Discrete Mathematics proving tautological implications argument proof argument

Discrete Mathematics proving tautological implications, argument, proof, argument form, rule of inference, constructing argument forms Liangfeng Zhang School of Information Science and Technology Shanghai. Tech University

Proof Methods

Proof Methods T T T F F F T T F F

Proof Methods

Applications

Tautological Implications

Proofs

Proofs

Proofs

Proofs

Proofs

Argument DEFINITION: An argument (论证) is a sequence of propositions • Conclusion(结论): the final proposition • Premises(假设): all the other propositions • Valid(有效): the truth of premises implies that of the conclusion • Proof (证明): a valid argument that establishes the truth of a conclusion EXAMPLE: a valid argument, a proof premise conclusion

Argument Form DEFINITION An argument form(论证形式) is a sequence of formulas • Valid: no matter which propositions are substituted for the propositional variables, the truth of conclusion follows from the truth of premises • rules of inference(推理规则): valid argument forms EXAMPLE: a valid argument form and an invalid argument form valid invalid

Rules of Inference Tautological Implication Rule of Inference Name Conjunction 合取 Simplification 化� Addition 附加 Modus ponens 假言推理 Law of detachment 分离律 Modus tollens 拒取式 Disjunctive syllogism 析取三段论

Rules of Inference Tautological Implication Rule of Inference Name Hypothetical syllogism 假言三段论 Resolution 消解(归结)

New Argument Forms

New Argument Forms 1 premise 2 rule of replacement applied to 1 3 premise 4 hypothetical syllogism applied to 2 and 3 5 rule of replacement applied to 4 6 premise 7 hypothetical syllogism applied to 5 and 6 8 rule of replacement applied to 7

Resolution

Resolution 1 2 3 resolution applied to 1 and 2 4 5 resolution applied to 3 and 4 6 7 contradiction