## How to "prove" any statement is logically true.

For the discussion of math. Duh.

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### How to "prove" any statement is logically true.

The assumption is the person questioning you cannot accept logical paradoxes. You tell them this:
"This statement is true if and only if {C}.", where {C} is the statement you want to prove is true.

Let the statement be A.

Suppose C is not true.
If A is true, then C must be true because A asserts that. A contradiction.
If A is not true, then A and C are both false. However, this satisfies the condition for A, making A true. A contradiction.

Suppose C is true.
If A is true, everything is consistent. (A and C are both true)
If A is false, everything is consistent. (exactly one of A and C are true)
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Carlington
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### Re: How to "prove" any statement is logically true.

This doesn't prove anything about any particular statement, though. It just describes how the statements are related. If you assume A then you've assumed C, and if you assume not A then nothing regarding C follows from that. If you don't assume anything about either A or C at all, then you need to prove the truth value of C in any case to say anything about the truth value of A.
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### Re: How to "prove" any statement is logically true.

To summarize the argument:

(1) If C is false, then A is not true, but it is also not false.
(2) Since everything (including A) must either be true or false then C must be true.

(1) is valid logic , but (2) is not: "This statement is false" is neither true nor false. The statement A is similarly self-referential and so can validly be neither true nor false, meaning we can deduce nothing about C's truthhood.

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### Re: How to "prove" any statement is logically true.

1. The original post is wrong.
2. Neither of these statements is true.
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### Re: How to "prove" any statement is logically true.

FancyHat wrote:Axiom 1: The original post is wrong.
Axiom 2: Neither of these statements is true.

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### Re: How to "prove" any statement is logically true.

This probably belongs in Serious Business; it's philosophical logic, not mathematics. Anyway, it looks to be a version of Curry's Paradox.
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Qaanol
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### Re: How to "prove" any statement is logically true.

You only need one direction (the “only if” implication). In other words:

“If this sentence is true, then Santa Claus exists.”

Case 1: Santa Claus exists.
We are done: Santa Claus exists.

Case 2: Santa Claus does not exist.
Now the sentence reduces to, “If this sentence is true, then false.”

Case 2a: The sentence is true.
A statement “If P then false” being true implies P is false.
But here P = “this sentence is true”, so P being false contradicts assumption 2a.

Case 2b: The sentence is false.
A statement “If P then false” being false implies P is true.
But here P = “this sentence is true”, so P being true contradicts assumption 2b.

The assumption that “Santa Claus does not exist” leads inexorably to a contradiction, so that assumption must be false. Therefore Santa Claus exists.
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### Re: How to "prove" any statement is logically true.

I thought of it as finding a solution pair (A,B) which satisfies all conditions.
The premise is that the only possible solutions are: {AB,~AB, A~B, ~A~B}

In my case:
If A, you know the solution is in this set {AB, A~B}.
The "evaluation" of A produces another constraint: The solution must be in {AB, ~A~B}
Given A, the solution must therefore lie in the intersection of the two sets = {AB}.
Likewise if ~A, solution is the intersection of {~AB, ~A~B} and {~AB, A~B} = {~AB}.

In Qaanol's case:
If A, you know solution is in {AB, A~B} intersect {AB, ~AB, ~A~B} = {AB}.
If ~A, you know solution is in {~AB, ~AB} intersect {A~B} = {}.

I think this is similar to quantum entanglement. Qaanol entangled A and B, "measured" A to be true, and concluded B is true. I entangled A and B but didn't "measure" A, deducing that B must be true no matter what you measure A to be.

Also, I don't think this is philosophy. I don't think there was any step that is considered "unreasonable" in the realm of mathematical proofs. I considered difference cases of the problem, rejected any cases that led to a contradiction, and showed that what remains is what needed to be proven.

For example: Prove a^2 + b^2 = 0 mod 2 (where a,b,n are integers), is only true if a = b mod 2.

1. You consider the case a != b mod 2 and show that it contradicts the premise.
2. You consider the case a = b mod 2 and show that it is consistent with the premise.
3. QED. There is no question of "but what if the equation is a paradox?". By showing there is a consistent result, you proved it is not a paradox.
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TheGrammarBolshevik
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### Re: How to "prove" any statement is logically true.

Cradarc wrote:Also, I don't think this is philosophy. I don't think there was any step that is considered "unreasonable" in the realm of mathematical proofs.

It's not that it's "unreasonable" in the realm of mathematical proofs (which is not what philosophy is, anyway). It's that the subject-matter - particularly the principle about truth, T⌈φ⌉ ↔ φ - isn't part of the subject-matter of mathematics.

Cradarc wrote:I considered difference cases of the problem, rejected any cases that led to a contradiction, and showed that what remains is what needed to be proven.

