[Last Call] Learn how to a build a cloud-first strategyRegister Now

x
Solved

# Prove this argument with rules of inference

Posted on 2009-04-25
Medium Priority
1,082 Views
How do I use the rules of inference to show that the following is logically valid?

[(~p V q) -> r] /\ [s V ~q] /\ [~t] /\ [p -> t] /\ [(~p /\ r) -> ~s] => ~q

I've used Matlab to check that it is actually valid, but I need to show a proof using the rules of inference.
0
Question by:rhianwen
• 3
• 3

LVL 85

Accepted Solution

ozo earned 2000 total points
ID: 24234482
suppose
[(~p V q) -> r] /\ [s V ~q] /\ [~t] /\ [p -> t] /\ [(~p /\ r) -> ~s]  /\ q

then
[(~p V true) -> r] /\ [s V false] /\ [~t] /\ [p -> t] /\ [(~p /\ r) -> ~s]
[ r] /\ [s] /\ ~p /\ [(~p /\ r) -> ~s]
[s] /\ [~s]
0

Author Comment

ID: 24234524
Why can you put the q on the end there? Also if its a contradiction doesn't that prove the argument invalid? I know the argument is valid because I used Matlab to construct a truth table to check it.
0

LVL 85

Expert Comment

ID: 24234531
[(~p V q) -> r] /\ [s V ~q] /\ [~t] /\ [p -> t] /\ [(~p /\ r) -> ~s]  /\ q
is invalid, so
[(~p V q) -> r] /\ [s V ~q] /\ [~t] /\ [p -> t] /\ [(~p /\ r) -> ~s] => ~q
is valid
0

Author Comment

ID: 24234724
So using the rules of inference I can say

H1     (~p V q) -> r
H2     s V ~q
H3     ~t
H4     p -> t
H5     (~p /\ r) -> ~s
----------------
~q

so

H3 & H4
~t /\ p -> t
=> ~p            (6)  (modus tollens)

H5 & (6)
~p /\ (~p /\ r) -> ~s
=> ~p /\ ~p -> (r -> ~s)               (exportation)
=>  ~p -> (r -> ~s)
=> r -> ~s                (7)

this is where I get stuck. In my head I can do the proof in like 4 steps but I'm having trouble laying it out in terms of the rules.

0

LVL 85

Expert Comment

ID: 24234841
6 & 1
=> r  (8)

8 & 7
=> ~s  (9)

9 & 2
=> ~q

0

Author Comment

ID: 24234859
Ahh awesome thankyou!

For some reason I was thinking I could only use each statement once so I was getting stuck with the 6 & 1 bit.
0

## Featured Post

Question has a verified solution.

If you are experiencing a similar issue, please ask a related question

Does your audience prefer people in photos or no people? How can you best highlight what you’re selling? What are your competitors doing, and what can you do that is different and unique from them?  Continue reading to learn how to make your images …
When crafting your “Why Us” page, there are a plethora of pitfalls to avoid. Follow these five tips, and you’ll be well on your way to creating an effective page.
The viewer will learn how to dynamically set the form action using jQuery.
Learn how to create flexible layouts using relative units in CSS.  New relative units added in CSS3 include vw(viewports width), vh(viewports height), vmin(minimum of viewports height and width), and vmax (maximum of viewports height and width).
###### Suggested Courses
Course of the Month18 days, 14 hours left to enroll

#### 834 members asked questions and received personalized solutions in the past 7 days.

Join the community of 500,000 technology professionals and ask your questions.