# 2 column proof - m ^ [(m^s)->r]^~r=>~s

A whale is a mammal (m). If a whale is a mammal and a spider has six legs (s), then a frog is a reptile (r). But, a frog is not a reptile. Therefore, a spider does not have six legs.

Work:

m = a whale is a mammal
s = a spider has six legs
r = a frog is a reptile

P1: m
p2: (m ^ s) -> r
p3: ~r
therefore:  ~s

Steps                 Reason
1.  m                 p1 - given
2.  (m ^ s) -> r      p2 - given
3.  ~r                p3 - given
4.  ???


Apologies. I'm trying to help my daughter with this and I don't know the next step. We are trying to do the conjunction of any 2 of 1-3, but that doesn't result in any of the theorems that her cheat sheet shows.

• It will help if you use proper formatting. This page includes information on how to use MathJax and $\LaTeX$ to make your math more readable. It would also help if you grammar check the post. "Is a wale is a mammal..." the first word is likely intended to be if instead. – JMoravitz Jun 28 '16 at 3:31
• As for the content of your question, recognize that the contrapositive of an if-then statement is logically equivalent to the original statement. $(m\wedge s)\Rightarrow r$ is logically equivalent to $\neg r \Rightarrow \neg(m\wedge s)$. Continue by using de'morgans to conclude $\neg r\Rightarrow (\neg m)\vee (\neg s)$. Since $m$ is true, $\neg m$ is false. Since $\neg r$ is true, it must be that $\neg s$ is true. – JMoravitz Jun 28 '16 at 3:34
• Thx @JMoravitz. I've also fixed my typo. Typing too fast ;) – ConsultUtah Jun 28 '16 at 3:38

## 1 Answer

If you want to prove that $s$ is false in some context, then you can (in that context) ask whether it is possible that $s$ is true or not. If there is no way that $s$ can be true in that context, then it must be that $s$ is false. (Related to this is proof by contradiction, in that the underlying reasoning is the same.) For concreteness, here is how the logical reasoning to answer your question would go: $\def\imp{\rightarrow}$

$m$.   [assumption]

$( m \land s ) \imp r$.   [assumption]

$\neg r$.   [assumption]

If $s$ then:

$m$.   [Something that is true is still true in more restricted conditions (here under "$s$ is true").]

$s$.   [We are working in the restricted context of "$s$ is true".]

$m \land s$.   [Therefore in this context $m \land s$ is true.]

$( m \land s ) \imp r$.   [As before, we're restating something already true outside the current context.]

$r$.   [By meaning of $\imp$.]

$\neg r$.   [Another restatement.]

Contradiction.   [This merely means that the current context is impossible.]

Therefore $\neg s$.   [The only possibility left is the opposite of the impossible context.]

• I'm assuming you know the meaning of the individual logical symbols. If not, please refer to math.stackexchange.com/a/1684208/21820. And please don't rely on "cheat sheets", because logical reasoning was designed to capture logical intuition, and never meant to be a meaningless game of symbols. If any step in the above explanation is not clear, ask away! Do not expect any less than a completely clear intuitive explanation from a competent teacher! – user21820 Jun 28 '16 at 9:59