OK, so you used deductive logic. Using deductive logic doesn't mean that you're doing math.
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Qaanol
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### Re: How to "prove" any statement is logically true.

We can in some sense say that the sentence is “half-true”. As in, let S = S→false, so that truth(S) = truth(S→false).

It seems intuitively plausible, given that truth(true→false)=0 and truth(false→false)=1, that truth(X→false)=1-truth(X). This entails truth(S)=1-truth(S) ⇒ truth(S) = ½.

So in a certain view, there is a resolution by eschewing the law of the excluded middle.
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### Re: How to "prove" any statement is logically true.

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### Re: How to "prove" any statement is logically true.

TheGrammarBolshevik wrote:
Cradarc wrote:Also, I don't think this is philosophy. I don't think there was any step that is considered "unreasonable" in the realm of mathematical proofs.

It's not that it's "unreasonable" in the realm of mathematical proofs (which is not what philosophy is, anyway). It's that the subject-matter - particularly the principle about truth, T⌈φ⌉ ↔ φ - isn't part of the subject-matter of mathematics.

Cradarc wrote:I considered difference cases of the problem, rejected any cases that led to a contradiction, and showed that what remains is what needed to be proven.

OK, so you used deductive logic. Using deductive logic doesn't mean that you're doing math.

I agree this isn't a math question. However, it is closely related to the principles that mathematics is founded on. I'm not making the claim that every statement is true, I'm showing that is possible to prove the claim without breaking any rules that apply to mathematical proofs.
If it were a philosophical issue, you can merely say reality doesn't support it and that is the end of it.
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Wildcard
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### Re: How to "prove" any statement is logically true.

Carlington wrote:This doesn't prove anything about any particular statement, though. It just describes how the statements are related. If you assume A then you've assumed C, and if you assume not A then nothing regarding C follows from that. If you don't assume anything about either A or C at all, then you need to prove the truth value of C in any case to say anything about the truth value of A.

This is exactly right.

But more to the point, there is a datum which is "true" (workable) both in philosophy AND in mathematical proofs, which is that if you make an assumption without realizing it, you are likely to get wrong answers. Your assumptions must be out in front of you, stated, where they can be tested or called into question, or else you will go on making invalid conclusions constantly.

@ original poster:
Your assumption is that everything that looks like a statement is a proposition. Except that you may not even know that a "proposition" by definition is something which is either true or false. The point is that assuming that something is a proposition, and therefore must be either true or false, and then demonstrating that in a certain case it cannot be either true or false, doesn't demonstrate anything at all. You never PROVED that what you stated even qualifies as a proposition.

Here are some examples of "logic" that you might enjoy. Work out the flaws if you can; it may assist your thinking. (Try to work out the EXACT logical flaws, not just hand-wave and say "that's ridiculous.")

"A fridge is something that always keeps things ice cold. But if I unplug this thing that you are calling a fridge, it doesn't keep the things in it ice cold. Therefore there is no such thing as a fridge."

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### Re: How to "prove" any statement is logically true.

"It is possible to refer to all non-self-referential sentences collectively in a single sentence."

Does that sentence refer to itself?

"Cradarc will never know that this sentence is true."

The rest of us can know, without paradox, that that sentence is true, because we can understand how Cradarc will be unable to take that sentence as true without then having to accept that it is therefore not true. But Cradarc will be trapped inside their own, personal paradox, forever... MWAHAHAHAHAHAAA!
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Qaanol
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### Re: How to "prove" any statement is logically true.

Wildcard wrote:Your assumption is that everything that looks like a statement is a proposition.1 Except that you may not even know that a "proposition" by definition is something which is either true or false.2 The point is that assuming that something is a proposition, and therefore must be either true or false, and then demonstrating that in a certain case it cannot be either true or false, doesn't demonstrate anything at all.3 You never PROVED that what you stated even qualifies as a proposition.4

1) Citation needed. Where does the OP make that assumption? Your claim is unfounded.

2) You have essentially restated the law of the excluded middle. This does not help in identifying statements to which it applies, especially when those statements are used as drop-in replacements for other statements in a standard method of proof.

3) Wow, you’re serious aren’t you? That train of thought—“Suppose ¬S. This implies X^¬X. Therefore S.”—is literally the format of every single proof by contradiction throughout mathematics. I have never seen such a proof that includes so much as a wisp of acknowledgement that any of its premises, or for that matter the statement X which actually gets contradicted, might not have a definite truth value.

4) Moreover, when you personally see a proof by contradiction, do you cry out, “This proof is incomplete, for it does not rigorously establish that the law of the excluded middle applies its premises and to the statement with which it derives a contradiction?”

