If this sentence is true....

[jedaisoul]

Gosh Jedaisoul, we seem to have concurrency problems all the time, we should live on different sides of the world!

Well, at least we are still talking...

 

Consider construct:

2) A <=> (A => :confused:

...as an axiom rather than as a consequence of the self-referential construct:

1) A = (A => :doh:

and regardless of what the devil A and B are. B can be calculated without further assumptions and is true; you need to suppose A = true only in the sub-calculation for (A => B) = true. This would not be disturbing because it follows from 2) as an axiom. What is disturbing is that it is consequential to the existence of statements of the form:

"If this statement is true, then B."

I have no training in formal logic, so statements like "B can be calculated without further assumptions and is true" are meaningless to me. It does not make sense to me that it should be true. Perhaps I need to do some serious reading about formal logic.

 

But anyway, I'm grateful that you have humoured me so far. Perhaps I may impose a little further... I'm still stuck on the view that the truth of the sentence "If this statement is true, then B." tells us nothing about the truth of B. So this may be crucial to my misunderstanding:

 

a) Do the words "this statement" above refer to:

i. "If this statement is true", or...

ii. "B", or...

iii. "If this statement is true, then B."

 

Hopefully, that may cause the "penny to drop". But if not...

 

B) Can you explain (without assuming any prior knowledge of formal logic on my part) why A <=> (A => B) means that B must be true?

 

c) Can you explain why the sentence "If this statement is true, then B." is of the form A <=> (A => B)?

 

Thanks.

[Qfwfq]

The answer to a) is iii

 

The answer to c) is that, if A is defined as X the two have the same value, which is to say that A <=> X, so A = X implies A <=> X. So:

 

If this sentence is true, then B.

 

is a case of A => B in which A stands for (hence coincides with) the whole thing. At this point replace the above X with (A => :confused:.

 

B = true isn't the meaning of construct 2), it is what follows from it. It is hard for me to know exactly what you lack in formal logic. Modus ponens is simply a name for the direct application of what we mean by an implication: If we know that A => B and we know that A = true, we can say B = true. Therefore in the calculus (A => :doh: = true and A = true leads to B = true. Another application of implication is called modus tollens: If we know that A => B and we know that B = false, we can say A = false. The argument essentially uses modus ponens. Taking the axiom:

 

A <=> (A => B)

 

and considering the hypothesis (only for the subargument):

 

A = true

 

using this and the <=> in the axiom in the forward sense (as =>), by modus ponens:

 

(A => B) = true

 

and so, again by modus ponens and the hypothesis:

 

B = true

 

so, by this subargument:

 

(A => B) = true

 

now applying modus ponens to this and the <=> in the backward sense: A = true

 

at this point, applying modus ponens again to the above two: B = true, QED.

[jedaisoul]

The answer to a) is iii

Thanks, I suspected that. I was assuming ii, which is why I came up with different answers.

 

...QED.

Thanks, I see your problem. Nasty. I'll think on this.

[jedaisoul]

The answer to c) is that, if A is defined as X the two have the same value, which is to say that A <=> X, so A = X implies A <=> X. So:

 

If this sentence is true, then B.

 

is a case of A => B in which A stands for (hence coincides with) the whole thing. At this point replace the above X with (A => :Glasses:.

Having given this some thought, I think the problem may lie with treating a conditional sentence the same as a statement. I would suggest that they are not the same. Take the sentence:

 

“If A then B”

 

A and B are statements. They may be true or false, dependent upon external facts.

 

“If A then B” is different. It can be formally true if it is correctly formed, irrespective of external facts. But to be logically valid requires:

i. A to be true, and...

ii. B to be true, and...

iii. There to be a consequential link between A and B.

 

For example take the conditional statement:

 

“If chickens can fly then cows are animals”.

 

“Chickens can fly” and “cows are animals” are statements, and both are externally verifiable as true.

Also “If chickens can fly then cows are animals” is formally true because it is correctly formed; irrespective of whether chickens can fly or cows are animals. But to be logically valid:

i. “chickens can fly” must be true (which it is).

ii. “cows are animals” must be true (which it is).

iii. There has to be a consequential link between chickens flying and cows being animals, (which there isn’t).

