to be meaningful, so that a convincing argument is required to overcome this assumption. A convincing M-argument would allow you to ignore the A-arguments, but we already know we're in paradox land and the Liar comes equipped with arguments on both sides. I'm not much moved by the 'apparently meaningful' argument, but I have to admit that many people are, you among them, so there's some reason to think the arguments against meaningfulness aren't that strong, or that the arguments for meaningfulness are just as strong.
So what about the A-arguments? Can we imagine a way to decide whether to assume meaningfulness or not without rehearsing the arguments for and against meaningfulness? What would we need for such a decision?
There's a common denominator between Moore's paradox and the liar. Do you see it too?
It's hard not to see it, but splitting the Liar in half allows you to assign truth values, so you can end up with a sentence that is arguably true but cannot be asserted in the first person in the present tense. (Moore had a nose for the problems indexicals raise.)
If you refer to some other sentence P, then there is the assumption that P can be either true, or false. For it to have that possibility, it must make a claim against reality. If you say, "That sentence is false," and "that" sentence is, "This sentence is false", its still just nonsense.
This sentence is false, does not make any sense. False in what way? Its like me saying, "This true is false". That's a contradiction, and not meaningful in any way.
As a general rule of thumb, if you run into a contradiction, it means you're doing something illogical. I think its easy enough to see that such a sentence is illogical. But feel free to show me otherwise. How can you make such a sentence have actual meaning? How is it not simply saying, "This true is false"?
I was thinking more along the lines of what one would say if they knowingly believed a falsehood, which of course cannot happen.
Why not? You are just restating your position, but you are not giving reasons for it. What the liar sentence claims is the truth value of a sentence, which all natural and many formal languages are equipped to do.
Quoting Philosophim
You can state, seemingly unproblematically, "X is false" for any number of X, including X that are sentences of a language. Why does it not make sense in this case? Again, I understand why you want to reach this conclusion, but you are not giving any reasons.
L = Liar sentence = This sentence is false
P = L is a proposition (i.e. it has a truth value)
1. If P Then either L Or ~L....premise
2. If L Then ~L........................the liar
3. If ~L Then L........................the liar
4. P..................................assume for reductio ad absurdum
5. L Or ~L................................1, 4 MP
6. L..................................assume for reductio ad absurdum
7. ~L........................................2, 6 MP
8. L & ~L..................................6, 7 Conj
9. ~L........................................6 to 8 reductio ad absurdum
10. ~L.............................assume for reductio ad absurdum
11. L........................................3, 10 MP
12. L & ~L..............................10, 11 Conj
13. L.....................................10 to 12 reductio ad absurdum
14. L & ~L..............................9, 13 Conj
15. ~P....................................4 to 14 reductio ad absurdum
The Liar Sentence isn't a proposition. The concept of truth, whatever it is, is not part of this argument and so, needn't be amended as Kevin Scharp recommends.
What you have here is in the form of a proof but of course it is not a proof because we're not talking about a formal system -- Tarski told us long ago that you just don't include predicates like '... is true' and '... is a proposition' in your language and you're fine. If you want them, they have to be in the next language up, the meta- to this object. What to do when it comes to natural languages -- there's the rub.
And while I sympathize with your approach, what happens with this sentence?
If it's true, then it can't be true because it can't take a truth value. If it's false, then it can take a truth value and so assigning it "false" in all models is fine. So it looks like the predicate '... is a proposition' should be fine and L' is necessarily false. Cool.
But then what about this one?
If it's true, it's either not true or it's not a proposition, so it's not a proposition, so it can't be true. If it's not true, it's true and a proposition.
This is the revenge paradox, and you could do the same thing with '... is meaningful' in my post above. In a sense, even formalizing truth the minimum amount that Scharp describes blocks you from formalizing other semantic predicates like '... is meaningful' or '... is a proposition' on pain of revenge. That would leave you with a formalized truth predicate you can't actually use in a formalized way. On the other hand, if you decide to give up on bivalence and roll the other semantic predicate you want into "truth", so you have three values, then you just get revenge immediately. This is the point @Kornelius is making here.
The Liar always argues its way out of any argumentative box you try to put it in. To deal with it once and for all you have to give up on formalizing your semantics even a little, formalize it differently from the way we have so far (and there are always such proposals around, some of which are nice), or protect the old semantics from having to deal with it in some other way. (Whether this last is even an option is unclear, and it's the approach I was suggesting be explored, just for funsies.)
If P is true, then the proposition ‘P is not true’ is NOT true.
If P = NotP (the liar sentence) then you go into a contradiction, because you postulate a contradiction to start with.
It is fundamentally the same as saying: « Let’s see what happens if we postulate that 1 is different from 1... Oh my god, arithmetics as we know them break down! Therefore arithmetics need to be replace by something else. »
I will respond to the second post you made. I believe you are reading too much into the inference. The inference from 1 to 2 is a simple syntactic substitution. For this reason, it is valid.
Let "The blue dog" be defined by the symbol B. Then we have the following:
1. Assume that: B walked to the park.
2. Then: The blue dog walked to the park (substitution).
Nothing more nefarious is going on in getting 2. from 1.
Notice, too, that a contradiction is in the form [math]P \wedge \neg P[/math]. The liar is not in this form. We need to reason to a contradiction. For this reason, certain minimal logic systems are endorsed as a solution to the paradox since the inferences to the contradiction will turn out to be invalid. But as far as I know, no logic system bars syntactic substitution.
Do you think there is a problem with syntactic substitution? Why would it fail in this case specifically, but work in all others? I mean maybe there is something to this, I just don't see it as harmful because I see the inference as entirely analogous to 1-2 here.
I think it does clarify the issue to separate M-type arguments from A-type arguments. I was largely making an A-type argument. That is, I was arguing without using a specific criterion of meaning. This is just to say that I was arguing by referencing speakers' intuitions and was anticipating with the other side may say to argue it is meaningless (i.e. ungrammatical, issue with self-reference). I think that (Li) is intuitively meaningful in this sense. Most competent speakers of English will not react to (Li) as they would to the sentence "the runner runs runningly run running". (Li) is [I]apparently[/i] meaningful, and I think we should not give up on giving a semantic value for this sentence without good arguments to establish that its apparent meaningfulness is illusory. It is in this sense that I think the burden is on the one who wishes to argue that the sentence is meaningless.
The Liar paradox shows that we cannot give a standard truth-conditional semantics for (Li) because something goes wrong with our concept of Truth. So the position this person is in is now to find an alternative semantics or fix the concept. I think the move to meaninglessness is not well-established by (Li) alone. I think the latter only establishes that the standard truth-conditional account with the standard concept of truth does not work for this sentence. You are right that there are two roads ahead: either it is meaningful or it is not, but the former position does not have the immediate burden of establishing that (Li) is meaningful, really their burden is to show how we can give a semantics for the sentence (and so explain why it is meaningful) or block the Liar paradox in another way (e.g. restricting our logic). In this sense, I think the claim that it is meaningless is counter-intuitive; the burden is on those who favour the latter view to show why it is meaningless without simply citing the Liar Paradox.
So far, the only arguments to that effect have been to claim that (Li) is analogous to obviously meaningless sentences in English (i.e. ungrammatical sentences). But (Li) is perfectly grammatical, so the analogy is a non-starter.
haha, I heard this song as a teenager too but never realized this.
Perhaps I am not following the thread here, but I think the argument you are marking is that (Li) is analogous to "The true is false". But I don't think it is. This is because the latter sentence is not grammatically correct. Truth is a property (a concept) of sentences, not a noun. But maybe I am missing something?
But you are absolutely right to ask about the way in which (Li) could be false. That, I think, is the point of the Liar Paradox. If our standard truth-conditional account doesn't work, then what do we do? Scharp's response is to say that the problem is in the concept of Truth, and he will eventually replace them with ascending truth and descending truth. (Li) then may be ascending false or descending false. I am not sure yet which one. Let me get back to you on that, but it would require some more discussion of how Scharp replaces the concept. Still, the idea here is that though the task is difficult (i.e. establish the way in which the sentence could be false or true), it does not mean that we cannot do it.
Thanks, everyone for your replies! I wasn't anticipating this many responses so quickly :)
No, I am saying: you are starting from an obvious contradiction. Li = not Li. It’s like basing arithmetics on 1=2...
Hmm, I think I understand your sentiment here, but (Li) is actually not the same as not-(Li). So it isn't a contradiction in this sense.
Any sentence P would be an obvious contradiction if it is in the form of, say, Q and not-Q. But (Li) is actually not in this form. To be in this form, (Li) would need to say: "This sentence is not true and it is not the case that this sentence is not true". Then I would agree with you, we just have an outright contradiction.
But this is not the case with (Li). Contradictions take on explicit syntactic forms, and you need to reason your way to this. That is the whole point of the Liar argument. It shows that reasoning from the assumption that (Li) is true leads to an outright contradiction, i.e., we can prove that (Li) and not-(Li) (which is basically what line 8 in the proof of the OP says --- to be more specific, it says something equivalent in the sense that it includes the truth-predicate, but we can get (Li) and not-(Li) by applying T-out on 8, but we don't need to since 8. itself is already a contradiction.)
To make this point a little clearer, imagine I define the following concept Down-True (T) such that:
[math] T(\phi)\rightarrow \phi [/math]
but the following fails (i.e. does not hold):
[math] \phi\rightarrow T(\phi) [/math]
*In fact this is what Kevin Scharp defines as descending truth.
Now I consider the following sentence:
(Li*) This sentence is not down-true
What do you make of it? Does it lead to a contradiction? What I am arguing here is that it is the constitutive principles of truth that make the liar sentence a problem, not the liar sentence itself. That is, there is something inconsistent about the constitutive principles of truth, not (Li) in itself. (Of course we could just say (Li) is fine and restrict our logic, but this also speaks against the idea that (Li) is an outright contradiction).
Natural languages include nonsense. That seems to me to be a good thing. We can formulate the liar paradox, talk about the little man who wasn't there, describe the colour of magic, even state that this piece of bread is the body of god...
Around the edge of the Map of Language, instead of "Here be Dragons", write "Here be nonsense".
I don't have Scharp's book, so I read some reviews. If he can develop a more formal schema by breaking the T-sentence into two sentences moving in opposite directions, we have an interesting formula that might show us something of use. But I'd put money on someone showing how this results in yet another bit of nonsense. Scharp has extended the map a little bit, but the dragons are still there.
You have at least two inferences here. You should make them explicit and state the justification for each step. In particular, your last step is Truth-Out, one of the constitutive principles mentioned in the OP. This is what allows you to make that inference. You have not justified step 1 to 2 yet.
