2. Propositional Logic Truth tables The lecture Truth values ! Propositional formulas, like " It is raining " The train is moving are true or false depending on the circumstances. One day it rains, another it does not. It may rain in Helsinki but not in Warsaw. ! A true propositional formula is said to have truth value 1. ! A false propositional formula is said to have truth value 0. Last Jouko Väänänen: Propositional logic viewed Truth values (Contd.) ! Propositional formulas of the simplest form p0, p1,... can have truth value 1 or 0 according to our choice. But if we give them truth values, then the truth values of formulas built from them such as p0 v p1 and p0 → p1 are completely determined. ! A choice of truth values for proposition symbols is called a valuation. Last Jouko Väänänen: Propositional logic viewed Valuation ! Valuations assign truth values 1 (true) or 0 (false) to proposition symbols. ! Valuations are the building blocks of truth tables, which are the main tool for analysing complicated propositional formulas. Last Jouko Väänänen: Propositional logic viewed Valuations mathematically defined ! Valuations map propositional symbols to the two element set {0,1} of truth values. Last Jouko Väänänen: Propositional logic viewed Valuation ! Valuations extend to all formulas by means of truth tables. ! The truth value v(A) of an arbitrary propositional formula A can be easily computed in terms of the truth values of the immediate subformulas of A. Last Jouko Väänänen: Propositional logic viewed Truth Tables ! Truth table = table of all possible valuations ! Negation Truth tables reflect our ! Implication intuition of the ! meaning of each Conjunction connective. ! Disjunction ! Equivalence Last Jouko Väänänen: Propositional logic viewed Truth table of Conjunction ∧ The only true case Last Jouko Väänänen: Propositional logic viewed Truth table of Disjunction The only false case Last Jouko Väänänen: Propositional logic viewed Truth Table of Negation Last Jouko Väänänen: Propositional logic viewed Truth Table of Implication The only false case Last Jouko Väänänen: Propositional logic viewed Interpretation of implication I ! Let p be the sentence “Jukka lives in Helsinki”. ! Let q be the sentence “Jukka lives in Finland”. ! Since Helsinki is in Finland, the formula p→q is true. ! This truth is based on Helsinki being in Finland, and is unaffected if Jukka in fact does not live in Helsinki. ! The only thing that would shatter the truth of p→q is if Jukka lived in Helsinki but not in Finland, in which case we would have to review our geography. Last Jouko Väänänen: Propositional logic viewed Interpretation of implication II ! Think of A and B as subsets of {0}. ! There are just two subsets: and {0}. ! Identify them with 0 and 1. ! Think of A→B as “A is contained in B” ! so 1st row is 1 → ! so 2nd row is 0 ! so 3rd row is 1 ! th so 4 row is 1. Last Jouko Väänänen: Propositional logic viewed Truth table of Equivalence Last Jouko Väänänen: Propositional logic viewed A truth table ! Every connective in a formula is the main connective of a subformula. We write under it the truth value of this subformula. p0 p1 (p0 p1) p0 ∧ p1) Last Jouko Väänänen: Propositional logic viewed A truth table ! Every connective in a formula is the main connective of a subformula. We write under it the truth value of this subformula. p0 p1 (p0 p1) p0 ∧ p1) Last Jouko Väänänen: Propositional logic viewed A truth table ! Every connective in a formula is the main connective of a subformula. We write under it the truth value of this subformula. p0 p1 (p0 p1) p0 ∧ p1) Last Jouko Väänänen: Propositional logic viewed A truth table ! Every connective in a formula is the main connective of a subformula. We write under it the truth value of this subformula. p0 p1 (p0 p1) p0 ∧ p1) Last Jouko Väänänen: Propositional logic viewed A truth table ! Every connective in a formula is the main connective of a subformula. We write under it the truth value of this subformula. p0 p1 (p0 p1) p0 ∧ p1) Last Jouko Väänänen: Propositional logic viewed A truth table ! Every connective in a formula is the main connective of a subformula. We write under it the truth value of this subformula. p0 p1 (p0 p1) p0 ∧ p1) Last Jouko Väänänen: Propositional logic viewed A truth table ! Every connective in a formula is the main connective of a subformula. We write under it the truth value of this subformula. p0 p1 (p0 p1) p0 ∧ p1) Last Jouko Väänänen: Propositional logic viewed A truth table ! Every connective in a formula is the main connective of a subformula. We write under it the truth value of this subformula. p0 p1 (p0 p1) p0 ∧ p1) Last Jouko Väänänen: Propositional logic viewed A truth table ! Every connective in a formula is the main connective of a subformula. We write under it the truth value of this subformula. p0 p1 (p0 p1) p0 ∧ p1) Last Jouko Väänänen: Propositional logic viewed A truth table ! Every connective in a formula is the main connective of a subformula. We write under it the truth value of this subformula. p0 p1 (p0 p1) p0 ∧ p1) Last Jouko Väänänen: Propositional logic viewed A bigger truth table p0 p1 p2 (p0 p1) (p1 p2)) (p0 p2)) Last Jouko Väänänen: Propositional logic viewed A bigger truth table p0 p1 p2 (p0 p1) (p1 p2)) (p0 p2)) Last Jouko Väänänen: Propositional logic viewed A bigger truth table p0 p1 p2 (p0 p1) (p1 p2)) (p0 p2)) Last Jouko Väänänen: Propositional logic viewed A bigger truth table p0 p1 p2 (p0 p1) (p1 p2)) (p0 p2)) Last Jouko Väänänen: Propositional logic viewed A bigger truth table p0 p1 p2 (p0 p1) (p1 p2)) (p0 p2)) Last Jouko Väänänen: Propositional logic viewed A bigger truth table p0 p1 p2 (p0 p1) (p1 p2)) (p0 p2)) Last Jouko Väänänen: Propositional logic viewed A bigger truth table p0 p1 p2 (p0 p1) (p1 p2)) (p0 p2)) Last Jouko Väänänen: Propositional logic viewed A bigger truth table p0 p1 p2 (p0 p1) (p1 p2)) (p0 p2)) Last Jouko Väänänen: Propositional logic viewed A bigger truth table p0 p1 p2 (p0 p1) (p1 p2)) (p0 p2)) Last Jouko Väänänen: Propositional logic viewed A bigger truth table p0 p1 p2 (p0 p1) (p1 p2)) (p0 p2)) Last Jouko Väänänen: Propositional logic viewed A bigger truth table p0 p1 p2 (p0 p1) (p1 p2)) (p0 p2)) Last Jouko Väänänen: Propositional logic viewed A bigger truth table p0 p1 p2 (p0 p1) (p1 p2)) (p0 p2)) Last Jouko Väänänen: Propositional logic viewed A bigger truth table p0 p1 p2 (p0 p1) (p1 p2)) (p0 p2)) Last Jouko Väänänen: Propositional logic viewed Inefficiency of Truth Tables ! Truth tables become eventually too large " n propositional symbols " 2n rows in the truth table " Truth table grows exponentially Last Jouko Väänänen: Propositional logic viewed.