Logic problems - not for the weak

[JadziaDax]

Symbolize the following sayings, revealing as much of their internal structure as possible and indicating the intended meanings of your abbreviations:

1. Everyone who believes in God obeys all of His commandments.

2. Everyone who has benefited from a scientific discovery owes money to some scientist or other.

3. If everyone has benefited from scientific discover or other, then some people haven't paid all of their bills.

[TIO]

Where the hell did those symantics come from? I've never seen those forms before.

[cowlick]

Okay - I don't feel like finding a way to post good notation for logical quantifiers and such.


There exists no X such that X beleives in God and does not follow God's commandments.
Let B(x) be the statement - X believes in God
Let F(x) be the statement - X follows God's commandments
Not(ThereExists(x) such that (B(x) and (Not(F(x)))
In EnglishEveryone who believes in God obeys all of His commandments

okay folks... you all talk about logic a lot... somebody else join in and prove it.

[H12]

Uh...truth be told, I'm confused. I don't understand where and how you're getting all those symbols, even in the examples.

[CSflim]

Originally posted by H12
Uh...truth be told, I'm confused. I don't understand where and how you're getting all those symbols, even in the examples.

It's Boolean Algebra. (Or whatever the non-Computer Science term for it is)
Of course you're confused! You can't expect somebody who has never heard of Calculus to understand what dy/dx means!

[meff]

OOUUCCHH my brain my brain!

*brain pours out of ears*

[dimbulb]

Originally posted by JadziaDax
Symbolize the following sayings, revealing as much of their internal structure as possible and indicating the intended meanings of your abbreviations:

1. Everyone who believes in God obeys all of His commandments.

2. Everyone who has benefited from a scientific discovery owes money to some scientist or other.

3. If everyone has benefited from scientific discover or other, then some people haven't paid all of their bills.

fun fun, just had to do this for my last assignment. frigging real analysis class. funfunfun ...

1. FOR ALL x element of X, P(x).
X = {y element of {all people}| y believes in God}
P(x): x obeys all of the commandments

[dimbulb]

ok let x.E.X represent x is an element of the set X

2.
FOR ALL x.E.X, THERE exist a y.E.Y s.t. P(x,y).
X = {y.E.{all people}|y has benefitted from scientific discoveries}
Y = {y.E.{all people}|y is a scientist}
P(x,y): x owes money to y.

[JadziaDax]

not quite, dimbulb... need to follow the original syntax.

[dimbulb]

isn't it possible to conduct the logic on the quantifier/statement level?

Not sure what was wrong with my interpretation in those terms. Not sure I fully understand your original syntax though. I can guess at it, but I fail to see the need to break it down so much.

[JadziaDax]

Because that is what her professor wants. He wants that syntax. And without that syntax, it's not correct.

I'm not the professor, I'm just trying to figure it out myself. I haven't seen this crap in over 10 years. But when your professor (who is giving you your grade) says "Do it like this", you don't argue it.

[SkanK0r]

So wait wait wait, you're having us do your homework!?

[QuasiMojo]

So like.....everyones a barber?

excuse me when I say...

WWHHat?!

everything breaks away at B

[JadziaDax]

Originally posted by SkanK0r
So wait wait wait, you're having us do your homework!?

What part of "I'm a teacher, not a student" don't you understand?

[QuasiMojo]

It's Boolean Algebra. (Or whatever the non-Computer Science term for it is)
Of course you're confused! You can't expect somebody who has never heard of Calculus to understand what dy/dx means!


__________________


Excuse me CS Lewis but there IS NO dy/dx.

__________________________________

aren't I clever?

)

[thechao]

Originally posted by JadziaDax
2. Everyone who has benefited from a scientific discovery owes money to some scientist or other.

3. If everyone has benefited from scientific discover or other, then some people haven't paid all of their bills.

Statement 2 is false. Money is a specific economic function for transfer of debt. Scientific discovery has nothing to do with scientists being paid, since a beneficiary does not necessarily imply a debt - called the "free lunch principle." It happens all the time in real life. By extension, Statement 3 is trivially true, since any statement which takes a contradiction as an assumption can lead to any conclusion.

[JadziaDax]

It's not a question of true or false...

Originally posted by JadziaDax
Symbolize the following sayings, revealing as much of their internal structure as possible and indicating the intended meanings of your abbreviations

[Grothendieck]

ok i'm taking it with me on my weekend vacation...

[nothingx]

Blah. I hate logic. If I tried, I could write it out, but I'm not even going to think about writing it in ASCII. I could do it in LaTeX, but then I gotta post it and all that other crap. I did this stuff in my second year in college. Oh yeah, and it would be a hell of a lot easier in first order predicate logic.

[Grothendieck]

moelester: nice to know i'm not the only one using latex. but i would say that this *is* first order predicate logic. just not standard notation -- Jadzia, is this some Principia Mathematica Whitehead/Russell notation?

Anyway, here's my try. I'm writing 3 for the "there exists" predicate. Plus I don't believe in square brackets and the like. And I'm too lazy for LaTeX too.
And I'm putting "." for "and", and "=>" for "leads to".