We’re starting to bump up against intuitionistic logic now, and I don’t think anyone wants to go there. The point is, how can we identify “in the wild” statements to which the law of the excluded middle does and does not apply?
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### Re: How to "prove" any statement is logically true.

As with most such cases, this is more a game of "fun with English" than something applicable to mathematics (still neat, though). The problem is that it confuses meta and object language. Truth is semantical, and there is no (and in most cases cannot) obvious way to introduce some T into the object language so that if [A] encodes the formula A into the object language, then T[A] precisely when A is, actually, true. You certainly cannot do this with first order arithmetic (nor many weaker systems). As far as I am aware, in all but the most artificial cases, if you can encode formulas in any reasonable way, you aren't encoding truth (yeah, that's not perfectly precise, but it's close enough). This is closely related to Godel's theorems (it's actually a result of Tarski's).

So, as far as mathematics goes, if you're doing almost anything meaningful, you don't have anyway of actually expressing the paradox you've outlined - and, if you could, somehow, express truth in some system, it's probably going to be that you can't express your paradox (I'm not sure on this point, this is getting away from what I actually study - and it's going to be a weird system, or one that does nothing but show that it does this...and I'm doubting you'll have negation; or that the system would be consistent if it all worked, etc.).

Anyway, long story short, there is nothing, at a glance, wrong with your actual statements, just that there is no way to ever actually have those be statements in any real theory.

-----
@Wildcard
Would you care to elaborate? I'm not familiar with anything you're saying (maybe you mean something different than what it reads, though?), and it does not seem to be part of any of the usual approaches to logic (again, maybe I'm reading something wrong). If you are coming from the point of constructivisim, intuitionistic logic, or some such, that's outside of my area, so I have no comment, but it does not appear to be mainstream mathematical notions of the things involved (if I misread you, I apologize)
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Wildcard
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### Re: How to "prove" any statement is logically true.

Qaanol wrote:We’re starting to bump up against intuitionistic logic now, and I don’t think anyone wants to go there. The point is, how can we identify “in the wild” statements to which the law of the excluded middle does and does not apply?
Interesting question; it's much more philosophical than mathematical. But there's a rather simple (philosophical) answer, which is that you don't need to identify statements "in the wild" regarding applicability of the law of the excluded middle. You can just assume it is true always...but then keep in mind that your conclusions drawn from that premise only apply in the "universe" wherein the law of the excluded middle applies to all data you have applied it to.

That's fine in mathematics, where objects or symbols only have exactly the properties we define them to have (i.e. a graph is either 4-colorable or it is not, by definition of 4-colorable) but in plain English as applied to life, people, or philosophy, you can make a statement such as "George is not a clown any more." Is it true or false? "Let us assume this statement is true. Then George must have stopped being a clown at some point. An exhaustive search of all records indicates that this event—ceasing to be a clown—has never taken place in George's life. Therefore George is a clown." Law of the excluded middle doesn't hold here because of additional layers of meaning called implications, connotations, implicit assumptions, etc. In my example the sentence in question is predicated on the assumption that George WAS a clown at some point. If this assumption is false, the sentence itself cannot be stated to be either "true" or "false", much like the question "Do you still beat your wife?" asked by an Inquisitor can find anyone guilty whether they answer "yes" or "no".

As I said, this is much more in the realm of philosophical logic than mathematics...but so is the original post, as pointed out by TheGrammarBolshevik already.

On a reread of the thread, the post I most agree with is the very first answer to the OP:
Carlington wrote:This doesn't prove anything about any particular statement, though. It just describes how the statements are related. If you assume A then you've assumed C, and if you assume not A then nothing regarding C follows from that. If you don't assume anything about either A or C at all, then you need to prove the truth value of C in any case to say anything about the truth value of A.

Edit: I actually didn't reread my earlier post before posting so I only just now noticed that I already quoted Carlington once before!

Also, @Forest Goose: I probably mislead you in my earlier post when I said that my point about assumptions is true (workable) both in philosophy AND mathematics. It is definitely a philosophical point I agree with, and it can be used in mathematics, but since different fields of mathematical logic approach the subject of truth very differently, it certainly does not apply broadly to all mathematics.

Clarified now?
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### Re: How to "prove" any statement is logically true.

Wildcard wrote:That's fine in mathematics, where objects or symbols only have exactly the properties we define them to have (i.e. a graph is either 4-colorable or it is not, by definition of 4-colorable) but in plain English as applied to life, people, or philosophy, you can make a statement such as "George is not a clown any more." Is it true or false? "Let us assume this statement is true. Then George must have stopped being a clown at some point. An exhaustive search of all records indicates that this event—ceasing to be a clown—has never taken place in George's life. Therefore George is a clown." Law of the excluded middle doesn't hold here because of additional layers of meaning called implications, connotations, implicit assumptions, etc. In my example the sentence in question is predicated on the assumption that George WAS a clown at some point. If this assumption is false, the sentence itself cannot be stated to be either "true" or "false", much like the question "Do you still beat your wife?" asked by an Inquisitor can find anyone guilty whether they answer "yes" or "no".

