| 21. | The only inference rule in the systems mentioned above is modus ponens, which is implemented by the cut rule.
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| 22. | However, they can be justified by checking that they are tautologies using truth tables and that modus ponens preserves truth.
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| 23. | However, conventional logic relies mainly on the rule modus ponens; thus conventional logic is " ponential ".
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| 24. | At a Modus Ponens and substitution proof you have an infinite number of choices for what you can substitute for variables.
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| 25. | The premises are taken for granted and then with the application of modus ponens ( an inference rule ) the conclusion follows.
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| 26. | Then negation introduction and elimination are just special cases of implication introduction ( conditional proof ) and elimination ( modus ponens ).
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| 27. | Without an inference rule ( like " modus ponens " in this case ), there is no deduction or inference.
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| 28. | It can also be shown that no pair of these schemata is sufficient for proving all tautologies with " modus ponens ".
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| 29. | Popular rules of inference in propositional logic include " modus ponens ", " modus tollens ", and contraposition.
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| 30. | A well-known axiomatization, comprising three axiom schemata and one inference rule ( " modus ponens " ), is:
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