Thread: propositional logic

Hi, Can someone take a look if I have worked through this correctly? thanks

Question 4.
~r -> p, (~p) v q, ~s -> (~p)^(~r), (~p) ^ r -> (~s) v t, ~q, therefore t.

1.~q premise
~q vp premise
therefore ~p disjunctive syllogism

2.~p from 1
~r -> p premise
therefore r modus tollens

3.~p from 1
r from 2
therefore ~p^r conjunctive addition

4.~p^r from 3
~s -> (~p)^(~r) premise
therefore s modus tollens

5.~p^r from 3
s from 4
(~p) ^ r -> (~s) v t therefore t disjunctive syllogism

thanks

(4) is an incorrect application of modus tollens.

From ~s->(~p^~r), to get s by modus tollens, you need to have ~(~p^~r). Which rules of inference are you using? Can you use De Morgan's Law? Also, in (1), I think you mean, ~p v q, instead of ~q v p. And finally, in step 5, you seem to be compressing to steps into one.

I don't know about others, but a lot of your OP is gibberish in my browser. Any chance you could re-type it using LaTeX? You can use \neg for negation, \vee for or, \land for and, \to for implication, etc.

Thanks quiney

So in 4
can I use -p^r as a premise?
-p^r is same as -(-rvp) DeMorgans


5. -rvp, -s --> -(-rvp) therefore s Modus tollens

???

Since you already have r, try using disjunctive addition to get (pvr). Then use De Morgan's and see what you can get from there.

( ) <- my reference**

1. premise
premise
disjunctive syllogism

2. from 1
premise
modus tollens

3. from 1
from 2
conjunctive addition

4.r from 2


5. from 3
I don't know how to use DeMorgans here to get
I understand DeMorgans to be

??

Thanks much for the clarification. It reads much better now.

A couple of comments.

1. Your step 5 should reference step 4, not step 3.
2. I'm confused as to what exactly your premisses are, and what your desired conclusion is. If you're trying to show from the following premisses:



,

then I think your proof has just managed to disprove it. Why? Because you've proven , as well as . Therefore, is false.

Originally Posted by dunsta

5. from 3
I don't know how to use DeMorgans here to get
I understand DeMorgans to be

You're not trying to get ~p & ~r, you're trying to get ~(~p & ~r). (p v r) is equivalent to ~(~p & ~r). Remember, De Morgan's works both ways.

Then, use modus tollens and your third premise and you're almost done.

I'm trying to prove t

I can see from here that means s is true, then -s or t, results in t is true i.e. proving the conclusion.

I too am confused as to how to get there using the basic inference rules, Ackbeet.

I think step 5 needs to show that s is true using then I can go on to use the premise from 3 with to show t,

which is what Im working on now with

5. from 4
DeMorgans

6. from 5

modus tollens

7. from 3
s from 6
premise
disjunctive syllogism

??

I'm trying to prove t

I can see from here that means s is true, then -s or t, results in t is true i.e. proving the conclusion.

I too am confused as to how to get there using the basic inference rules, Ackbeet.

I think step 5 needs to show that s is true using then I can go on to use the premise from 3 with to show t,

which is what Im working on now with

5. from 4
DeMorgans

6. from 5

modus tollens

7. from 3
s from 6
premise
disjunctive syllogism

??

You've got it. Just clean it up a bit, and, if you wouldn't mind, post the entire proof like this:

Premiss 1
Premiss 2
...
-------------
1.
etc.

n. , from ____.

Make sense?

Thanks Ackbeet and quiney, I really mean that.
As it's late here, and I can sleep well now this is solved, I will post the full solution 2mw if you like Ackbeet.

Thanks for taking the time to help.

You're welcome for whatever help I was. I do think posting the entire solution will do you good. You'll be able to see better the flow of the whole solution.