 

So the sentence “If chickens can fly then cows are animals” is more than just the juxtaposition of the two statements “chickens can fly” and “cows are animals”. It also implies a consequential link, which is not present. Therefore it is logically flawed.

 

Applying this to the sentence:

 

“If this sentence is true then the Earth exploded”.

 

It contains two statements “this sentence is true” and “the Earth exploded”. Both of which are factually false. But the sentence as a whole is formally true as it is correctly formed, (irrespective of the fact that both statements are false). But to be logically valid:

i. “This sentence is true” must be true (which it isn’t, because the Earth has not exploded).

ii. “The Earth exploded” must be true (which it isn’t).

iii. There has to be a consequential link between “this sentence is true” and “the Earth exploded”. In this case there is a consequential link, because the sentence would be true if the Earth had exploded.

So this fails, despite the consequential link, because the statements are factually untrue.

 

Putting this algebraically:

 

A => B is only valid if A and B are true, and there is a consequential link between A and B.

 

This has the benefit of not only excluding the self-referencing case (which is formally true), but also nonsense statements which are consequentially unrelated.

 

What do you think of the distinction I’m drawing between a formally true sentence, and whether it is logically valid? Also, could the requirement for B to be true be omitted from the test of whether the conditional sentence is logically valid, and still be effective?

[Qfwfq]

What do you think of the distinction I’m drawing between a formally true sentence, and whether it is logically valid?

OK your position, like some schools of logicians, differs from the standard. First, however, what you call "formally true" is usually considered a matter of "well formed" which is to say a syntactical matter in a formal system; specifically you are describing the requisite not for it to be an assert at all but instead for it to be an implication.

 

The requisites you pose for it being logically true are not standard and the first two are untenable:

Also, could the requirement for B to be true be omitted from the test of whether the conditional sentence is logically valid, and still be effective?

Both can and should be omitted.

 

A => B

 

I believe that according to any school of logicians, the above implication is considered true when, as I'll put it simply, "B is never false when A is true" and this does not require either A or B to be true; I don't know if your third requisite is shared by any schools and it isn't always easy to define formally. It is easier to affirm that, in a given case, B can be shown true without resort to A. This would be a matter of finding at least one argument (and beware of something in the argument being equivalent to using A), rather than of showing no argument exists. In practice one could only prove cases in which your third requisite is unfulfilled.

 

Two cases in which standard logic considers the implication true are:

1. A is identically false (e. g. "the moon is made of green cheese").
   
2. B is identically true (e. g. "cows are animals").
   

By identically I mean in every instance, unlike the case of "the apple is red". So an implication such as "If the moon is made of green cheese, then cows are animals." is a true one on both accounts (and therefore doubly true!!!). Any sane person will note how tremendously useful it is , and yet it is a true implication.

[jedaisoul]

OK your position, like some schools of logicians, differs from the standard. First, however, what you call "formally true" is usually considered a matter of "well formed" which is to say a syntactical matter in a formal system; specifically you are describing the requisite not for it to be an assert at all but instead for it to be an implication.

I agree.

 

The requisites you pose for it being logically true are not standard and the first two are untenable:Both can and should be omitted.

 

A => B

 

I believe that according to any school of logicians, the above implication is considered true when, as I'll put it simply, "B is never false when A is true" and this does not require either A or B to be true; I don't know if your third requisite is shared by any schools and it isn't always easy to define formally. It is easier to affirm that, in a given case, B can be shown true without resort to A. This would be a matter of finding at least one argument (and beware of something in the argument being equivalent to using A), rather than of showing no argument exists. In practice one could only prove cases in which your third requisite is unfulfilled.

Having given this further thought, I think that there are two "kinds" of statements, factual and conditional:

• A factual statement is of the form "A is B". E.g. "The moon is made of green cheese" or "Cows are animals".
  
• A conditional statement is of the form "If A then B". E.g. "If the moon is made of green cheese then cows are animals".

Hence I would suggest that there are two "kinds" of truth:

• Factual truth.
  
• Conditional truth.

As far as I can see, current formal logic does not distinguish between them. In statements of the form:

A => B, A can be factually or conditionally true (or both), and it seems to be treated exactly the same. If this is so, then this could be the root of the problem.