All to say. You are attempting to reason to a contradiction. You didn't actually get to an explicit contradiction (yet-- nowhere do you have (Li) as a statement, for example. You have (Li) is true, but you need an inference move from this to (Li)) and there are other moves you may be missing, but that's not really the point.
What did you think of the Liar involving down-true?
I think you are right: there is definitely nonsense in natural languages and a lot of it. I guess the question is whether the liar sentence itself is nonsense. We can say no to that one, even if we accept that there is a lot of nonsense around.
I'm not so sure. 'Whether an ideal speaker would consider P meaningful' looks to me like a separate criterion of meaningfulness -- that is, not just asking whether P is truth-apt.
Also, '... is meaningful' feels like a weasel-predicate. That is, '... is meaningful' deliberately avoids asking, for instance, 'What does it mean?' or 'What does it say?' Ask an average person about the Liar, and you can expect them to reply, 'Well that's stupid. It doesn't say anything.' (This applies to 'This sentence is true' as well!) Surely if you find a sentence meaningful, you could say what it means. There hasn't been 2000 years of debate over what the Liar means, what it says, because it evidently fails to quite mean anything.
Anyway, for the assumption angle, I was trying to think of something like that principle -- the name escapes me, has to do with entropy -- that there are always more ways of being wrong than right. Not exactly that, but something like it. A justification for assuming that a sentence, even a grammatical sentence, is nonsense until it is demonstrated that you can assign a truth value to it. (If it's a question or a command, say, so you fail to assign a value, then we need convincing arguments to bring them in anyway. For the 'always takes both values' case of the Liar, I can't imagine what a convincing argument would be -- we do generally prefer our sentences to take at most one truth value at a time -- but if there is one, then this approach fails anyway and we're back to the meaningfulness arguments.)
My immediate thought is that it does not follow from anything you wrote that truth is an inconsistent concept. In fact, I would - and have quite successfully - argued that truth is not a concept at all. What does follow from what you've shared is that that particular accounting practice cannot take proper account of correspondence.
You may have missed my point. So I'll rephrase:
Natural languages curls over on itself. That seems to me to be a good thing. We can formulate the liar paradox, talk about the little man who wasn't there, describe the colour of magic.
Around the edge of the Map of Language, write of "Here be Dragons" to mark the places where the linguistic topology becomes... uncanny. Indeed, it becomes so weird that even as you work out what is going on, it changes again.
The technical term you were searching for is "vague".
But there's no need for anything meta, right?
L = This sentence is false (the liar sentence)
P = L is a proposition
To tell you the truth, I fielded P because 1. it is an assumption made and 2. avoids self-referential gobbeldygook.
In a natural language, even in philosophical English, the object language and the meta-language are one. I do not understand your point.
Vagueness is better as it speaks to what we don't know. And even that to which may be unknowable or undifferentiated.
Uncanny just means strange and unsettling. Are you thinking it is a term logicians use?
These are equivalences, not inferences. They are just a different way to say the same thing. To say « P » or to say « P is true » is to say the same thing.
To take a trivial example, « I like popcorn » and « truly, I like popcorn » are saying the same thing. There are no inferences between one and the other. Seems to me this Kevin Scharp you are reading is not very sharp.
My bad. It's probably because I don't get you.
Anyway, here's what I think: To believe that we can assign a truth value to the liar sentence, it must be the case that we've made the assumption that it's a proposition. Thus:
L = This sentence is false (the liar sentence)
P = L is a proposition
Since I don't see any problems in the argument I presented, it follows that your objection must revolve around L and P. Can you proceed from there and please keep it simple for my sake. Thanks.
No, I definitely did not mean vague.
How does Scharp use these to solve the paradox?
I don't think so. On the surface, the meaning is clear: First, the sentence asserts that something (presumably a proposition) is false. That's not a problem, we generally understand such assertions. The proposition, rather than being quoted, is instead indicated - also not necessarily a problem. For example, "The second sentence on page 23 is false" would be a perfectly meaningful thing to say (but what if the sentence indicated happens to be that very sentence?) It is only once we trace the logical implications that we realize that something is wrong, but in order to be able to do that, didn't we have to first understand what the sentence is saying?
It's not just my objection; it's the revenge pattern: if you hope to avoid the Liar by using some other semantic predicate also available in your language, we'll just construct a new Liar-like sentence that is just as bad as the original.
'... is a proposition' is such a predicate: you absolutely can use it to block the Liar, just as you did, but it's no help with the revenge sentence, 'This sentence is not true or is not a proposition.' You might as well not have bothered.
But my sentences are not as convoluted as the above.
L = This sentence is false (liar sentence)
P = L is a proposition
Nothing can done to the liar sentence so that's that but there's no issue, as far as I can tell, with P - it looks perfectly ok.
Quoting Kornelius
Quoting Kornelius
What blocks:
9. (Li) is not true (8, truth conditions of conjunction).
?
Why can't it just be false?
On the one hand, sure, you can imagine having to explain to someone who doesn't get it what the problem here is, and that might feel like guiding them through a formal deduction.
On the other hand, the Liar is introduced as a premise, but then behaves like an inference rule. ('Given me, you may introduce not-me.') You have to understand the inference rule to use it, that is true. But that's understanding-how, not understanding-that. Inference rules are deliberately empty, have no 'that' content. They don't say anything themselves; it's premises that actually say stuff.
When we trace the logical implications of the Liar, we stop at contradiction, as @Kornelius did, because that's one of our things, a proof-doing practice. But I've also demonstrated it to people by starting at true and then saying, 'But if it's true, then it's false, but if it's false then it's true, but if it's true ...' and doing the same thing starting with false. You never 'finish' so you never get to a point where you've figured out what it says. The completion of the statement is always deferred.
I want to follow this but I can't.
Question motivated by the following insight: for everything else which is both True and False, the conjunction of True and False is simply False. Like "tautology AND contradiction" evaluates to "contradiction".
I'm pretty sure I'm being naive in some way, since evaluating the Liar as simply false is both an obvious approach and undermines a whole literature, but I'm sufficiently lazy that I'm hoping someone in thread can tell me precisely why I'm wrong.
Well, exactly, there is nothing wrong with the syntax of the sentence, it is only its semantic content that seems to be a problem. But in order to come to this conclusion you have to parse and interpret the sentence, which means that the sentence is ipso facto meaningful, at least on some level.
Hmm, I am not sure about this principle either. Perhaps you meant Russell's justification of the principle of simplicity, i.e. do not multiply entities beyond necessity because this increases the chances you are wrong. So maybe the idea is that we should multiple meanings beyond necessity. But I am not sure that makes it less likely we may be wrong. Saying that sentence P is meaningless when it is meaningful is a mistake (and conversely). So I am not sure how one assumption would be more conservative than the other in this sense.
Still, I do think you are right in pressing against the view I was arguing here, namely that we should assume it is meaningful. I want to clarify that I am not simply assuming this, but I think the sentence is intuitively meaningful. This is not an argument to its being meaningful, but it just raises the bar of proof for those who deem is meaningless.
Notice this is quite different from assuming a sentence or all sentences are meaningless until shown otherwise (or vice versa). Certain sentences are intuitively meaningful, some are not.
Most importantly, I am not saying that we need no argument as to what the meaning of (Li) is. The Liar paradox shows that we do need such an argument.
Is there an argument for thinking that (Li) is intuitively meaningless? Does it really sound like gobbly-gook? It can still be meaningless, of course.
Quoting Banno
Let me get back to you on this one. It will take a slightly longer post.
Quoting fdrake
Nothing.
And nothing blocks 10. (Li) is true (8, conjunction elimination).
In fact, any proposition classically follows from 8. We can prove anything.
For this reason, we cannot simply conclude that (Li) is not true, since we can conclude (Li) is true and, in fact, any proposition we wish.
But I am thinking this explanation may not be satisfactory. Did you have something else in mind?
Quoting Kornelius
Something like this; if you conclude that it's true and false, the conjunction of a truth and a falsehood is a false, so it's false. IE, it appears to be an instance of "X and not-X".
Ok, I see (I hope!). This is just the denial of dialetheia: there are no true contradictions. This seems right, but it doesn't follow from this that (Li) is false.
What we have is [(Li) is true and (Li) is not true] is not true.
Agreed. But by what rule of inference would it follow that (Li) is not true? That makes [(Li) is true and (Li) is not true] false just as much as (Li) is true.
But maybe I'm not following your suggestion?
EDIT:
Rather, what rule of inference would establish (Li) is false only
I almost gave 'Its syntax is fine' as an example of a rule for assuming a sentence is meaningful! But--
That's kinda like saying the semantic problems of a sentence are semantic problems.
Worse: doesn't this amount to saying that if a sentence has even a very severe semantic problem, such as being meaningless even if its syntax is fine, then you only find this out by looking at its semantics, and therefore it's meaningful?
But that's too general. You're right that we have to make enough sense of the Liar to understand that it empowers us to use it to infer 'I'm false' from 'I'm true' and vice versa. I can't dispute that.
But, as I tried to express above, that's pretty weird. If you think of the Liar as an introduction rule, then its semantic content is, in one sense anyway, syntactic. And that's all it has. But that's not supposed to happen. It's why we distinguish them.
I'm not sure it amounts to anything.
@SophistiCat and I have ended up having a discussion that's hard to distinguish from an argument about whether the Liar is meaningful. And there's just no point to that, because of revenge.
Even the other idea I had in mind amounts to the same thing. (If you ask what 3+5 is, there's a single right answer and an infinite number of wrong answers. This kind of thing shows up in a few places and there's a reason I'll bet @fdrake knows.) In essence it just amounts to having a designated value, and again we know this won't help because of revenge.
I would argue that nothing needs replacing, but an extension is required. A missing element of meaningful assertion has to be recognised.
The liar paradox arises because of self reference. The assertion lacks an object as it only involves the subject itself. There is no epistemic cut - a division in which a claim is being made about something and so the relationship is a semiotic or formal sign-based one.
A way to illustrate this is imagining a sheet of paper half red and half green. What colour is the boundary then separating the two halves? Is it red or is it green? Both or neither?
Logic collapses, the LEM can’t apply, because the boundary is not some third thing - a separate position from which the two surfaces could be described. The boundary can’t treat either as object, yet seems to treat both as object, while being in fact always subject in being the limiting part of both/either.
So an epistemic cut is a necessity. Semantics demands a world divided into subject and object. Then logic can work.
(Of course there is then the issue of whether the world actually “has” such divisions in and of themselves. Or whether the epistemic cut is something “we” add so as to render the world manageable by a logical calculus.)
