Are there more true statements or false statements?
May 21, 2022 11:01 AM Subscribe
My 11yo son asked me this at bedtime. Are there more true statements, or false statements?
At first I assumed that there were more false statements, because while there's only one correct answer for, say, 1+1=?, there are infinite wrong answers. But then we realized that if you change = to ≠ then it evens out. Is this answerable?
I think the numbers have to be equal because for every statement that can be made there is another statement that negates that one. Every true statement has a matching false statement and vice versa.
posted by Redstart at 11:21 AM on May 21, 2022 [6 favorites]
posted by Redstart at 11:21 AM on May 21, 2022 [6 favorites]
Suppose there is a statement, A. If this is a true statement, then the statement "A is untrue" is a false statement. If this is a false statement, then the statement "A is untrue" is a true statement. For every true statement, there is a corresponding false statement. For every false statement, there is a corresponding true statement. There is a one-to-one correspondence between true and false statements, and therefore, there are an equal number of true and false statements.
posted by CrunchyFrog at 11:22 AM on May 21, 2022 [12 favorites]
posted by CrunchyFrog at 11:22 AM on May 21, 2022 [12 favorites]
It's not so simple, bc there are infinitely many real numbers and infinitely many whole numbers, but there are a whole lot more real numbers than whole numbers!
A slightly better take is that for every true statement, there is a false statement, and vice versa. This is basically what you already got to: just put a negation symbol in front of the whole thing. This reasoning assumes there are exactly two truth values, which is not always a given, or even a good idea in some applications.
In real mathematical logic and philosophy of math (the kind only usually taught in grad school), we have to worry about what a statement even is, what is the universe of discourse etc. This is how Gödel got around to the incompleteness theorems, and showing that most of our frameworks allow for sightly problematic things like statements that are true but cannot be proven.
But I think for a kid of eleven years, the best tack is to set up a correspondence, that shows at least a rough sense of equality and a decent rationale. It's a great question! One that he can come back to many times over the years, if he wishes :)
posted by SaltySalticid at 11:23 AM on May 21, 2022 [20 favorites]
A slightly better take is that for every true statement, there is a false statement, and vice versa. This is basically what you already got to: just put a negation symbol in front of the whole thing. This reasoning assumes there are exactly two truth values, which is not always a given, or even a good idea in some applications.
In real mathematical logic and philosophy of math (the kind only usually taught in grad school), we have to worry about what a statement even is, what is the universe of discourse etc. This is how Gödel got around to the incompleteness theorems, and showing that most of our frameworks allow for sightly problematic things like statements that are true but cannot be proven.
But I think for a kid of eleven years, the best tack is to set up a correspondence, that shows at least a rough sense of equality and a decent rationale. It's a great question! One that he can come back to many times over the years, if he wishes :)
posted by SaltySalticid at 11:23 AM on May 21, 2022 [20 favorites]
CrunchyFrog has it. There are different sizes of infinity (for example, the set of whole number and the set of real numbers are both infinite, but the set of real numbers is much larger). We compare different sizes of infinite (or finite!) sets based on whether or not we can make one-to-one correspondences between two sets. If there is a one-to-one correspondence, then the sets are the same size; if not, they are different sizes. But while that is likely helpful background for you (look up Cantor’s diagonalization argument for more on different sizes of infinity), I think the correspondence argument in CrunchyFrog’s comment (without even mentioning or getting into the idea of infinity) is probably the most useful answer to give your kid.
posted by eviemath at 12:07 PM on May 21, 2022 [8 favorites]
posted by eviemath at 12:07 PM on May 21, 2022 [8 favorites]
I don't have (or recall) the formal training to answer this for sure, but I tend to think caek has it. There are infinite true statements and infinite false statements. So they're both infinity.
I'd be surprised if infinity doesn't always equal infinity.
posted by J. Wilson at 12:52 PM on May 21, 2022
I'd be surprised if infinity doesn't always equal infinity.
posted by J. Wilson at 12:52 PM on May 21, 2022
Does every statement P have exactly one negation "It's not the case that P"?
Among pairs of sentences and their negations, is it always the case that one is true and one is false (i.e. is the negation of a true statement always false and vice versa?).
If both answers are "yes", then yes.
Another answer:
There is a countably infinite number of each (which sort of follows from the ability to use connectives and iterative devices to make new sentences). So in that sense th esame.
posted by melamakarona at 12:53 PM on May 21, 2022 [2 favorites]
Among pairs of sentences and their negations, is it always the case that one is true and one is false (i.e. is the negation of a true statement always false and vice versa?).
If both answers are "yes", then yes.
Another answer:
There is a countably infinite number of each (which sort of follows from the ability to use connectives and iterative devices to make new sentences). So in that sense th esame.
posted by melamakarona at 12:53 PM on May 21, 2022 [2 favorites]
I like the answers involving one-to-one correspondence, but this isn't the only lens mathematicians use to answer questions about infinite sets!
For example, it's reasonable to say "One-third of the integers are multiples of 3, hence more integers are not multiples of 3 than are multiples of 3" (despite the fact that a one-to-one correspondence can be set up between multiples and non-multiples of 3). This statement can indeed be made rigorous in a particular sense. (Link goes to a fairly dense Wikipedia article, but the gist is that if we count up to some large number N and then stop, roughly one-third of the integers up to N will be multiples of 3, and this approximation gets more and more accurate the bigger N gets.)
I think your intuition that there are more false statements than true statements is valid in something like this sense: if you adopt a particular alphabet and consider all the statements that can be written using up to some fixed number of characters in that alphabet, there will probably be far more false statements among them than true statements. (And far, far more meaningless/ill-formed statements than true or false ones.) This might be provable for extremely simple alphabets (such as just digits, arithmetic operations, and equal signs).
posted by aws17576 at 12:57 PM on May 21, 2022 [4 favorites]
For example, it's reasonable to say "One-third of the integers are multiples of 3, hence more integers are not multiples of 3 than are multiples of 3" (despite the fact that a one-to-one correspondence can be set up between multiples and non-multiples of 3). This statement can indeed be made rigorous in a particular sense. (Link goes to a fairly dense Wikipedia article, but the gist is that if we count up to some large number N and then stop, roughly one-third of the integers up to N will be multiples of 3, and this approximation gets more and more accurate the bigger N gets.)
