/sci/ - Science & Math

[Anonymous No.12211810]

Is it true *today* that "There exists a student, such that if he passes this term, everyone will pass." It is certainly true that either everyone will pass, or there is someone who won't pass?

[Anonymous No.12211881]

>>12211810
If it is true, everyone will pass if he passes. If it isn't then everyone will fail.

[Anonymous No.12211882]

>>12211810
We won't know until after the fact.

[Anonymous No.12211894]

>>12211881
Not true. The condition states if he passes, everyone will pass, but the failure of one can still mean success for the others. Realistically if 100 people in class will pass including the student who fails, then his failure meets the criteria of not everyone passing. So the rest still might pass.

[Anonymous No.12211896]

>>12211894
Tl;dr between 0-99 people out of 100 would pass if he fails.

[Anonymous No.12211908]

>>12211810
that's just the gamblers fallacy written a different way

[Anonymous No.12211920]

who wins this? >>12211810

[Anonymous No.12211926]

Yes, there is some student who will have the lowest score in the course. If that student still passes then implicitly everyone will.

[Anonymous No.12211928]

>>12211810
If theres only one student, if he pass, everyone will pass
If there are two students, if one students pass and the other dont, only one may pass or everyone will pass

[Anonymous No.12211938]

>>12211928
in short, your chance of passing is still entirely dependent on you, otherwise, you rely on others passing and being the savior students

[Anonymous No.12212167]

>>12211810
If there is at least one student who does not pass, then the statement is true of that student because a false antecedent makes the statement true.
Otherwise, everyone passes, and the statement is true for all of the students.
So clearly, the answer is yes.

[Anonymous No.12212180]

>>12211810
Imagine: "The student" is the shittiest student with the lowest mark at the end. You don't have to know beforehand which student that is, just that after the exam there will be a most retarded student with the lowest mark.

[Anonymous No.12212191]

>>12211810
Assuming it's white to move the best move would be KF5, this would just get you in a position where black has a zugzwang. I think

[Anonymous No.12212213]

>>12211810
If everyone passes, you should ask for your money back. Grades aren't supposed to work that way.

[Anonymous No.12212224]

>>12211920
>>12212191
Whoever moves wins: move the kings to protect your pawns and there is nothing the other player can do.

[Anonymous No.12212306]

>>12212224
This. White to G5, or black to G4. Don't capture on the first move, force the other king to retreat and then capture.

[Anonymous No.12212356]

>>12211810
It's true in classical first order logic at least, yes.

For any predicate [math]\phi[/math],
[math] (\exists x. x=x) \implies \exists y. (\phi(y)\implies \forall z.\phi(z)) [/math]

In words,
"If there exists anything at all, then there exists a thing such that if it has the property [math]\phi[/math], then everything has the property [math]\phi[/math]"

Example:
"There is a table, such that if that table is a bird, then everything is a bird."

PROOF of the logical propositon:
Assume something exists.
Then for any predicate, it either holds for all terms in the universe of discourse, or there is something for which it doesn't hold.
In the first case, the property holds for everything and we're done.
In the second case, take the thing for which the property does not hold. Then assuming the property holds for it, we get a contradiction and everything follows from explosion. In particular, it follows that the property holds for everything.
QED

Of course, this proof uses LEM for an existence statement, uses explosion and uses material implication, so it's a non-relevant (in the sense of relevance logic) as you can get.
You can check out the Paradoxes of Material Implication Wikipedia article for more examples along those lines.

But to sum it up again in terms of the bird example: We know that your mom isn't a bird. So if we assume that your mom is a bird, we can prove everything.

It's the same thing with
>There exists a student, such that if he passes this term, everyone will pass.
If you use explosion on the student that fails the exam, you can formally derive stuff like "everybody passes", but the proof is not relevant in the relevance-logic sense.

[Anonymous No.12212369]

>>12211810
It's true in classical first order logic at least, yes.

For any predicate [math]\phi[/math],
[math] (\exists x. x=x) \implies \exists y. (\phi(y)\implies \forall z.\phi(z)) [/math]

In words,
"If there exists anything at all, then there exists a thing such that if it has the property [math]\phi[/math], then everything has the property [math]\phi[/math]"

Example:
"There is a thing, such that if that table is a bird, then everything is a bird."
PROOF (for the example, you can skip this and go on to the more formal proof below)
Not everything is a bird. We know that your mom isn't a bird. So if we assume that your mom is a bird, we can prove everything.

PROOF of the logical schema above:
Assume something exists.
Then for any predicate, it either holds for all terms in the universe of discourse, or there is something for which it doesn't hold.
In the first case, the property holds for everything and we're done.
In the second case, take the thing for which the property does not hold. Then assuming the property holds for it, we get a contradiction and everything follows from explosion. In particular, it follows that the property holds for everything.
QED

Of course, this proof uses LEM for an existence statement, uses explosion and uses material implication, so it's a non-relevant (in the sense of relevance logic) as you can get.
You can check out the Paradoxes of Material Implication Wikipedia article for more examples along those lines.


It's the same thing with
>There exists a student, such that if he passes this term, everyone will pass.
If you use explosion on the student that fails the exam, you can formally derive stuff like "everybody passes", but the proof is not relevant in the relevance-logic sense.

[Anonymous No.12212373]

>>12211810
It's true in classical first order logic at least, yes.

For any predicate ϕ,
(∃x.x=x) ⟹ ∃y. (ϕ(y) ⟹ ∀z.ϕ(z))

In words,
"If there exists anything at all, then there exists a thing such that if it has the property ϕ, then everything has the property ϕ"

PROOF of the logical schema above:
Assume something exists.
Then for any predicate, it either holds for all terms in the universe of discourse, or there is something for which it doesn't hold.
In the first case, the property holds for everything and we're done.
In the second case, take the thing for which the property does not hold. Then assuming the property holds for it, we get a contradiction and everything follows from explosion. In particular, it follows that the property holds for everything.
QED

Example:
"There is a thing, such that if that thing is a bird, then everything is a bird."
PROOF (same as above, just more specific case)
Not everything is a bird. We know that your mom isn't a bird. So if we assume that your mom is a bird, we can prove everything.

Of course, this proof uses LEM for an existence statement, uses explosion and uses material implication, so it's a non-relevant (in the sense of relevance logic) as you can get.
You can check out the Paradoxes of Material Implication Wikipedia article for more examples along those lines.

It's the same thing with
>There exists a student, such that if he passes this term, everyone will pass.
If you use explosion on the student that fails the exam, you can formally derive stuff like "everybody passes", but the proof is not relevant in the relevance-logic sense.

[Anonymous No.12212412]

>>12211810
We can certainly easily imagine scenarios where this works out. Finding them in real life is another issue