logic - Modus Ponens Proof - Mathematics Stack Exchange where an argument is sound when, from true premises, licences the derivation of a true conclusion This means that modus ponens is equivalent to : $\vDash ( (P \rightarrow Q) \land P) \rightarrow Q$, i e $ ( (P \rightarrow Q) \land P) \rightarrow Q$ is a tautology Thus, as said in the above answer, you can check it with a truth table
Is modus ponens a tautology? - Mathematics Stack Exchange 1 Modus ponens isn't that formula or any formula (and thus not a tautology), it's a rule A rule tells you one way of building a proof In fact, rules are part of the definition of what a proof is 1 (for a given logical system)
Modus ponens as a rule of inference vs. tautology 4 As I stated in my previous answer, the rule modus ponens is part of the definition of many common proof systems Let HPC stand for Hilbert's system for classical propositional logic whose only rule of inference is modus ponens and whose axioms are listed here after Frege's This is a well-defined mathematical object
Dictionary Meaning of Modus Ponens and Modus Tollens 7 I wanted to understand modus ponens and modus tollens better, and I searched for its dictionary meaning Wikipedia says that modus ponens is Latin for "mode that by affirming affirms" and that modus tollens is Latin for "mode that by denying denies"
Is Modus Tollens just an implied Modus Ponens on the Contrapositive? But if you treat modus tollens and contrapositive as axioms, modus ponens is a derived rule Hence modus ponens and tollens are equally derivable in a pure sense — the reason modus ponens is often listed first is because its implication aligns with the "arrow of time", our natural train of thought
Disjunctive syllogism vs. modus ponens - Mathematics Stack Exchange For systems of type (3), modus ponens and disjunctive syllogism would be different rules, simply because they involve different connectives Finally, for systems of type (4), I won't try to say anything because there are so many possibilities for doing "something else"
Deduction Theorem + Modus Ponens - Mathematics Stack Exchange 2 In order to verifiy if Peirce's law is sufficient, when added to Deduction Theorem and modus ponens, we can try to verify if the (complete) axiom system for propositional logic of Elliott Mendelson, Introduction to Mathematical Logic (4th ed - 1997) [page 35] can be derived under these assumptions