I think your intuition that there are more false statements than true statements is valid in something like this sense: if you adopt a particular alphabet and consider all the statements that can be written using up to some fixed number of characters in that alphabet, there will probably be far more false statements among them than true statements. (And far, far more meaningless/ill-formed statements than true or false ones.) This might be provable for extremely simple alphabets (such as just digits, arithmetic operations, and equal signs).
posted by aws17576 at 12:57 PM on May 21, 2022 [4 favorites]
Here's an interesting article on the question of whether there are different sizes of infinities: https://www.quantamagazine.org/mathematicians-measure-infinities-find-theyre-equal-20170912/.
posted by J. Wilson at 1:02 PM on May 21, 2022 [1 favorite]
posted by J. Wilson at 1:02 PM on May 21, 2022 [1 favorite]
We compare different sizes of infinite (or finite!) sets based on whether or not we can make one-to-one correspondences between two sets. If there is a one-to-one correspondence, then the sets are the same size; if not, they are different sizes.
This is why the set of integers is the same size as the set of even integers, even though it looks as if there ought to be only half as many of the latter (see also: Hilbert's Hotel).
All sets of countably infinite size are the same size by definition, and the definition rests on being able to show that there exists some systematic method for making one-to-one correspondences between elements of the first set and elements of the second that will provably, given sufficient time, eventually get around to linking any nominated element in either with some element in the other.
In the case of the set of integers and the set of even integers, one such systematic scheme is to link each member of the first set to the element of the second that has twice the first's value.
In the case of the set of true statements and the set of false statements, one such systematic scheme is as follows:
1. Pick a statement from the set of true statements that has not already been linked to one from the set of false statements.
2. Negate it.
3. If that negation doesn't already appear in the set of false statements then add it, because it demonstrably belongs there.
4. Link the true statement with its negation.
5. Pick a statement from the set of false statements that has not already been linked to one from the set of true statements.
6. Negate it.
7. If that negation doesn't already appear in the set of true statements then add it, because it demonstrably belongs there.
8. Link the false statement with its negation.
9. Go back to step 1.
Since every pass through this method links an unlinked statement from each set and never adds an unlinked statement to either, there are demonstrably no elements in either set that it will never in principle reach, and that's enough to establish a fully inclusive one-to-one correspondence between the two sets of statements and therefore show that they're the same size if they're not infinite, or define them as being the same size if they are.
posted by flabdablet at 1:07 PM on May 21, 2022 [2 favorites]
This is why the set of integers is the same size as the set of even integers, even though it looks as if there ought to be only half as many of the latter (see also: Hilbert's Hotel).
All sets of countably infinite size are the same size by definition, and the definition rests on being able to show that there exists some systematic method for making one-to-one correspondences between elements of the first set and elements of the second that will provably, given sufficient time, eventually get around to linking any nominated element in either with some element in the other.
In the case of the set of integers and the set of even integers, one such systematic scheme is to link each member of the first set to the element of the second that has twice the first's value.
In the case of the set of true statements and the set of false statements, one such systematic scheme is as follows:
1. Pick a statement from the set of true statements that has not already been linked to one from the set of false statements.
2. Negate it.
3. If that negation doesn't already appear in the set of false statements then add it, because it demonstrably belongs there.
4. Link the true statement with its negation.
5. Pick a statement from the set of false statements that has not already been linked to one from the set of true statements.
6. Negate it.
7. If that negation doesn't already appear in the set of true statements then add it, because it demonstrably belongs there.
8. Link the false statement with its negation.
9. Go back to step 1.
Since every pass through this method links an unlinked statement from each set and never adds an unlinked statement to either, there are demonstrably no elements in either set that it will never in principle reach, and that's enough to establish a fully inclusive one-to-one correspondence between the two sets of statements and therefore show that they're the same size if they're not infinite, or define them as being the same size if they are.
posted by flabdablet at 1:07 PM on May 21, 2022 [2 favorites]
SaltySalticid makes a lot of great points. Ignoring all of them and focusing on the main point that others are making, I have a question:
Is the negation of a false statement always a true statement?
I.e. are there any false statements which, when negated, produce either another false statement or an indeterminate statement?
If there are, it would break the one-to-one correspondence needed to establish equality of sizes.
posted by clawsoon at 1:29 PM on May 21, 2022 [3 favorites]
Is the negation of a false statement always a true statement?
I.e. are there any false statements which, when negated, produce either another false statement or an indeterminate statement?
If there are, it would break the one-to-one correspondence needed to establish equality of sizes.
posted by clawsoon at 1:29 PM on May 21, 2022 [3 favorites]
There are also interesting discussions to be had about whether 1+1 always equals 2 in the real world. For example, do you and your son get the same answer if you - working separately - add together the number of apples in the top row of this picture with the number of apples in the bottom row? If you get different results, which one of you is making a true statement?
posted by clawsoon at 1:34 PM on May 21, 2022 [1 favorite]
posted by clawsoon at 1:34 PM on May 21, 2022 [1 favorite]
I'd be surprised if infinity doesn't always equal infinity.
It doesn't. There are at least two non-equal infinities, one is countable and the other is not. But I don't think that's relevant to the (great) question and agree with the other answers that there are the same number of true and false statements.
posted by plonkee at 2:37 PM on May 21, 2022 [5 favorites]
It doesn't. There are at least two non-equal infinities, one is countable and the other is not. But I don't think that's relevant to the (great) question and agree with the other answers that there are the same number of true and false statements.
posted by plonkee at 2:37 PM on May 21, 2022 [5 favorites]
Do you have to be able to write out or say the statements, or simply conceive of them? Because if you're allowed to conceive of them, then for all real numbers r, "r = r" is a true statement and "r != r" is a false statement and thus the number of true statements and false statements would be uncountably infinite.
