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Research in Philosophy: Satisfiable and Valid Formula Derivable

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I would like to have these two questions answered for my own research in philosophy, if possible.

1) Is every satisfiable formula derivable? Why?

2) Is every valid formula derivable? Why?

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The solution discusses if every satisfiable/valid formula derivable and why.

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Please see attached response.

Here are your two questions:

1) Is every satisfiable formula derivable? Why?

2) Is every valid formula derivable? Why?

As posed, these questions are too vague to answer, so I will have to fill in some blanks. Whether the answer to question (1) is yes or no depends on what system of logic you are using. (In fact the answers to both questions depend on what system of logic you are using. Derivability is system-relative—that is, it only makes sense to ask, "is this formula derivable in such-and-such a system?" (I suspect that you are asking these questions with respect to propositional logic; if that is so, just read everything below that talks about PL and you can safely ignore everything that talks about QL.)

If you are using propositional/truth-functional logic, then the answer is yes. This is because propositional logic (PL) is "complete". "Completeness" is just the name for the fact that every satisfiable proposition in the language of PL is derivable. So I haven't explained anything yet, just given you the answer. The proof of completeness is actually complicated, but you can think of it this way. For a statement/formula to be satisfiable is just for there to be some assignment where it is satisfied. If you know truth-tables, a more concrete way of putting this is: for a statement to be satisfiable is just for it to have some truth table row where it is true.

So what does that mean? The statement in question is made up of atomic sentences, right? These are just letters (maybe A, B, C, or P, Q, R, etc.) in the ...

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