It would be interesting to compare it to Kripke's truth, which renders the liar paradox neutral by making it underivable.
Nah I think you were following it fine. It's simply a case of my suggestion not being very good!
Hi Tim, Interesting comment. I watched that video. I had always thought that very fast particles increased in gravitational mass. But apparently not so. As you seem to know a bit about this, can I ask a question that's a bit off the subject of the OP?
I had assumed that, if you lift a weight against the force of gravity, its gravitational potential energy increases by mgh and therefore its mass increases by mgh/c^2. Is that the case? Whatever provided the energy would have a corresponding decrease in mass to compensate.
Likewise, if you accelerate a particle, you add energy, 0.5mv^2. Does its mass increase by 0.5mv^2/c^2?
Thanks Tim. I did look at some of his other videos. Being from Fermilabs, I tend to think he's quite possibly right. I'm on a physics forum. I might ask the question there.
The usual one: if P entails a contradiction, P is false.* If you want, you could say the problem is that both the Liar and its negation are false, which in turn violates the LEM.
Quoting fdrake
Disagree.
* It's not uncommon to just define [math]\neg P[/math] as [math]P\to\bot[/math]. Even in systems that don't, this is an introduction rule for [math]\neg[/math].
Hmm, the issue is that both (Li) and not-(Li) entail a contradiction. So the simple reductio argument to conclude not-(Li) won't suffice. Since not-(Li) also entails a contradiction, then it is not the case that not-(Li). In other words, by your own reasoning, it is false that (Li) is false. But notice that the rule you mention here doesn't get us from [(Li) is true and (Li) is not true] is false to (Li) is not true. If a conjunction is false, either of its conjuncts could be false, so the inference here is invalid. The reductio, however, works both ways and is actually formalized in the proof I outlined in the OP.
Your last suggestion is a viable one (though not necessarily one I agree with). We could think that the Liar Paradox shows that LEM is not valid. However, the reasoning in the Liar paradox is actually intuitionistically valid, so we need more than this. I am no expert on intuitionistic logic and less so on minimal logics. From the literature, I know that only very minimal logics would not allow the Liar reasoning to go through.
Quoting Srap Tasmaner
Right, but in the Liar we also have [math] \neg P \rightarrow \bot [/math] so we cannot simply conclude [math]\neg P[/math], no?
Sorry -- knew that would be confusing. It's valid because it doesn't matter that (Li) is part of the expression (Li) & ~(Li); what matters is that that's a contradiction and it's false, and therefore one of the premises from which we derived it is false. Since there was only one premise...
Here's your original presentation:
Quoting Kornelius
What I was pointing out is just step (5) in your (or Scharp's) presentation. The inference rule I referred to is just the reductio rule.
What we really want is for steps 5-8 to be a separate argument, using [ T-in ] instead of [ T-out ] and showing that ~(Li) is also false. By continuing on past the reductio, we just rederive the premise in step 1, rederive step 4 as step 8, all of which still only gets you ~(Li) because it's the only premise.
Quoting Kornelius
This presentation only shows, with some repetition, that the Liar is false. (If the idea was to sweep up intuitionist logic along with classical logic, by avoiding appeals to the LEM and to double negation elimination, it does not appear to me successful.)
That assumption is discharged at step 5. in the proof. The reasoning from step 5 to 8 is what we get when we have (Li) is false as a theorem. We proved it outright, it is not within the scope of a premise/assumption. I actually like natural deduction systems for this reason! :grin:
This is just to say that if we accept that (Li) is false, then we're accepting a contradiction because it implies a contradiction. We can outright prove that (Li) is false (step (5)) and a contradiction follows outright. The contradiction is not within the scope of step 1. since it was already discharged.
As far as I know, the proof outlined here is intuitionistically valid but I might be wrong. I am quite certain it is though, we never made use double negation elimination or LEM.
Ugh, the Liar is such a pain. I was about to slap my forehead and say you're absolutely right, but are you? I'm genuinely not sure now. The discharge step would have this form:
4.5. P ? P & Q
Can we say that P has been discharged if it's on the right hand side here? That doesn't seem right.
(This is the consideration I dismissed, in so many words, above. 'It doesn't matter that (Li) is part of the expression (Li) & ~(Li).' Geez.)
Whether we can discharge (Li) this way truly doesn't matter though because (4.5) is just P ? Q, which is clearly fine, and we can have that even before getting to 4:
3.5. True( Liar) ? ~True( Liar )
and that's just equivalent to ~True( Liar ).
So, yes, no question, by the time we get to applying [T-in], (1) has been discharged, either by forming the contradiction, as you did, or by forming (Li)?~(Li).
And so you are, finally, absolutely right we get ~(Li) as a theorem and then (Li) as a theorem from that, so we end up proving outright (Li) & ~(Li). Thanks for setting me straight!
(Not sure which of the logical equivalences I used are intuitionist-safe, but I so don't care at the moment.)
Haha, that makes two of us!
This discussion has been great; it made me think about it in greater detail than I had.
But honestly, I think the proof is much easier to see if it is presented in a natural deduction system. I hadn't given much thought to the assumption's being discharged, but that would be obvious if we used the former.
Quoting Srap Tasmaner
:rofl: Well, in any case, we would surely need independent arguments to adopt a non-classical logic.
Something I'm missing about the Liar...
"(Li) is happy" doesn't cause any such issue. We don't spend hours working out what "This sentence is happy" could possibly mean because sentences aren't the sorts of things which can be happy.
But sentences aren't the sorts of things which can be true either. Beliefs can be true, propositions can be true, mathematical equations can be true...sentences themselves can't. It's like saying "This horse is true", I don't know what it would even mean?
Maybe something analogous to "hammers are for hammering", "this coin is worth two cents", "this note is a middle-C", etc.
Quoting Ash Abadear
Not following you at all I'm afraid. Any chance of an expansion?
Quoting bongo fury
Thanks. So if the sentence is a token for the proposition, then is the proposition about the sentence (token) or itself?
The token is about the things it is about. The proposition is an unnecessary abstraction?
But in the Liar, the sentence is about the sentence, so if it were to act as a token it could only be for some kind of assertion or proposition. There's no real state of affairs it's referring to. That's why I thought it was strange to give it a truth value.
So is a token the sort of thing that can be 'true'?
In a word, truth is dynamic, not static. The need to embrace truth, from our self-awareness (conscious existence), as you so well suggested, represents another paradox. The first one you mentioned from the liar's paradox (propositions of self-reference) not only represents paradox itself, but perhaps more importantly incompleteness; hence, dynamic (see Gödel and Heisenberg uncertainty/incompleteness theorem).
Thinking itself (about truth) requires the passage of time (dynamic). And is almost yet another 'dualistic', metaphysical kind of question... .
To that end, one could argue that truth is both dynamic and static, just like one could make the case for truth being both subjective and objective.
Thanks you two!
Sentences are precisely the things that can be true or false. The truth predicate applies to sentences (or propositions). It does not apply to any other object.
Quoting Isaac
Not quite (I think), unless something has changed. We often speak this way, but I think what we mean is that the content of a belief is true. And the content of a belief is a proposition (usually expressed by a sentence). Since a belief is technically a mental state (or a propositional attitude), I don't think truth technically applies, but we say a belief is true as a shorthand for the content of the belief is true. I might be mistaken here, but I don't attach much importance to it.
Quoting Isaac
Quite right, the sentence has no meaning because it has no semantic value. It is not a well-formed sentence because truth does not apply to objects like horses.
The Truth predicate perfectly well applies to things like (Li) because the latter is a sentence.
Quoting 3017amen
Hmmm, I'm not seeing the immediate connection between the Liar Paradox and the incompleteness theorems, but maybe there is an interesting one. Could you elaborate?
I've been thinking, if we're going to (what we needn't) treat True(...) as a predicate applicable to sentences, then we should also be able to talk about classes. (There are of course problems with enumerating sentences, but in natural language we do use class talk that might present technical problems or resist formalization.)
The Liar, then, claims -- speaking a little loosely here -- that it is not true, which is equivalent to claiming that it is true iff it is false, claims thus to be a member of a class that is by definition empty, and claims further that said class is not empty because it itself is a member. Thus the Liar is false.
'This is the prettiest square circle anyone has ever drawn.'
Sure great question! And welcome to the forum by the way.
Mathematician and logician Kurt Godel (and Alan Turing/Turing machine and Bertrand Russell--you can Google that if you will) explored the concepts of infinity versus finitude as it relates to mathematics (deduction).
Without getting into the technical details (which we can if you want) Godel parced the relationship between the description of mathematics and mathematics itself. Basically, he was known to have labeled mathematical propositions combining a sequence of propositions into corresponding natural numbers that form associated labels. Logical operations about mathematics were made to correspond to mathematical operations themselves. The idea linked the self-referential concept of Godel's proof by identifying the subject with the object.
Accordingly, self-referential paradoxes is the appropriate analogy. As mentioned in the OP, the liar's paradox is an unresolved paradox that of course is self referential. It's undecidable, and it's incomplete. And it's based upon a priori logico deductive reasoning.
Quoting Kornelius
I'll return the favor and ask you for clarification of your foregoing quote. How are you suggesting there are no undecidable propositions? (How could this be?)
An inscriptionalist only wants to add to that that turning out to be false can just mean (for this particular claim) being rejected by the system of token-producing agents, who will judge deduction to be an inappropriate treatment for the sentence (i.e. any token of it), and will thus invalidate it in respect of its constituting a licence to print unlimited copies (of itself and other consequent sentences).The speakers will, in other words, put a brake on application of logic to this particular bit of natural language.
By way of gossip, I'm almost certain Godel himself said as much, and may even have suggested he wasn't thrilled about how close the arguments were. To be googled.
"This statement is false" and "This statement is true yet unprovable" are both self-referential. Beyond that, there's not much in common.
Quoting Isaac
Sentences include questions, commands, and so on; Of these, only statements can be true or false.
Quoting Isaac
"The present King of France is bald" is another statement to which we cannot (easily) allocate a truth value. They are not uncommon. In contrast, "This sentence is false" allocates a truth value to itself. You're right to think along these lines, and indeed Kripke's solution can be understood as a formalisation of that idea.
https://en.wikipedia.org/wiki/Diagonal_lemma?wprov=sfla1
https://thephilosophyforum.com/discussion/comment/342838
And obviously what I had in mind when I said 'presupposition'. Now consider statements that have presuppositions that aren't contingently but necessarily false. It's not the statement itself you need to deny, but the presupposition.
Could you rephrase the question?
Here.
It seems to me, to invoke the Bard, "much ado about nothing" or "full of sound and fury, signifying nothing"; and only the anally retentive will be bothered by it.