But most of those real numbers are irrational, and infinitely long to write out, so obviously if the statements need to be writable or speakable this example isn't meaningful.
posted by potrzebie at 3:27 PM on May 21, 2022 [1 favorite]
But most of those real numbers are irrational, and infinitely long to write out, so obviously if the statements need to be writable or speakable this example isn't meaningful.
posted by potrzebie at 3:27 PM on May 21, 2022 [1 favorite]
If a statement has any numeric content,* then there will be an infinity of false statements that correspond to it. For example, if "John is 47 years old" is true, then "John is x year old", for any value other than 47, is false. Not only are there an infinity of false statements, but because "John is x years old" is false for any real number (e.g., "John is pi years old" or "John is e years old"), there are an uncountable infinity of possible falsehoods. So not only are there more falsehoods than truths, but the cardinality of falsehoods is much greater than the cardinality of truths. If there is 1 truth, there are ℵ1 falsehoods. If there are ℵ0 truths, there are ℵ2 falsehoods. I believe I can make a case for there being ℵ2 potentially true statements in English, so one would expect ℵ4 false statements. That's a lot of falsehood!!
* ...or can be made to have numeric content, by switching something countable for something numeric, such as swapping letters for numbers.
posted by ubiquity at 3:53 PM on May 21, 2022
* ...or can be made to have numeric content, by switching something countable for something numeric, such as swapping letters for numbers.
posted by ubiquity at 3:53 PM on May 21, 2022
What shall we do with the uncountably infinite number of statements that don’t correspond* to any state of affairs against which we could verify* truth or falsity?
“The king of France is bald” is a famous example.
* “correspond” and “verify” are both loaded with assumptions that make the OP’s question even more interesting than it already seems
posted by rd45 at 4:21 PM on May 21, 2022
“The king of France is bald” is a famous example.
* “correspond” and “verify” are both loaded with assumptions that make the OP’s question even more interesting than it already seems
posted by rd45 at 4:21 PM on May 21, 2022
I was going to make a similar argument, ubiquity, but then I realized that the cardinality is the same. All your true and false statements are of the form "John is x years old", and so they map to the real numbers. You're just re-using the same set of real numbers, minus one statement, for each true statement.
posted by zompist at 4:23 PM on May 21, 2022
posted by zompist at 4:23 PM on May 21, 2022
Borges’ story “The Library of Babylon” is about this predicament
posted by rd45 at 4:27 PM on May 21, 2022 [1 favorite]
posted by rd45 at 4:27 PM on May 21, 2022 [1 favorite]
What shall we do with the uncountably infinite number of statements that don’t correspond* to any state of affairs against which we could verify* truth or falsity?
I think for this question we ignore them. We're interested in true statements and false statements, not statements that don't fit neatly into either category.
posted by Redstart at 4:28 PM on May 21, 2022 [3 favorites]
I think for this question we ignore them. We're interested in true statements and false statements, not statements that don't fit neatly into either category.
posted by Redstart at 4:28 PM on May 21, 2022 [3 favorites]
Is the negation of a false statement always a true statement?... clawsoon.
Yes, unless you allow for multiple truth values in some way. The key term there is the Law of the Excluded Middle. This is of course a 'law' by fiat, the rejection of which is Many-Valued Logic.
This is why the hard core answer is "it depends on what you mean by 'true' and 'false' and 'statement' :)
posted by SaltySalticid at 5:18 PM on May 21, 2022 [4 favorites]
Yes, unless you allow for multiple truth values in some way. The key term there is the Law of the Excluded Middle. This is of course a 'law' by fiat, the rejection of which is Many-Valued Logic.
This is why the hard core answer is "it depends on what you mean by 'true' and 'false' and 'statement' :)
posted by SaltySalticid at 5:18 PM on May 21, 2022 [4 favorites]
J. Wilson, in that Quanta article, note that the mathematicians proved two particular infinities equal but the whole thing is framed between
posted by clew at 5:35 PM on May 21, 2022
the infinite size of the natural numbers and the larger infinite size of the real numbersSome infinities are definitely larger than others.
posted by clew at 5:35 PM on May 21, 2022
Get yourself a copy of any book by Douglas Hofstadter and you will quickly (or perhaps slowly) see that this is a much more difficult question to answer than it appears. There are some great Martin Gardner books that are accessible to an 11-year-old with an interest in logic and puzzles. This question could lead to a lot of fun and wondering.
posted by rikschell at 5:41 PM on May 21, 2022
posted by rikschell at 5:41 PM on May 21, 2022
I think that it would be fun and more educational to use these answers with your son to estimate answers.
posted by theora55 at 5:56 PM on May 21, 2022
posted by theora55 at 5:56 PM on May 21, 2022
If a statement has any numeric content,* then there will be an infinity of false statements that correspond to it. For example, if "John is 47 years old" is true, then "John is x year old", for any value other than 47, is false.
But those aren't the statements being paired in the one-to-one correspondence folks have proposed. The true statement "John is 47 years old" is paired with the false statement "John is not 47 years old", while for each n not equal to 47, the false statement "John is n years old" is paired with the true statement "John is not n years old".
posted by eviemath at 6:14 PM on May 21, 2022 [1 favorite]
But those aren't the statements being paired in the one-to-one correspondence folks have proposed. The true statement "John is 47 years old" is paired with the false statement "John is not 47 years old", while for each n not equal to 47, the false statement "John is n years old" is paired with the true statement "John is not n years old".
posted by eviemath at 6:14 PM on May 21, 2022 [1 favorite]
Statements with quantifiers get tricky. Given that "some Mefites are drunk" is true, what is its negation? It's not "some Mefites are not drunk", since that is also true. (There is a way to properly negate the statement, but I leave that as an exercise for the reader.)
posted by zompist at 6:25 PM on May 21, 2022
posted by zompist at 6:25 PM on May 21, 2022
The negation of "some Mefites are drunk" would normally be "no Mefites are drunk".
More precisely: using the standard universal and existence quantifiers ("for all" and "there exist", respectively), the first statement would be "there exists x such that (x is a Mefite and x is drunk)".
While the negation could be phrased in English as "there does not exist...", in order to minimize types of quantifiers used it would more typically (although more convolutedly to the average reader) be phrased in logic as "for all x such that x is a Mefite, x is not drunk".
posted by eviemath at 7:20 PM on May 21, 2022 [1 favorite]
More precisely: using the standard universal and existence quantifiers ("for all" and "there exist", respectively), the first statement would be "there exists x such that (x is a Mefite and x is drunk)".