 

I.e. (A => :) => B

• Is conditionally true if A => B is conditionally true, but B is factually false.
  
• Is factually true if A => B is conditionally true, and B is factually true.

If this distinction is valid then, as well as using the words "conditionally true" and "factually true", we need symbols to distinguish conditional truth from factual truth. So I suggest that .t. and .f. be used to symbolise conditional truth, and .T. and .F. be used to symbolise factual truth.

 

Does that make sense?

[Qfwfq]

Er, what you call conditional statement is usually called an implication. I don't see reason for two kinds of truth, nor get what you mean by:

In statements of the form:

A => B, A can be factually or conditionally true (or both),

For instance I don't see how one would manipulate these truth values and what modus ponens and modus tollens would be like in a logic of this type.

[jedaisoul]

Er, what you call conditional statement is usually called an implication. I don't see reason for two kinds of truth, nor get what you mean by:...

For instance I don't see how one would manipulate these truth values and what modus ponens and modus tollens would be like in a logic of this type.

Hi Q, I intend to reply to your comments, but I'm busy reading up on formal logic, so I can (hopefully) use the correct symbols in the correct way.

 

At present I'm working my way through the truth tables, trying to put them into words, so I can understand them. For example, in the truth table for A -> B or A => B:

A	B	A => B
i.	T	T	T
ii.	T	F	F
iii.	F	T	T
iv.	F	F	T

I understand i, ii, and iv, but why in line iii is A => B True when A is False and B is True? I understand that if it were False, then the logic would be identical to A = B (equates), which is pointless, but can you put this relationship in to words, so I can understand it?

 

Anyway, I'm convinced there is something in the idea of two forms of truth, a conditional truth ( A -> ;) and a factual truth (A), which underlies the problem when a conditional expression is used in place of a factual one. So I will reply as soon as I can express it better...

 

Terry

[Qfwfq]

but why in line iii is A => B True when A is False and B is True?

The meaning of A => B poses no restriction on the value of B for instances in which A is false, therefore it has the same value in iii and iv.

[jedaisoul]

The meaning of A => B poses no restriction on the value of B for instances in which A is false, therefore it has the same value in iii and iv.

Thanks for the reply. That makes sense, of sorts, but if it poses no restriction, surely the result should be Undefined, rather than True?

 

I'm not necessarily expecting a reply to that question. It's something I'm going to have to think about to make sense of.

 

Thanks again...

[Qfwfq]

...but if it poses no restriction, surely the result should be Undefined, rather than True?

It's the value of B that's uninfluential (not undefined, it's an input parametre of the table). The two values of A => B are hence the same and are true.

 

Another way to put it: an instance in which A is false cannot be a counterexample to the implication.

[jedaisoul]

It's the value of B that's uninfluential (not undefined, it's an input parametre of the table). The two values of A => B are hence the same and are true.

 

Another way to put it: an instance in which A is false cannot be a counterexample to the implication.

Thanks, I understand now. How I would put it is that if A is False then B can be True or False, therefore A => B is True for both values of B.

 

On the wider question, the more I read about formal logic, the more I come to the conclusion that it does not correspond to verbal logic. As you said, in the truth table the value of B is an input. But in verbal logic (If A then :hihi: it is an output. That is a fundamental difference of meaning of an inference. Hence the problem when you try to treat real world situations by formal logic.

 

In which case I think you were right in the first place, formal logic is useless (for that purpose).

 

Which brings me back to my idea about two "types" of truth. I'm not sure how far I'm going to get, but I'm going to try to "symbolize" verbal logic. I'll try to make it correspond to formal logic as far as possible, but I doubt that is wholly possible. Any advice would be most welcome...

 

Terry

[jedaisoul]

OK, here's a start at symbolizing verbal logic...

 

Truth Table (Verbal)

 

“If A Then B” (A -> :hihi: Truth Table

[u]A	A -> B	B[/u]
T	t	T
T	f	U
F	t	F
F	f	U

“If This Sentence is True Then B” (A =>  -> B Truth Table

[u]A => B	A	B[/u]
t	T	T	
t	F	t
f	T	U
f	F	U

 

Notes:

T = Factually True

F = Factually False

t = Conditionally True

f = Conditionally False

U = Undefined

 

You will notice that:

1. Unlike conventional truth tables, B is the output (right hand column).
   
2. A is an input.
   
3. A -> B is also an input i.e. whether B is actually dependent on A.

The latter is a conditional truth as it does not tell us whether B is true or not, it just tells us whether the relationship is true. So, to distinguish it from factual truth, it is sybolized by t and f (lower case). If the relationship A -> B is false, B is always undefined.

