The number of absolutely true statements

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Discussion Overview

The discussion revolves around the nature and existence of absolutely true statements, exploring whether there are infinitely many, none, or a different quantity. Participants engage with logical implications, paradoxes, and the relationship between language and truth, touching on theoretical and philosophical aspects.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that if there is at least one absolutely true statement, then there must be infinitely many, using a recursive argument.
  • Others argue that claiming there are no absolutely true statements leads to a contradiction, suggesting that there must be infinitely many true statements.
  • A participant questions whether an "undeterminable amount" necessarily equates to "infinitely many."
  • One viewpoint suggests that the cardinality of true statements may align with the cardinality of natural numbers, but this is critiqued as lacking insightful conclusions.
  • Another participant reflects on the nature of language and its relationship to logic, suggesting that statements may not hold meaning outside their linguistic context.
  • There is a discussion about the implications of self-referential statements and their role in induction systems.
  • A later reply introduces the idea of non-Boolean logic, proposing that both "there is at least one absolutely true statement" and "there are infinitely many true statements" could coexist.
  • One participant presents a complex analogy involving entropy and information theory to illustrate the relationship between truth and structure.

Areas of Agreement / Disagreement

Participants express a range of views, with no clear consensus on the existence or quantity of absolutely true statements. Multiple competing perspectives remain, particularly regarding the implications of language, logic, and the nature of truth.

Contextual Notes

Some arguments rely on specific definitions of truth and language, and there are unresolved assumptions about the nature of statements and their meanings. The discussion includes various logical frameworks that may not align with traditional Boolean logic.

phoenixthoth
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there are either infinitely many or none.

proof:
it suffices to prove that if there is at least one, then there are infinitely many. suppose there is at least one absolutely true statement.

absolutely true statement number 2:
there is at least one absolutely true statement.

absolutely true statement number 3:
there are at least two absolutely true statements.

for n>3, absolutely true statement number n:
there are at least n-1 absolutely true statements.

QED
 
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If you say that there are NO absolutlely true statements than that statement is not absolutlely true, implying that you can not know the number of absolutely true statements.

It is a non-sensical unsupportable argument.
The lingual equivalent of an optical illusion.
Such as:
"This statement is a lie."
 
correct.

note: the statement was this:
there are either infinitely many or none.

there was not an effort to claim which one was the actual case.

if the statement "there are no true statements" leads to a contradiction, then perhaps it is not the case that there are no true statements. then by the above claim, there are infinitely many.

another way to say that is if logic is not a banality, then there are infinitely many true statements.
 
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Originally posted by phoenixthoth
there are either infinitely many or none.

Good point.
However, does "undeterminable amount" necessarily equate to "infinitely many"?
 
the cardinal number of the set of true statements is alph_null, the same as the cardinal number of the set of natural numbers, the same as the cardinal number of all finite statements in a language with finite alphabet/character set. in that sense, there are as many true statements as there are statements.

however, this is a rather retarded proof because it doesn't give any insightful true statements.
 
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Originally posted by phoenixthoth
however, this is a rather retarded proof because it doesn't give any insightful true statements.

Another good point!
 
it does give me a bit more faith that it is conceivable that an absolutely true statement can be expressed in language.

in a book I'm reading, the example was this statement:
I

to even say "I am" detracts from the truth slightly in the author's opinion. this is sort of equivalent to "God exists" in the sense that it is of the form "x is." in that sense, the author would have to agree that "God exists" is not absolutely true.

i have noticed what appears to be a fatal flaw.
"there is at least one true statement" may be the ONLY true statement. i was assuming that the true statement referred to by what's in the quotes was a statement other than itself. oops.

wait. if it is the only true statement, then this is also true:
there are at least two true statements. here they are:
1. there is one true statement
2. there is at least one true statement

but then we have a contradiction between the true statements "there are at least two true statements" and "there is one true statement." this contradiction implies that the assumption "it (there is at least one true statement) is the only true statement" is wrong. therefore, there are at least two true statements.

