Necessary truth (or logical truth) is commonly described as something that is true "in all possible worlds". One might say that we can realize its truth through logic instead of through experience.
If Caesar crossed the Rubicon, then someone has [crossed the Rubicon].
This is necessarily true. It is not logically possible that Ceasar has done something that nobody has done.
Wood is a light, durable substance useful for building things.
Elephants dissolve in water. If you put an elephant in water, it will disintegrate.
Neither of these statements are necessarily true. Even in our world, there is wood that is not durable or useful for building things.
And perhaps there is a world out there where elephants are made out of sugar. That's a bit silly. But my point is that we only know that elephants can survive in water because we have seen it or trust people who claim that they have, not because of logic.
M1 All people are mortal.
M2 Socrates is a person.
M3 Socrates will never die.
M4 Socrates is mortal.
$M3$ and $M4$ are logically inconsistent, so they are not jointly possible. $M1$ and $M2$ together are logically equivalent to $M4$, so $\{M1,M2,M3\}$ are also not jointly possible. So the maximal subsets that are jointly possible are $\{M1, M2, M4\},\ \{M1,M3\},\ \{M2,M3\}$. Again, in logic we are free to imagine a world in which people are not mortal or a world in which Socrates is not a person unless we have taken those claims as hypotheses.
Best Answer
I much prefer Magnus’ treatment over the second one. Logic doesn’t care about whether some statement is actually true or not, let alone whether we know or believe it to be true. Logic only cares about implication: it asks: If we consider such-and-such to be true, what else would be true.
In fact, in logic we can work with completely non-sensical statements like ‘Hupflubbers are brig’ and ‘Nothing that is brig is shomsooza’, because logic will now tell you that it logically follows from those statements that ‘No hupflubbers are shomsooza’.
Now, what is a potential worry is cases of ambiguity and vagueness, especially as we try to apply logic to real life. For example, if I use ‘bank’ to refer to a financial institution in one sentence, then we should not combine that with some other sentence where I use ‘bank’ to refer to a river’s edge. And in real life, we may declare that ‘John is a tall person’ and that ‘Susan likes tall people’, but we should not be so quick to infer that Susan likes John: the context under which John is considered ‘tall’ may be quite different from the contect under which Susan likes ‘tall’ people and besides, does Susan literally like all tall people?!
But like I said, these considerations are mainly for when you try to apply logic to real life. As an abstract and idealized branch of mathematics, all that matters for logic is that the expressions have the abstract form of a statement … which is exactly what allows us to do formal logic.
And while the second source does allude to the potential issues of vagueness and ambiguity, I really don’t like how it makes it sound as if it is about the difference between facts and opinions. Yes, as explained above, vagueness and ambiguity can hinder the application of logic to real life. But, as long as we are clear on our terms, then why does it matter whether something is fact or opinion? If the house is beautiful, and if I like beautiful things, then as long as the two sentences agree on their use of ‘beautiful’, we can infer that I like the house. That is, we can do logic on statements that are opinions just as much as on statements that are facts.
In fact, the line between ‘fact’ and ‘opinion’ is a lot more blurry than we like to think anyway. Is it a fact that Pluto is not a planet? Or even that $1$ is not a prime? Remember Pluto used to be a planet, and $1$ used to be a prime. And it wasn’t as if we suddenly realized that we were wrong about these beliefs. We didn’t say: ‘Oh shoot! And here I thought $1$ was a prime, but it turns out I was wrong about it! Dang!’ No, it simply was that we felt it was more useful and practical to regard Pluto as something other than a full-fledged planet. So the point is that one can have interminable discussions about what is ‘fact’ and what is ‘opinion’ or what would even make something ‘true’, but fortunately, as explained earlier, in the abstract context of mathematical formal logic we can completely sidestep such discussions: it just doesn’t matter!
Indeed, as should be clear from what I have been saying, the goal of logic is not to ‘establish truth’. As such, I find the second source confusing, if not inaccurate.