That's rather glib, actually:

Stopping requires that a thing started for stopping to be true; to still be doing something requires that it began to be true - these are fairly indisputable, the problem is that we generally assume the starting is a given, when it isn't we add an explanation. I don't think anyone would object to, "No, I never started beating my wife.". You point this very thing out when you talk about implications and connotations, however, none of this entails that the statement lacks a truth value, only that what is falsifying it is something that isn't normally objected to.

Imagine we found out that water, really, was H20X (X some magical special thing made of weird matter), if you were drinking a glass of water and someone said, "Having some H20?", the correct answer is, "No", that that would be confusing doesn't modify the logical nature of the statement - if bystander Joe assumes that you aren't drinking water as a result, that's only because Joe is incorrect about what water is, not because there is no truth value.

Moreover, there are genuine notions that cannot be captured in object language, just as there are functions that cannot be computed, and others that cannot be proved, and etc. However, none of that, in anyway, entails that anything in the object language cannot have a truth value given a semantic interpretation - indeed, that's another good point, mathematical statements, syntactically, don't even have a truth value. In the language of groups, "the group is abelian" has no truth value (nor does any other such) till it is attached to an actual thing that it is speaking of.
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### Re: How to "prove" any statement is logically true.

Well, I would say that this is a philosophical proof that paradoxes are possible. However, I would not say that this is practical proof that paradoxes are possible.

You know, there is an old saying that in theory, theory is the same as practice, but, in practice, they are completely different.

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### Re: How to "prove" any statement is logically true.

waltmck wrote:Well, I would say that this is a philosophical proof that paradoxes are possible. However, I would not say that this is practical proof that paradoxes are possible.

You know, there is an old saying that in theory, theory is the same as practice, but, in practice, they are completely different.

If this was an actual paradox, it would be a major problem; you can't salvage naive set theory by just saying, "meh unrestricted comprehension is just theoretical". At any rate, there is no paradox here, just wrongness, albeit of a somewhat interesting variety and not of a stupid kind, but wrongness nonetheless.
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### Re: How to "prove" any statement is logically true.

It is an actual paradox; as I said above, it's the Curry Paradox (but with the conditional upgraded to a bionditional for no reason).
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### Re: How to "prove" any statement is logically true.

TheGrammarBolshevik wrote:It is an actual paradox; as I said above, it's the Curry Paradox (but with the conditional upgraded to a bionditional for no reason).

It's not a paradox that works in mathematics, in the sense that it is not something that can be stated in a working system (of sufficient strength) - so it's a mathematical paradox in the sense that the barber paradox is a disproof of just such barbers.
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### Re: How to "prove" any statement is logically true.

It's a linguistic "paradox", but not a mathematically logical one. Mathematically, the argument is that (A≡(A≡C))→C is a tautology in propositional logic, and indeed it is. (And as GB points out, (A≡(A→C))→C is also a tautology.) End of story.

In "What is the Name of This Book?", Smullyan explains the paradox in the form of a retelling of Portia's test from The Merchant of Venice. In it, Portia leads her suitor into a room with three caskets: gold, silver, and lead. The suitor is told that one of the caskets contains a dagger and the other two are empty. If the suitor can avoid choosing the casket with the dagger, he will win her hand in marriage. Each of the caskets has an inscription on it.

Gold: The dagger is in this casket.
Silver: This casket is empty.
Lead: At most one of these statements is true.

Which casket should the suitor choose?

Here is how Smullyan imagined him reasoning.
Spoiler:

And here is what happened when he made that choice.
Spoiler:
He opened the lead casket to find the dagger inside it! Portia opened up both the gold and silver caskets to show that they were empty, verifying that everything she told him about the test was accurate.

Why did that happen?
Spoiler:
Because you can write anything you like on a casket in the real world without causing a paradox! The suitor's error was in assuming that since the lead casket's statement was grammatically correct and seemed logistically well-formed, that it must be either true or false. Of course, as more easily seen in the Cretan paradox or the Jourdain double-card paradox, the Law of the Excluded Middle is not a valid assumption to make about sentences in natural languages.

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### Re: How to "prove" any statement is logically true.

Tirian wrote:It's a linguistic "paradox", but not a mathematically logical one. Mathematically, the argument is that (A≡(A≡C))→C is a tautology in propositional logic, and indeed it is. (And as GB points out, (A≡(A→C))→C is also a tautology.) End of story.