I didn't read through the whole thread, so apologies if someone has already made this point.
This sentence has five words
Hence, self-reference is not the whole of the problem.
Ah. You lost me in referring to a reference to a reference to another reply... it wasn't so obvious.
So what do we conclude here? Quoting Srap Tasmaner
Which statement and which presupposition?
This sentence is true.
This sentence is about itself.
This sentence is a sentence.
This sentence is not that sentence.
Need I go on?
This sentence is not about itself.
This sentence is not a sentence.
And lest you think the problem is "not",
This sentence is that sentence.
"This sentence is true" is nonsense; it is like saying "This car is true".
"This sentence is about itself" just says that the sentence refers to itself, which seems unproblematic.
"This sentence is a sentence" again refers to something grammatical, beyond mere semantics, and thus is not self-referential in the same way as the "Liar".
"This sentence is not that sentence" again refers outside itself, so is unproblematic.
Quoting Janus
Hmmm.
Also I would make the further point that the "This sentence is about itself" is unproblematic because it doesn't really say anything about itself; it is kind of a non-statement.
Perhaps then it is the combination of mere self-referentiality with a truth claim (which to be coherent requires pointing to something else) that leads to contradiction and/or meaninglessness.
That, in a nutshell, is Kripke's response.
https://plato.stanford.edu/entries/truth-axiomatic/#KripTheo
Edit: I took a look at the linked article, but unfortunately it was so technical I lacked the background to unravel it without significant effort.
Janus I'm not splitting hairs but I'm not sure that's exactly correct. Saying " this sentence is true" is actually true , I think, because it's a statement about itself.
Similarly, " this statement is a lie" is referencing the statement itself too. And so if the statement is true, then it is false; and if it's false, then it's true.
Other examples are:
Socrates: 'What Plato is about to say is false.'
Plato : 'Socrates has just spoken truly.'
And:
"Janus cannot prove this statement to be true."
It basically means that there will always exist certain true statements that cannot be proved to be true.
I don't see what you think is true here.
Quoting 3017amen
Here you have two different sentences which are referring to each other; which seems to be just a more elaborate form of self-referentiality, so I don't see why the same would not apply as with the "Liar".
Quoting 3017amen
Can't see a problem with that one, either.
Quoting 3017amen
That's an entirely different matter. No empirical statement that happens to be true can be proven (in the deductive sense) to be true. How would you prove that water boils at 100 degrees C, for example?
The idea would be something like this:
The Liar does not allow us to accept or reject it, and that's somewhat reminiscent of what goes wrong with
There are in fact two claims here (following Russell's analysis just for the moment to make the point); because of scope issues, we can end up, or seem to end up, denying the predication when what we want to deny is the existential claim that the there is one such individual who is the present king of France. Later work calls such implicit claims 'presuppositions'. (The usual criterion for A being a presupposition of B is that B and its negation both imply A.)
Now consider a claim like this:
Agreement is wrong, and disagreement is wrong, but not because of anything like a vacuous singular term or a definite description, none of that. So what is wrong? In predicating 'tall' of your ideas, there is a presupposition -- or anyway, something like that -- that 'tall' could reasonably be applied or not to ideas. And this is what you want to deny -- not that this predication is, say, mistaken, that sadly your ideas are only of average height. What we need to deny is that 'tall' is even on the table for ideas.
Now the Liar.
The Liar predicates falsehood of itself. Thus the Liar presupposes (or something) that it is the sort of sentence that can be true or false. Not all can. The Liar certainly looks like one that can, no question. Does that turn out to be the case? If not, what we want to deny is not the Liar's predication, but the Liar's presupposition.
(Before this approach occurred to me, I was thinking about the Liar implicitly claiming that it is possible to be a member of the class of sentences that are true iff they are false, and that it is a member, and that this necessarily empty class has a member and is not empty. I might still end up liking that more, not sure. It's more machinery.)
What we need to say of the Liar is that it's not even wrong. And the way we get to do that is by taking away its claim to have predicated anything at all, by taking away the presuppositions upon which that predication is, well, predicated.
What Colour Are Numbers?
Quoting Srap Tasmaner
So the present King of France's being bald, and his not being bald, presuppose there being a present King of France. thanks for clarifying this use.
And for any sentence A, being true, or being false, presupposes... and here I lose track.
I want to write "there being a sentence", but that would be true of "this sentence is false"; after all, if it's not a sentence, what is it? and if we are to discount it as a sentence, then why is that not just special pleading? Is there more reason to discount it than just that it is annoying?
Edit: perhaps this: what is presuposed by "this sentence is false" that is not presupposed by "this sentence is true"?
You might not thank me when you get around to tackling the article. It presupposes much.
Presuppositions were introduced by @Srap Tasmaner, for whom I have hight regard; but I do not yet see how they help here - so best address those questions to Srap.
This seems to indicate an interesting approach. If the Liar were changed to "This proposition is false" we might ask "which proposition?". There is a sentence there, but is there a proposition?
So what does it mean when I say my friend Jack is a 'true gentleman'?. What am I seeking if I seek the 'truth'? How is the 'true cost' of something different from its apparent cost? What sentence am I aiming for I I try to be 'true to my cause'? When we ask "Is the story of Jesus's resurrection true?" would it matter what the actual sentences constituting that story were, or are we asking something more general?
Quoting Kornelius
When I yell some obscenity after stubbing my toe, what is the proposition that constitutes the belief which drove that behaviour?
Quoting Banno
Interesting. Is there any paper in which this is expounded?
Edit - found your link - thanks.
Yeah. As has been said - more authoritatively than I put it, the ability to evaluate the truth value of a proposition seems to depend on the nature of the proposition. It does seem rather odd to me to persist in having truth as a property of certain propositions without noting the features which those propositions share and how such shared features allow truth evaluation where lack of such denies it.
It seems that the sorts of propositions which allow truth evaluation are those which are about beliefs, and propositions which do not are those which do not reference a belief. So it's far more productive to talk about the truth of the belief rather than the truth of the proposition which references it.
I'm also partial to Wittgenstein's idea that we know the meaning of a proposition when we know what would necessarily be the case if it were to be true. Applying that criterion to the "Liar" (assuming for the sake of argument that it could be called a proposition) makes it looks meaningless.
Yes. I tend to go further to say that we need know how we would behave if it were the case. I can't form a belief about whether it is the case that "this sentence is false" because I don't know what I would do differently if it were.
Self-referentiality may seem like a problematic concept, but Gödel, Tarski, and Carnap have shown that it is possible to construct a self-referential sentence (or something close enough) inside arithmetic. That is, given a sufficiently strong arithmetic theory T and a coding schema "...", we have the following:
Fixed-point or diagonal theorem: If P(x) is a formula of the language of T with only "x" as its free variable, then there is a sentence F in the same language such that T proves that F is equivalent to P("F").
In other words, for any property P, if T is a sufficiently strong arithmetical or syntactic theory, then there are sentences of the language which ascribe P to themselves (or close enough). Hence, unless syntax or arithmetic is an incoherent enterprise, self-referentiality can't be the problem.
"This sentence is false" is incapable of being false. It is also incapable of being true. What makes it so puzzling is the mistaken presupposition that it is even capable of being either.
Seems I agreed with some of those who are more qualified on that basic point.
And what would you do differently as a result of "This sentence is true"?
It is also, presumably, meaningless?
Still keen.
Here's roughly what I'm thinking at the moment.
The Liar purports to predicate falsehood of itself, but as asserting and denying the Liar come to the same thing this is no real predication at all, but only a sort of pretense.
If we force the Liar into the same sort of form Russell analysed, we get for the logical form a somewhat better result than we might have expected -- something like this:
Better because, if we make the case that the Liar does not predicate but only purports to, then that description is indeed vacuous, and for the very good reason that just as we cannot consistently affirm or deny the Liar, neither can the Liar itself. If it admits it's not really predicating, we're done; if it lies to itself, and relies on that lie, it fails.
I'm still going round and round on this, but that's where I am at the moment.
ADDENDUM:
I was thinking of definite descriptions but I suppose you could do something like
Then you deny that this class has any members.
Either way, you end up claiming, as with the present king of France, that the sentence is false not because it's true but because no sentence actually does predicate falsehood of itself.
It's a puzzle, to be sure.
So the first test I'd advocate would be, does that work for other formulations - such as "This sentence is true".
So would one make the case that this sentence does not predicate but only purports to?
Well, one might, but it's not so clear as to why. Yet if we do not, then it seems we are engaged in special pleading, for we do it to the Liar, but not to this other, very similar, sentence.
It's a curious issue.
I'm thinking that "This sentence is true" is correct relativism, while "This sentence is false" is improper relativism
What is relativism here?
I believe it's a an infinite thought that is not spurious.
Yes. I couldn't hold the belief that "this sentence is true" because there's nothing I'd be inclined to do differently to if it were false. Part of that is meaninglessness (lack of meaningful referent), but part is the self-referentiality. It goes nowhere. "This sentence has five words" is also self-referential, but is affects other beliefs (how easily I could fit it on a page, what number I'd reach if were to count each word out cardinally, how much ink I should need to write it...), so it's not semantically closed.
As a third problem (not that we need a third reason why it's nonsense), such a closed self-referential sentence gives us no context. Is the sentence 'true' like an arrow, 'true' like a true-gentleman, 'true' like straight ruler, 'true like the capital of France being Paris? Is the sentence 'false' like 2+2=5, or 'false' like "make one false move and the bomb will go off".
And... what is an infinite thought?
I'm not following at all.
Well when I read the two paradoxes you presented, my mind did something infinite each time. One positive, the other negative. Btw, is "This sentence is true" your personal creation or did you read it somewhere?
Well how come it's finished then?
How can someone enter eternity you might ask in that case.
It doesn't seem like the sort of thing I'm likely to ask, no.
It's pretty standard to contrast this with the Liar.
"Who? Huh? The potential to be does that" responds Hegel
"Then the world is inferior to potentiality!" says Heidegger
"Potential is the slave of the actual. " Hegel
"That sounds backwards." Heidegger
" Because you need to see mind as superior to world. " Hegel
"Then potential is subservient to mind!!" Heidegger
" No. Mind is in a different place" responds Hegel
"Then, so what is real?" Gasps Heidegger
"The world!" Hegel
"How?" Heidegger
" Because this sentence is true" chimes in Banno
The prosentential theory also throws out 'This sentence is true', I believe, on the grounds that this is only purported anaphora; being your own antecedent, you cannot inherit your content from anywhere, so you have no content.