one can still progress to infinity.

either it is the case or it is not the case that the ONLY two true statements are these:
1. there is at least one true statement
2. there are at least two true statements

but if there are only two true statements, then we can add a third one: there are two true statements. having a third one condradicts the assumption that there are only two. therefore, there are at least three true statements.

suppose n>3.
if there are only n-1 true statements, we can add an nth true statement: there are n-1 true statements. having an nth one contradicts the assumption that there are only n-1 true statements. therefore there are at least n true statements.

run this through for all n and you've got infinitely many very uninsightful true statements.

i think you can also prove there are infinitely many false statements. let A be any statement. then A&notA is false. since there are infinitely many statements, there are infinitely many false statements.
 
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I'd need to spend a lot more time thinking about this before I could say something for certain...but the first thing that has occurred to me is "Is it not possible that you hzve focussed way too much on language (words, statements...etc) and completely missed the point of logic only really meaning something when applied to reality?"

I mean, 'there are at least n true statements' doesn't mean anything outside of hte language it is spoken in, and language isn't really a thing: it is a system of representing things. Don't confuse the things being represented, with the stuff doing the representing (the words).
 
that was you 666th post. how cool!

I mean, 'there are at least n true statements' doesn't mean anything outside of hte language it is spoken in, and language isn't really a thing: it is a system of representing things. Don't confuse the things being represented, with the stuff doing the representing (the words).
one comment. there is a map:
words in language A <---> meaning <---> words in language B

assuming that the meaning in question is expressible in both language A and language B, it's the same meaning independent of language. at least, independent of languages that can express that meaning.

it may be the case that a statement is absolutely true IF AND ONLY IF it occurs on the list of statements i gave earlier. in that case, the meaning of the statements is almost nil, if not completely banal.

another direction.
if i replace 'there are at least n true statements' with any statement, don't your comments still apply? i guess I'm saying i don't buy the statement that it doesn't mean anything outside of hte language it is spoken in.
 
  • #10
Hi phoenixthoth,

When we can say "there is at least one true statement" I think we automatically in some induction system where the above sentence is the first object, and our own cognition is the induction's engine, which has the property of self reference cybernetic system.

Any event in such a system is "no event" or "infinitely many events".

In any research, the relations between the object and the subject and their properties, can't be ignored.

What do you think?
 
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  • #11
Here is a great circle...
"There is not a single true statement" If this is true then of course it is not.

John
 
  • #12
If "There is not a single true statement" it implies that "" XOR "there are infinitely many true statements" XOR "there are finitely many true statements".

But what about a non-Boolean logic?

In non-Boolean logic there can exist simultaneously two opposite states like:

A) "there is at least one absolutely true statement".

B) "there are infinitely many true statements".


In this case we can get a complementary object, which is the result of an association between infinitely many (points-like) singletons {.,.,.,.,.} and a one continuous (line-like) element {________}.

Through this point of view, there are connections between structure's symmetry-degree and information's clarity-degree.

High Entropy means maximum level of redundancy and uncertainty, which are based on the highest symmetry-degree of some system.

For example let us say that there is a piano with 3 notes and we call it 3-system :

DO=D , RE=R , MI=M

The highest Entropy level of 3-system is the most left information's-tree, where each key has no unique value of its own, and vice versa.
Code: 
<-Redundancy->
    M   M   M  ^<----Uncertainty
    R   R   R  |    R   R
    D   D   D  |    D   D   M       D   R   M
   {.,  .,  .} v   {.,  .,  .}     {.,  .,  .} 
    |   |   |       |   |   |       |   |   |
3 = |   |   |      {|___|_} |      {|___|}  |
    |   |   |       |       |       |       |
   {|___|___|_}    {|_______|}     {|_______|}
    |               |               |
An example of 4-notes piano:

