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

[dybmh]

Any way, it might be helpful to read this:

Truth table - Wikipedia

en.wikipedia.org

From your link:

You're using the wrong truth table for the statement "All the Jews I know are atheists".

Ax. dog(x) and known(x) ---> brown(x)

"All dogs I know are brown"

Really this should be "(dog(x) and known(x) and brown(x))"

Notice the "ands". It's only true if all three conditions are met. That is logical and demonstrable. That's why it is a better method.

 

[dybmh]

The truth is it's a nonsensical statement.

What should be done with a nonsensical statement? Should it be considered true? Should all statements, nonsense or not, be assumed to be true unless it is proven false?

 

[Polymath257]

That is incomplete. Not P --> ( P --> Q ) is incomplete. Not P --> ( P --> ( Q AND/OR Not Q )) is complete.

Both are well formed logical statements. And both are true.

Then evaluating "true/false" using "only" the truth table is a poor method.

Sorry, but that is how it goes with propositional logic.

Not if it's noon.

If the car DOESN'T start and it IS noon, then how is the implication true?

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

"the car doesn't start and it's noon" is true. "IF the car starts THEN it's noon." is true.

The implication is true because F-->T doesn't test the implication, so it is not false.

Agreed. That's the 2nd row of the truth table.

No, it is that last two rows of the table. In both of those rows, p is F and is thereby irrelevant to testing the implication.

This only works with the fourth row where both p is false and q is false. If p is false and q is true, that is a valid counter-example.

No, it is a counter-example to the equivalence, not the implication. But in the discussion here, only the implication is required.

"In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy."

Counterexample - Wikipedia

en.m.wikipedia.org

Yes, so if p-->q and not q, then we can conclude not p. In more detail,
( (p-->q) and not(q) )--> (not p)
Now look at the truth table for this when q is T and p is either T or F. You will find that F-->T AND F-->F are both true implications.

Would you please reconstruct the following statement into an implication?

"All the Jews I know are atheists." Please try to make into something that is not too awkward. I think when you do that, no one will object to it, because when the statement is CHANGED into an "If ... Then" statement, it accurately communicates ignorance and becomes less absurd.

Ax ( known(x) and jew(x) ) --> atheist(x)

So, for all x, if x is known and x is a jew, then x is an atheist.

If it's OK to make changes to the statement to reduce it's absurdity, then it should be OK to make similar ( yet opposite ) changes to the statement to increase its absurdity.

"All I know are ... " =/= "I don't know any ..." makes a similar change, in scope, to the statement as the addition of "IF ... THEN".

Once again, you are saying = and I am claiming --> (more, accurately, <--).

I disagree that if the hypothesis is false that it implies anything, unless everything is assumed to be true unless it is proven false. That's not an implication, that is extreme optimism. I am an optimist, I know optimism when I see it. Optimism has a place and purpose and is useful. Evaluating true/false is not one of them.

Once again, sorry, but that is how the logic works. I challenge you to find *any* truth table for implication that doesn't have this.

You have agreed that the statement: "All the dogs I know are brown" is intentionally not talking about "dogs" if the speaker correctly states they do not know any "dogs".

It actually goes further than this, doesn't it?

The statement: "I don't know any dogs" was translated into: Ax not(dog(x) and known(x)).

not ( dog and known ) = ( not dog ) AND/OR ( not known ).

In logic, the default is and/or. But yes.

This is the negation of a conjunction. I won't be rude and call it "Basic Logic". But it is demonstrably true.

Yes, it is basic logic. So, at this point, we have
Ax ( ( not dog(x) ) or (not known(x) )

In other words, everything is either not a dog or is not known.

So, when a person says "All the dogs I know are brown" AND "I don't know any dogs". They are not talking about dogs, or they are not talking about knowledge, or they are not talking about dogs AND knowledge.

This produces 3 and ONLY 3 possible meanings:

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.

The statement

Ax not(dog(x) and known(x) )

is logically equivalent to

not (Ex (dog(x) and known(x) )

So, there is no thing that is both known and a dog.

 

[Polymath257]

From your link:

View attachment 77204

You're using the wrong truth table for the statement "All the Jews I know are atheists".

