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However, the fewer the axioms, the more complicated the arguments are, and the more difficult they are to establish the truth of them.This is done in the implication introduction rule.The assumption x is discharged in the application of this rule.General Strategy, break down what you need to prove according to the introduction rule.We need a deductive system, which will allow us to construct proofs of tautologies in a step-by-step fashion.The proof rules we have given above are in fact sound and complete for propositional logic: every theorem is a tautology, and every tautology is a theorem.Some sources call it the axiomatic method.Testing whether a proposition is a tautology by testing every possible truth assignment is expensivethere are exponentially many.Because it has no premises, this rule is an axiom : something that can start a proof.Rule of Sequent Introduction, let the statements P_1, P_2, ldots, P_n be conclusions in a proof, on various assumptions.Modus Tollendo Ponens (1 quad If we can conclude phi lor psi, and we can also conclude neg phi, then we may infer psi.For a disjunction "A lor B it may not be possible because sometimes you can prove neither "A" nor "B in which case you need to go by contradiction, which is to assume neg ( A lor B ) and obtain a contradiction, from which.Proof by Cases If we can conclude phi lor psi, and: (1 quad By making the assumption phi, we can conclude chi (2 quad By making the assumption psi, we can conclude chi then we may infer chi.Neg ( r imp p ).One builds a proof tree whose root is the proposition to be proved and whose leaves are the initial assumptions or axioms (for proof trees, we usually draw the root at the bottom and the leaves at the top).Rule of Conjunction If we can conclude both phi and psi, we may infer the compound statement phi land psi.Then a proof can be found for any substitution instance.Soundness and Completeness A measure of a deductive system's power is whether it is powerful enough to prove all true statements.It embodies proofs by contradiction.Modus ponens is an elimination rule for.

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