What's the proper way to prove P$\rightarrow$Q is not tautologically equivalent to Q$\rightarrow$P

Question

I want to rigorously show that [$(p \rightarrow q)\equiv(q \rightarrow q)$] is not true.

$Let A =(p \rightarrow q), B =(q \rightarrow q)$

I tried to prove this by supposing $A$ and showing that $B$ does not follow.

[$(p \rightarrow q) \rightarrow (q \rightarrow p)$]

I expanded that but it didn't $= 0$

I was told to look at Reductio Ad Absurdum, but I don't see how that applies in this case.

This is a real-world example. This is not homework.

Show a value assignment for $P$ and $Q$ where the two formulas have different truth values. — hmakholm left over Monica, Sep 24 '16 at 20:46
Also, please expand on how this can be a "real-world example". — hmakholm left over Monica, Sep 24 '16 at 20:47
@egreg It's possible to show that A above is not equivalent to B above by a similarly method used to show that (P -> Q) is not equivalent to (Q -> P). Namely, you assume the second thing and then show that the first thing can evaluate to falsity. — Doug Spoonwood, Sep 24 '16 at 21:17
@DougSpoonwood Sure, but the title and the question don't agree anyhow. — egreg, Sep 24 '16 at 21:21
@Bluebaron: It still says $Q\to P$ in the title but $q\to q$ in the body. — hmakholm left over Monica, Sep 24 '16 at 21:38

Martin Argerami · Accepted Answer · 2016-09-24T21:04:33.540

1

The most straightforward way is to construct the truth tables and see that they are not equal. Or you could show values of $P$ and $Q$ that make them different (which would amount to mention the specific line of the truth tables where they differ).

If $P$ is true and $Q$ is false, then $P\to Q$ is false, while $Q\to P$ is true.

Edit: Changed $Q\to Q$ to $Q\to P$, to go with the title.

edited Sep 24 '16 at 21:04

answered Sep 24 '16 at 20:54

Martin Argerami

205,756

Nope. I screwed up that comment, too. I meant for $P \rightarrow Q$. – Bluebaron Sep 24 '16 at 21:12

score -1 · Answer 2 · answered Sep 24 '16 at 21:18

-1

Assume $B$. Show that $A$ could be false.

By a similar method you can show that if B holds true, then (p $\rightarrow$ r) could be false.

answered Sep 24 '16 at 21:18

Doug Spoonwood

11,211

What's the proper way to prove P$\rightarrow$Q is not tautologically equivalent to Q$\rightarrow$P

2 Answers2