No, I am not. The correct translation is
if x is a jew and x is known, then x is an atheist.
It is an implication, not a conjunction.

"All dogs I know are brown"

Really this should be "(dog(x) and known(x) and brown(x))"

No, that is incorrect.
That is saying that everything is a dog and is known and is brown.

Notice the "ands". It's only true if all three conditions are met. That is logical and demonstrable. That's why it is a better method.

The 'ands' are incorrect here. If I say that all dogs that I know are brown, it is an implication, not a conjunction of all three. To say
Ax dog(x) and known(x) and brown(x)
says that everything is all three. And that is not what is intended.

Instead, you want to say that all dogs that I know (the hypothesis) are brown (the conclusion).

 

[Alien826]

OH! The definition of an empty-set as a subset is a contradiction? I have been saying this for weeks. In fact, "empty-set" IS "contradiction".

One day when walking on the stair
I met a man who wasn't there.
He wasn't there again today.
Oh how I wish he'd go away!

 

[Polymath257]

One day when walking on the stair
I met a man who wasn't there.
He wasn't there again today.
Oh how I wish he'd go away!

Yesterday upon a stair
I met a man who wasn't there
He wasn't there again today
I think he's from the CIA

 

[Alien826]

Any proposition.

In essence, a falsehood implies anything.

Should that be anything but p? Or is that implied in q (that is q is "anything but p")? Otherwise aren't we first saying not p, then p (as a part of q)?

 

[dybmh]

@Polymath257 ,

Please answer the following question:

Do you think it is a good idea to assume every statement is true until it is proven false?

Are people guilty until proven innocent?

We can go around and around in circles about how "All dogs I know are brown" should be translated into logical notation. But at the end of the day, it all comes down to this:

Do you think it is a good idea to assume every statement is true until it is proven false?

I asked, and you have declined to answer, which usually means it proves the oppostion.

I am claiming that this method being used to evaluate "All the dogs I know are brown" is not useful.

 

[Polymath257]

@Polymath257 ,

Please answer the following question:

Do you think it is a good idea to assume every statement is true until it is proven false?

No. But that is irrelevant to the logical question here.

Are people guilty until proven innocent?

No. But that is irrelevant to the logical question here.

We can go around and around in circles about how "All dogs I know are brown" should be translated into logical notation. But at the end of the day, it all comes down to this:

Do you think it is a good idea to assume every statement is true until it is proven false?

No. But that is irrelevant to the logical question here.

I asked, and you have declined to answer, which usually means it proves the oppostion.

I am claiming that this method being used to evaluate "All the dogs I know are brown" is not useful.

And that is wrong. The *logic* is simply different than you think it is. The only thing I can suggest is to go take a class in basic logic (propositional and quantifier).

 

[Polymath257]

Should that be anything but p? Or is that implied in q (that is q is "anything but p")? Otherwise aren't we first saying not p, then p (as a part of q)?

Nope. if p is false, then p-->p is still true. But, in the case that p is false, it is ALSO true that p-->not(p).

In fact, it is a logical truth that not(p)->(p->not(p)). In fact, a more general statement is true: q->(p->q).

 

[Polymath257]

Here's another link to read. This is about the term 'vacuously true', which happens in the situation of p-->q when p cannot happen.

Vacuous truth - Wikipedia

en.wikipedia.org

This has direct relevance to the current discussion.

 

[dybmh]

No. But that is irrelevant to the logical question here.

No. But that is irrelevant to the logical question here.

No. But that is irrelevant to the logical question here.

What is the purpose of logic?

And that is wrong. The *logic* is simply different than you think it is. The only thing I can suggest is to go take a class in basic logic (propositional and quantifier).

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?

to be a member of the empty set is a contradiction. It is always false. But then, for the empty set to be a subset of any set requires a false to imply anything.

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"

Here's another link to read. This is about the term 'vacuously true', which happens in the situation of p-->q when p cannot happen.

Vacuous truth - Wikipedia

en.wikipedia.org

This has direct relevance to the current discussion.

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.

What is useful about about a vacuous truth?

