Logical Equivalences with English Statements

Question

Showing two statements $p$ and $q$ are logically equivalent is to show $p \Longleftrightarrow q$. I understand this, however I think when looking at english statements showing whether or not they are logically equivalent becomes a little fuzzy when considering the nuances of words and their fundamental definitions.

I got into a healthy debate as to whether or not the two following statements were equivalent.

"There is no gravity on Earth."

"Humans are not held down by gravity."

Any help is appreciate and I just like to see other peoples' reasoning. For the record I argued against these two statements being logically equivalent.

Answer 1

See Logical equivalence :

In logic, statements $p$ and $q$ are logically equivalent if they have the same logical content. That is, if they have the same truth value in every model.

Thus, the concept applies to formulas, like $p \to q$ and $\lnot p \lor q$ in classical propositional calculus (see the correspondig truth tables).

A statement of natural language, like e.g. "There is no gravity on Earth" has a specific truth value and it makes little sense to consider "its tuth value in different interpretations".

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Note. An interpretation of a formula of classical propositional logic is a function that maps each propositional symbol to one of the truth values True and False. This function is known as a truth assignment or valuation function.

Thus, two propositional formulas are logically equivalent iff in every truth assignment they have the same truth value.

Answer 2

I agree with you that these are not logically equivalent. For example, we can imagine a logically possible world where there is no gravity on Earth, and where humans don't live on Earth but some other planet where there is gravity. Then the first statement is true, but the second statement is false, and so these are not logically equivalent.