Does "All I know are ..." = "I don't know any ..."?

[Alien826]

You should have given credit to William Hughes Mearns who wrote it.

I would have if I had known it. I was just quoting it from memory, and I had the first and last lines wrong too!

 

[Alien826]

You mean, "take a leap of faith"?

I have learned from experience that disagreeing with @Polymath257 on the subject of mathematics is a fool's errand.

I'll add that mathematical rules are not always subject to intuitive examination. For example, raising a number to the power of zero equals one. There's a perfectly good proof that this true, but now try to imagine multiplying something by itself zero times. Doesn't make sense, right?

 

[dybmh]

I have learned from experience that disagreeing with @Polymath257 on the subject of mathematics is a fool's errand.

I'll add that mathematical rules are not always subject to intuitive examination. For example, raising a number to the power of zero equals one. There's a perfectly good proof that this true, but now try to imagine multiplying something by itself zero times. Doesn't make sense, right?

It's easily resolved from the perspective of "it's just a rule".

Dear Dybmh,

"All Jews I know are atheists" AND "I don't know any Jews" IS ridiculous. It's not actually true. It's not true by any stretch of the imagination. It's actually false, but, because of the rules of logic that I choose to use, it's easier to pretend that statements like this are vacuously true on occasion. Traffic engineers might use this technique sometimes to predict and prevent jam-ups during busy periods. And sometimes in solid-state electronic design, this is useful. And really, it's nothing more than playing devil's advocate. Which can be useful as long as it's balanced with a saint. But once it's taken out of those contexts, it's not useful and can be harmful.

This isn't something that's proveable. You don't have to believe me that it's true. People who try to prove it, always come up short, because, it's just a rule."

 

[Zwing]

It's easily resolved from the perspective of "it's just a rule".

Often the case in maths.

 

[Polymath257]

And this is the root cause of the problem. The principle of explosion is required as a result of the defintion. Unsound arguments are accepted not rejected in order to protect the contradiction which is built into the defintion.

When discussing the topic, either intentionally or unintentionally, incomplete statements are made which obscures that a contradiction is being accepted as truth.

Standard logic has the principle of explosion. If you want to do paraconsistent logic, you can avoid that principle. But you won't be doing standard logic that s accepted by pretty much everyone.

What is the purpose of logic?



The way implication is being defined is such that: "everything is true unless it is proven false." You admit that this is not a good idea. You have also admitted that if contradictions are being used in logic it is absurd and should be rejected. You have admitted that non-sequiturs are not filtered out using the method you are employing. You have admitted the claim describes something cannot exist.

Strike 1: "everythinng is true unless it is proven false"
Strike 2: "Contradictions are being accepted as true"
Strike 3: "non-sequiturs are not rejected"

What is useful about the method you are employing?



This is a perfect example. Translation of the above: "it IS a contradiction ... But I need that contradiction to be true in order match the defintion"



I've read it many many times.

From the link: "Such statements are considered vacuous truths, because the fact that the antecedent is false prevents using the statement to infer anything about the truth value of the consequent."

No inference can be made about the truth value.

OF THE CONSEQUENT. But the implication is true. The difference is between the truth value of p-->q and that of q.

What is useful about about a vacuous truth?

As the article points out, it is frequently the beginning of the induction to a general result. Many results in set theory start with a proof about the empty set and then show how to build larger sets from that.

"Both are true" is incomplete. "Both are vacuous truths" is better.
"Both are vacuous truths and neither are ACTUALLY true." is complete.

Vacuously true implications are true. They are a specific type of true statement.

"'every x is a y' is true if there are no x's" is incomplete. "'every x is a y' is vacuously true if there are no x's" is better.
"'every x is a y' is vacuously true if there are no x's, but not ACTUALLY true." is complete.

Vacuously true implications are still true.

These have direct relevance to the discussion. The discussion is incomplete without them.

