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What is precise definition of proposition (logic) ? In the textbooks they define :

"a declarative sentence that is either true or false."

But for me this definition is rather imprecise. for example : $x>2$ Is It a proposition? and why? in my idea $x>2$ Is't proposition ; because can be both true and false .

Almot1960
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    $x>2$ can't be both true and false simultaneously for the same real value of $x$. – Mike Pierce Dec 19 '18 at 17:13
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    $x>2$ is not a proposition because it involves a variable. To make it a proposition, we can either add a quantifier in front, or specify a value for the variable to take. – Sambo Dec 19 '18 at 17:18
  • I guess if $x$ is already fixed, then $x>2$ is a proposition. For example: Let $x^2=5$ and $x>0$. Proposition: $x>2$. However $x>2$ on itself is probably not a proposition. – SmileyCraft Dec 19 '18 at 17:18
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    @SmileyCraft I agree that if $x$ is already fixed (say, replace it with the number $\pi$), then $x>2$ is a proposition. But I think the example you give has an implicit quantifier, and you're saying something like "for every $x$: if $x^2=5$ and $x>0$ then $x>2." – Mark S. Dec 19 '18 at 17:56
  • If you want a "formal" treatment, I suggest you look here: https://plato.stanford.edu/entries/logic-classical/ (especially sections 2 and 4). – ryan221b Dec 19 '18 at 18:13
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    Propositions are usually defined gramatically, and the exact grammar depends on which logic you are using. – DanielV Dec 19 '18 at 18:28
  • See also the post Formal definition of proposition. – Mauro ALLEGRANZA Dec 20 '18 at 08:05

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