1

enter image description here

I don't understand why A isn't a valid consequence of (~Q) and Q=>P. I think the first line that I highlighted is the textbook's proof of why A isn't a valid consequence of (~Q) and Q=>P, since they're both true but A is false. However, on the second highlighted line, we see that (~Q), Q=>P and A are all true, which to me would make A a valid consequence of (~Q) abd Q=>P, yet it's not.

While writing, I just noticed that for the proposition ~P and Q=>P, there's only one value that makes them both true, and since A is also true for this particular value, that would make A a valid consequence of the ~P and Q=>P proposition. However, for ~Q and Q=>P, there are TWO values that make both of them true, but for one of the case the value for A is false, so by convention people would say that ~Q and Q=>P isn't a valid consequence to avoid any "contradiction". Is my interpretation correct?

Long Vuong
  • 518
  • 1
    Notice the word "every" in the definition of valid consequence: every assignment that makes $P_1,\dots,P_n$ all true also makes $A$ true. So if even one assignment makes $P_1,\dots,P_n$ all true but makes $A$ false, then you don't have a valid consequence, no matter what any other assignments do. – Andreas Blass Sep 24 '17 at 01:07
  • @AndreasBlass just to confirm, in the first highlighted line, If A was true, THEN A would be a valid consequence of ~Q and Q=>P ? – Long Vuong Sep 24 '17 at 01:10
  • 1
    Right; if that truth value were T and nothing else were changed, then you'd have a valid consequence. – Andreas Blass Sep 24 '17 at 01:12

1 Answers1

2

You just need to understand the definition of valid consequence. $A$ is a valid consequence of $\lnot Q$ and $Q\Rightarrow P$ if it is true under every assignment of truth values for which $\lnot Q$ and $Q\Rightarrow P$ are true. So the first highlighted row automatically means that $A$ is not a valid consequence. It doesn't matter if there are some assignments for which all three are true. You don't even need to look at any other lines.

The fact that there is a row in which all three are true means that $A$ is consistent with $\not Q$ and $Q\Rightarrow P.$ Valid consequence is a stronger condition than consistency.

spaceisdarkgreen
  • 58,508