Lecture2
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This lecture defines tautologies, contradictions, and logical equivalence. It also defines valid and fallacious arguments. Students completed exercises to prove De Morgan's laws, that the negation of a negation is the original proposition, and to identify fallacies of denying the hypothesis, affirming the conclusion, and applying modus tollens.
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The document discusses different types of logical statements and their truth values. It defines simple and compound statements, truth tables, negation, conjunction, disjunction, implication, converse, and biconditional. Examples are provided to illustrate each concept, such as truth tables showing the possible truth value combinations of statements connected by logical operators like "and", "or", and "if-then".
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Lecture2
- 1. Lecture 2 • Definition: A Tautology is a proposition that is always true • Definition: A Contradiction is a proposition that is always false • Definition: p and q are equivalent iff p q is a tautology iff p and q have the same truth table. Notation p q Thursday, January 17, 13
- 2. • Class Exercise, Prove De Morgan’s Laws ¬(p q) ¬p ¬q ¬(p q) ¬p ¬q • Class Exercise, Prove: ¬(¬p) p Thursday, January 17, 13
- 3. • Definitions: An argument is a sequence of propositions. All but the final proposition in the argument are premises and the final proposition is the conclusion • Definition: An· argument is valid iff (p p ) > 1 2 p ·· qn is a tautoloty. Otherwise it is called a fallacy • Lemma: If and argument is valid and all premises are true, then the conclusion is true (if a premise is false, the conclusion might be false) Thursday, January 17, 13
- 4. • Exercise: Fallacy of denying the hypothesis ((p q) ¬p) ¬q • Exercise: Fallacy of affirming the conclusion ((p q) q) p • Exercise: Modus Tollens ((p q) ¬q) ¬p Thursday, January 17, 13