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

Question

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 03:31
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 03:34

score 1 · Answer 1 · edited Apr 13 '17 at 12:21

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 http://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 09:59

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

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