In "What is the Name of This Book?", Smullyan explains the paradox in the form of a retelling of Portia's test from The Merchant of Venice. In it, Portia leads her suitor into a room with three caskets: gold, silver, and lead. The suitor is told that one of the caskets contains a dagger and the other two are empty. If the suitor can avoid choosing the casket with the dagger, he will win her hand in marriage. Each of the caskets has an inscription on it.

Gold: The dagger is in this casket.
Silver: This casket is empty.
Lead: At most one of these statements is true.

Which casket should the suitor choose?

Here is how Smullyan imagined him reasoning.
Spoiler:

And here is what happened when he made that choice.
Spoiler:
He opened the lead casket to find the dagger inside it! Portia opened up both the gold and silver caskets to show that they were empty, verifying that everything she told him about the test was accurate.

Why did that happen?
Spoiler:
Because you can write anything you like on a casket in the real world without causing a paradox! The suitor's error was in assuming that since the lead casket's statement was grammatically correct and seemed logistically well-formed, that it must be either true or false. Of course, as more easily seen in the Cretan paradox or the Jourdain double-card paradox, the Law of the Excluded Middle is not a valid assumption to make about sentences in natural languages.

This is a really good way of putting it. It took me a while to think about, but this is a really good way of showing that philosophy does not necessarily apply to real life.

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### Re: How to "prove" any statement is logically true.

I just think the story reminded me of this more than revealing anything about the nature of paradoxes. Maybe I'm missing something.
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### Re: How to "prove" any statement is logically true.

Forest Goose wrote:It's not a paradox that works in mathematics, in the sense that it is not something that can be stated in a working system (of sufficient strength) - so it's a mathematical paradox in the sense that the barber paradox is a disproof of just such barbers.

Sure, it's not a mathematical paradox in the sense you describe (why would anyone have a working system that's susceptible to such a paradox?) But you said above that it's not an "actual paradox" and that it's "just wrongness." Work on the Curry Paradox in other areas of logic shows that this simply is not the case.
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### Re: How to "prove" any statement is logically true.

Totally agree with TGB. Consider that we still call Russel's paradox a paradox, despite the fact that it is resolved in much the same way that a mathematical version of Curry's paradox is resolved, by not allowing unrestricted comprehension.
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### Re: How to "prove" any statement is logically true.

It's just a matter of semantics, I think. Websters defines "paradox" as "an argument that apparently derives self-contradictory conclusions by valid deduction from acceptable premises". So there is room for subjectivity there; an argument that is "apparent" and "acceptable" to GB might seem patently flawed to FG. Perhaps we can all agree that this is a form of what mathematicians call Curry's Paradox and that Curry's Paradox is not an unchecked attack on the consistency of mathematics.

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### Re: How to "prove" any statement is logically true.

I'd accept that it is a paradox in the broad, more philosophically normal, sense of, "arrives at a seeming contradiction"*, with the distinction that an antinomy is a more genuinely problematic case. But, in the context of this thread, and the op, I was going with the more casual sense of using it for a genuine problem since it felt as if that was the question under discussion (,or it is the interesting question, at least, in terms of mathematics, I think).

But, really, it is a question of what term we are using, not of content at this point - the statement I was trying to make was, "In so far as mathematics, this amounts to nothing", specifically aimed at the post about theory and practice, since this has no bearing on either of those as pertains to mathematics (whatever one takes as delineating those two within mathematics). The content wasn't about if it was a paradox, but if it had any bearing on anything mathematically relevant (since it is in the mathematics forum).

*The Barber of Seville is a falsidical paradox, it seems to arrive a contradiction, but it is false as the assumption of the barber cannot work; for example.
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### Re: How to "prove" any statement is logically true.

Forest Goose wrote:Truth is semantical, and there is no (and in most cases cannot) obvious way to introduce some T into the object language so that if [A] encodes the formula A into the object language, then T[A] precisely when A is, actually, true. You certainly cannot do this with first order arithmetic (nor many weaker systems). As far as I am aware, in all but the most artificial cases, if you can encode formulas in any reasonable way, you aren't encoding truth (yeah, that's not perfectly precise, but it's close enough).

You say all these things as if they were…true.
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Forest Goose
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### Re: How to "prove" any statement is logically true.

Qaanol wrote:You say all these things as if they were…true.

Are you disputing that they are or can you encode the op's argument into firsst order logic in a system that contains arithmetic? I'm not sure the point you're trying to make, these concepts are well defined and it's a fairly old established result (unless you mean that I'm stating it/representing it poorly).
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Qaanol
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### Re: How to "prove" any statement is logically true.