DO=D , RE=R , MI=M , FA=F
Code: 
------------>>>

    F  F  F  F           F  F           F  F
    M  M  M  M           M  M           M  M
    R  R  R  R     R  R  R  R           R  R     R  R  R  R
    D  D  D  D     D  D  D  D     D  R  D  D     D  D  D  D
   {., ., ., .}   {., ., ., .}   {., ., ., .}   {., ., ., .}    
    |  |  |  |     |  |  |  |     |  |  |  |     |  |  |  |
    |  |  |  |    {|__|_}|  |    {|__|} |  |    {|__|_}|__|_}
    |  |  |  |     |     |  |     |     |  |     |     |
    |  |  |  |     |     |  |     |     |  |     |     |
    |  |  |  |     |     |  |     |     |  |     |     |
   {|__|__|__|_}  {|_____|__|_}  {|_____|__|_}  {|_____|____}
    |              |              |              |

4 =
                                   M  M  M
          R  R                     R  R  R        R  R
    D  R  D  D      D  R  D  R     D  D  D  F     D  D  M  F
   {., ., ., .}    {., ., ., .}   {., ., ., .}   {., ., ., .}    
    |  |  |  |      |  |  |  |     |  |  |  |     |  |  |  |
   {|__|}{|__|_}   {|__|}{|__|}    |  |  |  |    {|__|_}|  |
    |     |         |     |        |  |  |  |     |     |  |
    |     |         |     |       {|__|__|_}|    {|_____|} |
    |     |         |     |        |        |     |        |
   {|_____|____}   {|_____|____}  {|________|}   {|________|}
    |               |              |              |


    D  R  M  F
   {., ., ., .}    
    |  |  |  |
   {|__|} |  |
    |     |  |
   {|_____|} |
    |        |
   {|________|}
    |
 
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  • #13
Any tautology over a sufficiently small set is true. 1 = 1 is a true statement.
 
  • #14
Originally posted by phoenixthoth
note: the statement was this:
there are either infinitely many or none.

there was not an effort to claim which one was the actual case.

It doesn't matter, since the "truth" could be that there is a finite number of true statements, and that yours is not one of them.
 
  • #15
Originally posted by selfAdjoint
Any tautology over a sufficiently small set is true. 1 = 1 is a true statement.

I've wondered about that. How do you logically deduce that this is the case? And after having done so, how do you call this conclusion "true" when deductive logic is itself incomplete?
 
  • #16
1=1 is a necessary truth because we made it so. It is a definition. We said that 1 = 1...and that is how it is true. There need not be a proof of it. It is defined thus.

Necessary Truth.

Phoenixthoth
I didn't mean to say that our words have no meaning, but more importantly, their meaning only exists in the things they are representing. By saying things like 'there is a statement' doesn't really mean anything because those words don't refer to anything. They aren't grounded in anything real. We are losing site of reality in amidst all of these words.

Words are only here to give us a way of communicating ideas about reality that we experience. We have to try our best not to confuse those words with reality. (IMO)
 
  • #17
Absolutely true statement:
there are infinite number of absolutely useless statements. (including this one)

this sums it up :wink:
 
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  • #18
there are probably infinitely many false statements as well.

the flaw in this whole absolute truth thing is that it rests on logic which is not provably absolutely true. i can define 1=1 and i can define 1!=1. of course, one theory is a whole lot more useful than the other, and to say that "1!=1 is true" is pretty banal and absurd, but i can do it if i want.

revised statement: if you accept the rules of logic then there are infinitely many true statements.
 
  • #19
It doesn't matter, since the "truth" could be that there is a finite number of true statements, and that yours is [sic] not one of them.
if there are finitely many true statements that means there are either none or at least one.

already saw why it can't be none assuming non-contradiction.

then isn't the statement "there is at least one true statement" true?

if so, assuming logic, then there are infinitely many:
"there are n true statements (n>1)" is true.
then, "there is at least one true statement" follows. the the argument above gives infinitely many true statements, assuming mutual exclucivity.
 
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  • #20
Originally posted by phoenixthoth
revised statement: if you accept the rules of logic then there are infinitely many true statements.

