Download Logic in Proofs (Validity of arguments) Modus ponens logical

Logic in Proofs (Validity of arguments)
Modus ponens logical implication [p v (p 6 q)] Y q
means the proposition [p v (p 6 q)] 6 q is a tautology.
For compound propositions P and Q, [P v (P 6 Q)] 6 Q is also a tautology.
Modus tollens logical implication [(p6q) v ¬ q] Y ¬ p
means the proposition [(p 6 q) v ¬ q] 6 ¬ p is a tautology.
For compound propositions P and Q, [(P 6 Q) v ¬ Q] 6 ¬ P is also a tautology.
A theorem is a statement of the form “If H, then C”
where H is a set of hypotheses and C is a conclusion.
A formal proof is a sequence of propositions P1, P2, ..., P n, C in which each proposition is either
a hypothesis,
a tautology,
or a consequence of previous members of the chain by using an allowable rule of inference.
Rules of inference (logical rules) are based on logical implications of the form
H1vH2v...vHm YQ. If the Hi have already appeared in the chain and H1vH2v...vHm YQ is true,
then Q can be added to the chain.
Some common rules of inference are:
Addition: P Y (P w Q)
Simplification: (P v Q) Y P
Modus Ponens: [P v (P 6 Q)] Y Q
Modus Tollens: [(P 6 Q) v ¬ Q] Y ¬ P
Disjunctive Syllogism: [(P w Q) v ¬ P] Y Q
Hypothetical Syllogism: [(P 6 Q) v (Q 6 R)] Y (P 6 R)
Conjunction: P v Q Y (P v Q)
Substitution Rule (a) [Names don’t matter in a tautology (only the form)!]
If all occurrences of some variable in a tautology are replaced by the same (possibly compound)
proposition, then the result is still a tautology.
Substitution Rule (b) [Equivalences do not change truth value!]
If any (compound) part of a proposition is replaced by an equivalent form, then the original
proposition’s truth value is unchanged.
[Consider Examples 1 through 10 on pages 78 through 84.
Also consider the Theorem and Corollary on page 84.]