Principle of explosion - Wikipedia

en.m.wikipedia.org

"In classical logic, intuitionistic logic and similar logical systems, the principle of explosion (Latin: ex falso [sequitur] quodlibet, 'from falsehood, anything [follows]'; or ex contradictione [sequitur] quodlibet, 'from contradiction, anything [follows]'), or the principle of Pseudo-Scotus (falsely attributed to Duns Scotus), is the law according to which any statement can be proven from a contradiction. That is, once a contradiction has been asserted, any proposition (including their negations) can be inferred from it; this is known as deductive explosion."​

​

"Due to the principle of explosion, the existence of a contradiction (inconsistency) in a formal axiomatic system is disastrous; since any statement can be proven, it trivializes the concepts of truth and falsity"​

From falsehood ---> anything is disasterous. It trivializes truth and falsity.

Sorry, but intuitionist logic isn't the general accepted logic. In fact, intuitionist logic has proven to be so stilted as to be useless in most cases.

"This is a well-known, though somewhat counter-intuitive aspect of logic" is incomplete.
"This is a well-known, though somewhat counter-intuitive aspect of logic which trivializes truth and falsity." is complete.

Trivializes.

Trivialism - Wikipedia

en.m.wikipedia.org

Trivialism is the logical theory that all statements (also known as propositions) are true and that all contradictions of the form "p and not p" (e.g. the ball is red and not red) are true. In accordance with this, a trivialist is a person who believes everything is true.​

​

The consensus among the majority of philosophers is descriptively a denial of trivialism, termed as non-trivialism or anti-trivialism.[3] This is due to it being unable to produce a sound argument through the principle of explosion and it being considered an absurdity (reductio ad absurdum).​

​

The concensus is the principle of explosion is unable to produce a sound argument. "A falsehood implies anything" is unable to produce a sound argument.

Simply false. ALL of modern math is based on it.

Not P ---> ( P ---> ( Q AND/OR Not Q ) is true )

( Q AND/OR Not Q ) is true

Trivialism = all contradictions of the form "p and not p" (e.g. the ball is red and not red) are true

Nobody is claiming trivialism. But that is not the issue here.

You wanted a good reason for the truth table for implication. Let's start with the proof by contradiction:

( (p-->q) and not(q) ) --> not(p)

In other words, if you assume something is true and it leads to a result that you know is wrong, then the original assumption was wrong: the thing was actually false.

Now, a logically true statement has to be true no matter what truth values the pieces have. For example, p or not(p). if p is T, then not(p) is F and the or statement is T. But, if p is F, then not(p) is T and again the or statement is T.

I think can agree on the truth table for and:

p | q | p and q
T | T | T
T | F | F
F | T | F
F | F | F

Now, it q is T, then not(q) is F and the expression (p-->q)and not(q) must be F. So in the implication ( (p-->q) and not(q) ) --> not(p)

we have, in part

p| q| (p-->q) and not(q) | not(p)
T| T | F | F
F| T | F | T

Hence for ( (p-->q) and not(q) ) --> not(p) to be logically true, we need the last implication to be T when the left is F and the right is either F or T.

And that is the proof that the truth table for implication must have a false implying anything. In particular, the implications
F-->F and F-->T must both be T.

One thing you seem to be having trouble with (and it is common to do so) is that the truth of an implication p-->q is different than the truth values of either p or q in general. The question is whether a deduction is valid or not, not whether the conclusion is true (if the hypothesis is false, the deduction itself can be valid, but the conclusion can be either T or F).

 

[Polymath257]

Let's look at the so-called truth table for implication. Row 3.

p | q | p --> q
F T T

This is complete rubbish.

No, it is precisely what it MUST be for logic to work correctly.

p = car starts
q = it's noon

Test #1: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is considered true?

Yes

Test #2: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is still considered true?

Yes.

Test #3: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is still considered true?
Test #4: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is still considered true?
Test #5: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is still considered true?
Test #6: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is still considered true?

Really? It doesn't matter how many times the test fails, this so-called truth table has chosen to ignore all those failures? It shouldn't ignore any counter-examples. But it ignores them all.

Total trash. Don't urinate on my back and say it's raining.

But in those cases the *implication* is, in fact, true.

What is useful about this method other than shielding an axiom from being challenged?

It is the basis of proof by contradiction. It is the basis of using a contrapositive to prove things. It is easy enough to prove that the truth table must be what it is for simple logical statments to be valid.

How about 3 reasons? I've given 3 reasons why it's not useful. You said "No. It's useful".

1) It makes the statement ( (p-->q) and not(q) ) --> not (p) a logical truth, which is should be. It is the principle of proof by contradiction.

