The predicate “is true” usually gets applied to a sentence with a subject and predicate. The classic example is “Snow is white”. As Tarski says, “‘Snow is white’ is true if and only if snow is white”.
English allows us to pretend we’re applying the words “is true” to a noun, for example “Islam is true”. But this confuses Tarski: “Islam is true if and only if Islam” is nonsense. So we should properly understand “Islam” in this sentence as a stand-in for various sentences lumped under the name Islam, for example “Allah is God”, and “Mohammed is His prophet.” When we do this, the statement “Islam is true” unpacks to “‘Allah is God’ is true, and ‘Mohammed is His prophet’ is true.” This fits nicely in Tarski form: “Islam is true if and only if Allah is God and Mohammed is His prophet.”
So the general idea is that you can’t use a truth-function to evaluate the truth of a noun until you unpack the noun into a sentence.
Now consider the sentence “This sentence is true”. It Tarski-izes to “This sentence is true if and only if this sentence”, which doesn’t work. To make it work, we have to unpack the noun “this sentence” into a sentence. “This sentence” unpacks to the sentence to which it refers: “This sentence is true”. So the unpacking ends with:
“‘This sentence is true’ is true.”
The second round of unpacking ends with:
“″This sentence is true’ is true’ is true.”
And so on, with each unpacking just adding one more “is true” after it without making it any less packed. Trying to unpack the noun fully will lead to infinite regress; stopping at any point will mean you’re trying to run a truth predicate on a noun.
What can be said about a truth predicate can also be said about a falsehood predicate, so the Liar Sentence just returns “invalid argument for function”, the same as if you pointed to a dog and said “That dog is false!”
The other sentences mentioned as contrasts don’t have this problem. “This sentence is in English” also requires a sentence as an argument. It gets one: “The sentence ‘This sentence is in English’ is in English” is a perfectly valid sentence. It’s not necessary to evaluate the truth of the sentence in the middle (its English-ness isn’t related to whether it’s true or false), so we can leave that one unevaluated and just evaluate the frame sentence, which evaluates the inner sentence’s Englishness, which comes out as true.
I’m comfortable (mostly, it’s a bit of a bullet bite) saying ‘all sentences are either true or false’ doesn’t have a truth value, since to determine one you have to reference the sentence itself and that function doesn’t terminate. You can say in English or a Meta-language that all well-formed formulas in some system are either true or false. But you can’t say this in the object language.
Did you intend to note that “this sentence is either true or false” is a true sentence (for most methods of evaluation) that can’t be evaluated by Yvain’s fairly straightforward approach? Because that’s definitely interesting (thanks Jack).
Just not messing with recursion, in general, is a fairly old solution and not very satisfying. I blame Yvain’s writing ability for leading 9 people astray :D
I take it the problem is that it doesn’t unpack even though it does have a truth value. Or at least it isn’t obvious how to unpack it. It’s a false negative candidate.
So the point is that it is a sentence that demonstrates a problem with using unpackability as a requirement for qualifying as meaningful English? That seems reasonable.
Note that the universal “All sentences are either true or false” also, doesn’t appear to meet the unpackability requirement, though I’m not confident I know how to make a Tarski sentence out of that.
Well the dominant strategy in most of these attempts is to deny that all sentences are true or false; some sentences fail to return a truth value because they are meaningless/non-terminating/take invalid arguments etc.
I really like this. It’s an intuitive model of reference in the language, and most importantly it rules out self-reference for an actual reason (never unpacks).
EDIT:
Trying to unpack the noun fully will lead to infinite regress
I wonder if you couldn’t do something with that infinite regress. Maybe that’s something interesting in a formal language—doing calculus on infinite recursion? If that’s even possible.
That’s not the kind of paperclips maximizing that I like. That instrument should be melted down to make numerous smaller paperclips. It is not maximising usefulness by being an instrument.
On the opening night of Paperclip Maximiser, have a metalsmith set up a furnace and proceed to fashion the orchestra members’ instruments, one by one, into paperclips.
If it makes clear that that’s what the reference is to, that would be great! A lot of people trivialise the paperclip maximiser as a mere “thought experiment”, when actually there are real, conscious, sentient beings out there that are really like that and whose interests need to be accounted for.
(I’d say something about how I had a mental image of a dog being untrue in the sense of being somehow unfaithful, but I was laughing at this sentence before that occurred to me.)
What about this:
The predicate “is true” usually gets applied to a sentence with a subject and predicate. The classic example is “Snow is white”. As Tarski says, “‘Snow is white’ is true if and only if snow is white”.