My point is that formal logical systems, first-order or otherwise, are irrelevant to the topic at hand. Truth is a valid notion in natural language, and you yourself tacitly utilize it even while decrying its inability to be captured by formal systems.
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Forest Goose
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### Re: How to "prove" any statement is logically true.

Qaanol wrote:My point is that formal logical systems, first-order or otherwise, are irrelevant to the topic at hand. Truth is a valid notion in natural language, and you yourself tacitly utilize it even while decrying its inability to be captured by formal systems.

No, I saw a post about proving anything in a mathematics forum and responded to that as if it were about mathematics and regarding the mathematical validity of that - this isn't the "natural language and philosophy" forum, why would I respond to it as if it were nonmathematically meant?
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Qaanol
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### Re: How to "prove" any statement is logically true.

It is already evident that you recognize “logically true” and “derivable as a theorem in first-order logic” are not synonyms. The OP did not ask how to derive any statement as a theorem in first-order logic.

If the logic you are using cannot handle statements which talk about their own truth, then that logic is insufficient for analyzing natural-language statements. The topic of discussion here is not a nail that first-order logic can hit.
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Forest Goose
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### Re: How to "prove" any statement is logically true.

Qaanol wrote:It is already evident that you recognize “logically true” and “derivable as a theorem in first-order logic” are not synonyms. The OP did not ask how to derive any statement as a theorem in first-order logic.

If the logic you are using cannot handle statements which talk about their own truth, then that logic is insufficient for analyzing natural-language statements. The topic of discussion here is not a nail that first-order logic can hit.

...So, your criticism is what? It appears to be, "You discussed this mathematically in a mathematics forum, but it should be discussed philosophically, in this nonphilosophy forum!". The mathematical result, on its own, is interesting and certainly related (so topical, accurate, and interesting...), but I apologize to you, random internet person, for bringing up mainstream mathematics in a mathematics forum, I, obviously, should have abandoned standard mathematical logic and approached it in an entirely nonmathematical way.

Do you have some form of mathematical criticism to add, by the way, or are you really just upset that I approached this in a, despite being topical, way that you didn't like?
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arbiteroftruth
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### Re: How to "prove" any statement is logically true.

FG: The problem isn't that you're discussing it mathematically. The problem is that you're equating "can't be formalized in the standard logic of mathematics" with "can't be formalized".

Consider a three-valued logic with the values being T, F, and P (for True, False, and Paradox), and a connective ":=" declaring an atomic proposition symbol on the left side as being defined by the well formed formula on the right side, and implying they are equal if a consistent assignment of a classical truth value exists, and implying the atomic proposition being defined has a truth value of Paradox otherwise (in which case it is no longer necessarily logically equal to the defining formula). Let there also be a singular connective "\" indicating "is true".

Truth tables:

~T=F
~P=P
~F=T

\T=T
\P=F
\F=F

T&T=T
T&P=P
T&F=F
P&T=P
P&P=P
P&F=F
F&T=F
F&P=F
F&F=F

Other connectives are defined with & and ~ in the usual way.

The liar paradox can then be formalized as L:=~L, and we find that neither T nor F are consistent truth values for L, so it must be P, at which point the definition is no longer relevant, although in this case the definition also evaluates to P. The modified liar's paradox for three-valued logic, "this sentence is not true", is formalized as L:=~\L. Again we find that neither T nor F are consistent, so we assign a value of P. When L=P, ~\L=T, so unlike the first example L and its definition still don't agree, but in this system a definition only implies equality if the proposition being defined has a value other than P.

The paradox under discussion can be represented as A:=A->C. If C is true, then T is a consistent value for A, while F is not, so A is definitively true. If C is false, then neither T nor F are consistent values for A, so A is Paradox. In this case, defining A->C as ~(A&~C) and using the truth tables above, when A=P and C=F, A->C = P.

We can also formalize the dagger puzzle that Tirian posted.

G="the dagger is in the gold casket"
S="the dagger is in the silver casket"
L="the dagger is in the lead casket"

We are given that there is exactly one dagger, and it is in one of the three caskets. (GvSvL)&~((G&S)v(G&L)v(S&L))
The statement on the gold casket is G. The statement on the silver casket is ~S.
The statement on the lead casket is A:=~((G&~S)v(G&A)v(~S&A))
Let us assume that statements of physical reality about the dagger have classical truth values. So G, S, and L are all either true or false, and not paradox.

