Propositions 2/57 A proposition is a statement that is true or - PowerPoint PPT Presentation

Propositions 2/57 A proposition is a statement that is true or false. Examples: ◮ grass is green Propositional Logic ◮ 5 > 3 Lectures 1 and 2 (Chapters 2–7) ◮ 5 < 3 ◮ grass is green and roses are blue ◮ if x > 1 , then x 2 � = x Non-examples: ◮ What time is it? ◮ Don’t look back! / department of mathematics and computer science / department of mathematics and computer science Abstract propositions: vocabulary Abstract propositions: grammar 3/57 5/57 Inductive definition of abstract propositions: We want to study logic without being distracted by the 1. (B ASIS ) every proposition variable is an abstract proposition; concrete contents of propositions! 2. (S TEP ) 2.1 if P is an abstract proposition, then so is ( ¬ P ) ; 2.2 if P and Q are abstract propositions, Proposition variables: Examples: then so are ( P ∧ Q ) , ( P ∨ Q ) , ( P ⇒ Q ) , and ( P ⇔ Q ) . a , b , c , . . . ◮ grass is green a ( ¬ is unary ; ∧ , ∨ , ⇒ , and ⇔ are binary ) ◮ 5 > 3 Connectives: a ∧ ’and’ ◮ 5 < 3 Examples: Non-examples: a ∨ ’or’ a a ∧ ◮ grass is green and ¬ ’not’ b ⇒⇒ a a ∧ b roses are blue ⇒ ’if . . . then . . . ’ ( ¬ a ) a ¬ b ◮ if x > 1 , then x 2 � = x a ⇒ ¬ b ⇔ ’if and only if’ (( ¬ a ) ∧ b ) ((( ¬ a ) ∧ b ) ∨ b ) / department of mathematics and computer science / department of mathematics and computer science

Abstract propositions: grammar Omitting parentheses: 6/57 7/57 Inductive definition of abstract propositions: We want to omit as many parentheses from abstract propositions as possible, but without causing ambiguity. 1. (B ASIS ) every proposition variable is an abstract proposition; 2. (S TEP ) 1. Outermost parentheses can always be omitted 2.1 if P is an abstract proposition, then so is ( ¬ P ) ; 2. We agree on the following priority schema: 2.2 if P and Q are abstract propositions, ¬ then so are ( P ∧ Q ) , ( P ∨ Q ) , ( P ⇒ Q ) , and ( P ⇔ Q ) . NB: since ¬ has highest priority, parentheses ∧ ∨ around a negation may always be omitted. We ( ¬ is unary ; ∧ , ∨ , ⇒ , and ⇔ are binary ) ⇒ may, e.g., also omit the parentheses from ¬ ( ¬ a ) . ⇔ Show that ((( ¬ a ) ∧ b ) ∨ b ) is indeed an abstract proposition: Examples: 1 a 1 2 (( ¬ a ) ∨ ( ¬ b )) ( ¬ a ) ∨ ( ¬ b ) ¬ a ∨ ¬ b � � 2.1 ( ¬ a ) 1 1 2 b (( ¬ a ) ∧ b ) ( ¬ a ) ∧ b ¬ a ∧ b � � 2.2 (( ¬ a ) ∧ b ) 1 1 ? ? b ( ¬ ( a ∧ b )) ¬ ( a ∧ b ) ¬ a ∧ b ¬ a ∧ b NO! � � � 2.2 ((( ¬ a ) ∧ b ) ∨ b ) / department of mathematics and computer science / department of mathematics and computer science Negation Conjunction 8/57 9/57 A conjunction P ∧ Q is A negation ¬ P is true if P is true and Q is true; true if P is false; false in all other cases. false if P is true. Truth table for ∧ : Truth table for ¬ : P Q P ∧ Q 0 = false 0 0 0 P ¬ P We first list all possible 1 = true 0 1 0 0 1 combinations of assignments to P 1 0 0 1 0 and Q . 1 1 1 / department of mathematics and computer science / department of mathematics and computer science

Disjunction Inclusive vs. Exclusive Or 10/57 11/57 A disjunction P ∨ Q is Examples: true if P is true, or Q is true, or both; false otherwise. ◮ Inclusive: Can you show me a passport or drivers license? Truth table for ∨ : ◮ Exclusive: Do you want peanut butter or jam on your sandwich? P Q P ∨ Q 0 0 0 0 1 1 Exercise: 1 0 1 Give abstract proposition that ‘behaves’ as exclusive or of a and b 1 1 1 (notation: a ⊕ b ). / department of mathematics and computer science / department of mathematics and computer science Implication Biimplication 12/57 13/57 A biimplication P ⇔ Q is An implication P ⇒ Q is true if P and Q have the same truth value; true if whenever P is true, then also Q is true; false otherwise. false otherwise. Truth table for ⇔ : Truth table for ⇒ : When is P ⇒ Q clearly not true? P Q P ⇒ Q P Q P ⇔ Q Now consider n > 2 ⇒ n + 1 > 2 . 0 0 1 Clearly, this implication is true for every n . 0 0 1 0 1 1 0 1 0 n = 1 : both n > 2 and n + 1 > 2 false. 1 0 0 1 0 0 n = 2 : n > 2 false, n + 1 > 2 true. 1 1 1 n = 3 : both n > 2 and n + 1 > 2 true. 1 1 1 / department of mathematics and computer science / department of mathematics and computer science