While the negation could be phrased in English as "there does not exist...", in order to minimize types of quantifiers used it would more typically (although more convolutedly to the average reader) be phrased in logic as "for all x such that x is a Mefite, x is not drunk".
posted by eviemath at 7:20 PM on May 21, 2022 [1 favorite]
I'm not convinced that the argument from negation is correct. In many symbolic logic systems, a negated statement is always longer than the original ("A implies B", written A→B is 3 symbols; the negation ¬(A→B) is 5 symbols), but the number of 5-symbol statements is much larger than the number of 3-symbol statements. So all you've proven is that the number of true (N+2)-symbol statements is at least as big as the number of N-symbol false statements, but there are exponentially more (N+2)-symbol statements, many of which turn out to be false.
To tie it back to an example above, it's like counting the number of 1-digit numbers (10) and comparing them to the number of 2-digit numbers that are multiples of 3 (34). By this incorrect argument, there are more multiples of 3 than there are numbers, which I hope we can agree is not true.
Instead, the approach of looking at all statements of length-N, according to whether they're true or false, seems like it could lead to an actual answer but still it will depend on your alphabet and grammar. For instance, if it's digits, "+" and "=", then you can see right away by counting that most sentences are false, and some are nonsense ("0=7+" for instance is a nonsense sentence). If you include "≠" as a symbol alongside "=" then the by counting you can see that for any given statement length there are equal numbers, since you've got a non-length-increasing negation.
posted by the antecedent of that pronoun at 12:57 AM on May 22, 2022 [1 favorite]
To tie it back to an example above, it's like counting the number of 1-digit numbers (10) and comparing them to the number of 2-digit numbers that are multiples of 3 (34). By this incorrect argument, there are more multiples of 3 than there are numbers, which I hope we can agree is not true.
Instead, the approach of looking at all statements of length-N, according to whether they're true or false, seems like it could lead to an actual answer but still it will depend on your alphabet and grammar. For instance, if it's digits, "+" and "=", then you can see right away by counting that most sentences are false, and some are nonsense ("0=7+" for instance is a nonsense sentence). If you include "≠" as a symbol alongside "=" then the by counting you can see that for any given statement length there are equal numbers, since you've got a non-length-increasing negation.
posted by the antecedent of that pronoun at 12:57 AM on May 22, 2022 [1 favorite]
For any given order of infinity, a finite multiple times that cardinal number is equal to itself, however. Infinities don’t behave like normal finite numbers.
posted by eviemath at 4:07 AM on May 22, 2022 [1 favorite]
posted by eviemath at 4:07 AM on May 22, 2022 [1 favorite]
the number of 5-symbol statements is much larger than the number of 3-symbol statements
Assumes facts not in evidence; the number of well-formed n+k-symbol statements is not necessarily greater than the number of well-formed n-symbol statements.
posted by flabdablet at 6:26 AM on May 22, 2022
Assumes facts not in evidence; the number of well-formed n+k-symbol statements is not necessarily greater than the number of well-formed n-symbol statements.
posted by flabdablet at 6:26 AM on May 22, 2022
The number of true statements of the form "John is r years old", where r is putatively allowed to be any real number, is also dubious from the get-go since some amount of time will always elapse during the evaluation of the statement.
Integers are an abstraction of counting, designed in such a way as to be applicable to any counting problem without their own properties getting in the way. Similarly, the reals are an abstraction of measurement. In practice, any actual measurement is best represented by three real numbers (lower bound, upper bound, confidence level), which is why error bars are a thing.
It's usually possible to evaluate whether or not two specified real numbers are equal with some rigor, just by comparing and evaluating their specifications. Evaluating whether or not some measurement pertaining to a physical object is equal to some specified real number, in any sense that creates the kind of countability difficulties exposed by Cantor's diagonal argument? Perhaps not.
posted by flabdablet at 6:44 AM on May 22, 2022 [1 favorite]
Integers are an abstraction of counting, designed in such a way as to be applicable to any counting problem without their own properties getting in the way. Similarly, the reals are an abstraction of measurement. In practice, any actual measurement is best represented by three real numbers (lower bound, upper bound, confidence level), which is why error bars are a thing.
It's usually possible to evaluate whether or not two specified real numbers are equal with some rigor, just by comparing and evaluating their specifications. Evaluating whether or not some measurement pertaining to a physical object is equal to some specified real number, in any sense that creates the kind of countability difficulties exposed by Cantor's diagonal argument? Perhaps not.
posted by flabdablet at 6:44 AM on May 22, 2022 [1 favorite]
This is why the hard core answer is "it depends on what you mean by 'true' and 'false' and 'statement' :)
Indeed, SaltySalticid. And maybe that's the best way to answer your son's question: with more questions about what exactly one means by 'true', 'false', and 'statement'. Then maybe read Logicomix together, which is a wonderful graphical novel that uses the biographies of logicians to explain why people have worked so hard on questions just like this one, and gives a popular science introduction to foundational questions in mathematics and logic.
Anyway, in classical logic, you can observe the difference between tautologies, contradictions, and contingent statements. Tautologies are always true no matter how you interpret the symbols. Contradictions are always false no matter how you interpret the symbols. Contingent statements have a truth-value that depends on how the symbols are interpreted. Many of the statement-counting arguments above would develop a correspondence between contradictions and tautologies: the negation of a tautology is always false, the negation of a contradiction is always true.
Even this is a rich field for discussion: which symbols are 'logical' and which symbols are about 'context'? Can you tell apart contingent statements from the other types together? Restricting attention to contradictions and tautologies relative to a fixed logical language, do those sets have the same size?
Many seemingly non-contingent statements like "for all x, for all y: x * y = y * x " are actually contingent because there are plenty of reasonable mathematical contexts where multiplication is non-commutative (x and y are n x n matrices, for example). So you need to figure out what it means to interpret a statement to start counting, and then decide what to do with contingent statements.
