Associated Topics || Dr. Math Home || Search Dr. Math

### Invalid Logic Argument

```
Date: 9/9/96 at 19:44:13
From: Anonymous
Subject: Invalid Logic Argument

Dr. Math,

Show that the following argument is not valid:

[p=>r],[(not r) or s], [q or (not s)], [p and (not t)] yields [q=>t]

My logic was that if I want the result to be true and then have a
false premise, then the argument would be invalid.  Because I was
trying to make everthing true, I started with the conjunction to
simplify things a bit.

I started with the statement [p and (not t)].  Both [p] and [not t]
must be true for this premise to be true.

When [not t] is true [t] is false.

The only way for the conclusion [q=>t] to be true while [t] is false
is for [q] to be false.

If [q]is false and I want [q or (not s)] to be true, then {not s}
must be true, making [s] false.

If [s] is false and I want [(not r) or s] to be true, then [not r]
must be true, making [r] false.

If [r] is false and I want [p=>r] to be true, then [p] must also be
false, BUT in the first step [p] is true.  [p] cannot be both true and
false; therefore the argument is not valid.

My instructor marked off the whole problem because I did not make
[q=>t] false as my first step. I would like to know if I am correct.

Thanks a whole bunch in advance,  Kara Archer
```

```
Date: 9/10/96 at 1:38:30
From: Doctor Mike
Subject: Re: Invalid Logic Argument

Hello Kara,

spelling if that is not exactly right), or in English Reduce to
Absurdity.  That is, assume the opposite of what you want to prove,
then logically derive an impossibility, and conclude that assumption
is false.

If the instructor does not actually see a step of the form "Assume
...... is ....." then you have not shown beyond a shadow of a doubt
that you fully understand it.  This is appropriate at the stage of an
instructor testing comprehension of a new and tricky concept.  You had
all the details, but they were not tied up in a nice neat package.
Neat packages are easier to grade.

My chosen way to show I understand this logic concept would be:

1. I want to show this is NOT a valid argument, so I will assume
that it IS a valid argument.
2. Therefore if all 4 premises are true, then q=>t is also true.
3. Exactly your logic, resulting in (p and (not p)), a contradiction.
4. My explicit assumption in (1) can't be true, so the argument
is invalid.

The instructor can see all he/she needs to in order to give full
credit and everybody is happy.

By the way, there is a direct proof. If you just assume the 4
premises, then you can conclude that t is false and the others are all
true.  Then the conclusion q=>t could not follow since it would be
true=>false.

I hope this helps out.

-Doctor Mike,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Logic

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search