I am studying symbolic logic, and the textbook I am using Understanding Symbolic Logic by Virginia Klenk says that the proof method and the truth table method will always yield the same results, for first order logic. It says that in arithmetic or math, truth tables and the proof method yield different results. I am having trouble imagining how a truth table would even be used to prove things in math. And what is an example where the truth table method in arithmetic or math in general differs from the results of using the proof method in that same subfield of math?
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1@ThomasAndrews I would even say that truth-tables don't make a whole ton of sense inside first-order logic; they're really only useful in the setting of propositional logic (unless we shift attention to "infinite truth-tables," which is stretching the notion a bit). – Noah Schweber Mar 18 '22 at 17:14
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@NoahSchweber yep, that was what I meant, too, but had a brain free and wrote first order when I meant propositional logic. – Thomas Andrews Mar 18 '22 at 17:26
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IF you are referring to the statement at page 113, it seems to me that the author is introducing the propositional rules... – Mauro ALLEGRANZA Mar 18 '22 at 17:42
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And IF the statement about "truth table in mathematics" alludes to incompleteness, it seems to me a WRONG way to refer to the well-know "divergence" between truth and provability in formal system for e.g. arithmetic. See page 193 for a more "reasonable" statement. – Mauro ALLEGRANZA Mar 18 '22 at 17:44
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And see page 193: "In predicate logic we can't use truth tables per se..." – Mauro ALLEGRANZA Mar 18 '22 at 17:48
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Turth tables don't help with mathematics. – ryang Mar 08 '23 at 17:17