The size of your context ends up being really important, too. It turns out that if your context (universe of discourse) is finite, we have Zero-One laws that say: for each statement of First-Order logic, it is either almost always true or almost always false when interpreted in a finite context. I actually don't know what the ratio of true to false statements would depend on, if you then let statements "count" as true if they are true with high probability and vice-versa for false. These slides state Zero-One laws in a somewhat comprehensible away, and also link to some delightful books about logic. This article is the most readable explainer for Zero-One laws I could find. Caution: it is actually not that readable.
If you wanted to work out this question over infinite structures, you would have to pick a specific infinite set and vocabulary --- like the Natural Numbers, or the Real Numbers. Then you're in a specific context, so every sentence gets a truth-value. Then you try and do the correspondence argument. I'm pretty sure that because the languages are countable it would go through.
posted by kitten_hat at 8:51 AM on May 22, 2022 [4 favorites]
Indeed, SaltySalticid. And maybe that's the best way to answer your son's question: with more questions about what exactly one means by 'true', 'false', and 'statement'. Then maybe read Logicomix together, which is a wonderful graphical novel that uses the biographies of logicians to explain why people have worked so hard on questions just like this one, and gives a popular science introduction to foundational questions in mathematics and logic.
Anyway, in classical logic, you can observe the difference between tautologies, contradictions, and contingent statements. Tautologies are always true no matter how you interpret the symbols. Contradictions are always false no matter how you interpret the symbols. Contingent statements have a truth-value that depends on how the symbols are interpreted. Many of the statement-counting arguments above would develop a correspondence between contradictions and tautologies: the negation of a tautology is always false, the negation of a contradiction is always true.
Even this is a rich field for discussion: which symbols are 'logical' and which symbols are about 'context'? Can you tell apart contingent statements from the other types together? Restricting attention to contradictions and tautologies relative to a fixed logical language, do those sets have the same size?
Many seemingly non-contingent statements like "for all x, for all y: x * y = y * x " are actually contingent because there are plenty of reasonable mathematical contexts where multiplication is non-commutative (x and y are n x n matrices, for example). So you need to figure out what it means to interpret a statement to start counting, and then decide what to do with contingent statements.
The size of your context ends up being really important, too. It turns out that if your context (universe of discourse) is finite, we have Zero-One laws that say: for each statement of First-Order logic, it is either almost always true or almost always false when interpreted in a finite context. I actually don't know what the ratio of true to false statements would depend on, if you then let statements "count" as true if they are true with high probability and vice-versa for false. These slides state Zero-One laws in a somewhat comprehensible away, and also link to some delightful books about logic. This article is the most readable explainer for Zero-One laws I could find. Caution: it is actually not that readable.
If you wanted to work out this question over infinite structures, you would have to pick a specific infinite set and vocabulary --- like the Natural Numbers, or the Real Numbers. Then you're in a specific context, so every sentence gets a truth-value. Then you try and do the correspondence argument. I'm pretty sure that because the languages are countable it would go through.
posted by kitten_hat at 8:51 AM on May 22, 2022 [4 favorites]
You're right, I left it as an assumption that there are necessarily more syntactically valid statements of length (n+k) then of length n. My gut says this is true for usual ways of writing formal logic or arithmetic, but I have no proof.
posted by the antecedent of that pronoun at 9:45 AM on May 22, 2022
posted by the antecedent of that pronoun at 9:45 AM on May 22, 2022
People more knowledgeable have given better answers above to a really fun question.
One other random idea for continuing the discussion in the future might be to use concrete examples to ask the kid to ponder what restrictions one can place on the question that might make it more or less tractable. [edit: to be clear, I don't mean "tractable, more or less"] e.g., "Are there more true or false statements one can make about the number of pencils in this house?" "Are there more true or false statements one can make about this specific pencil?" What if you throw out all statements with the words "not" or an "untrue" or unequal" in them? What if you say that all false statements about a particular dimensional or existential property of the pencil count as only one statement? etc.
posted by eotvos at 12:26 PM on May 22, 2022
One other random idea for continuing the discussion in the future might be to use concrete examples to ask the kid to ponder what restrictions one can place on the question that might make it more or less tractable. [edit: to be clear, I don't mean "tractable, more or less"] e.g., "Are there more true or false statements one can make about the number of pencils in this house?" "Are there more true or false statements one can make about this specific pencil?" What if you throw out all statements with the words "not" or an "untrue" or unequal" in them? What if you say that all false statements about a particular dimensional or existential property of the pencil count as only one statement? etc.
posted by eotvos at 12:26 PM on May 22, 2022
It's the sort of question that mathematicians would ask. Maybe your 11 yr old has a penchant for math.
posted by jouke at 12:42 PM on May 22, 2022 [1 favorite]
posted by jouke at 12:42 PM on May 22, 2022 [1 favorite]
Hofstadter's Gödel, Escher, Bach might be a bit much for an 11-year old, but it might be fun to go through it together. It's a book that rewards re-reading, and is well worth revisiting as you grow older. I've been re-reading parts of it since high school, and I get something different out of it each time. It's a fun exploration of logic, undecidable statements, recursion, and music (among many other things).
Mathematically, given statements of finite length and any finite system of logic, you will have a set of true statements, a set of false statements, and a set of undecidable statements where you cannot use the system of logic to determine their truth value. While we can pair up statements with their negation, I'm not sure if this provides a true 1 to 1 mapping between the set of true statements with the set of false statements. This is leaving my areas of mathematical study, but I'm not sure one way or the other if there exist true or false statements whose negation might be undecidable. The Wikipedia page on the Law of the excluded middle seems to indicate that this could be the case in some systems: "Many modern logic systems replace the law of excluded middle with the concept of negation as failure. Instead of a proposition's being either true or false, a proposition is either true or not able to be proved true."
Another way to approach this is to play with the idea that determining whether a statement is true or false can have different meanings depending upon if we're talking about mathematical statements and formal logic, or spoken language. From here you can get into empiricism. Is "the cat is on the table" true or false? Let's look at the table. Do we both agree on our observation? What if the cat is behind a centerpiece and you can't see it but your kid can? What if we ask the cat, even though cats are notorious liers?
Cat, are you on the table?