That is incomplete. Not P --> ( P --> Q ) is incomplete. Not P --> ( P --> ( Q AND/OR Not Q )) is complete.

Both are well formed logical statements. And both are true.

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

The statement 'every x is a y' is true if there are no x's

"'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.

This [vacuous truth] has direct relevance to the current discussion.

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.

In essence, a falsehood implies anything. This is a well-known, though somewhat counter-intuitive aspect of logic.

"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.

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

 

[Heyo]

One day when walking on the stair
I met a man who wasn't there.
He wasn't there again today.
Oh how I wish he'd go away!

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

 

[dybmh]

for the empty set to be a subset of any set requires a false to imply anything.

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.

 

[Heyo]

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.

What is useful about about a vacuous truth?





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



"'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.



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.



"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.

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

You are arguing against a well established subset of logic. Aristotle is dead (for 2344 years), you can't argue with him and you can't argue against a system that has survived for so long. Take it as a brute fact.
There are other subsets of logic which aren't binary.

 

[dybmh]

You are arguing against a well established subset of logic.

So what? Is this an untouchable institution? Does such a thing exist? Should such a thing exist?

Aristotle is dead (for 2344 years), you can't argue with him and you can't argue against a system that has survived for so long.

Aristotle's on my side on this one

"In classical logic, trivialism is in direct violation of Aristotle's law of noncontradiction"

Trivialism - Wikipedia

en.m.wikipedia.org

Take it as a brute fact.

You mean, "take a leap of faith"?

There are other subsets of logic which aren't binary.

I have brought 3 good reasons to evaluate the statement in a different manner.
I have brought an easy logical demonstrably effective alternative. ( Contradictions are always false. Look for contradictions first. )

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.

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

And in an implication, a false implies anything.

for the empty set to be a subset of any set requires a false to imply anything.

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

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

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.

 

[Heyo]

So what? Is this an untouchable institution? Does such a thing exist? Should such a thing exist?



Aristotle's on my side on this one

"In classical logic, trivialism is in direct violation of Aristotle's law of noncontradiction"

Trivialism - Wikipedia

en.m.wikipedia.org

You mean, "take a leap of faith"?



I have brought 3 good reasons to evaluate the statement in a different manner.
I have brought an easy logical demonstrably effective alternative. ( Contradictions are always false. Look for contradictions first. )

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.

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





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

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

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.

"You are shown a set of four cards placed on a table, each of which has a number on one side and a color on the other. The visible faces of the cards show 3, 8, blue and red. Which card(s) must you turn over in order to test that if a card shows an even number on one face, then its opposite face is blue?"

Please try to solve this. Then try to solve it using your revised Aristotelian logic.

(Puzzle taken from Wason selection task - Wikipedia)

 

[dybmh]

"You are shown a set of four cards placed on a table, each of which has a number on one side and a color on the other. The visible faces of the cards show 3, 8, blue and red. Which card(s) must you turn over in order to test that if a card shows an even number on one face, then its opposite face is blue?"

Please try to solve this. Then try to solve it using your revised Aristotelian logic.

(Puzzle taken from Wason selection task - Wikipedia)

It's an interesting puzzle. I'm not sure that it relates.

I'll give it a shot, but, to be clear, are you saying that I'm going to need either: a vacuous truth, accepting a contradiction as true, or the 3rd row of the truth table to solve the puzzle?

If not, I'm not seeing any obvious contradictions in the puzzle. And I'm not sure that it's solveable. And I'm OK with that. If there WAS an obvious contradiction, THEN it would be relevant.

So, is it relevant or not?

 

[dybmh]

"You are shown a set of four cards placed on a table, each of which has a number on one side and a color on the other.

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...

The visible faces of the cards show 3, 8, blue and red. Which card(s) must you turn over in order to test that if a card shows an even number on one face, then its opposite face is blue?"

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

Please try to solve this. Then try to solve it using your revised Aristotelian logic.

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

(Puzzle taken from Wason selection task - Wikipedia)

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.

 

[Zwing]

Yes or No? Does "All I know are ..." = "I don't know any ..."

Not at all. "All I know are ..." indicates that you know several of type A, and none of any other type. "I don't know any ..." indicates that you don't know any of type A.