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Here is a concrete example. Let our base system be Euclidean geometry without the parallel postulate. Then, in our ordinary understanding, the parallel postulate is true (it is true in the standard Euclidean plane) but its negation is possible (because there are other planes that satisfy the rest of the axioms of Euclidean geometry, but not the parallel ...

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You may consider the table of truth. Consider any false sentence $P$, a sentence $Q$ and the sentence $P\to Q$. The truth table is: $$\begin{array}[t]{c|c|c|} P & Q & P\to Q \\ \hline F & T & T\\ F & F & T\\ \end{array}$$ Here's an example: Consider the axioms: "Every triangle is not a real number" and "Geometry is a field of ...

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Something may be logically possible while it is actually false. It is possible you might have met the queen of England at a football match last Sunday.   Have you actually done so? It is not possible you might have met the queen of England at a football match while she was actually sailing on a royal yacht.   That would be a contradiction. ...

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