Who, me? On this table? But I'm not supposed to be on the table, so clearly I'm not on the table. QED, bitches. [baps centerpiece onto the floor]
And so on. I agree with everyone that this can be a fun ongoing topic to explore.
posted by indexy at 1:57 PM on May 22, 2022 [2 favorites]
Mathematically, given statements of finite length and any finite system of logic, you will have a set of true statements, a set of false statements, and a set of undecidable statements where you cannot use the system of logic to determine their truth value. While we can pair up statements with their negation, I'm not sure if this provides a true 1 to 1 mapping between the set of true statements with the set of false statements. This is leaving my areas of mathematical study, but I'm not sure one way or the other if there exist true or false statements whose negation might be undecidable. The Wikipedia page on the Law of the excluded middle seems to indicate that this could be the case in some systems: "Many modern logic systems replace the law of excluded middle with the concept of negation as failure. Instead of a proposition's being either true or false, a proposition is either true or not able to be proved true."
Another way to approach this is to play with the idea that determining whether a statement is true or false can have different meanings depending upon if we're talking about mathematical statements and formal logic, or spoken language. From here you can get into empiricism. Is "the cat is on the table" true or false? Let's look at the table. Do we both agree on our observation? What if the cat is behind a centerpiece and you can't see it but your kid can? What if we ask the cat, even though cats are notorious liers?
Cat, are you on the table?
Who, me? On this table? But I'm not supposed to be on the table, so clearly I'm not on the table. QED, bitches. [baps centerpiece onto the floor]
And so on. I agree with everyone that this can be a fun ongoing topic to explore.
posted by indexy at 1:57 PM on May 22, 2022 [2 favorites]
unless you allow for multiple truth values in some way
For statements like "The interior angles of a triangle add up to 180 degrees"?
posted by HiroProtagonist at 7:50 PM on May 22, 2022 [1 favorite]
For statements like "The interior angles of a triangle add up to 180 degrees"?
posted by HiroProtagonist at 7:50 PM on May 22, 2022 [1 favorite]
The process I sketched above would pair that up with "The interior angles of a triangle do not add up to 180 degrees", which is false.
The simple fact that I might find it easier to think of more false statements than true ones doesn't stop me from constructing a corresponding true statement for each of them via negation. Any of the infinite variety of false variants like "The interior angles of a triangle add up to 179.9 degrees" would be paired up with its on own negatio (e.g. "The interior angles of a triangle do not add up to 179.9 degrees", which is true) and maintain count equality that way.
Ah, I hear you say, but what about contexts like non-Euclidean geometry, where the interior angles of a triangle add up to sums that depend on the sizes and potentially positions of the triangles and the curvature of the space in which they're embedded?
Doesn't really change anything. If you're going to make statements about a triangle in such a space - statements that are tightly specified enough to have an unambiguous truth value - then their negations will have the opposite truth value. If you're going to make statements about all triangles within such a space, there will still be ways to negate them so that they yield the opposite truth value.
Unless you're going to disallow negation itself as an operation guaranteed to yield some definite truth value when applied to a statement that already has one, per the excluded-middle variant logics discussion above, then this kind of pairing will always work and the number of true and false statements will remain equal.
posted by flabdablet at 4:05 AM on May 23, 2022
The simple fact that I might find it easier to think of more false statements than true ones doesn't stop me from constructing a corresponding true statement for each of them via negation. Any of the infinite variety of false variants like "The interior angles of a triangle add up to 179.9 degrees" would be paired up with its on own negatio (e.g. "The interior angles of a triangle do not add up to 179.9 degrees", which is true) and maintain count equality that way.
Ah, I hear you say, but what about contexts like non-Euclidean geometry, where the interior angles of a triangle add up to sums that depend on the sizes and potentially positions of the triangles and the curvature of the space in which they're embedded?
Doesn't really change anything. If you're going to make statements about a triangle in such a space - statements that are tightly specified enough to have an unambiguous truth value - then their negations will have the opposite truth value. If you're going to make statements about all triangles within such a space, there will still be ways to negate them so that they yield the opposite truth value.
Unless you're going to disallow negation itself as an operation guaranteed to yield some definite truth value when applied to a statement that already has one, per the excluded-middle variant logics discussion above, then this kind of pairing will always work and the number of true and false statements will remain equal.
posted by flabdablet at 4:05 AM on May 23, 2022
I'm sure that paradoxes enter into this discussion, but I'm not sure how.
Which column does "this statement is false" go into?
posted by clawsoon at 8:48 AM on May 24, 2022
Which column does "this statement is false" go into?
posted by clawsoon at 8:48 AM on May 24, 2022
Same bucket as "this statement is true" - statements that simply don't have a truth value.
Personally I think of your example as distilled Satanism and mine as distilled Christianity.
posted by flabdablet at 11:01 AM on May 24, 2022
Personally I think of your example as distilled Satanism and mine as distilled Christianity.
posted by flabdablet at 11:01 AM on May 24, 2022
On interior angles, this is actually one classical motivation for spherical geometry and hyperbolic geometry! Both those violate the 'law' being evoked, which is only in fact applicable to Euclidean geometry. And if you want your 'statements' to apply across such limitations, then it would be reasonable to allow such a statement about interior angles of a triangle to be neither true nor false, because it depends on the scoping conditions, and if you allow for any geometry, then neither 'true' nor 'false' suffices.
Not sure if anyone is still reading, feel free to DM me for additional reading and citations along these lines :)
posted by SaltySalticid at 5:38 PM on May 26, 2022
Not sure if anyone is still reading, feel free to DM me for additional reading and citations along these lines :)
posted by SaltySalticid at 5:38 PM on May 26, 2022
Is there any possibility that no non-tautological statement is completely true?
posted by clawsoon at 5:35 AM on May 27, 2022
posted by clawsoon at 5:35 AM on May 27, 2022
Is there any possibility that no non-tautological statement is completely true?
