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are modus ponens and tollens axioms if not how can they be derived or proved?

if I am good at math then I am good at logic, I am good at math, therofe I am good at logic

this seems evident but is there any proof for them?

looking at the truth table for implication we see p->q is true when P is true and Q is true

SirMrpirateroberts
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Modus ponens/tollens are rules of inference; we accept that arguments with such a form are valid. Before we accept these rules of inference, what would one mean by a "proof"? If by "proof" we mean a valid argument (a sequence of statements with truth values, perhaps), what are the rules for determining whether or not an argument is valid? We need to decide on some ground rules before we can start making arguments. This is usually were modus ponens comes in. In some logical systems, modus tollens can be deduced from modus ponens. But, generally speaking, modus ponens and modus tollens are not derived from even simpler principles; they are asserted as valid argument forms.

C. Caruvana
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  • so they are axioms, I thought they can be proof by truth tables – SirMrpirateroberts Jun 05 '23 at 02:05
  • what about the law of simplification p and q therefore p,q – SirMrpirateroberts Jun 05 '23 at 02:34
  • @SirMrpirateroberts I would ask what you mean by "proved by truth tables" without having some definitions already in place. For example, why is "False => True" taken as True? This is a convention based on Boolean logic, I think. If you have more truth values than just True/False, things can get tricky. See https://math.stackexchange.com/questions/2376048/ – C. Caruvana Jun 05 '23 at 02:35
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    @SirMrpirateroberts Also, a 'proof' by truth-table for Modus Ponens looks like this: "If you analyze Modus Ponens using a truth-table, and find that there is no row where the premises are true but the conclusion false, then Modus Ponens is valid. Now, when we do the truth-table analysis, we find that there is no row with true premises and false conclusion. Therefore, Modus Ponens is valid". Note that this proof uses Modus Ponens itself! Point is: we can;t keep on asking for justification. At some point we'll just need to accept some things. – Bram28 Jun 05 '23 at 18:05