Modal Logic: Possibility, Necessity, and Incoherent Worlds

In the language of modal logic, a necessary fact is a fact that is true in every possible universe. For example, the fact is that 1+1=2 is true in every universe, or the fact that a and not a cannot be true at the same time, or that there are infinitely many prime numbers. 

The skeptic might think that in another universe, we may define numbers differently, and thus come to the conclusion that 1+1=3. But what I am referring to is the deeper meaning behind what is expressed by variables. What the sentence refers to is a true statement, regardless of the semantics or referential language used. 

I came across an argument that there necessary truths cannot possibly exist, because we do not have access to other universes. For example, in another possible universe, numbers may not exist at all! This skeptic believed that because we were cut off from the experience of another universe, that we do not know whether necessary facts could possibly exist across all possible worlds. 

But good thing that we do not need the existence of numbers to see that necessary facts exist.  All we need is to demonstrate that one necessary fact exists! 

 Okay. Whew. What is the fact? Let's take the law of non-contradiction: "A"and "not A" cannot be true at the same time. It's commonly taken to be a fact in this universe. But why can we apply it to all other possible universes? Well, that's because to say that one is a possible universe, one is implicitly admitting that this other-universe abides by the laws of logic. The term possible is a modal term. 

Thus, a possible universe that does not abide by the laws of logic is not a possible universe at all, it is an incoherent one. Therefore, it is an impossible universe, and by definition does not exist.  

Therefore, every possible universe has at least one true, necessary fact: that being that the said universe is possible.  

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