If we suppose A is true, there exists one consistent assignment of truth values. G=F, S=T, and L=F. Thus if A is true, the dagger is in the silver casket.
If we suppose A is false, again there exists one consistent assignment of truth values. G=T, S=F, L=F. Thus if A is false, the dagger is in the gold casket.
If we suppose A is paradox, we must remember that a proposition being defined is paradox only if no consistent classical value exists. There is only one assignment of truth values that does this. G=F, S=F, L=T. Thus if A is paradox, the dagger is in the lead casket. And we arrive at the correct result that reasoning about what's written on a casket doesn't actually force anything about the physical reality of where the dagger is.

Forest Goose
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### Re: How to "prove" any statement is logically true.

I never said it can't be formalized, in general. Indeed, Curry's original paper, and the one by Kleene & Rosser that it is named after, go right on ahead and deal with just such formalizations - this was a problem with Church's original system. All that aside, it is pretty damn obvious that I'm talking about modern mainstream mathematics, the kind used in almost all mathematics done nowadays - especially the less esoteric stuff...and even then...

As for your system: what exactly is ":=" is it a connective? A relation? Syntactic? Semantic? How do valuations work for your "logic" - the consequence relations too, how do they work?

It seems like a half shot at something like Priest's Logic, which, honestly, I'm not sure why you don't just go with that (or some other variation on paraconsistent multivalent propositional calculus), what you have now only resembles a logic in how you've written it, but it doesn't seem to have any of the things a logic does; better to go with something studied and working.
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Qaanol
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### Re: How to "prove" any statement is logically true.

It’s rather like those arguments that arise over improper sums. You know, when someone asks, “What is 1-1+1-1+1-1+…? And how about 1+2+3+4+5+6+…?”

Invariably, a person who has studied infinite series will say, “The definition of an infinite sum is the limit of the partial sums, and the partial sums here don’t converge, so there is no answer.”

Then someone else will come along and say, “If we use Cesàro summation, the first one is ½ but the second one still diverges.”

And the second person will say, “But that’s not what summation means, so whatever you’re doing it isn’t summation.”

Then a fourth person will arrive and say, “Actually, if we use zeta regularization, then the second sum is −1/12. This comes up a lot in physics and it gives the right answer.”

And perhaps a fifth person will say, “You know that Ramanujan summation handles both these series, and gives the same answers you’ve gotten piecemeal, right?”

But the second person will say, “You’re all just making up new meanings for summation, that’s not really math!”

It definitely is math, and inventing new systems and methodologies to extend old ones past their breaking points is an extremely common part of mathematics. That’s how we got the gamma function from the factorial, the complex log from the real logarithm, and things like the Dirac delta which don’t fit the strict definition of “function” but nonetheless work as intended.

Anyway, we can draw a direct parallel between “If this statement is true then Q” and “1-1+1-1+1-1+…”:
Spoiler:
Let P = “If this statement is true then Q”. This is equivalent to P = (P→Q). So let’s substitute:
P = (P→Q)
P = ((P→Q)→Q)
P = (((P→Q)→Q)→Q)

Jumping to the “last line” we get an infinitely-nested implication:
P = (((((⋯→Q)→Q)→Q)→Q)→Q)

The standard approach to infinitely-nested operations is to take the limit of partial applications:
P1 = (Q→Q)
P2 = ((Q→Q)→Q)
P3 = (((Q→Q)→Q)→Q)
P4 = ((((Q→Q)→Q)→Q)→Q)

And this sequence can be analyzed, because (Q→Q) = True and (True→Q) = Q.

So we have:
P1 = T
P2 = Q
P3 = T
P4 = Q

Now the partial-implications are {T, Q, T, Q, …}, while the partial sums of “1-1+1-1+…” are {1, 0, 1, 0, …}.

When Q=False, and we let T=1 and F=0, then the parallel is exact. Even when Q is not False, we still get a result of:
P = (1+Q)/2

Now our logic admits non-integer values. Does that make it “non-mathematical”? Of course not!

Fractions are still numbers even though they aren’t integers, algebraic numbers are still numbers even though they aren’t fractions, transcendental numbers are still numbers even though they aren’t zeros of polynomials over ℚ, complex numbers are still numbers even though they have imaginary components, and reasoning with non-binary truth values can still be logic!
Last edited by Qaanol on Thu May 14, 2015 7:18 pm UTC, edited 1 time in total.
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arbiteroftruth
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### Re: How to "prove" any statement is logically true.

Forest Goose wrote:I never said it can't be formalized, in general. Indeed, Curry's original paper, and the one by Kleene & Rosser that it is named after, go right on ahead and deal with just such formalizations - this was a problem with Church's original system. All that aside, it is pretty damn obvious that I'm talking about modern mainstream mathematics, the kind used in almost all mathematics done nowadays - especially the less esoteric stuff...and even then...

Okay, but then it's not clear why you're acting like this paradox is only a philosophical matter and not a mathematical one. "Not mainstream mathematics" =/= "not mathematics".

