My nephew and the "This sentence is false" paradox

[Daklu]

Last year I posted an exchange I had with my teenage nephew, B, regarding the concept of "fairness."

He spent the night with us last night and somehow the topic of paradoxes came up. He claimed the sentence, "This sentence is false" is a paradox. I replied that isn't necessarily true. Naturally, he disagreed and wanted me to tell him how it can not be a paradox. I declined and instead challenged him to a wager of his choosing.

He agreed to be my "indentured servant" if I could show how that sentence is not a paradox. He of course threw in as many lawyerly clauses as he could think of. I had to use the exact same words in the exact same order with no additional words in the sentence. The kitchen was kind of messy, so I agreed. Then I said,

"This sentence is false?"

and explained because the sentence is not asserting a truth value it is no longer a paradox.

To his credit, he didn't argue the point and agreed it was not a paradox. Turns out he deftly left himself a backdoor with the 13th amendment which, in his words, "prohibits indentured servatude." (I believe it only prohibits involuntary indentured servatude, but I let that pass.) In response I invoked the universal invariant "might is right" and gave him a noogie instead.

I really have to say I enjoy watching teenagers develop their thinking and reasoning skills.

[Aristos Queue]

If he's gotten through a first year of high school algebra, it's time to put "Godel, Escher, Bach" into his hands.

Give him my favorite variant of this quest. "Some books have their own titles somewhere in their text. For example, Frank Herbert's Dune has the word "Dune" actually in the story. Now, suppose I write a new book... this book is an index. It lists all the books in the world that have their titles in their text. I want this book to be a complete list. Should the book include its own title or not?"

As for the "This sentence is false" paradox, there are other ways to handle this. Here's one that I like which differentiates syntax from semantics:

"Let 'false' be a variable equal to the constant true. This sentence is 'false'."

[Daklu]

Wow, your experience with high school algebra must have been way different than mine. I read through part of the book (the preview on Amazon must contain 80% of the book's pages) and I think it would be a rare 14 year old that would understand or enjoy that book. At $13 it's cheap enough for me to get it with my next Amazon order, but it's scope seems to be beyond a little paradoxical thinking.

Should the book include its own title or not?

Why is this your favorite variant? What do you find appealling about it?

In some ways it is the opposite of a paradox. The paradoxical equivalent would be to write a book containing a list of all books that do not have their names in their titles. No answer is correct. With your variant both answers are correct. Hmm... it could be used as a way to introduce them to the concept of how a "right answer" often depends on the context in which it's applied.

Here's one that I like which differentiates syntax from semantics:

I hadn't considered explicitly separating syntax from semantics, but now that you mention it I can see that's what I did. By altering my tonal inflection I changed the semantics without changing the syntax. I suppose any "disproof" of the paradox would have to separate syntax and semantics to some extent, since the semantics of the original statement is indeed a paradox.

(I imagine it's possible to harden the original statement against these kinds of attacks, but "This abstract concept I have in mind that represents an asserted truth value in a system of mutually exclusive truth values is in fact the opposite of the asserted truth value" doesn't roll off the tongue as easily.)

[Val Brown]

(I imagine it's possible to harden the original statement against these kinds of attacks, but "This abstract concept I have in mind that represents an asserted truth value in a system of mutually exclusive truth values is in fact the opposite of the asserted truth value" doesn't roll off the tongue as easily.)

Except that all such syntaxes -- and semantics -- rely on a temporality "within them" that resolves the paradox a la G. Spencer Brown's "Laws of Form" (one of my favorite books). That's the easiest way to notice the resolution of seemingly paradoxical statements (that aren't) as well as Godel's statement. Notice the statement (or process) reenter itself and, as it does so, it can appear to oscillate in its value.

And, yes, it sounds like I also had a different experience of mathematics back in High School than what many did -- Calculus in grade 9 to start with.

[crelf]

Punctuation is part of the sentance, therefore your assertion is incorrect.

More importantly: what was your side of the wager? ie: what did you promise to do/not do should you be unable to prove?

[asbo]

Punctuation is part of the sentance, therefore your assertion is incorrect.

I'm in this camp. Given that the kid was keen enough to throw in protective clauses, I'm surprised he let this slide. I've gotten into arguments over less.