Computing truth table Tautology 14/57 15/57 Example: An abstract proposition is a tautology if its column in a truth Computing the truth table of ¬ ( a ⇒ b ) : table exclusively consists of 1 s. a b a ⇒ b ¬ ( a ⇒ b ) 0 0 1 0 Examples: 0 1 1 0 1 0 0 1 a b b ⇒ a a ⇒ ( b ⇒ a ) 1 1 1 0 ◮ a ∨ ¬ a 0 0 1 1 ◮ a ⇒ a 0 1 0 1 ◮ a ⇒ ( b ⇒ a ) 1 0 1 1 1 1 1 1 ◮ . . . / department of mathematics and computer science / department of mathematics and computer science Contradiction Contingency 16/57 17/57 An abstract proposition is a contradiction if its column in a truth table exclusively consists of 0 s. An abstract proposition is contingent if it is not a tautology, nor a contradiction. Examples: Examples: ◮ a ∧ ¬ a ◮ ( a ⇒ b ) ∧ ( a ∧ ¬ b ) 1. a ◮ . . . 2. a ⇒ ¬ a 3. a ∧ b a b a ⇒ b ¬ b a ∧ ¬ b ( a ⇒ b ) ∧ ( a ∧ ¬ b ) 4. a ∨ b 0 0 1 1 0 0 0 1 1 0 0 0 5. ¬ a ⇒ ( b ∧ c ) 1 0 0 1 1 0 1 1 1 0 0 0 / department of mathematics and computer science / department of mathematics and computer science

Equivalence Equivalence 18/57 19/57 Example: Combine truth tables for ¬ ( a ⇒ b ) and ¬ ( ¬ a ∨ b ) : a b a ⇒ b ¬ ( a ⇒ b ) ¬ a ¬ a ∨ b ¬ ( ¬ a ∨ b ) All tautologies are equivalent. 0 0 1 0 1 1 0 We introduce an extra symbol True to denote an arbitrary tautology. 0 1 1 0 1 1 0 1 0 0 1 0 0 1 All contradictions are equivalent. 1 1 1 0 0 1 0 We introduce an extra symbol False to denote an arbitrary contradiction. Note: the columns for ¬ ( a ⇒ b ) and ¬ ( ¬ a ∨ b ) are identical. Not all contingencies are equivalent. Abstract propositions with identical columns in a combined truth table are said to be equivalent. / department of mathematics and computer science / department of mathematics and computer science Abstract propositions: grammar (extended) Notation 20/57 21/57 From now on, we shall consider True and False officially as part of val If P is equivalent to Q , then we write P = = Q . the vocabulary of abstract propositions. val Inductive definition of abstract propositions: Note: = = is not part of the vocabulary of the language of abstract propositions; it is a meta-symbol . 1. (B ASIS ) True and False are abstract propositions, and every proposition variable is an abstract proposition; So: 2. (S TEP ) val a ⇒ b = = ¬ a ∨ b 2.1 if P is an abstract proposition, then so is ( ¬ P ) ; � �� � � �� � abstr. prop. abstr. prop. 2.2 if P and Q are abstract propositions, � �� � then so are ( P ∧ Q ) , ( P ∨ Q ) , ( P ⇒ Q ) , and ( P ⇔ Q ) . meta-formula ( ¬ is unary ; ∧ , ∨ , ⇒ , and ⇔ are binary ) / department of mathematics and computer science / department of mathematics and computer science

Proving equivalences Commutativity, Associativity 22/57 23/57 val Suppose we want to prove that ¬ ( Q ⇒ R ) = = ¬ R ∧ Q . Commutativity: Associativity: How did we proceed (until now)? val val P ∧ Q = = Q ∧ P ( P ∧ Q ) ∧ R = = P ∧ ( Q ∧ R ) val val P ∨ Q = = Q ∨ P ( P ∨ Q ) ∨ R = = P ∨ ( Q ∨ R ) We would prove the equivalence using truth tables: P ⇔ Q val ( P ⇔ Q ) ⇔ R val = = Q ⇔ P = = P ⇔ ( Q ⇔ R ) Q R Q ⇒ R ¬ ( Q ⇒ R ) ¬ R ¬ R ∧ Q 0 0 1 0 1 0 val val NB: P ⇒ Q = � = = Q ⇒ P NB: P ⇒ ( Q ⇒ R ) = � = = ( P ⇒ Q ) ⇒ R 0 1 1 0 0 0 1 0 0 1 1 1 1 1 1 0 0 0 P Q P ⇒ Q Q ⇒ P P Q R P ⇒ ( Q ⇒ R ) ( P ⇒ Q ) ⇒ R 0 1 1 0 0 1 0 1 0 The columns of ¬ ( Q ⇒ R ) and ¬ R ∧ Q are identical, so it holds that NB: In view of Associativity, we shall write P ∧ Q ∧ R to denote both val ¬ ( Q ⇒ R ) = = ¬ R ∧ Q . ( P ∧ Q ) ∧ R and P ∧ ( Q ∧ R ) . / department of mathematics and computer science / department of mathematics and computer science Be Careful! Idempotence, Double negation 24/57 25/57 Are the abstract propositions Idempotence: val ( a ∧ b ) ∨ c NB: P ⇒ P = � = = P P ∧ P val = = P val P ∨ P val P ⇔ P = � = = P and = = P a ∧ ( b ∨ c ) equivalent? ‘It is not that I don’t like spinach’ Double Negation: NO! (see p. 19 in your book) (NB: in propositional logic the intended nuance ¬¬ P val = = P cannot be captured.) This shows that the (remaining) parentheses in these abstract propositions are important. In fact, they make the difference! / department of mathematics and computer science / department of mathematics and computer science