Forest Goose wrote:As for your system: what exactly is ":=" is it a connective? A relation? Syntactic? Semantic? How do valuations work for your "logic" - the consequence relations too, how do they work?

arbiteroftruth wrote:Consider a three-valued logic with the values being T, F, and P (for True, False, and Paradox), and a connective ":=" declaring an atomic proposition symbol on the left side as being defined by the well formed formula on the right side, and implying they are equal if a consistent assignment of a classical truth value exists, and implying the atomic proposition being defined has a truth value of Paradox otherwise (in which case it is no longer necessarily logically equal to the defining formula).

That is, given a statement of the form "A:=(wff)", if it is possible to assign A a value of true and have the wff on the right side also evaluate to true, or it is possible to assign A a value of false and have the wff on the right side also evaluate to false, then one may infer that A has one of the classical truth values and is logically equal to the wff. If (and only if) both classical truth values lead to A and the wff having different truth values from each other, one may infer that A has a value of paradox.

Forest Goose wrote:It seems like a half shot at something like Priest's Logic, which, honestly, I'm not sure why you don't just go with that (or some other variation on paraconsistent multivalent propositional calculus), what you have now only resembles a logic in how you've written it, but it doesn't seem to have any of the things a logic does; better to go with something studied and working.

What I've proposed is exactly Priest's Logic, with the addition of the ":=" connective for constructing definitions and potentially causing and identifying paradoxes.

You seem to expect everything that belongs in a math forum to be a fully-formed rigorous statement. But that reduces the math forum to an archive of short math papers, rather than a place for discussion of mathematical concepts. I suspect prolonged interactions with Treatid have made you reflexively hostile to the casual discussion of half-formed mathematical ideas.

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### Re: How to "prove" any statement is logically true.

Okay, but then it's not clear why you're acting like this paradox is only a philosophical matter and not a mathematical one. "Not mainstream mathematics" =/= "not mathematics".

That's true, but the part of mathematics that this might belong to is something symbol dense and not something most people, even in mathematics, know about - any general handling of this is going to end up citing papers or philosophical.

That is, given a statement of the form "A:=(wff)", if it is possible to assign A a value of true and have the wff on the right side also evaluate to true, or it is possible to assign A a value of false and have the wff on the right side also evaluate to false, then one may infer that A has one of the classical truth values and is logically equal to the wff. If (and only if) both classical truth values lead to A and the wff having different truth values from each other, one may infer that A has a value of paradox.

I understand what you're trying to say, the way you're saying it just rubs me wrong (mathematically, it doesn't in the general sense of aggravation) - but, as you mention subsequently, yes, I am generally suspicious, but not because of treatid. I've found for every person starting such a discussion with the intent to discuss the mathematics of it, genuinely, I've encountered about a thousand cases that end up with either bad philosophical debates about mathematics or someones grand theory that lets them do X, Y, and Z, none of which make any sense.

What I've proposed is exactly Priest's Logic, with the addition of the ":=" connective for constructing definitions and potentially causing and identifying paradoxes.

Your connective doesn't seem like a connective to me (and a few other things about it), is "(~/(L := /L) & (L:=(L -> ~L))) := L" a wff? If you put L := ~/L, what does "L := ~/L" valuate as?

You seem to expect everything that belongs in a math forum to be a fully-formed rigorous statement. But that reduces the math forum to an archive of short math papers, rather than a place for discussion of mathematical concepts. I suspect prolonged interactions with Treatid have made you reflexively hostile to the casual discussion of half-formed mathematical ideas.

I would like an archive of short mathematics papers, it tends to be how I, generally, see math - and most of why I can't do physics problems, despite knowing all the mathematics involved in it. That said, I'm very accustomed to anything touching the topic of logic and foundations to end up as either philosophical debate or insane conclusions that are not supported by anything - I don't mean to say that would happen here, per se, it's just what I've experienced as the norm a lot of places (I imagine this is how physics people feel when they encounter a thread about "Dark Energy and Blackholes", a lot of the people starting such threads have a lot of nonsense to say, generally, it's not shocking they are a bit more on their guard, so to speak).

All of that said, I still stand by what I've said: with regards to what is generally termed mathematics, this paradox is a nonstarter; and, personally, I think that, by itself, is sufficiently interesting to be worth saying. (You'll notice that I'm not trying to argue that other people can't put forward their own answers - if you want to discuss things related to Priest, etc., I'm not going to bitch and moan - you want to toss around the natural language and philosophy ball, I'm not going to bitch and moan - so, I'm a bit perturbed when my answer, which isn't false, is contentious since it doesn't attempt to preclude anyone else from tossing around half-formed mathematical ideas if they so chose).
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