CS636 Final Term Highlighted Handouts by Phistanphobia

TAUTOLOGIES AND CONTRADICTIONS

• Some propositions P(p, q, . . .) contain only T in their evaluation or, in other words, they are true for any truth values of their variables. Such propositions are called tautologies.
• Analogously, a proposition P(p, q, . . .) is called a contradiction if it contains only F in its evaluation or, in other words, if it is false for any truth values of its variables.

ARGUMENTS

• An argument is an assertion that a given set of propositions P1, P2,...,Pn, yields another proposition Q.
• An argument provides support or evidence in favor of one of the others.
• An argument P1, P2, ..., Pn Ͱ Q is said to be valid if Q is TRUE whenever all the premises P1, P2,...,Pn are true.
• Eg: valid argument p, p → q Ͱ q

ARGUMENTS

• Now the propositions P1, P2, . . . , Pn are true simultaneously if and only if the proposition P1 ∧ P2 ∧ . . . Pn is true.
• Thus the argument P1, P2, . . . , Pn Ͱ Q is valid if and only if Q is true whenever P1 ∧ P2 ∧ . . . ∧ Pn is true, or, equivalently, if the proposition (P1 ∧ P2 ∧ . . . ∧ Pn) → Q is a tautology.