[Qfwfq]

As you said, in the truth table the value of B is an input. But in verbal logic (If A then :hihi: it is an output.

B becomes an "output" or result (by modus ponens) when A => B is asserted true. Asserting the implication true strikes out row ii from your truth table and, in the only remaining row having A true (i), B is true.

 

In which case I think you were right in the first place, formal logic is useless (for that purpose).

Well, it isn't really useless, it is just disturbing that this argument exists and some folks would like to shed light on the matter. It is a more serious problem than "de contradictione sequitur quodlibet", which only means one should never admit contradicting axioms.

 

In terms of your truth table, Curry's argument boils down to i being the only row in which the first and third columns have the same value. It can however be argued verbally, as I did earlier in the thread, only this may be harder to follow because words are only words.

 

So, does anybody think it would make sense to tweak modus ponens in such a way that:

 

A <=> (A =>

 

being true does not imply B being true? I have however been trying other interesting considerations, e. g. on the meaning of the troublesome self-referential structures.

[Kriminal99]

Ahem, the problem Krim is in the foundations of logic itself, I'm seeking a discussion on how the trouble could be avoided. Note that the sentence does not assert its own truth, it only refers to it, while determining an assert's truth through external means is not the concern of logicians such as Curry or Tarski. Tarski's approach is also by means of avoiding self-reference, I ask whether it is necessary and sufficient. Unfortunately the argument is rather subtle and folks here seem to have difficulty appreciating it.

 

I started this thread in a jocular manner but let's get down to business; although the Stanford article spells the argument out in detail it does it in a less simple way, especially with the list it uses, let me do it more plainly. First we must show the implication is true, which can be done by supposing:

 

Hyp: The sentence is true.

 

and showing that the implied clause follows as consequence. As the sentence is the implication, this is true by the hypothesis itself, therefore by modus ponens it does follow:

 

Th: Planet Earth exploded.

 

Now showing that the thesis follows from the hypothesis means the implication is true, not just under the hypothesis. This requires no assumption, it is sufficient that a sentence may talk about its own truth. Using again the fact that the sentence is the implication (the other way around) and modus ponens trivially completes the argument.

 

Logically, this is no self-contradiction. It only makes logic useless and of course implies contradiction indirectly, by the fact that the argument works for every implied clause (and hence for its negation as well).

 

If this sentence is true, then God exists.

 

If this sentence is true, then God does not exist.

 

Absolutely catastrophic, if the problem isn't somehow avoided.

 

I'd rather say meaningless because 'an apple' is not a well-formed assert (subject without predicate).

 

Not every apple is red; not true and therefore false by tertium non datur. "A zebra has stripes" is true (according to experience) but the same would hold if at least one zebra had no stripes. From the logician's point of view, these are not determined without some axiom about apples or zebras.

 

Free choice of axiom, without which it is not determined. Neither choice is inconsistent, a formal system may include either.

 

I do not believe the problem is in the foundations of logic itself, unless the foundations of logic include the means by which we define words. I am aware of the "avoiding self reference" approach, but I think it is incorrect. You only need to avoid self reference when doing so violates the REAL definition of the involved concepts.

 

And by real, I mean a specific type of algorithm that gives a type of definition that solves all confusion and logical paradoxes of any kind.

 

At this point I have decided to stop using every message board on the internet as my main discussion medium and am focusing on finishing a book about it, but I will give an idea as to what type of definition I am referring to by comparing your responses to how I was really looking at the claims.

 

There are two ways to read "An apple is red". I did not mean it in the "Apples are red" sense. This would require a single counter example. It was meant to refer to a specific apple that would require further knowledge to determine the truthfulness of the statement.

 

You seemed to be aware of this when you said that it wasn't "a well formed assert". Tell me, where did you get the information that told you that true false statements require asserts that are different than that? Did you need to read something to know it? Or would you have known it anyways?