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For $x$ in the set of real numbers

If $x^{2} > 0$ then $x > 0$

I am unsure whether this is a proposition. If $x^2 > 0$ is true then $x > 0$ is false and hence the statement is false. If $x^2 > 0$ is false $(x^2 = 0)$ then $x > 0 $ is false and hence the statement is true. This means there is no unique truth value for this statement and is why I think it wouldn't be a proposition.

Am I correct in thinking this?

mrbennyboy98
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  • If $x^2>0 $ then $x>0$, this statement is clearly false. I can't seem to understand the discrepancy. –  Nov 12 '18 at 14:11
  • The proposition is silent in the case $x^2≤0$. It only addresses the case $x^2>0$, in which case it is certainly false. – lulu Nov 12 '18 at 14:12
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    What's your definition of "proposition"? – 5xum Nov 12 '18 at 14:16
  • My definition of a proposition is a statement that can either be true or false but not both. The statement in question can be both true and false and therefore wouldn't be a proposition. The question made no mention of predicates or quantifiers and was given as written in my question. – mrbennyboy98 Nov 12 '18 at 14:26
  • @mrbennyboy98 We are used to interpreting $x$ as a variable. In that case the statement can be looked at as a bunch of propositions (one for each $x\in\mathbb R$). But maybe (I am not sure) here it must be looked at as a constant. Then it is true or false, and not both. – drhab Nov 12 '18 at 14:32
  • @drhab I agree with you that this statement should be looked at as a bunch of propositions. However we hadn't yet done this in class. So if we could consider interpreting x as a constant then what would you say the truth value of the statement would be? – mrbennyboy98 Nov 12 '18 at 14:41
  • @lulu Why would the proposition be silent in the case x^2 <= 0 ? x^2 can be zero. – mrbennyboy98 Nov 12 '18 at 14:44
  • The proposition only makes a claim about what happens when $x^2>0$. It says nothing at all about what happens when that assumption does not hold. – lulu Nov 12 '18 at 14:48
  • @lulu Would one not use the truth table for implication to determine the truth value of the statement? i.e. consider what happens when x^2 > 0 is true and what happens when x^2 > 0 is false. – mrbennyboy98 Nov 12 '18 at 14:54
  • @mrbennyboy98 for that I would first translate it to the statement: $x^2\leq0\text{ or }x>0$ and then discern $3$ cases: If $x=0$ then the statement is true. If $x>0$ then the statement is true. If $x<0$ then the statement is false. – drhab Nov 12 '18 at 14:58
  • @drhab Thankyou for the reply. Your answer and Henning Makholm's have just about cleared this question up for me. – mrbennyboy98 Nov 12 '18 at 15:03

1 Answers1

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The claim is not a claim of the form that is considered in propositional logic -- it belongs squarely in predicate logic.

Some authors of introductory texts go to the trouble of defining a quasi-formal concept of what the word "proposition" means. Usually not much is actually done with this concept and it is forgotten about completely when you get to define propositional logic formally. Often it looks like the main purpose of offering the definition is to attempt to explain why it's called "propositional" logic.

Unless you anticipate being asked "is such-and-such English sentence a proposition?" in an exam, I would not worry about a particular author's definition of the word. It is not going to be important.

hmakholm left over Monica
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