2) It makes the statement (p and (p--> q) ) --> q a logical truth, which it should be (being the basic principle of deduction).

3) It makes the statement the statement ( (p-->q) and (q-->r) ) --> (p-->r) a logical truth, which it should be (the principle of chaining deductions).

3 reasons please?

@Heyo, you're welcome to respond too. Can you come up with 3 reasons for this? I'll grant you that it's useful in comedy, but that's not math or logic.

Well, we can go through the axioms for logic if you want a derive that not(p)--> (p-->q), in other words, if p is false, then it implies anything.

Do you really want a course in formal logic?

 

[Polymath257]

Often the case in maths.

Actually, very seldom. usually the rules have a proof, but the proof requires a knowledge of logic, which is sadly lacking in many people.

 

[Polymath257]

It's easily resolved from the perspective of "it's just a rule".

It's a necessary rule for standard patterns of deduction to be valid.

 

[Polymath257]

I have learned from experience that disagreeing with @Polymath257 on the subject of mathematics is a fool's errand.

I'll add that mathematical rules are not always subject to intuitive examination. For example, raising a number to the power of zero equals one. There's a perfectly good proof that this true, but now try to imagine multiplying something by itself zero times. Doesn't make sense, right?

Actually 0^0 as a limit can be *anything*. It is convenient to have 0^0 =1 in combinatorics since it gives the number of functions from the empty set to itself (exactly one--the empty function).

 

[dybmh]

OF THE CONSEQUENT

"All the Jews I know are atheists" is actually "All the Jews I know are ( atheists or not )"

"are atheists or not" is the consequent and it is being considered true.

Nobody is claiming trivialism.

Denying it makes sense, but I don't think it's actually true.

Vacuously true implications are true. They are a specific type of true statement.

Is a vacuous truth actually true?

Actual means it describes reality, it describes something that exists.

Vacuously true implications are still true.

Why not answer the questions completely? Why repeated omit parts of the answer?

"every x is a y' is true if there are no x's" is incomplete. "'every x is a y' is vacuously true if there are no x's" is better.
"'every x is a y' is vacuously true if there are no x's, but not ACTUALLY true." is complete.

And I didn't see an answer to the question: what is the purpose of logic? This is important, because, you said it was irrelevant to the topic that assuming everything is true is a poor method.

All the dogs I know are brown.
All the dogs I know are brown.
All the dogs I know are brown.

How are any of these true?

Show me a dog that I know that is not brown.

This is a poor method.

I can easily say, "Show me a dog that you know that is brown."

I can't show you one that isn't brown.
You can't show me one that is brown.

So, how are any of those statements be actually true if it says dog, but it cannot be a dog, and it says know, but it can't be known?

 

[dybmh]

It's a necessary rule for standard patterns of deduction to be valid.

Let's look at the so-called truth table for implication. Row 3.

p | q | p --> q
F T T

This is complete rubbish.

p = car starts
q = it's noon

Test #1: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is considered true?
Test #2: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is still considered true?
Test #3: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is still considered true?
Test #4: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is still considered true?
Test #5: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is still considered true?
Test #6: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is still considered true?

Really? It doesn't matter how many times the test fails, this so-called truth table has chosen to ignore all those failures? It shouldn't ignore any counter-examples. But it ignores them all.

How is this useful? How is this a standard pattern of deduction? It's the devil's advocate saying "try it again tomorrow. Maybe it'll work. Heeheeheehee."

 

[Polymath257]

Let's look at the so-called truth table for implication. Row 3.

p | q | p --> q
F T T

This is complete rubbish.

p = car starts
q = it's noon

Test #1: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is considered true?
Test #2: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is still considered true?
Test #3: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is still considered true?
Test #4: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is still considered true?
Test #5: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is still considered true?
Test #6: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is still considered true?

Really? It doesn't matter how many times the test fails, this so-called truth table has chosen to ignore all those failures? It shouldn't ignore any counter-examples. But it ignores them all.

How is this useful? How is this a standard pattern of deduction? It's the devil's advocate saying "try it again tomorrow. Maybe it'll work. Heeheeheehee."