My little thing sees them differently. The argument is more or less:
If a sentence could assert its own falsehood,
and if S were such a sentence,
then there would be no difference between affirming and denying S,
therefore there would be no difference between S asserting its own falsehood and not,
therefore S can only purport to assert its own falsehood,
therefore no sentence can assert its own falsehood.
I find the no-content approach pretty persuasive, but I like that this approach recognizes that the mess we're in with 'I'm false' is different from whatever is odd about 'I'm true.'
In his book (p. 81), Scharp mentions that the following triad is inconsistent for a logic L:
(i) L accepts modus ponens and conditional proof;
(ii) L accepts the standard structural rules for derivability (in particular, it accepts cut and contraction);
(iii) The theory consisting of capture (from S infer T("S")) and release (from T("S") infer S) is non-trivial in L.
Scharp argues that the culprit is (iii). But, as Ripley argues in his review of Scharp's book, it may be that the culprit is (ii). In order to understand what is going on, it helps to recast your derivation in terms of the sequent calculus. For those who don't know, the sequent calculus is a calculus that instead of operating with sentences, operates with sequents, or sequences of sets of sentences. The basic idea is this: we interpret a sequent S : R as saying that the disjunction of the sentences in R is derivable from the conjunction of the sentences in S. So it allows us to study the structural properties of the derivability relation. Here are a couple of important rules (I'll use S, R as variables for sets of sentences and A, B, C for sentences):
Structural Rules
Weakening: From S : R, infer S, A : R; from S : R, infer S : A, R (i.e. if a disjunction of a set of sentences is derivable from S, it is derivable from S and A; if a disjunction of a set of sentences is derivable from S, then adding a further disjunct preservers derivability);
Cut: From S : A, R and S', A : R', infer S, S' : R, R' (i.e. if A implies B and B implies C, then A implies C);
Contraction: From S, A, A: R, infer S, A : R; from S : A, A, R, infer S : A, R (i.e. we can reuse premises during a derivation).
Identity: A : A can always be inferred.
Rules for negation:
~L: From S : A, R, infer S, ~A : R (if A v B and ~A, then B);
~R: From S, A : R, infer S : ~A, R (if A & B implies C, then B implies ~A or C)
Rules for truth:
Capture: From S, A : R, infer S, T("A") : R;
Release: From S : A, R, infer S : T("A"), R.
Moreover, a contradiction is symbolized in this system by the empty sequent, : .
Using these rules, we can show that the liar implies a contradiction as follows:
L : L (Identity)
T("L") : L (Capture)
: ~T("L"), L (~R)
: L, L (Definition of L)
: L (Contraction)
L : L (Identity)
L : T("L") (Release)
L, ~T("L") : (~L)
L, L : (Definition of L)
L : (Contraction)
And from : L and L : , we may derive, by cut, : .
This derivation is obviously more complicate, but, on the other hand, it makes clear what are the structural principles involved in Scharp's (ii): Cut, Identity, and Contraction (it also has the advantage of making clear that conjunction is not involved, so that the only logical connective involved is negation). Now, Identity is unimpeachable. What about Cut and Contraction? Now, it is well known that Cut and Contraction are strange rules. In particular, they are the only rules whose premisses are more complicated than the conclusion. In particular, they are the only rules that allow for a formula to "disappear" from the conclusion (that the interaction of Cut, Contraction, and quantification produces anomalies is well-known. Cf., among others, the comments from Jean-Yves Girard on Contraction in his The Blind Spot and this interesting paper by Carbone and Semmes). So we know from logical investigations alone that there are problems with Cut and Contraction.
But there is more. Cut and Contraction are not just responsible for the Liar. As Ripley notes in his review of Scharp (linked above), they are also responsible for a whole host of paradoxes. So, if we get rid of those, we get rid not only of the liar, but also of those other beasts as well (Ripley elaborates a bit in this paper). So why is Scharp so sure that the inconsistent concept is truth? Maybe the inconsistent concept is validity or derivability, if we think of Cut and Contraction as constitutive of those...
Fair point.
Maybe I'm being naive or missing the point, but I use the word "truth" pretty much as it is used in a court of law. When you swear to tell the truth, the whole truth, and nothing but the truth? Basically you are saying that your words and sentences will - to the best of your ability - describe facts. I'm not super knowledgeable about all the different schools of philosophy, but I'm pretty certain that this is some variation of the Correspondence Theory.
So when you say "This sentence is false"? In order for for this sentence to have any meaning, the pronoun "this" must refer to some statement that makes a factual assertion about reality/existence/the universe/etc. In this case, no such assertion is being made, hence the sentence is meaningless and cannot take a truth value.
This would equally apply to many variations.
"This sentence is true"
"The sentence 'Quadruplicity drinks procrastination' is false".
etc
Let us suppose you are right and the Liar is meaningless. This raises the question: why is it meaningless? Let us suppose, for definiteness, that the liar is "This sentence is not true". It is composed of meaningful parts meaningfully put together. That is, "This sentence", "is not" and "true" are each meaningful expressions and the sentence is grammatical. So why does it fail to be meaningful?
One possible answer to this is 's: the problem is with self-reference. Self-reference is a meaningless construction. This is tempting, but, as I have pointed out in my first post in this thread, self-reference is built in our best syntactic and arithmetic theory. So unless we are also willing to throw out arithmetic, self-reference must be considered unimpeachable. But if it is not self-reference, then what is the culprit?
I have little knowledge of syntactic and arithmetic theory, so I would appreciate it if you could explain how self-reference is built into them.
I haven't said that self-reference always leads to meaninglessness, or that it alone is the problem in the case of the Liar. I suggested earlier that maybe self-reference coupled with a truth claim could be the problem.
@Banno then said this was in a nutshell Kripke's response, and linked an SEP article, which I lack the predicate logic background to understand without considerable work, so I'm still in the dark as to the tie-in with Kripke.
I don't have much to add besides what I have already mentioned in my first reply to you. Take arithmetic, for instance. It is not difficult (though it is laborious) to show that it can code any syntactic notion. In particular, given any reasonable alphabet and vocabulary, it is possible to code it using numbers (this should be obvious in this digital age). Less trivially, it is possible to code any syntactic operation using predicates about numbers (this is called arithmetization of syntax). Importantly, the operation of replacing a variable x in a formula P(x) by another term, say n, is also expressible in the language of arithmetic. Using these ingredients, we can build, for any formula P(x), a sentence S such that S is equivalent to P("S"), where "S" is the code for S. So S effectively says of itself that it has P. This is called the fixed-point or diagonal lemma in most treatments (for more details, cf. Peter Smith's Gödel book, esp. chaps. 19-20).
Since this construction uses only typical arithmetical resources (addition, multiplication, numerals), it follows that any arithmetic theory will contain self-reference. Hence, self-reference, by itself, is not to blame for the problems resulting from the liar. You now say that the problem is with self-reference coupled with truth. That's not exactly right, since we can construct paradoxes without the use of self-referentiality (see Yablo's Paradox), but let us suppose you are right for the sake of the argument. The question then arises about what it is about truth that, when coupled with self-reference, generates the paradox. That this is a problem specific to truth is clear from the fact that other notions, when coupled with self-reference, do not generate any paradox (e.g. provability). And here we are back to square one.
I wonder if you could clarify something for me: when you talk about getting rid of Cut and Contraction, is the idea that we can build useful formal systems (good enough for mathematics) that are not prey to the Liar -- though natural languages may have to struggle on without relief -- or is it that the sequent calculus, say, or some other system, represents the system that underlies the informal* reasoning we do in natural languages? (I associate -- loosely and perhaps incorrectly! -- the first view with Frege and the second with Montague.)
Are we to say, 'If only Epimenides (and everyone since) hadn't used Cut and Contraction!' or 'At least we don't have to worry about that in our new and cleaner system'?
* Perhaps "unformalized" is better -- the whole point is that such reasoning may have an underlying system we would consider strongly analogous to the formal systems we develop.
A bit of both, I suppose. I'm following Scharp (and Ripley's account of Scharp), here (this is not meant as an endorsement of his position, I'm just trying to explore it). If I got the gist of his position right, he argues that the constitutive principles of truth, as present in our ordinary practice, are inconsistent. Ripley then presents the following suggestion: perhaps it is not our practices regarding truth that are inconsistent, but rather our ordinary practices regarding validity or perhaps derivability which are inconsistent. Now, once we identify a certain conception as inconsistent, either we accept the inconsistency and attempt to live with it (a dialetheist would perhaps adopt this view), or else we may try to reform it (this is Scharp's position). One way of reforming it is by jettisoning the principles that got us in a pickle in the first place. If you accept Ripley's diagnosis, then that means creating a new formal system that gets rid of Cut (I think you can keep Contraction if you get rid of Cut).
So it is both "If only Epimenides (and everyone since) hadn't used Cut and Contraction!" (the diagnosis part) and "At least we don't have to worry about that in our new and cleaner system!" (the prognosis).
The position is very reminiscent of Carnap's ideal of explication. According to Carnap, it is common for an ordinary concept to be vague and imprecise. This is not troubling---indeed, it may be an advantage---in our day-to-day dealings, but it may hamper scientific progress. Thus, one of the main tasks for the philosopher is to create precise (typically formalized) analogues of those ordinary concepts which are important for science. These analogues need not capture every feature of their ordinary counterparts---if they did, they would not be as precise as we need! But they should capture enough for the scientific purpose at hand. This activity of engineering precise concepts for the needs of science is called by Carnap explication. My point is that, as far as I can see, both Scharp and Ripley are engaged in explication.
You're equivocating definitions of meaningful and truth. There's two potential conversations here and you can't bridge them in the way you're attempting to.
There the rules of one or more formal systems in which truth means something like [coheres with the rules] and meaningful means something like [uses the accepted syntax of the system]. In mathematics 2+2=4 is true because is coheres with any expression of the same form in the rest of the system and it is meaningful because it uses accepted syntax (in the way 2%4=& wouldn't be).
There are then the uses of the terms in ordinary language and the psychological states and behaviours associated with the beliefs they (sometimes) represent. Here 'truth' might be something more like {works out as I expected when I act as if it were the case}, and meaningful is more like {I know what to do about what you've said - it has some consequence on my behaviour or mental state that is predictable from the expression}.
The two are barely related - we simply do not consult formal systems of the type you describe to infer either truthfulness or meaningfulness, it's nothing more than a game. It's like discussing why the Bishop cannot move from c1 to g1 in chess by invoking it's restriction to diagonal movement in the rules and then expecting the result to have an effect when laying the pieces out before the game commences.