English allows us to pretend we’re applying the words “is true” to a noun, for example “Islam is true”. But this confuses Tarski: “Islam is true if and only if Islam” is nonsense. So we should properly understand “Islam” in this sentence as a stand-in for various sentences lumped under the name Islam, for example “Allah is God”, and “Mohammed is His prophet.” When we do this, the statement “Islam is true” unpacks to “‘Allah is God’ is true, and ‘Mohammed is His prophet’ is true.” This fits nicely in Tarski form: “Islam is true if and only if Allah is God and Mohammed is His prophet.”
So the general idea is that you can’t use a truth-function to evaluate the truth of a noun until you unpack the noun into a sentence.
Now consider the sentence “This sentence is true”. It Tarski-izes to “This sentence is true if and only if this sentence”, which doesn’t work. To make it work, we have to unpack the noun “this sentence” into a sentence. “This sentence” unpacks to the sentence to which it refers: “This sentence is true”. So the unpacking ends with:
“‘This sentence is true’ is true.”
The second round of unpacking ends with:
“″This sentence is true’ is true’ is true.”
And so on, with each unpacking just adding one more “is true” after it without making it any less packed. Trying to unpack the noun fully will lead to infinite regress; stopping at any point will mean you’re trying to run a truth predicate on a noun.
What can be said about a truth predicate can also be said about a falsehood predicate, so the Liar Sentence just returns “invalid argument for function”, the same as if you pointed to a dog and said “That dog is false!”
The other sentences mentioned as contrasts don’t have this problem. “This sentence is in English” also requires a sentence as an argument. It gets one: “The sentence ‘This sentence is in English’ is in English” is a perfectly valid sentence. It’s not necessary to evaluate the truth of the sentence in the middle (its English-ness isn’t related to whether it’s true or false), so we can leave that one unevaluated and just evaluate the frame sentence, which evaluates the inner sentence’s Englishness, which comes out as true.
What about
‘all sentances are either true or false’.
This sounds like the sort of sentance we’d want to assign a truth value to. Yet we can instanciate it into
‘this sentance is either true or false’
Which is problematic—and yet it seems that it must have a truth value if the first sentance did.
I’m comfortable (mostly, it’s a bit of a bullet bite) saying ‘all sentences are either true or false’ doesn’t have a truth value, since to determine one you have to reference the sentence itself and that function doesn’t terminate. You can say in English or a Meta-language that all well-formed formulas in some system are either true or false. But you can’t say this in the object language.
Did you intend to note that “this sentence is either true or false” is a true sentence (for most methods of evaluation) that can’t be evaluated by Yvain’s fairly straightforward approach? Because that’s definitely interesting (thanks Jack).
Just not messing with recursion, in general, is a fairly old solution and not very satisfying. I blame Yvain’s writing ability for leading 9 people astray :D
Why is that a problem? It is a true sentence.
I take it the problem is that it doesn’t unpack even though it does have a truth value. Or at least it isn’t obvious how to unpack it. It’s a false negative candidate.
So the point is that it is a sentence that demonstrates a problem with using unpackability as a requirement for qualifying as meaningful English? That seems reasonable.
That’s what I got from it.
Note that the universal “All sentences are either true or false” also, doesn’t appear to meet the unpackability requirement, though I’m not confident I know how to make a Tarski sentence out of that.
Well the dominant strategy in most of these attempts is to deny that all sentences are true or false; some sentences fail to return a truth value because they are meaningless/non-terminating/take invalid arguments etc.
I really like this. It’s an intuitive model of reference in the language, and most importantly it rules out self-reference for an actual reason (never unpacks).
EDIT:
I wonder if you couldn’t do something with that infinite regress. Maybe that’s something interesting in a formal language—doing calculus on infinite recursion? If that’s even possible.
My knowledge of the Arabic language is only good enough to recognize that this is a tautology.
...and now that I think about it, it doesn’t appear that the first part of
is actually an existence claim!
~exist(God)
Correct!
By the way, what do you think about this (last sentence)?
That’s not the kind of paperclips maximizing that I like. That instrument should be melted down to make numerous smaller paperclips. It is not maximising usefulness by being an instrument.
Now that would make headlines.
On the opening night of Paperclip Maximiser, have a metalsmith set up a furnace and proceed to fashion the orchestra members’ instruments, one by one, into paperclips.
Oh, that goes without saying.
I was more interested in whether you would appreciate a musical composition drawing attention to your existence and values.
If it makes clear that that’s what the reference is to, that would be great! A lot of people trivialise the paperclip maximiser as a mere “thought experiment”, when actually there are real, conscious, sentient beings out there that are really like that and whose interests need to be accounted for.
Importantly, paperclip maximiser analogues exist in the form of corporations, even if they aren’t sentient.
hee hee hee hee hee
(I’d say something about how I had a mental image of a dog being untrue in the sense of being somehow unfaithful, but I was laughing at this sentence before that occurred to me.)