Modulo non-standard definitions of “true” and of “tautological”, no. There are all sorts of complex implications that rely on other statements being true but don’t include those other statements directly, and thus are not tautological. Eg. think of combining “If x is an apple then it is a fruit” with “x is an apple” to conclude that it is true that “x is a fruit” (except more complicated): it’s not just tautologically true that “x is a fruit”, you need to know the other details in order to conclude that. For a slightly meatier example, consider “There exists a cardinal number larger than aleph0” (aleph0 is the cardinality of the natural numbers, and is the smallest order of infinity). This is true (with the standard set of set theory axioms and logic), but needs a proof. That’s what mathematicians do, is come up with the explanations for why non-tautologically statements are true (or why other statements are false, or why the truth value of something is undecidable). If the only true statement we’re tautologically true, the entire discipline wouldn’t exist (not as an academic discipline, at least - there would be nothing to study/no new knowledge to create).
posted by eviemath at 5:52 AM on May 27, 2022
Modulo non-standard definitions of “true” and of “tautological”, no. There are all sorts of complex implications that rely on other statements being true but don’t include those other statements directly, and thus are not tautological. Eg. think of combining “If x is an apple then it is a fruit” with “x is an apple” to conclude that it is true that “x is a fruit” (except more complicated): it’s not just tautologically true that “x is a fruit”, you need to know the other details in order to conclude that. For a slightly meatier example, consider “There exists a cardinal number larger than aleph0” (aleph0 is the cardinality of the natural numbers, and is the smallest order of infinity). This is true (with the standard set of set theory axioms and logic), but needs a proof. That’s what mathematicians do, is come up with the explanations for why non-tautologically statements are true (or why other statements are false, or why the truth value of something is undecidable). If the only true statement we’re tautologically true, the entire discipline wouldn’t exist (not as an academic discipline, at least - there would be nothing to study/no new knowledge to create).
posted by eviemath at 5:52 AM on May 27, 2022
Is there any possibility that no non-tautological statement is completely true?
"Something's going on" is the only one I've never been able to justify contradicting, not even when recalling states of consciousness only barely capable of formulating it.
posted by flabdablet at 1:48 AM on May 28, 2022
"Something's going on" is the only one I've never been able to justify contradicting, not even when recalling states of consciousness only barely capable of formulating it.
posted by flabdablet at 1:48 AM on May 28, 2022
That’s what mathematicians do, is come up with the explanations for why non-tautologically statements are true
I had to think through what I was trying to say a bit more. I was thinking about the real world, rather than the mathematical world: Can we make any completely true statements about the real world? Given the limitations of our senses and brains and whatnot? Given the way that we make objects out of processes and always talk about incredibly simplified versions of what exists?
posted by clawsoon at 12:01 PM on May 28, 2022
I had to think through what I was trying to say a bit more. I was thinking about the real world, rather than the mathematical world: Can we make any completely true statements about the real world? Given the limitations of our senses and brains and whatnot? Given the way that we make objects out of processes and always talk about incredibly simplified versions of what exists?
posted by clawsoon at 12:01 PM on May 28, 2022
Implicit in that clarification is the false idea that math is not “the real world”. Perhaps the phrase you’re looking for is “about empirical phenomena”? But even then, are apples and fruits not empirical?
posted by eviemath at 6:49 AM on May 29, 2022
posted by eviemath at 6:49 AM on May 29, 2022
the false idea that math is not “the real world”
and the also-false idea that math is non-tautologous.
Math is wholly and entirely an exploration of the consequences of its own essentially arbitrary definitions. It's about as tautologous an enterprise as it's possible to imagine.
The fact that so many of those definitions and the patterns formed by their consequences map reasonably tidily and even somewhat comprehensively to observables that exist independent of math is a large part of what makes the enterprise useful, but it's not what defines it.
posted by flabdablet at 9:15 AM on May 29, 2022
and the also-false idea that math is non-tautologous.
Math is wholly and entirely an exploration of the consequences of its own essentially arbitrary definitions. It's about as tautologous an enterprise as it's possible to imagine.
The fact that so many of those definitions and the patterns formed by their consequences map reasonably tidily and even somewhat comprehensively to observables that exist independent of math is a large part of what makes the enterprise useful, but it's not what defines it.
posted by flabdablet at 9:15 AM on May 29, 2022
But even then, are apples and fruits not empirical?
Apples are a perfect example of what I'm thinking of. Apples and fruits are ill-defined oversimplifications which make it easier for us to deal with the real world. An effect of the ill-definedness of apples is that no method can be devised to reliably count them (see my link above for an example), which makes reliably doing math with them difficult.
posted by clawsoon at 12:13 PM on May 29, 2022 [1 favorite]
Apples are a perfect example of what I'm thinking of. Apples and fruits are ill-defined oversimplifications which make it easier for us to deal with the real world. An effect of the ill-definedness of apples is that no method can be devised to reliably count them (see my link above for an example), which makes reliably doing math with them difficult.
posted by clawsoon at 12:13 PM on May 29, 2022 [1 favorite]
You could be a brain in a vat fed false input, so nothing "empirical" is absolutely known. But there's always Descartes' answer: "I'm thinking" is true even if you are a brain in a vat.
Now, modern neuroscience would doubt that "I" is an an actual coherent thing... but you can't support skepticism about the outside world via facts about that same outside world. If you're a brain in a vat, "modern neuroscience" is not real and can't prove that your notion of "I" is wrong.
I suspect though that a brain in a vat can know quite a lot of things. E.g., I can say "I am using the English language", and point out quite a lot of non-tautological facts about English lexicon and grammar.
posted by zompist at 3:55 PM on May 29, 2022
Now, modern neuroscience would doubt that "I" is an an actual coherent thing... but you can't support skepticism about the outside world via facts about that same outside world. If you're a brain in a vat, "modern neuroscience" is not real and can't prove that your notion of "I" is wrong.
I suspect though that a brain in a vat can know quite a lot of things. E.g., I can say "I am using the English language", and point out quite a lot of non-tautological facts about English lexicon and grammar.
posted by zompist at 3:55 PM on May 29, 2022
If you're a brain in a vat, "modern neuroscience" is not real and can't prove that your notion of "I" is wrong.
If you're a brain in a vat that's genuinely skeptical then you'll avoid assuming that the referent of "I", in and of itself, admits of being unambiguously delineated in any way whatsoever, and will therefore assiduously avoid constructing any ontological scheme that requires you to do so.