The solution to the liar in formal systems is perhaps a fascinating subject to those invested in those systems, but it has no bearing on the solution to the liar in ordinary language. You cannot invoke a system most people do not understand to explain why most people do not use sentences like the liar in their day-today speech - unless you're suggesting that we are all superlative logicians in our subconscious.
But would "This sentence is true" have elicited any different consequence?
The sentence itself, is in fact true, because it's a sentence.
Quoting Janus
Yes, correct.
Quoting Janus
Sounds kind of like Kantian things-in-themselves... .
I thought I was clear that the problem has nothing to do with self reference. Rather is because the sentence is incoherent - it makes no sense.
it is possible to construct sentences that - while grammatically correct - have no semantic meaning.
[i]Quadruplicity drinks procrastination.
Colorless green dreams sleep furiously.
The unambiguous zebra promoted antipathy.
Etc[/i]
We all immediately recognize that these sentences are composed of words which have clear common use definitions, yet we all immediately recognize that under the clear common use definitions of the words these are nonsense sentences.
So now the question is - are such sentences true or false? I am not highly knowledgeable about all the different philosophical movements, but I believe that there are two schools of thought on this topic.
One school of thought basically says (and stealing a Star Trek reference here) "Dammit Jim, quadruplicity does not drink procrastination. That sentence is false"
The other school of thought says that you cannot assign a truth value to such utterances.
I go with that second line of thinking.
But it is not like the examples you give of nonsense.
"... is not true" is just the sort of thing we say about sentences, and it is here said, with the usual meaning, about a sentence. If it doesn't make sense in this case, why not?
I'm repeating myself, but I'll try again. When I use the words "truth" or "true" I am using them in the same sense as used in a court of law. A sentence is true if and only if it describes a fact/event. A sentence is false if it describes an fact/event that could have happened but did not.
The cat is on the mat. This sentence is either true or false depending on where the cat happens to be physically located.
The cat undermined indecisiveness Under the standard/common'/dictionary definitions of the words, this sentence is neither true nor false since it does not assert anything that could be a fact/event.
A sentence has to make a potentially factual assertion in order to take a truth value.
This sentence is false. <-- This sentence just to the left does not make a factual assertion. It does not take a truth value.
The cat is on the mat. <-- This sentence just to the left makes a factual assertion. It is either ture or false.
I hope this helps I don't know if I can make it any clearer.
I would say it's advisable to read the thread, hence:
Mathematician and logician Kurt Godel (and Alan Turing/Turing machine and Bertrand Russell--you can Google that if you will) explored the concepts of infinity versus finitude as it relates to mathematics (deduction).
Without getting into the technical details (which we can if you want) Godel parced the relationship between the description of mathematics and mathematics itself. Basically, he was known to have labeled mathematical propositions combining a sequence of propositions into corresponding natural numbers that form associated labels. Logical operations about mathematics were made to correspond to mathematical operations themselves. The idea linked the self-referential concept of Godel's proof by identifying the subject with the object.
Accordingly, self-referential paradoxes is the appropriate analogy. As mentioned in the OP, the liar's paradox is an unresolved paradox that of course is self referential. It's undecidable, and it's incomplete. And it's based upon a priori logico deductive reasoning.
BTW, as you alluded, this is philosophy; not the law of contracts.
You said that the sentence "This sentence is not false" was meaningless. I then asked, supposing it meaningless, why it is meaningless. Notice that there is an important respect in which it differs from your other examples of meaningless sentences: whereas your other examples all display violations of thematic relations (and, if you like generative grammar, theta roles), "This sentence is not true" does not display such violation. So there must be some other reason why it is meaningless. You then say that it does not describe a state of affairs. Well, the sentence "the cat is on the mat and the cat is not on the mat" also does not describe a possible state of affairs, yet it is clearly meaningful---in fact, it is false. So what is the difference between that sentence and the liar?
I take it that there are two ways of interpreting your objection to the use of formal systems. One is a ban on the significance of formal systems tout court---they are merely a game. The other is a ban on formal systems as tools for interpreting ordinary practices. That is, perhaps formal systems have their uses in sharpening our concepts or in helping to predict a given phenomenon, or whatever, but not in understanding ordinary practices. I will consider first the stronger reading, which is I think more easily disposed of, and then I will offer some remarks as to why I think you're mistaken even on the second reading.
The basic thrust of my argument in favor of formal models is this. The world is complex, and to understand its structure in its entirety is a hopeless endeavor. Fortunately, science has provided us with a very useful paradigm for making progress, namely to understand one bit at a time. Think of Galileo's inclined plane, here. By abstracting away from complexities such as friction, etc., he was able to isolate the effects of gravity on the fall of objects. Similarly, by abstracting away from, say, the pragmatic aspect of communication, we may usefully isolate important aspects of a given concept. Now, one may say that this is only possible in physics because physical phenomena are much simpler. I think this is a misconception produced by the tremendous success of idealization in physics; in truth, physical phenomena are extremely messy, and it is only our idealization practices that introduce some order into this chaos (cf. the work of Nancy Cartwright in this regard).
Hence, formal models can be extremely useful in understanding a concept, if only because their simplicity provides a good testing field for our hypotheses. As Timothy Williamson says:
The case of the concept of truth is one of the examples adduced by Williamson to illustrate this claim. As he puts it earlier in the essay:
So I hope it is clear that formal methods have their place in understanding the concept of truth. Let us then turn to the question of whether formal methods have their place in understanding our ordinary practices. Again, I think the answer is yes. Specifically, I think it's highly plausible that ordinary reasoning conforms to the Cut and Contraction rules. Obviously, this does not mean that, when people reason, they consciously employ the formalism of the sequent calculus! But it does mean that this formalism aptly describes their practices.
In order to understand how this can be so, it is useful to recall here Sellars's distinction between pattern-governed behavior and rule-obeying behavior. Both types of behavior occur because of rules, but only the latter occurs because the agent has a conscious representation of the rule. Here is one of Sellars's examples of pattern-governed behavior that is not rule-obeying behavior: the dance of bees. In order to indicate the position of a given object of interest, bees developed a complicated dance that codifies this direction for the other bees. This gives rise to a norm of correctness for the dance: the dance is correct if it indeed points in the direction of the object. If the bee performs the dance, and the dance leads nowhere, clearly something has gone wrong. In Millikan's helpful terminology, that is because it is the proper function of the dance to indicate the object, so that, if it is not so indicating, it is failing its purpose. Notice that this does not require anything spooky, just natural selection, and notice also that although we can clearly describe the dance in normative terms by employing a normative vocabulary, obviously the bees can do no such thing (the question of whether or not the bees must have a conceptual representation of space in order to perform such a dance is a separate and more difficult question. For a surprisingly good case for answering it in the affirmative, cf. Carruthers, "Invertebrate concepts confront the generality constraint (and win)").
So my claim is that our ordinary reasoning practices are, in this respect at least, much like the bee dance. They are pattern-governed behavior, that is, a behavior that happens because it has been selectively reinforced (either through natural selection, if such reasoning is innately specified, or through socialization; here, game theory can provide some nice formal models of how this can happen without a conscious effort by the agents), but not because the agents are aware of the rules governing their behavior. These rules are, however, implicit in our practices, and the role of logical vocabulary is (among other things) to make them explicit, since it is only by making them explicit that they become subject to rational evaluation.
These glimpses into your views of philosophy -- or at least glimpses of how many contemporary academic philosophers view their work, perhaps you among them -- are very helpful.
If philosophy is to be not just a sort of maternity ward for the sciences, and not their handmaiden, but itself a science (if not the queen), then it's the science of -- ? Concepts?
You've addressed this sort of question at length, but doesn't this strike you as an odd thing to say:
I don't think my eyebrows would have shot up if he had said, 'Far more is known in 2007 about modeling truth in certain widely used formal systems than was known in 1957.'
After all, here we are discussing a book published five or six years after that pronouncement, which proposes that the concept of truth Williamson refers to is inconsistent and ought to be scrapped.
That needn't give one pause, buy can we say there is more to philosophy being a science than philosophers proceeding as if it is? Is there a pudding you could point to in which we would find the proof?
I'm happy that my posts have been helpful, though I'm not too sure if I'm representative of academic philosophy---I'm finishing my PhD in a third-world university, after all, with little or no contact with the big players.
As for your assessment of Williamson's claim, I personally think Scharp's books is exemplary of the trend he was discussing. Knowledge about a concept can include knowledge that a concept is inconsistent, after all! And Scharp's discussion is thoroughly informed by the relevant technical literature, so much that Ripley can actually point out that, according to his own light, perhaps it is not the concept of truth that is inconsistent, but the concept of derivability or validity. Notice that Ripley's position, according to which the problem is not with the concept of truth but with our reasoning practices would be difficult even to formulate, let alone emerge as a serious contender in this debate were it not for the formalism of the sequent calculus. So this whole debate surrounding Scharp's book can be considering the pudding, if you will.
If that's not enough, I think there are at least two more interesting formal results that should be considered in this debate (and that, given Williamson's reference to Halbach, it is plausible to hold that it is what he had in mind). After Tarski formulated his T-schema (itself a formal achievement!), radical deflationism appeared to be almost inevitable. For suppose that there is something substantive about the predicate "... is true". Then, presumably, it contributes something to the truth-conditions of the sentences in which it appears. But, by Tarski's T-schema, for any sentence S, "S" is true iff S, whence the truth conditions of "'S' is true" are the same of S, so the predicate can't contribute anything to the truth conditions of the sentences in which it appears. By modus tollens, there is nothing substantive about truth.
That would appear to be the end of the story, but Tarski and Gödel proved further that it is impossible to add a truth-predicate to a theory and preserve consistency. This seems weird: if truth is not substantive, how can the addition of a truth predicate generate a contradiction? Anyway, perhaps there is something funny about the interaction of the truth-predicate with other formal devices, so here is a proposal: just take as many instances of the T-schema as are consistent, and this will fix at least the extension of the predicate. This proposal was one of the first versions of minimalism: the truth predicate is entirely exhaustible by the maximally consistent set of instances of the T-schema. Unfortunately, Vann McGee showed that there are many maximally consistent sets of instances of the T-schema, so this procedure will not uniquely pin down the truth predicate.
At the same time, there is a growing suspicion that perhaps there is something substantive, after all, to the truth predicate. This suspicion is buttressed by the following formal result: even adding a truth-predicate that obey very minimal compositional principles to a theory is sufficient to obtain a stronger theory. That is, if the truth predicate were indeed non-substantial, we would expect that its addition to a theory would not result in new theorems being proved. But that is precisely what happens. In particular, the consistency of the old theory can be proved in the new theory. So it does seem that there is something to truth, after all. Part of the problem seems to be that the truth predicate is not exhausted by the T-schema, but can also function as a device for generalization (i.e. "Everything she said is true"), which can only be eliminated through infinitary resources (an infinite disjunction "Either she said P and P, or she said Q and Q, etc."---though do note that it's not entirely clear that even this disjunction exhausts the truth-predicate in its generalization function).