[Daklu]

Punctuation is part of the sentance, therefore your assertion is incorrect.

Interesting counter-argument. I'll counter-counter with...

Punctuation is an artifact of written communication. It does not exist in verbal communication. As our conversation was verbal and not written, there is no punctuation to be included as part of the sentence.

what was your side of the wager?

Nothing. He never asked for anything.

I noted it at the time, but didn't really think about it until now. It was actually pretty clever on his part to not make me stake something. I had information he wanted. Though we ostensibly had a wager, he figured out how to get the information at no cost to himself by virture of the 13th amendment. Had he required that I stake something on the wager there's always the chance I'll think the risk is too great and not enter the wager, thus eliminating the opportunity for him to get the information. Pretty astute for a 14 year old boy.

...or maybe he just forgot.

[crelf]

Punctuation is an artifact of written communication. It does not exist in verbal communication. As our conversation was verbal and not written, there is no punctuation to be included as part of the sentence.

That's like saying that a spoken conversation doesn't have any words or sentances either - just sounds. Except that both parties understand the communication because they have a common frame of reference of, um, a structured lanuguage of words in sentences.

Anyway, as we're debating definitions, we should have references (emphasis added). From http://dictionary.re...browse/sentence :

sen·tence

noun

1.

Grammar . a grammatical unit of one or more words that expresses an independent statement, question, request, command, exclamation, etc., and that typically has a subject as well as a predicate, as in John is here. or Is John here? In print or writing, a sentence typically begins with a capital letter and ends with appropriate punctuation; in speech it displays recognizable, communicative intonation patterns and is often marked by preceding and following pauses.

Even if you argue that punctuation doesn't change a sentence (which it does), your example changed the predicate, which is half of the makeup of the sentence in question.

[mike5]

I hadn't considered explicitly separating syntax from semantics, but now that you mention it I can see that's what I did. By altering my tonal inflection I changed the semantics without changing the syntax. I suppose any "disproof" of the paradox would have to separate syntax and semantics to some extent, since the semantics of the original statement is indeed a paradox.

Ha, the "classic" mistake. Goedel's incompleteness proof shows exactly that it is impossible to separate the syntax from semantics, even for something as simple as adding natural numbers.

Br, Mike

[Daklu]

I strongly suspect a semi-competant 8th grader could out-argue me in any discussion of grammatical construction. My knowledge is exhausted just beyond noun, verb, and adverb. The debate is kind of interesting though so I'll give it another shot. It won't surprise me in the least if you find my counter-counter-counter-counter argument unsatisfying.

That's like saying that a spoken conversation doesn't have any words or sentances either - just sounds.

I disagree they are similar. The concept of a word is fundamentally different from the concept of punctuation. Words have concrete representations in both written and spoken english. In writing, a word requires some combination of letters. In speech, it requires some combination of sounds. In neither case does the word require other grammatical structures around it for it to be recognized as a word. Punctuation does not share this property. It has a concrete representation in writing but it does not have one in speech.

If I show you a card with "car" printed on it, you will recognize it as a word.

If I say "car" to you, you will recognize it as a word.

If I show you a card with a semicolon on it, you will recognize it as a semicolon.

What is the verbal equivalent of a semicolon? A slight pause? If I walk up to you, pause briefly, then turn around and leave are you going to think, "oh, that was a semicolon?"

punctuation

noun

1. the practice or system of using certain conventional marks or characters in writing or printing in order to separate elements and make the meaning clear, as in ending a sentence or separating clauses.

(Emphasis mine.)

There isn't a verbal equivalent of a semicolon. Spoken english doesn't have "punctuation." It only has speech patterns. When reading aloud written punctuation is communicated to listeners by modifying speech patterns in the form of pauses, tonal inflection, and volume changes rather than saying "semicolon," "exclamation point," or "question mark." However, having punctuation and speech patterns serve the same purpose (to help communicate meaning) in their respective mediums is not sufficient to claim equivalence between the two concepts. Therefore, likening the statement "punctuation does not exist in verbal communication" to the statement "spoken conversation doesn't have any words" is invalid.