1. (x)((Px . Rx) => (y)((Cy . Gy) => Oxy))

P means "person", R means religous, C means commandment, G, means of Godly origin, O means obey.

2. (x)((Px. (3y)(Sy . Bxy)) => (3y)(S'y . Oxy))

S means scientific disovery, S' is scientist, B is benefit, O is owe.

3. ((x)(Px => (3y)(Sy . Bxy)))=>((3y)(Py.~ ((z)((B'z.Dyz)=>P'yz))))

P, B, S as in 2, B' is a bill, D means debt, P' means pay a bill.

All this with the proverbial grain of salt. Typos are hard to avoid without TeX.

[David2000]

This is precisely why computers will never have human intelligence--they think in symbols and geegaws!

[JadziaDax]

Originally posted by Grothendieck
Jadzia, is this some Principia Mathematica Whitehead/Russell notation?

Honestly couldn't tell you where this horrible notation was coming from. The page that the problems and examples are on is a photocopy from the text and all the book information is missing. When this particular student tried to get a copy of the book (to find a better explanation of what was going on), she was denied. The professor for the class is a total prick.

Here's the scenario so you can understand a little better. This student needs this one class to graduate. She had to miss time in the class due to severe illness, and she called the professor to see if there was any way she could still get credit for the course (being that it's the ONLY course she needs). His answer was to give her a total of 9 problems from some text that is nothing like what they went over in class.

So, she took the challenge and struggled with these problems for quite a while. Then, she decided to try to get others to help her. She's tried everyone she knows all over the country. No one had a clue. Her aunt, the secretary for my building, asked if I could help out. I looked at the papers and was about to go out of my mind. I couldn't even find a text in my personal library to help with this notation. I had, nonetheless, written something down and decided to bring the problems here to get another opinion.

There are six more problems (but I don't believe they are of this nature).

So, I thank everyone who participated and gave these horrendous problems a try.




Oh, btw, this is for a course in law school, if you can believe that.

[dimbulb]

geeez, this professor sounds like a total asshole. So there's no information whatsoever on which book this came from?

i'm trying to figure it out..... but here's something i found.
http://www.jwrider.com/lib/logicnotation.htm
turns out that the symbol i thought mean "subset" turns out to mean imply...... aahhhh...things may not look so bleak after all.....

[dimbulb]

ok, here's my translation of statement A:

FOR ALL x, such that x is an element of P, THEN THERE EXISTS y, such that y is both an element of P, and an element of {y: x knows y}

so
(x)[........] means for all x, the following in the brackets holds
reversed C means "implies" or THEN or ==>

Py = set of all Persons
Kxy = set of all persons that know x.

does it make sense? or am i blowing smoke out of my ass??

[dimbulb]

more on notation, the "dot" should mean the intersection of 2 sets.....

so according to how i understand it, the first question is....

1. (x)[(Px * Bx)==> (x)(Px*Cx)]

If x is an element of the intersection of the set of all ppl and the set of all Believers in God, then x is an element of the set that is the intersection of the set of all people, and the set of obeyers of the Commandments

Px={x: x is a person}
Bx={x: x believes in God}
Cx={x: x obeys all commandents}

[Grothendieck]

Originally posted by dimbulb
more on notation, the "dot" should mean the intersection of 2 sets.....

dimbulb: The notion of "logic" in the standard foundations of mathematics is a precursor to the notion of "set theory". Logic can be done thinking only about writing symbols on a piece of paper in a coherent way (i.e. following metamathematical rules).

In logic, you have things called "terms" and "statements", without any "surrounding set theory". Thus the notation (x) just means "for all x", not "for all x in a given set", cf. the fact that there is no set mentioned. The dot means "and", its as simple as that.

Part of the mathematical discipline of "model theory" is finding models for logic in set theory. This means replacing statements like "Px" by explicit statements about explicit sets. "For all x" gets replaced by a statement about all sets in a given "big set" which we call universes. We need these universes to avoid set theoretical paradoxes, the best known of which is the set containing all sets which do not have themselves as elements. Does that set contain itself as an element?

Nobody really wants to know about these things though, except professionals, do they? I agree with David2000 on that one...

[dimbulb]

well, i'm not trying to be the expert here. I'm not a mathematician, and have no foundation in logic. I'm just trying to help out. hmm, actually, i'm an engineer trying to convert myself to an applied mathematician, but thats besides the point. Whatever i do know is from my own readings, and sadly, its obviously not very complete. As my prof says, 4 years of an education as an engineer has screwed my math foundation, if i did have one in the first place.

I do know about models and whatnot though, and about Russell's paradox and whatnot. oh wells, i'm glad that whatever i'm going to do, won't involve too much of delving into this logic "foundation". Think i can simply start from the second floor, as opposed to the basement of mathematics....
And if you think about it, and intersection is some sort of AND. heh...

[Grothendieck]

And I'm not trying to be clever, just trying to avoid misconceptions, and help those interested sort things out. They abound in the "foundations of mathematics".

You won't have to delve deeply. Mathematics, and foundations of mathematics, have parted ways as scientific disciplines.

Yes of course and is the same as an intersection. Analogies is what maths is about.

And your prof is wrong. Ask him about the mathematician called Jean Leray