Not much better, phoenixthoth. Accepting logic is actually what lead me to the conclusion, previously, that "there are an infinite amount of true statements or there are no true statements" is paradoxical. The "rules of logic" are what don't allow this statement, ITFP.
 
  • #21
Originally posted by phoenixthoth
if there are finitely many true statements that means there are either none or at least one.

already saw why it can't be none assuming non-contradiction.

then isn't the statement "there is at least one true statement" true?

if so, assuming logic, then there are infinitely many:
"there are n true statements (n>1)" is true.
then, "there is at least one true statement" follows. the the argument above gives infinitely many true statements, assuming mutual exclucivity.

But saying "there is at least one true statement" could be false. If there are no true statements, then this is not a true statement, and we are fine, aren't we?
 
  • #22
Originally posted by Another God
1=1 is a necessary truth because we made it so. It is a definition. We said that 1 = 1...and that is how it is true. There need not be a proof of it. It is defined thus.

Necessary Truth.

You're just saying that it's an axiom. This may be so, but doesn't logic require that all statements be tested?
 
  • #23
as posted at least twice now...

"If there are no true statements, then this is not a true statement, and we are fine, aren't we?"

if there are no true statements, then the statement "there are no true statements" is true. but since there are no true statements, it is false. so it's both true and false. assuming the law of noncontradiction, this means it is not the case that there are no true statements. therefore, there is at least one true statement. hence, "there is at least one true statement" is true.
 
  • #24
Originally posted by phoenixthoth
as posted at least twice now...

"If there are no true statements, then this is not a true statement, and we are fine, aren't we?"

if there are no true statements, then the statement "there are no true statements" is true. but since there are no true statements, it is false. so it's both true and false. assuming the law of noncontradiction, this means it is not the case that there are no true statements. therefore, there is at least one true statement. hence, "there is at least one true statement" is true.

Hold on a second, I'd considered that, and I have reason to believe that it is not so strong an objection as it may seem:

If I say "there are no true statements", and it turns out that there really are no true statements, then my statement cannot be true... that is your objection, right? Well, think about it, my statement isn't "true" it's "trueandfalse". You see, I never said "there are no statements that are both true and false", I just said there were none that were completely true.

Does this work?
 
  • #25
the whole argument is embedded in a two-valued logic system. in my own investigations of russell's paradox, under certain assumptions, i can find a statement with truth value 0.5 (if truth values are in the interval [0,1] where 0 reduces to F and 1 reduces to T in two-valued logic). one can ask how large is the set of statements having truth value x where x&isin;[0,1]. I'm not sure. in a three-valued logic system, if the truth value of "there are no true statements" is the third value, then it's not clear to me how there would be infinitely many true statements. in a two-valued logic system, "there are no true statements" can't be true and sine there are only two values, it is false. hence, there is at least one true statement and "there is at least one statement" is true. it seems that if we embed math in a logic with at least three truth values, then there can be a set containing all sets as elements without russell's paradox. there seems to be a tradeoff. when more truth values are allowed, there is more freedom to create objects which may contradict someone's "common sense."
 
  • #26
I see where you're going with this. Moving away from a true/false binary logic to a "gray area" where there is true,false, and truefalse. The ironic part is that real life is full of these "truefalse" answers, but math cannot accommodate it the way it should.

Or am I missing the point?
 
  • #27
fuzzy logic and/or three-valued logic may accommodate it.
 
  • #28
There are no absolute truths in the universe, that's the only absolute truth there is! swallow that one, all you paradox freaks!
 
