Logic buffs: Is this a tautology: (Got it, how about a new challenge?)

[Noriaki]

Got it, how about a new challenge

See my posts below about the BONUS questions

Doing excercises in practice for the midterm...but I have no solutions...and me and my friend are arguing about how this should be....

Any how, I have to translate this from natural language to formal logic and then use the Gensen proof system to decide if it's a tautology or not.

The attack will succeed only if the enemy is taken by surprise or the position is weakly defended. The enemy will not be taken by surprise unless he is overconfident. He will not be over confident if the position is weakly defended. Therefore the attack will not succeed.

A - the attack will succeed
S - the enemy will be taken by surprise
W - the position is weakly defended
C - the enemy will be overconfident


I'm going to use ~ for not. And I'll just use the words AND and OR because I don't have the proper symbols at my disposal. -> will do for implication.

We agree here:
First Sentence (S OR W) -> A
Third W -> ~C
Fourth ~A

First point of contention is how to translate the second sentence...it says Not S Unless C...
We're not totally sure how to translate that..

I think the logical content of Unless is an or...so it's ~S OR C
I know there is some extra implied meaning with unless, but I still think that it's an or...
Either they won't be suprised, or he will be overconfident.
It's not saying the exact same thing, but the logical content is the same I think.

He disagrees...but doesn't really have an alternative to offer.


The second disagreement is how to join them together.

I think it's First Sentence AND Second AND Third IMPLIES Fourth.
He thinks we should AND them all together...

Any other ideas....

 

[kev0ut]

the first one is wrong it should be:

A -> (S or W)

when they say ONLY IF it is reversed from what it would be if it were IF. (The attack succeeding implies that S or W happened)

The 2nd one:

S -> C

(The enemy being taken by surprised implies that the enemy was overconfident)

 

[FrontlineWarrior]

if not overconfident, then not taken by surprise
~P --> ~Q
which is to say
P --> Q
which translates to
if overconfident, then taken by surprise
which seems false.

the correct way to do it is as follows:
if not taken by surprise, then not overconfident
~Q --> ~P
which equals
Q --> P
which translates to
if taken by surprise, then overconfident
which sounds right.

 

[d0ofy]

I agree with kev0ut with regards to the first line being wrong. I think it should be A -> (S OR W)

On the Unless line, it can be done 3 different ways.
1) ~S OR C (It can be either case, or none of them, but never can both happen, satisfying the UNLESS)
2) S -> C
3) ~C -> ~S

 

[FrontlineWarrior]

edit: wrong conclusion. I didn't see that it was supposed to be ~A!!!

forgot what a tautology is, but the argument doesn't prove A. you need further information on whether C or ~C. my vague memory tells me this is what a tautology roughly means. i have no idea what a gensen proof is.

1. A -> (S v W)
2. S -> C
3. W -> ~C
4. C (~~C)
5. thus ~W
6. thus S
7. thus C, but already assumed C in (4)
note doesn't prove A, because even if ~W, there's still S, and even if there was no S, it would still be denying the antecedent.

or
1. A -> (S v W)
2. S -> C
3. W -> ~C
4. ~C
5. thus ~S
6. thus W
7. thus ~C, but already assumed in (4)

 

[Noriaki]

Ahh A -> (S OR W) perhaps that's why we can't get this thing to work...

A only if S OR W...
Yeah I think you guys are right....



<< On the Unless line, it can be done 3 different ways.
1) ~S OR C (It can be either case, or none of them, but never can both happen, satisfying the UNLESS)
2) S -> C
3) ~C -> ~S >>

Of course!
DUH! Damn I'm dumb right now...

P -> Q is logically equivilant to ~P OR Q

Obviously those are all the same...maybe it's time to go to bed and worry about this in the morning hehehe
thanks


A tautology means that it's ALWAYS true.
I'm pretty sure it's not...in which case all I really need is a counter example.

 

[d0ofy]

A tautology is when no matter what the circumstances are, you always get True.

 

[FrontlineWarrior]

fallacy - (noun) 1. the belief that your manhood is bigger than it is
nomical fallcy - (noun) 1. the belieft that naming your manhood will help it grow.

 

[FrontlineWarrior]

sorry for confusing. i'm having mental diarrhea.

 

[kev0ut]

im really glad im done with logic. it was quite useless. just like every other subject i've ever had to suffer thru

 

[FrontlineWarrior]

logic was pretty easy. we had the watered down version
i wish i still remembered the less obvious stuff. it sometimes comes in handy.
but overall i like the easy A. hehehe

 

[kev0ut]

me and my friend had to figure all this crap out for ourselves because our prof was a 70 year old mumbling idiot and nobody could understand him. i ended up getting a B because his tests were so rediculous.

have you ever tried reading a logic book? it makes my head spin. i think im going to vomit from just thinking about it.

 

[Noriaki]

Thanks guys

We got it, it was the A -> (S OR W) part that fixed it.

We interpret the sentence as:

([A -> (S OR W)] AND [~S OR C] AND [W -> ~C]) -> ~A

And this can be shown as not being a tautology (I have two, use S, A, C true, W false or W, A true, S, C false and the statement fails, so it's not a tautology), which I think is correct. Becuase the sentence doesn't guarantee an outcome I don't think. They could win, or they could lose. It's not guaranteed though, so it's not a tautology.

 

[Noriaki]

<< me and my friend had to figure all this crap out for ourselves because our prof was a 70 year old mumbling idiot and nobody could understand him. i ended up getting a B because his tests were so rediculous.

have you ever tried reading a logic book? it makes my head spin. i think im going to vomit from just thinking about it. >>

We don't have a textbook
Our prof doesn't really like any particular one. He uses bit of a couple, I have a good prof thankfully.
This stuff is confusing enough as is...

 

[Noriaki]

Here's one to try if you are feeling keen:

BONUS QUESTION. A propositional formula is "flakey" iff it is logically equivalent to a formula with strictly fewer variables. How many flakey formulas are there involving the variables A, B, and C? (1 mark).

Taken from an assignment in the class.
It's only one lousy mark, but that's not really the point now is it, it's a bonus challenge

He'll give us the answer to this in a couple days, I'll get back to anyone that wants to know


Here's another neat one, from my last assignment.

BONUS QUESTION A formula is "monotonic" iff it is logically equivalent to one which does not use not, implication or bijection (if and only if). How many logically different monotonic formulas are there using only the variables A and B? (1 miserable mark).
Clarification: How many forumlas are there using only the connectives AND and OR using two variables.

TRUE is a formula by itself, as is FALSE, as is P as is Q. Of course this only leaves 2 P AND Q, P OR Q for a grand total of six.

Ok that's not to terribly hard...but try it with P, Q and R rather than just P and Q.
That was Bonus #2 and he made it worth not even 1 miserable mark.



This i can give you an answer for, but I'll let you play with it

Remember FALSE, TRUE, P, Q, R are 5 formulas.
Now build more, using only the 3 variables and the connectives AND, OR.

 

[Noriaki]

PM me for the answer to the second bonus (it's from the previous assignment)

If you feel like you are doing my work for me, don't answer the first one (the flaky one) until tommorow afternoon since this assignment is due at 2:30 (Yes I have an assignment due the class before the midterm :|)

I don't want you to do my work for me, I just want to see what you guys think of the answer.