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I am confused if the statement "$p$ implies $q$" logically equivalent to the statement "$p$ implies only $q$"?

Assuming that the two said statement is logically equivalent, then the truth value of the statement ...

"If $a^2=b$ and $b>0$, then $a=\sqrt{b}$."

... is false. Since a can be equal to $a=\sqrt{b}$ OR $a=-\sqrt{b}$, not only $a=\sqrt{b}$.

AYA
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    What would it mean to say that $p$ implies $\textit {only} ;q$? $p$ implies $p$ for instance. – lulu Jul 11 '20 at 12:12
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    "$p$ implies $q$" is a well-defined logical expression. "$p$ implies only $q$" is meaningless. Even with good will, I can't guess. –  Jul 11 '20 at 12:13
  • @lulu Sorry for the confusion, Kindly check the additional context I added for the question. – AYA Jul 11 '20 at 12:19
  • @YvesDaoust Sorry for the confusion, Kindly check the additional context I added for the question. – AYA Jul 11 '20 at 12:19
  • Your example does not make sense. "$a$ greater than zero" is not compatible with "$a=-\sqrt b$". –  Jul 11 '20 at 12:19
  • As has been remarked ,the statement "$p$ implies $\textit {only},q$" has no meaning. As to the given context, $a$ can not be $-\sqrt b$ because $a$ is assumed to be positive. – lulu Jul 11 '20 at 12:20
  • Sorry, I copied the wrong statement. Please see the revised context. – AYA Jul 11 '20 at 12:23
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    In that case the desired claim is indeed false. $(-1)^2=1$ but $-1\neq \sqrt 1=1$. – lulu Jul 11 '20 at 12:24
  • I don't know if I understand exactly what you are saying. Could "If I'm Canadian then I live north of Mexico" and "If I'm Canadian then I live north of the USA" be a couterxample? – MasB Jul 11 '20 at 12:47

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They are not equivalent. For example the statement "If it's raining, then I'll use an umbrella" makes sense but "If it's raining, then only I'll use an umbrella" is false since others may use an umbrella as well.

CyclotomicField
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    But the "then only" could be translated "then only will I use an umbrella". – MasB Jul 11 '20 at 14:06
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    @BernardMassé that's still false however as I can us an umbrella when it's sunny to stay in the shade or poke people. Interpreting it either way leads to the same problem. – CyclotomicField Jul 11 '20 at 15:07
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$$a^2=b>0\implies a=\sqrt b$$ is a false statement, full stop.

$$a^2=b>0\implies a=\sqrt b\lor a=-\sqrt b$$

is a true statement, full stop.

You never assume that "there could be other predicates but they are missing" or anything of this kind. If you want to express that "$a=\text{only}\sqrt b$" with the ulterior motive that "$a=-\sqrt b$" could have been possible, you write $a=\sqrt b$, and there is no need to mention $a\ne-\sqrt b$.