  • #29
suppose "there are no absolute truths" is an example of an absolutely true statement.

the following are also:
"there is at least one absolutely true statement." note that since this is different from "there are no absolute truths", the total number of absolute truths is > 1.

let n be the number of absolutely true statements. so far, n>1 because it's at least 2:
1. "there are no absolute truths"
2. "there is at least one absolutely true statement."

now there are three choices:
1. n<2 (not possible by the list above)
2. n=2
3. n>2

suppose n=2. then a third absolute truth is "there are exactly two absolutely true statements," so our list becomes:
1. "there are no absolute truths"
2. "there is at least one absolutely true statement."
3. "there are exactly two absolutely true statements."

but wait, now n=3!=2 (more precisely, n>2 and not n=2). hence choice 2 isn't possible.

hence n>2.

let k be a number that is at least 2. we know that for k=2, n>k is true (base case). now we wish to show the indcution step: n>k implies n>(k+1). this, combined with the base case, will show that n is infinite.

suppose n>k (where k is at least 2). then we have the following k truths:
1. "there is at least one absolutely true statement." (ie "n>0")
...
k. "there are at least k absolutely true statements." (ie "n>(k-1)")

if these were the only absolute truths, then n=k. but n>k, by induction, so we can add another truth:
(k+1). "there are at least k+1 absolutely true statements." (ie "n>k")

now there are three choices:
1. n<(k+1) (not possible by the list of k+1 truths above)
2. n=(k+1)
3. n>(k+1)

suppose n=(k+1). then a (k+2) absolute truth is "there are exactly (k+1) absolutely true statements," so our list becomes:
1. "there is at least one absolutely true statement." (ie "n>0")
...
k. "there are at least k absolutely true statements." (ie "n>(k-1)")
(k+1). "there are at least k absolutely true statements." (ie "n>k")
(k+2). "there are exactly (k+1) absolutely true statements." (ie "n=(k+1)")

but wait, we have k+2 absolute truths though n=(k+1). hence, choice 2 is not possible and so it must be 3: n>(k+1). the induction is complete.

hence there are infinitely many absolute truths.

more generally, if there is at least one absolute truth, then there are infinitely many.
 
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  • #30
Originally posted by phoenixthoth
suppose "there are no absolute truths" is an example of an absolutely true statement.

the following are also:
"there is at least one absolutely true statement." note that since this is different from "there are no absolute truths", the total number of absolute truths is > 1.

let n be the number of absolutely true statements. so far, n>1 because it's at least 2:
1. "there are no absolute truths"
2. "there is at least one absolutely true statement."

now there are three choices:
1. n<2 (not possible by the list above)
2. n=2
3. n>2

suppose n=2. then a third absolute truth is "there are exactly two absolutely true statements," so our list becomes:
1. "there are no absolute truths"
2. "there is at least one absolutely true statement."
3. "there are exactly two absolutely true statements."

but wait, now n=3!=2 (more precisely, n>2 and not n=2). hence choice 2 isn't possible.

hence n>2.

let k be a number that is at least 2. we know that for k=2, n>k is true (base case). now we wish to show the indcution step: n>k implies n>(k+1). this, combined with the base case, will show that n is infinite.

suppose n>k (where k is at least 2). then we have the following k truths:
1. "there is at least one absolutely true statement." (ie "n>0")
...
k. "there are at least k absolutely true statements." (ie "n>(k-1)")

if these were the only absolute truths, then n=k. but n>k, by induction, so we can add another truth:
(k+1). "there are at least k+1 absolutely true statements." (ie "n>k")

now there are three choices:
1. n<(k+1) (not possible by the list of k+1 truths above)
2. n=(k+1)
3. n>(k+1)

suppose n=(k+1). then a (k+2) absolute truth is "there are exactly (k+1) absolutely true statements," so our list becomes:
1. "there is at least one absolutely true statement." (ie "n>0")
...
k. "there are at least k absolutely true statements." (ie "n>(k-1)")
(k+1). "there are at least k absolutely true statements." (ie "n>k")
(k+2). "there are exactly (k+1) absolutely true statements." (ie "n=(k+1)")

but wait, we have k+2 absolute truths though n=(k+1). hence, choice 2 is not possible and so it must be 3: n>(k+1). the induction is complete.

hence there are infinitely many absolute truths.

more generally, if there is at least one absolute truth, then there are infinitely many.

Hi phoenixthoth, Have you tested this with kinesiology?
 

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