So at least two prima facie plausible positions (minimalism and a certain naive deflationism) have been refuted by formal considerations. As a result, we have gained a deeper understanding of the truth-predicate and how to handle it. We know now that it is not just an innocuous predicate and that it is tangled up with all sorts logical considerations. Of course, that is not to say that all questions have been settled, very far from it (there are still minimalists and deflationists around, after all). But it is to say that we have a deeper understand of what is in question when discussing truth.
Incidentally, I don't think boundary policing ("Is philosophy a science?") is much helpful. Philosophy is whatever is practiced at philosophy departments. In many cases, this involves a lot of interdisciplinary work (with cognitive scientists, linguists, physicists, medical doctors, etc. etc.), so that the boundaries are not very sharp. In other cases, it is more abstract, perhaps more reflective, and so clearly more distant from whatever it is we consider to be science. But good philosophy is good philosophy, and I don't see much value in pushing one conception of what philosophy should be over others.
A sentence is just a string of words; how could a string of words be true or false? I think it is more in keeping with what is commonly meant to say that sentences express propositions, and that it is propositions which may be true or false. I say this because a propositions can be expressed in many different ways (sentences).
Quoting 3017amen
I'm not seeing the connection.
Because it describes something, as in declarative sentences. And of course in this instance, if it makes a declaration about itself, it has the potential to become an unresolved paradox.. The liars paradox in neither true nor false. It's based on a priori and logical deductive reasoning. Kind of like mathematics (Godel/Heisenberg etc.).
One of the downsides of so-called pure objective reasoning...
I agree with that. "This sentence is false" becomes problematic only if we reason that it follows that it must be true that it is false, or in other words that its negation must be true. If we do that then we become stuck in a vicious circle of contradiction; if it is true then it is false, if it is false it is true,,,
But if we just ignore its truth claim in the first instance, seeing that it refers to no possible state of affairs which could make it true or false; then we simply step aside and the whole absurd logical machinery rolls past without touching us.
It's not a problem with "objective reasoning"; because this so-called proposition has no coherent object.
I don't think so Janus, hence:
Socrates: What Plato is about to say is false.
Plato: Socrates has just spoken truly.
Consider this: Socrates: What Plato said is false
Plato: Socrates has spoken truly
In this case there could be a coherent object in the statement of Plato's being referred to (which we have not seen) and we are not able to make any assessment as to whether both are correct in their agreement that the statement was false until we know what that statement is.
This is next door to the view I've come to.
Certainly the Liar appears, or attempts, or purports to predicate falsehood of itself. But there is no way for it to predicate falsehood of itself without also predicating truth of itself -- an instance of predicating Fx and ~Fx at the same time in the same sense. Not only is that a contradiction -- which just leads us back around the loop if we're only concerned with truth value -- it's just not predication. So I think we view the Liar as infelicitous, a misfire, an attempt at predication that fails.
I keep thinking that the heavy logico-semantic approaches take the Liar at its word -- that because it purports to have predicated falsehood of itself, that's what it has done.
That leaves lots to think about, because this way of looking at it doesn't exactly explain the Liar -- on this view, explaining exactly how and why it fails to say what it's trying to. It can't be said -- and I'd like a cleaner way of saying that too -- so we really already know that it's going to fail, but there are different ways of failing to do something impossible, and I'd like a clearer view of what happens here.
Thanks for indulging my questions -- I hope it's also of value to you to formulate your views for us to read. (I have some conflicting allegiances, so it is indeed helpful to get another's perspective.)
I adhere to a correspondence account (not theory, mind) of truth. I see Tarski's T-sentence as a minimalist presentation of the logic of correspondence. I don't believe we can analyze and explain just how correspondence works (hence no theory) but we know it does because all our practices are founded on it. I tend to think of the logic of truth as being the other face of the logic of actuality.
If we can't say just how truth works, then it would seem to follow that we cannot precisely explain how cases like the Liar fail. We can see that there is no state of affairs to which they refer, though, and realize that there is something missing that is not missing form ordinary propositions. I am satisfied with that and prefer to turn to other things. I realize that others may not be so satisfied, though. :smile:
Yeah that would be me. For instance, I'm not convinced that '... is true' is a predicate at all, so the scheme I presented there is only a nod toward what's really happening. Lots to think about.
Neither really, just that (as you later allude to) as models they need to have something to hook them back to the thing they're modelling. In models of physics, for example, that's experimental hypothesis testing. I have no objection to models in principle (I'm pretty much a model-dependent realist so models are my building blocks of reality - quite important). I also have a great deal of sympathy for the work of Nancy Cartwright as a consequence. It's just that logical models of things like truth make what, to me, is a massive assumption about the way language works in psychology which is largely unsupported by the evidence in that field.
When we assign a truth value to a proposition, the assumption goes, we're performing some analysis of the syntax, the semantics of the actual proposition and the result is some binary value (true/false). But this is not what we see happening in the brain, nor do we infer it from behavioural experiments.
When reading a sentence with a semantic mismatch "my dog is house" fro example, there are N400 elevations associated with language comprehension which do not trigger responses in higher cortices. We dismiss, or flag such sentences as being 'untrue' without recourse to any logical processing whatsoever. that a dog can't be a house i just part of what learning how to use the words 'dog' and 'house' entail, it's has nothing to do with the logical truth value of the proposition - these processes are virtually identical to those seen with syntactic mismatches "my dog is quietly".
Semantic mismatches produce the same neural responses here regardless of the truth of the proposition being assessed.
There was a key study done in Germany a few years ago which presented subjects with opposing groups of sentences to assess - true/matched "Africa is a continent"; true/mismatched "Saturn is not a continent"; false matched "Saturn is not a planet" and false/mismatched "Africa is not a planet". What they found was that the truth evaluation method used was context dependent. Those tasked with evaluation (rather than sorting) engaged a different process despite earlier experiments showing the n400 response to sematic mismatching.
Basically sentences which are meaningless by lack of sematic matching are processed as such prior to, and independent of their truth evaluation. "This sentence is false" doesn't strike us a odd because we don't know how to evaluate the truth condition. It strikes us as odd because there's a semantic mismatch (sentences alone aren't the sorts of things that can be false).
Truth evaluation (as opposed to sematic mismatch assessment) is a complex process. Fluent sentences (presented in say a clear, high contrast font) are more likely to be assessed as true, even by experts in the subject of the sentence, than ones in a low contrast font. Even at expert, highly specific knowledge assessment, the linguistic aspects of the sentence (fluency, grammar, prosody, source...) play a part in truth evaluation.
Finally, even when we arrive at a fairly uncomplicated truth-evaluation outside of pure linguistic comprehension, we find that semantic processing is embodied almost entirely. Sensory-motor, auditory, visual, olfactory, etc... are engaged in the processing of state evaluation in concepts pertaining to those relevant cortices. "this sentence is true/false" would not be processed in the same way as ""my dog is green" is true/false" because the very processing mechanism for the truth evaluation of "my dog is green" relies on the visual cortex's model of 'dog' and 'green' not as logical concepts but as exterior world states to which there is an appropriate response.
Basically assessing the truth value of sentences is a context dependent involving syntactic and semantic assessments, socially mediated source judgements, emotional valence, and embodied response rehearsal. We're basically classifying these propositions on the basis of what we'd do about them, not on the basis of their logical coherence.
Logical models may well be a very positive tool in some areas, I'm not trying to dismiss their utility entirely, but when dealing with something like our mental processing, they have to be indexed to what's actually happening.
How could a word denote an object? How could a coin have a value? How could a hammer have a purpose? How could a note be a quarter-note?
By convention / a game of pretend.
Quoting Janus
Unnecessary platonism.
Actually, you can. it's done quite often in everydayness when people banter about. Even though the direct object is not explicit, it's implied as being an indirect object.
Perhaps more importantly, self-reference is about the knowing of the subject/person itself, and because we don't know the nature of our own existence, such paradox exists. Think of it as if there was another language that could possibly have the capacity to unpack such a paradoxical statement. As it stands, our language, being part of a temporal condition, precludes such resolution.
Also, keep in mind that because a particular string of vocabulary is incorrect syntax for one language, it does not mean that it's incorrect for another. "Car on part" may be incorrect syntax for English, but is correct syntax for French (not to mention unusual syntax for lyrics, poetry, computer language, etc.).
Quoting Janus
But that's not a self-referential declaration. In that transaction, it leaves out the concept of self-reference by not using the word 'about'. Which is to say that the statement would be about someone else, the indirect object. And so all you have there is just ordinary semantics.
But back to the OP, all this basically means is that there will always exist certain true statements that cannot be proved to be true (just like in mathematics/Gödel).
Yes, of course, obviously the meaning of words, what they refer to, is established by convention, by praxis. How else?
Quoting bongo fury
You're misunderstanding; I am not promoting platonism. The word 'sentence', in conformity with conventional usage, can be taken to mean either "a string of words" or something like 'a statement' or 'a proposition'.
The logic behind what I said is simple; the same proposition can be expressed in many different sentences, and when we say a sentence is true the meaning of 'sentence' as 'proposition' is the appropriate one. No platonism required.
If you make a guess as to what someone will say, and claim that it will be true or false, then you have imagined a proposition which contains objective content which may be checked for its truth or falsehood.
In other words you have not merely to claim that what someone is about to say will be true or false, but to state what you think they are about to say, and why you think it is true or false; if you are not merely playing the fool, that is.
Quoting 3017amen
No empirical statements (propositions proper) can be proven to be true, but we can, in principle at least, check to see if they are.
Also I think you are misusing Gödel. His Incompleteness Theorem applies only to a certain kind of arithmetic, unless I am mistaken. And even there the claim is that there are true statements, within the system, that cannot be proven from within the system. They may be proven from without though. So, the basic idea here, and anyone who understands this better than I do may correct me, is that systems of this kind will be either incomplete or inconsistent.
The statements themselves are a priori just like mathematical truth's. That's why it's analogous to Gödel (Gödel did the same thing in his experiment). They are essentially a priori constructed sentences that reference themselves.
:100: :up:
Quoting Janus
I don't want to be the Spanish Inquisition, but you did seem to think it absurd that true and false could apply to strings of words; and while this might have been for completely other reasons than the concreteness of words, or belief in beliefs and other mental or otherwise abstract entities (propositions and states of affairs) as the only rightful subjects for such predication, such beliefs do seem rife in this thread, and they seem to me an unnecessary cause of confusion.