Already answered

 

[Polymath257]

"All the Jews I know are atheists" is actually "All the Jews I know are ( atheists or not )"

No, it is not.

"are atheists or not" is the consequent and it is being considered true.

No, that is not the consequent of the original statement.

Denying it makes sense, but I don't think it's actually true.



Is a vacuous truth actually true?

Actual means it describes reality, it describes something that exists.

It is logically true.

Why not answer the questions completely? Why repeated omit parts of the answer?

I don't. You add unnecessary things and claim them to be the same. They are not.

"every x is a y' is true if there are no x's" is incomplete. "'every x is a y' is vacuously true if there are no x's" is better.

That just states a particular aspect of why it is true.

"'every x is a y' is vacuously true if there are no x's, but not ACTUALLY true." is complete.

It is just as true of an implication as any other.

And I didn't see an answer to the question: what is the purpose of logic? This is important, because, you said it was irrelevant to the topic that assuming everything is true is a poor method.

All the dogs I know are brown.
All the dogs I know are brown.
All the dogs I know are brown.

How are any of these true?



This is a poor method.

I can easily say, "Show me a dog that you know that is brown."

Which is irrelevant to determining the truth of the statement 'All the dogs that I know are brown'.

I can't show you one that isn't brown.
You can't show me one that is brown.

So, how are any of those statements be actually true if it says dog, but it cannot be a dog, and it says know, but it can't be known?

OK, you just need to take a class in this. It is too much to try to do in a forum such as this.

 

[Polymath257]

Let's look at the so-called truth table for implication. Row 3.

p | q | p --> q
F T T

This is complete rubbish.

p = car starts
q = it's noon

Test #1: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is considered true?
Test #2: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is still considered true?
Test #3: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is still considered true?
Test #4: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is still considered true?
Test #5: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is still considered true?
Test #6: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is still considered true?

Really? It doesn't matter how many times the test fails, this so-called truth table has chosen to ignore all those failures? It shouldn't ignore any counter-examples. But it ignores them all.

But the test never fails. In each case, the implication is true because the hypothesis is false. It really is that simple.

How is this useful? How is this a standard pattern of deduction? It's the devil's advocate saying "try it again tomorrow. Maybe it'll work. Heeheeheehee."

It is useful because it is a logical consequence of the standard axioms of logic. It is necessary to show that those standard techniques (like argument from contradiction) actually work.

 

[Polymath257]

Let's look at the so-called truth table for implication. Row 3.

p | q | p --> q
F T T

This is complete rubbish.

p = car starts
q = it's noon

Test #1: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is considered true?
Test #2: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is still considered true?
Test #3: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is still considered true?
Test #4: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is still considered true?
Test #5: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is still considered true?
Test #6: car doesn't start AND it's noon. "IF the car starts THEN it's noon" is still considered true?

Really? It doesn't matter how many times the test fails, this so-called truth table has chosen to ignore all those failures? It shouldn't ignore any counter-examples. But it ignores them all.

How is this useful? How is this a standard pattern of deduction? It's the devil's advocate saying "try it again tomorrow. Maybe it'll work. Heeheeheehee."

Maybe this will work. Is the following implication true or not?

(x>5) --> (x>2)

In other words, if x is more than 5, does that imply that x is more than 2?

I hope you agree that this is a valid implication. it is true for every number x.

Now, what happens when you take x=3? you get that F--> T is true.

What happens when you take x=1? you get that F--> F is true.

Remember that the implication is *always true*.

 

[Heyo]

First of all, we don't know how many colors are possible. If the colors are not limited to only read and only blue, then the puzzle is not solveable.

Second, we don't know that the cards have an established pattern. It could be that there is no pattern and it's completely random. Without a pattern, it is unsolveable.

Assuming only red and only blue, and that even numbers are restricted to one color, and odd numbers are restricted to another color...



Any card will do, if there is a consistent pattern, and only red or blue colors are on the cards.



Seems like a pretty simple application of the law of the excluded middle as long as some rather strong assumptions are made.



Let's see how I did.

Hmmmm.... they say I got it wrong. I'll need to read the entire page and think about it. Thanks.

