prop - Introduction to Propositional Logic Prepared by...

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1 Introduction to Propositional Logic Prepared by: Kendra Cooper Contents Introduction Basic Truth Tables Tautology, Contradiction, Logical Equivalance Laws of Logic Principle of Duality Inverse, Converse, Contrapositive, Logical Implication Rules of Inference Building a Valid, Logical Argument Translating English into Propositions Problems Introduction A proposition is a statement that is either TRUE or FALSE. A primitive proposition is one that cannot be broken down into simpler statements. For example: p:The formula for water is H 2 O. q: Elvis is alive. Propositions can be combined into compound statements using logical connectives . The logical connectives include: Conjunction (AND) Disjunction (OR) Negation (NOT) Implication Biconditional Exclusive or (XOR)
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2 Basic Truth Tables Use 0 to represent FALSE, 1 to represent TRUE. A truth table all possible values for a statement Conjunction p q p q 0 0 0 0 1 0 1 0 0 1 1 1 Disjunction p q p q 0 0 0 0 1 1 1 0 1 1 1 1 Negation p ¬ p 0 1 1 0 Implication p q p q ¬ p q 0 0 1 1 0 1 1 1 1 0 0 0 1 1 1 1 Biconditional p q p q (p q) (q p) 0 0 1 1 0 1 0 0 1 0 0 0 1 1 1 1 Exclusive OR (XOR) p q p q (p ∧¬ q) ( ¬ p q) 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 0
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3 General Truth Tables Use 0 to represent FALSE, 1 to represent TRUE A truth table all possible values for a statement The statement can be a composite statement (p q) r p q r p q p q r 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 1 0 1 1 1 1 1 Number of Rows in the Truth Table Question: What is the relationship between the number of unique propositions in the statement and the size (i.e., number of rows) in the truth table? Answer: We have seen that: - When there is one unique proposition in the statement, there are two rows in the truth table. - When there are two unique propositions in the statements, there are four rows in the truth table. - When there are three unique propositions in the statement, there are eight rows in the truth table. In general, if there are n unique propositions in the statement, there are 2 n rows in the truth table. Truth tables do not scale well (e.g, five propositions require thirty two rows).
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4 Tautology, Contradiction, Logical Equivalence Tautology If the statement is always TRUE, then this is called a tautology (T 0 ). For example, p ∨¬ p p p ¬p 0 1 1 1 Contradiction If the statement is always FALSE, then this is called a contradiction (F 0 ). For example, p ∧¬ p P p ¬p 0 0 1 0 Logical Equivalence Two statements, s 1 and s 2 , are logically equivalent if and only if s 1 has the same truth values as s 2 for all choices of truth values for their primitive components In other words, the biconditional connective is a tautology This is written as: s 1 s 2 For example, p q ¬ p q the values in the truth table match Proof: p q p q ¬ p q (p q) ( ¬ p q) 0 0 1 1 1 0 1 1 1 1 1 0 0 0 1 1 1 1
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