Since that's pretty much all of them, the only remaining option is to conclude that all knowledge that requires making distinctions between anything and anything else is best considered to be a collection of working assumptions with varying degrees of dubiety, commit to ongoing as-required curation and maintenance of that collection, stop worrying about whether the collection you're currently working with is "true" in any deeper sense than being demonstrably useful and acceptably non-contradictory, and learn to enjoy rather than fear the astonishment that's part and parcel of having even quite major revisions made to it in the light of ongoing experience.
posted by flabdablet at 10:30 PM on May 29, 2022
If you're a brain in a vat that's genuinely skeptical then you'll avoid assuming that the referent of "I", in and of itself, admits of being unambiguously delineated in any way whatsoever, and will therefore assiduously avoid constructing any ontological scheme that requires you to do so.
Since that's pretty much all of them, the only remaining option is to conclude that all knowledge that requires making distinctions between anything and anything else is best considered to be a collection of working assumptions with varying degrees of dubiety, commit to ongoing as-required curation and maintenance of that collection, stop worrying about whether the collection you're currently working with is "true" in any deeper sense than being demonstrably useful and acceptably non-contradictory, and learn to enjoy rather than fear the astonishment that's part and parcel of having even quite major revisions made to it in the light of ongoing experience.
posted by flabdablet at 10:30 PM on May 29, 2022
That’s a very expansive definition of tautology, flabdablet.
posted by eviemath at 4:27 AM on May 30, 2022
posted by eviemath at 4:27 AM on May 30, 2022
It is indeed.
My preference for it is rooted at least in part in an instinctive dislike and failure to see the usefulness of the proposition that mathematical ideas and theorems have some kind of existence prior to and independent of the thinkers who first conceive of them.
The best theorems are blindingly obvious, so much so that they do feel as if they'd just been hanging about in some kind of ether waiting to be discovered, but that's only ever the case after they've been formulated. I'm quite confident that it will never stop being possible to formulate countless blindingly obvious mathematical truths that humankind has somehow managed to spend its entire history hitherto completely unaware of.
I have personally experienced an almost overwhelming sense of having encountered a thing that had spent forever waiting to be discovered until somebody finally noticed it was there. In my case it happened the first time I listened to Dark Side of the Moon; Breathe, in particular, felt necessary in a way that very little other music ever has. I still get echoes of that feeling every time I sit down and play that album through again. But nobody makes any serious argument that Dark Side is anything but an original work by four talented British musicians, and I've never found good reason to think of what mathematicians work with in any other light.
Some thinkers draw a hard distinction between the tautologous ("analytical") truth of certain propositions in symbolic logic and some other kind of truth that's held to apply to non-trivial mathematical theorems. I've yet to see any such distinction maintain its integrity or usefulness against rigorous examination of its edge cases so I can't see the point. To me, a sound mathematical proof demonstrating that a theorem follows from the definitions of its underlying suppositions belongs in the same category of things as an analytically true proposition in logic and I see no need for any interpretation more mystical or mysterious than that.
posted by flabdablet at 5:11 AM on May 30, 2022
My preference for it is rooted at least in part in an instinctive dislike and failure to see the usefulness of the proposition that mathematical ideas and theorems have some kind of existence prior to and independent of the thinkers who first conceive of them.
The best theorems are blindingly obvious, so much so that they do feel as if they'd just been hanging about in some kind of ether waiting to be discovered, but that's only ever the case after they've been formulated. I'm quite confident that it will never stop being possible to formulate countless blindingly obvious mathematical truths that humankind has somehow managed to spend its entire history hitherto completely unaware of.
I have personally experienced an almost overwhelming sense of having encountered a thing that had spent forever waiting to be discovered until somebody finally noticed it was there. In my case it happened the first time I listened to Dark Side of the Moon; Breathe, in particular, felt necessary in a way that very little other music ever has. I still get echoes of that feeling every time I sit down and play that album through again. But nobody makes any serious argument that Dark Side is anything but an original work by four talented British musicians, and I've never found good reason to think of what mathematicians work with in any other light.
Some thinkers draw a hard distinction between the tautologous ("analytical") truth of certain propositions in symbolic logic and some other kind of truth that's held to apply to non-trivial mathematical theorems. I've yet to see any such distinction maintain its integrity or usefulness against rigorous examination of its edge cases so I can't see the point. To me, a sound mathematical proof demonstrating that a theorem follows from the definitions of its underlying suppositions belongs in the same category of things as an analytically true proposition in logic and I see no need for any interpretation more mystical or mysterious than that.
posted by flabdablet at 5:11 AM on May 30, 2022
Counterpoint: that we have to construct mathematics is exactly why it is not all tautological. Saying that all math is tautological seems to me to be the consequence of that Platonic approach that claims that all ideas already have their own existence, that leaves no room for choice or perspective in definitions of new mathematical objects, among other problems.
Second counterpoint: the cultural notion that you express that the best mathematical ideas should be obvious in retrospect aligns with a culture that minimizes the joy and role of productive struggle and of process in math. Which leads to situations like one of my (very smart, famous within the field, and actually one of the more caring teachers as well) grad school profs saying, as part of the career advice us grad students were given, that it was better to get a job that involved some teaching (as opposed to what some other profs held up as the ideal of not doing any teaching or caring about teaching) … because even as a top of his field mathematician, he often felt like a failure when he was stuck on a problem, and the teaching helped remind him of his personal value. Which is seriously unhealthy. (A better viewpoint on the state of being stuck.)
posted by eviemath at 5:51 AM on May 31, 2022
Second counterpoint: the cultural notion that you express that the best mathematical ideas should be obvious in retrospect aligns with a culture that minimizes the joy and role of productive struggle and of process in math. Which leads to situations like one of my (very smart, famous within the field, and actually one of the more caring teachers as well) grad school profs saying, as part of the career advice us grad students were given, that it was better to get a job that involved some teaching (as opposed to what some other profs held up as the ideal of not doing any teaching or caring about teaching) … because even as a top of his field mathematician, he often felt like a failure when he was stuck on a problem, and the teaching helped remind him of his personal value. Which is seriously unhealthy. (A better viewpoint on the state of being stuck.)
posted by eviemath at 5:51 AM on May 31, 2022
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posted by caek at 11:15 AM on May 21, 2022 [6 favorites]