1/5/2011 PROPOSITION A proposition makes a claim (either an assertion or denial) that may be either true or false. It must Unit 1B have the structure of a complete sentence. Propositions and Truth Values NOTE: A proposition has one of two possible truth values: T = true or F = false NOTATION: Propositions are sometimes denoted by lower-case letters: p, q, r, s, t, . NEGATION (OPPOSITES) TRUTH TABLE The negation of a proposition p is another A truth table is a table with a row for each proposition that makes the opposite claim of p. possible set of truth values for the propositions being considered. NOTATION: The negation of p is written as not p or ~ p. TRUTH TABLE FOR p AND not p DOUBLE NEGATION not p The double negation not not p (~ ~ p) and p p ~ p have the same truth value. T F not p nottt not p p ~ p ~ ~ p FT TFT FTF 1 1/5/2011 LOGICAL CONNECTORS CONJUNCTIONS When we join two propositions, say p and q, We will consider three logical connectors. with the logical connector and , the resulting •And compound proposition p and q is called a conjunction. •Or • If . then . LOGIC OF A CONJUNCTION DISJUNCTION The conjunction of p and q [ p and q ] is true When we join two propositions, say p and q, only if both p and q are true. with the logical connector or , the resulting compound proposition p or q is called a p q pandqp and q ddsjuctoisjunction. TTT TFF FTF FFF TWO TYPES OF “OR” THE LOGIC OF DISJUNCTIONS There are two types of “or”: The disjunction of p and q [ p or q ] is false only if both p and q are false. •An inclusive or means “either or both.” p q porqp or q •An exclusive or means “one or the other, TTT but not both.” TFT FTT NOTE: In logic we always use the inclusive or, FFF unless told otherwise. 2 1/5/2011 LOGIC OF A CONDITIONAL CONDITIONAL PROPOSITION PROPOSITION A statement of the form if p, then q is called a The conditional if p, then q is true in all cases conditional proposition (or implication). except when p is true and q is false. p q If p, then q . Proposition p is called the hypothesis (or TTT antecedent). TFF Proposition q is called the conclusion (or FTT consequent). FFT ALTERNATIVE PHRASINGS OF VARIATIONS ON THE CONDITIONALS CONDITIONAL Name Form Example The following are common alternative ways of Conditional if p, then q If you are sleeping, then stating if p, then q: you are breathing. p is su ffici ent p will l ead t o q p iliimplies q Converse if q, then p If you are breathing, then for q you are sleeping Inverse if not p, If you are not sleeping, q is necessary q if p q whenever p then not q then you are not breathing. for p Contrapositive if not q, If you are not breathing, then not p then you are not sleeping LOGICALLY EQUIVALENT STATEMENTS Exercise: Make a truth table for the variations Two statements are logically equivalent if they on the conditional. share the same truth values: if one is true, so is that other; and if one is false, so is the other. NOTE: A conditional statement: if p, then q and its contrapositive: if not q, then not p are logically equivalent. 3.