Quoting Janus
By platonism I mean commitment to abstract entities, and so I would be glad to withdraw the charge (of an excess in this tendency) if it turned out you just meant to recognise a potential fuzzy set of (the "many different") sentences that were rough paraphrases of each other. Have true apply to any and all paraphrases by implication whenever they applied to one. But it seems clear you want to warn me that people round these parts believe in propositions as a separate class of entity. Well that's what I meant by "unnecessary platonism".
Talking of paraphrase, here are some nominalist paraphrases of "this sentence is not true":
Where the subject of the sentence is the phrase "this sentence", and the predicate is the remaining word string.
I wouldn't say propositions are " a separate class of entity". More modestly I would say that a proposition is the semantic content of some sentences or statement. (I write "some" to indicate that this is not to say that there are no non-propositional sentences).
Correct; hence:
Amen: What Janus is about to say is false.
Janus: Amen has just spoken truly.
Yet, the sentences, in themselves, are coherent and complete. In other words, they are not sentence fragments lacking both subject/predicate.
And so there will always exist certain true statements that cannot be proved to be true.
Yet another mystery in life :snicker:
False modesty. :wink: Nowt so abstract as "semantic content".
I haven't said semantic content is "abstract'.
A concrete example, then?
Even taking "proposition" as a term of art, it's not at all clear that this is what we ascribe truth to. Some concept of force, and in particular assertoric force, [s]send[/s] seems to be required.
Ah, propositions not abstract enough...
So, examples please of sentences (or if you must, propositions) that are truth-apt only when asserted?
Quoting bongo fury
Is the first somehow unable to be false?
I'm not sure what you mean here. When I make a statement about some aspect of how things are, I am proposing that things are as my statement asserts, no? So in that sense a proposition is an assertion (at least it would be insofar as I would be asserting that it is true), or so it seems to me.
I only mean that we might want a more, let's say, "neutral" way of describing the semantic content of a statement, so that we can bring out the relationships between
and so on. The content of these is, if not quite identical in every case, closely related. A state of the cat being on the mat is claimed to hold, is asked about, is wished for, is demanded, and so on. We comfortably assign meaning to questions, commands, and so on, but truth and falsity only to assertions, that is, to claims that said state is realized in the world.
(There's clearly some close connection between grammatical mood and force, but it's not absolute, since you can, for instance, quite readily ask a question -- speak with interrogative force -- using a sentence in the indicative mood. Think of a detective going over a witness's statement:
D: 'And at that point you saw the man carrying a small ostrich.'
W: 'That's right.'
D: 'Did he see you?'
W: 'No, I don't think so. He was having some trouble with the ostrich.'
The detective is not asserting that the witness saw a man carrying a small ostrich, but asking the witness to confirm that the detective has understood them correctly.)
you are thinking that the term 'proposition' is too strong, given that the detective is not proposing that W saw a man carrying a small ostrich, but rather asking whether he did?
The bare phrase 'A man carrying a small ostrich' then, we should think of as, what? A description of a state of affairs, perhaps?
Sure, something like that, and that would be the semantic content. "Proposition" is probably mostly used to pick up content + assertoric force, but there's no need to assume or to accept the ambiguity if we can just say so in so many words. And the point, again, is the minimal one that truth does seem to have something to do with assertion. Just what exactly, I'm incapable of saying.
(( This ended up very long -- apologies. ))
Having pressed that distinction I'm going to muddy the waters a bit. Quite a bit.
There's a certain way of speaking about sentences I find quite natural but have been agonizing over in my contributions to this thread. (I also find it a little surprising that no one has called me on it -- if tgw were still here, I think he would have.)
I have described the Liar as purporting to predicate falsity of itself but failing to. I posted comments along those lines several times, and each time I had to decide whether to bother about the little pedant on my shoulder chastising me: 'Sentences do not attempt, do not purport, and so on; a speaker uttering the sentence with assertoric force would be attempting or purporting, and so on.' This is not a minor quibble: Austin, for instance, claimed that it is the historical stating of a sentence -- the speech act which ordinary usage might pick out with words like "He asserts that ..." or "She is claiming that ..." -- which is true or false, not the sentence itself, and certainly not the meaning of the sentence.
Even if Austin's view strikes you, as I think it does most, as wrong, there is some appeal to the idea that a sentence can only be true or false relative to a particular occasion of (perhaps hypothetical) utterance, since what a sentence means, if it is not a tautology, is in quite obvious ways dependent to some degree or other upon those circumstances (of time, place, environment, and of course language), and for sentences involving indexicals or anaphora, as the Liar does, that degree might be considerable. But situation semantics is not my interest in this little post.
A natural thought is that, while it is the sentence itself that is the truth-bearer, we take asserting that sentence as a sort of prerequisite for the assertibility of judgments of truth and falsity. Thus, in telling a story, or reading aloud, or going over a witness's statement, we are not taken to have made an assertion, to have ourselves made any claim to truth, and so in turn the audience is not asked or expected to endorse what we say or not. Insofar as the speaker makes no claim, the audience is not asked or expected to either. Except when they are: you might repeat another's claim, neither explicitly giving nor withholding your endorsement, but to submit it to your audience's judgment. But then we have the original speaker's claim on the table, if not yours. Inverted commas may remove the assumption that the speaker is making an assertion, but leave intact the assumption that the original speaker was. -- But that's all on the side of assertibility, and it still seems clear that whether invited to or not, the hearer of an indicative sentence is always at liberty to judge it true or false; it's just that their judgment may be inappropriate or inconsequential.
And I'm finally getting to the point I actually want to raise: there is a way of talking about sentences that takes the sentence itself as its own speaker. 'What does this sentence here say?' 'What is question no. 3 asking for?' 'This paragraph claims just the opposite.' 'The sign says you have to wait here.' 'The instructions tell you what to do if it doesn't work.' There's even pleonastic speech:
[quote=Woody Guthrie]As I went walking I saw a sign there
And on the sign it said "No Trespassing."
But on the other side it didn't say nothing,
That side was made for you and me.[/quote]
We can take all of this as just a casual way of talking to be analysed away: that to describe a sentence as "saying" something is just to say that it means what a speaker would mean if they spoke that sentence -- a kind of metonymy, in which we attribute to the sentence an intentionality that properly only belongs to the (perhaps hypothetical) speaker, as if the sentence "borrows" its apparent capacity to mean something from its (perhaps hypothetical) speaker, when in truth it's just a sound or a mark, an inert object.
But if we're also going to distinguish, as it seems we very often need to, between sentence-meaning and speaker-meaning -- between "what the words say" and what the speaker "meant by saying those words" -- we might begin to see the point of imagining the "borrowing" going the other way: when we mean something by saying some words, perhaps it is we as speakers who are borrowing the capacity of words to mean something, a capacity which we lack not being signs or symbols but persons. (We certainly produce signs and symbols, but if they have meaning, is it because we imbue them with what we mean, or do we produce them because they already have meaning? Is what we mean the same kind of thing?) There would be some sense, then, to the widespread persistence of idioms which treat sentences as their own speakers, despite everything else which tells us that this is plainly false.
Which brings us back around to the problem of semantic content and assertion. There is a sense in which we naturally read indicative sentences as asserting themselves. (Frege's original version of the Begriffsschrift, if I recall correctly, had a "judgment-stroke", a symbol to indicate that an expression was being asserted, but later versions of the predicate calculus dropped it as unnecessary -- assertion is taken as built-in.) There is a natural reading of the Liar as saying that it's false, "saying" in some "full blooded" sense, asserting its falsity just as much as we would be if we sentient speakers were to assert, 'The sentence 'This sentence is false' is false.'
Bare unspoken sentences that implicitly assert themselves are quite handy for doing logic, of course. It's practically the whole point, to divorce what is said from the person who happens to be saying it; except when you can't, because of indexicals and anaphora, for instance, and then you need quite a bit more machinery than you get from Frege to start making sense again. But there is another point to looking at indicative sentences this way: an assertion is a claim to truth. Who is to sit in judgment of that claim? The speaker is supposed, or assumed, to have already judged a sentence true, so if anyone is to judge the claim, it will have to be someone else; as the speaker, you have already cast your vote. What would be the point of you voting again, by endorsing your own claim? If you tried to pass off your vote in favor of the claim as separate from it, as an additional independent vote, you would be doing something illegitimate. Like everyone else, you get one vote, not two. Thus it is when a sentence makes semantic claims about itself. Sentences like 'This sentence is meaningful/meaningless' or 'This sentence is true/false' appear to be doing something which may be impossible -- cf. that a picture is (or isn't) an accurate representation of something cannot be part of that picture -- but I for one have a strong sense that it is at least illegitimate. The Liar has already cast the vote that all indicative sentences cast for their own truth; it does not get an extra vote to declare itself false as well.
How exactly does the liar sentence require mucking about with any other kinds of sentence than the declarative, assertoric kind?
The only kind, after all, that we ordinarily expect to divide up into true and false instances?
And why, when we do happen to be focussed on that kind, the sudden skepticism about them so dividing, and the addiction to complicating the issue by introducing beliefs, assertions, propositions, attitudes etc?
A nominalist suspects that the motivation is a mystical fascination with abstractions (e.g. "the cat's being on the mat") and the possibility of grasping them; where one ought to be content to point the predicate "is on the mat" at a cat. (Or the predicate "is on" at a sequence of two appropriate objects.)
Quoting Srap Tasmaner
I'd be thrilled if anyone in this thread were prepared to dissolve statements, assertions, beliefs, propositions and truths into one colour. But belief in the abstractions and the supposed distinctions is no doubt too dyed into the wool.
Quoting Srap Tasmaner
Quoting bongo fury
We are token-producing agents, in the material world.
Quoting Janus
https://en.m.wiktionary.org/wiki/there%27s_nowt_so_queer_as_folk
Quoting bongo fury
Right, I realize now you meant to write 'nowt' and that it means much the same as "nothing" or 'nought'. But my question was really about what point the whole sentence was intended to make.
I think it likely that around half of what I posted in this thread is dead wrong; I just don't know which half.
This one:
Quoting bongo fury
... or of combining them with mysterious forces.
Sure, if it works, why not. My interruptions were just a shout out for the more down to earth option.
Yes. It is the same with Russell's Paradox. The paradox begins with "The set..." but the thing under discussion is not a set at all (the paradox shows it can't be). Yet, it exists as something that is not a set. In fact, the paradox arises out of the assumption that a non existent set is a set; the paradox is in the way the proposition is stated: "The set..."