From http://dictionary.re...browse/sentence :

I'm going to extrapolate a bit here, because the definition doesn't support what I think you're trying to say it supports. Please correct my misconceptions of your argument. If I understand correctly, you're saying tonal inflections are part of the unique identity of a sentence. In other words, changing the inflections produces a sentence that is uniquely different from the original sentence, so when I used a rising inflection on the final word of "this sentence is ^false" I have created a new sentence instead of restating the original one and lose the bet. Is that accurate?

The definition you supplied simply defines what a sentence is. It does not (nor does it attempt to) explain what properties of a sentence define its unique identity. There are no rules for evaluating sentence equivalency in that definition. I claim intonation changes are insufficient to establish the uniqueness of a sentence. Consider hearing the following sentences: (I use the carot '^' to indicate rising pitch while speaking.)

"Did you get money from the bank" (spoken monotonically)

"Did you get money from the ^bank" (spoken as a question)

Both sentences have the same words and the same meaning--I want to know if you got money from the bank. Is the difference in intonation sufficient reason to consider them unique sentences? I don't think it is. If I read the Gettysburg Address at a public gathering using different tonal inflections than Lincoln did, is it reasonable to claim my speech is unique? Again, I don't think so. Most people would correctly say the sentences I spoke were the same that Lincoln did.

Even if you argue that punctuation doesn't change a sentence (which it does)

I won't argue against it as the question is irrelevant unless you can convince me punctuation marks exist in spoken language.

your example changed the predicate

Like I said, grammar isn't my strong point, so you'll have explain how the predicate changed. It looks to me like the predicate is "is false" in both cases.

[jcarmody]

There isn't a verbal equivalent of a semicolon. Spoken english doesn't have "punctuation." It only has speech patterns. When reading aloud written punctuation is communicated to listeners by modifying speech patterns in the form of pauses, tonal inflection, and volume changes rather than saying "semicolon," "exclamation point," or "question mark."

Just for some off-topic fun, I'd like to point out that this has been attempted, sort of.

[crelf]

I strongly suspect a semi-competant 8th grader could out-argue me in any discussion of grammatical construction. My knowledge is exhausted just beyond noun, verb, and adverb.

I'm certianly not a lingustic expert either, and when I know I can't speak with authority, I refer to someone who can, hence my dictionary quote for the definition of "sentence". Unless we limit the debate to the learned's definitions as opposed to our own limited understanding of what we think it might be (or, more possibly, what we would like it to be), I think it's futile.

Just for some off-topic fun, I'd like to point out that this has been attempted, sort of.

Ahhh Victor, thanks for the afternoon laugh!

[Val Brown]

Ha, the "classic" mistake. Goedel's incompleteness proof shows exactly that it is impossible to separate the syntax from semantics, even for something as simple as adding natural numbers.

Br, Mike

Ha, the "classic" mistake. Goedel's incompleteness proof shows exactly that it is impossible to separate the syntax from semantics, even for something as simple as adding natural numbers.

Br, Mike

Mine is/was a classic mistake -- I hit Post when I "only" meant to paste. But I do wonder if you've read "Laws of Form". It does resolve the Godel's incompleteness assertion.

And, actually spoken languages doesn't have punctuation as does written language and that's has been part of the real challenge of live, duplex language translators and "speakers" being implemented in computers. The artifice is quite good and getting better by leaps and bounds, but until "pregnant pauses" and such can be implemented for receptive and passive speech, the distinction will remain.

[Daklu]

@Val Brown

Just curious... what do you mean by "duplex language translators?"

[Daklu]

Incidentally, "this statement is false" is known as the Liar Paradox and many people much smarter than me have proposed solutions to it.

[asbo]

Just curious... what do you mean by "duplex language translators?"

If I had to guess, out of your native tongue to x while simultaneously into your native tongue from x.

[jcarmody]

I'm happy to call this type of paradox "undefined" and move on, just like when a math operation results in NaN. I should probably just read more.

A little more reading, this time on Russel's Paradox, makes me question what the implications are to the mathmeticians trying to resolve it. Is it purely academic?

[Aristos Queue]

No, it isn't purely academic. Research in this area has lead to new ways of defining data types in programming languages, meta typing in ways that we as humans seem to generally recognize but have been hard to formalize in ways a computer can understand and optimize for compilation. It is an area full of results that have no practical application (so far) and in that sense often appears to be purely academic, but knowing which branches will pay dividends ahead of time is an unsolved problem (and suffers from the Halting Problem).