Question

Rules of inference: Can I use a single premise twice to arrive at an argument or a conclusion?

Answer 1

Yup. But it's rare.

Answer 2

Okay, thanks. I was trying to solve this problem, but no luck
http://prnt.sc/d4g86p

Answer 3

oh this problem still? I forgot to get back to you; had a lot of hw sorry

Answer 4

let me take anothe rlook

Answer 5

So last time you used material implication to turn the implications into or statements right? what did you get?

Answer 6

Yes. I got q v (r -> ~s)
And p v r

But to be honest, I found that when I turned them into statements that it was harder to solve. I just kept getting "Stuck"

Answer 7

well you didn't go all the way for q v (r -> ~s)... reduce it further!
as for number 4, try de morgans...

Answer 8

de morgans on ~(p v w) I mean

Answer 9

But won't I need to have r -> ~s) separate from the q inorder to reduce it further?

Answer 10

q v (r -> ~s) = q v (~r v ~s)... but we know that the disjunctive v (or) function is associative, so we simply have q v ~r v ~s, right?

Answer 11

Ahh, alright. Let me try working it out again

Answer 12

yeah... also a remark... using material implication and converting to disjunctive v isn't necessary, but even though it's slower, I like to do it as a form of canonical and methodical problem solving for these types of problems... that way it's easier to catch mistakes

Answer 13

in fact, it's possible to reason out the problem with modus tollens and a single application of de morgans on the fourth statement.

Answer 14

Essentially if you memorize the material implication law you will NEVER have to work around in circles worrying about modus tollens and modus ponens and syllogisms. All you need is the basics of conjunctive and disjunctive logic, which is extremely nice.

Answer 15

Ahhh, so true!

Answer 16

It will also help later on if you take classes like digital logic design or equivalent, where you are required to optimize boolean expressions via means such as karnaugh maps... implications are a moot point by then usually and it's easier to be able to think in AND and OR logic gates

Answer 17

I've used simplification on ~p ^ ~w and was left with ~p

Answer 18

mm you will need ~w later... it'd be smarter to separate them as
1. q v ~r v ~s
2. p v r
3. ~q v w
4. ~p
5. ~w
Take it from there :)

Answer 19

So I can separate ~p ^ ~w into ~p and ~ w?

Answer 20

Like as two separate arguments?

Answer 21

well you can use simplification on it both ways. ~p^~w implies by definition ~p and ~w holds... really it is implied, when you have multiple statements, that they are anded together.... like we are actually saying:

( q v ~r v ~s) ^ (p v r) ^ (~q v w) ^ (~p^~w)
which you can see equals
( q v ~r v ~s) ^ (p v r) ^ (~q v w) ^ (~p) ^ (~w)
due to the law of associativity

Answer 22

Oh okay. Let me try to solve it now

Answer 23

I think I got it to work now! But I realised that if I just left them as implications, that it was easier. Thanks so much!

Answer 24

yeah... either way works really... the material implication method is slower, but when you get a complicated chain of implications that seem to be circular, it's a godsend

Answer 25

:)