Ex falso quodlibet (principle of explosion) establishes that you can't learn about the premise from a true consequence.
That is one of most common logical errors that is made when solving that puzzle.
"if a card shows an even number on one face, then its opposite face is blue"
does not say that if the colour side is blue number is even. It also doesn't say that if the colour side is not blue that the number is odd.
The symbol for implication is an arrow, it only works in one direction.

 

[mikkel_the_dane]

Ex falso quodlibet (principle of explosion) establishes that you can't learn about the premise from a true consequence.
That is one of most common logical errors that is made when solving that puzzle.
"if a card shows an even number on one face, then its opposite face is blue"
does not say that if the colour side is blue number is even. It also doesn't say that if the colour side is not blue that the number is odd.
The symbol for implication is an arrow, it only works in one direction.

@Polymath257 @dybmh

Okay, all 3 of you, we now based on the everyday world, observe by testing different people for the ability to think, that there are at least the following categories of humans.
Those who can't do certain tasks in logic and math.
A subset of those will claim that the only truth there is the correspondence theory of truth or other claim to the effect of all truth is independent of brains.
Those who can do logic and math as such.
A subset of those conflate logical truth with correspondence truth.
Those who understand that is 2 different kinds of truth in play here.

And then we go meta-cognition and ask if there is one coherent method for any version of truth, that applies to all of the everyday world, for which you get a positive answer and the answer is so far no.

Now all of these debate of truth, proof, evidence and all the other variants of the correct positive answer have one problem in common. They are all cognitive and here is the test of the limit of cognition. Can I think differently than you in some cases and get away with it. And the answer is yes as longs we can both act subjectively and get away with it.

So here it is for the problem of logic, math and in effect philosophy as rational.
If the everyday world is in the strong sense rational, then I can't be irrational and answer no. But I have just done so. And those people, who don't catch that they in their brain compare the world is rational versus the world is not rational will claim, it is wrong. But it is wrong due to how they think. And in effect they conflate all with the law of non-contradiction as per Aristoteles:
"It is impossible that the same thing belong and not belong to the same thing at the same time and in the same respect."

In modern terms for the same limited place and time for the same thing, it can't belong and not belong to the same thing and in same respect.
That is all good and fine, but you and I are not the same thing and if I can act different, you can't make it the same, because it is not the same.
That is the everyday limit of logic no matter how rational you are. And I have for falsify in the general sense, shown you the falsification of everything can be added up to be the same. I just act differently and I have done so in this post.

So now I swear as a former professional. <Beep> learn the limit of any sufficiently complex plan!!! It WILL break down the moment it comes into contact with the rest of humanity, if one or more humans can act different.
So for the <beep> plan to end all plan. It goes like this. If we just think and act the same, it will work. Well, yes, it is a beautiful idea but it doesn't survive contact with the rest of humanity.
I am in practice as realistic as it goes.

 

[Polymath257]

Ex falso quodlibet (principle of explosion) establishes that you can't learn about the premise from a true consequence.
That is one of most common logical errors that is made when solving that puzzle.
"if a card shows an even number on one face, then its opposite face is blue"
does not say that if the colour side is blue number is even.

Good so far.

It also doesn't say that if the colour side is not blue that the number is odd.

Actually yes, it does. That is the contrapositive, which is equivalent.

What it does NOT say is 'if the face is odd, then the color is not blue'.

The symbol for implication is an arrow, it only works in one direction.

 

[Polymath257]

Humans are notoriously bad at logic.

And this is the winning statement of the thread.

 

[Polymath257]

First of all, we don't know how many colors are possible. If the colors are not limited to only read and only blue, then the puzzle is not solveable.

Yes, it is. And with exactly the same answer.

Second, we don't know that the cards have an established pattern. It could be that there is no pattern and it's completely random. Without a pattern, it is unsolveable.

Irrelevant to the problem.

Assuming only red and only blue, and that even numbers are restricted to one color, and odd numbers are restricted to another color...

Not the condition of the problem.

Any card will do, if there is a consistent pattern, and only red or blue colors are on the cards.



Seems like a pretty simple application of the law of the excluded middle as long as some rather strong assumptions are made.

Nothing is required other than the statements in the problem itself.

Let's see how I did.

Hmmmm.... they say I got it wrong. I'll need to read the entire page and think about it. Thanks.

Yes, think it through. The problem, as stated, is solvable with no additional assumptions.