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There are two statements P and Q, where P is $x^2-3x+2 =0$ and Q is $x=1$ or $x=2$ or $x=3$.

The first statement, where $x$ $\epsilon$ $\Bbb R$, says

$x^2-3x+2 =0$ $\Rightarrow$ $x=1$ or $x=2$ or $x=3$

and the second statement, where $x$ $\epsilon$ $\Bbb R$, says

$x=1$ or $x=2$ or $x=3$ $\Rightarrow$ $x^2-3x+2 =0$

I am struggling to understand why the first statement is true whilst the second statement is deemed to be false.

The closest relation to a question such as this that I understand is as follows:

$x^2-3x+2=0$ $\Rightarrow$ $x=1$ or $x=2$ is a true statement

but $x^2-3x+2=0$ $\Rightarrow$ $x=1$ is a false statement as we know there are two solutions to the quadractic ie. $(x-1)(x-2)=0$ which implies $x=1,2$ . Therefore the second implication is incorrect as it missed out the solution $x=2$.

The meaning of the word 'or' is also a bit dubious in my mind so my logic behind the inital question was that either of $x=1, x=2$ or $x=3$ could be solutions to statement P, thus $x=1$ or $x=2$ could be solutions but $x=3$ need not be a solution since the word 'or' has been used but applying this sort of logic to the second statement above doesn't help convince me that it is false.

user258521
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1 Answers1

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The implication can be understood as a conditional claim/promise. $ \def\imp{\Rightarrow} $

"$A \imp B$" means that if "$A$" is true then "$B$" is true. It does not at all guarantee that $A$ is the 'most general' condition possible. Neither does it guarantee that $B$ is the 'most informative' conclusion possible.

Clearly the first statement is true because if $x^2-3x+2 = 0$ then one of 3 cases must hold, namely $x=1$ or $x=2$ or $x=3$. Yes the conclusion did not exclude the impossible case of $x=3$ (under the given condition), but the implication is still a true conditional claim.

Also, the second statement is false because in the situation that $x=3$, the condition holds (because it only requires one of the three cases to hold) but the conclusion is false. Hence the implication is a false conditional claim since it fails in at least one situation.

See this post for a more detailed explanation of vacuous truths and one useful way to understand them via game semantics.

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user21820
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  • Note also that you should not ever think about statements in terms of "solutions", whatever you might mean by that. In classical logic, every statement is either true or false, and has nothing to do with solutions. In particular "solution to a statement" is a meaningless phrase. – user21820 Oct 16 '16 at 12:45
  • Could you represent this via a truth table? – user258521 Oct 16 '16 at 13:28
  • @user258521: I could but what I describe here is the more fundamental aspect. Truth tables can only help you for propositional logic and no further. – user21820 Oct 16 '16 at 13:47
  • @user258521: And I presume you already know the truth table for implication? You could just compute the truth values of the condition and conclusion in each example you present, and check that they agree with the explanation I gave. Namely, the conditional claim/promise is false only when the condition is true but the conclusion is false. – user21820 Oct 16 '16 at 13:53
  • You are correct; I was confusing this with a different question in the context of functional equations. I apologize – William Balthes May 07 '17 at 06:11
  • @WilliamBalthes: Okay sure; thanks for responding and so I'll delete my comment. – user21820 May 07 '17 at 07:17