Announcements For The Logic of Boolean Connectives Truth Tables, Tautologies & Logical Truths. Outline. Introduction Truth Functions

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1 Announcemens For The Logic o Boolean Connecives Truh Tables, Tauologies & Logical Truhs 1 HW3 is due nex Tuesday William Sarr William Sarr The Logic o Boolean Connecives (Phil ) Rugers Universiy 1/33 William Sarr The Logic o Boolean Connecives (Phil ) Rugers Universiy 2/33 Ouline Inroducion Truh Funcions 1 Inroducion & Review 2 Truh Tables 3 Logical Truhs & Tauologies Las class, we learned he meaning o,, in erms o ruh uncions We also saw ha ruh uncions allowed us o do somehing useul: Figure ou he ruh value o a complex senence rom he ruh values o is aomic pars, and vice versa For example, we know ha ( Cube(b)) is rue in a world where neiher a nor b are cubes, since: I and Cube(b) are alse hen Cube(b) is alse I Cube(b) is alse, hen ( Cube(b)) is rue William Sarr The Logic o Boolean Connecives (Phil ) Rugers Universiy 3/33 William Sarr The Logic o Boolean Connecives (Phil ) Rugers Universiy 5/33

2 Inroducion Truh Tables Review The Boolean Connecives These calculaions ones like he one we jus wen hrough are a bi clunky Truh ables provide a much cleaner mehod As i urns ou, consrucing ruh ables will also provide us wih a helpul way o undersand 3 core logical conceps: 1 Logical Consequence 2 Logical Truh 3 Logical Equivalence Today, we ll learn all abou ruh ables & hese applicaions! Truh Table or P P rue alse alse rue Truh Table or P Q P Q rue rue rue rue alse alse alse rue alse alse alse alse Truh Table or P Q P Q rue rue rue rue alse rue alse rue rue alse alse alse lips he value akes he wors value akes he bes value William Sarr The Logic o Boolean Connecives (Phil ) Rugers Universiy 6/33 William Sarr The Logic o Boolean Connecives (Phil ) Rugers Universiy 7/33 Sep 1: The Reerence Columns Sep 2: Inside Ou We are going o consruc a ruh able or (1) Firs, some columns: 1 A column or each aomic sub-senence o (1), called a reerence column 2 A column or (1) isel Truh Table or (1) Now, we ill he reerence column w/ruh values One row or each unique logical possibiliy Truh Table or (1) Nex, we ill in he column beneah he innermos connecive : In he irs row, is so is In he second row, is so is William Sarr The Logic o Boolean Connecives (Phil ) Rugers Universiy 10/33 William Sarr The Logic o Boolean Connecives (Phil ) Rugers Universiy 11/33

3 Sep 3: The Main Connecive Truh Table or (1) Lasly, we ill in he columns beneah he ouermos connecive : In he irs row, is and is, so heir disjuncion is In he irs row, is and is, so heir disjuncion is This column liss every logically possible ruh value or (1) How o Consruc a Truh Table or P 1 Reerence Columns: Draw a column or each aomic sub-senence o P, hese columns are called he reerence columns and are illed wih every possible combinaion o ruh-values or he sub-senences 2 Inside Ou: Draw a column or P isel. Then ill in he column below P s innermos connecive. Repea or he nex innermos connecive, unil you ge o he main connecive. 3 Main Connecive: Fill in he column under he main connecive. This row liss he possible ruh values o P William Sarr The Logic o Boolean Connecives (Phil ) Rugers Universiy 12/33 William Sarr The Logic o Boolean Connecives (Phil ) Rugers Universiy 13/33 Reerence Columns There is More o I... When you have more han one aomic sub-senence, illing in he reerence columns requires more hough Remember ha each row o he reerence columns liss a unique logical possibiliy Also remember ha here is supposed o be a row or every unique possibiliy Okay, well how many rows would we need or a ormula wih 2 aomic sub-senences? Reerence Columns How Many Rows? Le s igure i ou: We have 2 aomic sub-senences Each can have 2 dieren ruh values (,) So here 2 2 = 4 possible combinaions o ruh values and aomic sub-senences Thereore, a able or a ormula wih 2 aomic sub-senences needs 4 rows You Always Need 2 n Rows In general, i here are n aomic sub-senences o P hen here will be 2 n possible assignmens o ruh values o hose aomic sub-senences, in which case he ruh able or P should have 2 n rows. William Sarr The Logic o Boolean Connecives (Phil ) Rugers Universiy 15/33 William Sarr The Logic o Boolean Connecives (Phil ) Rugers Universiy 16/33

4 Anoher Example Sep 1 Le s consruc a ruh able or: (2) ( Cube(b)) We need 2 reerence columns: In hem, we need a row or each o he 4 logical possibiliies can be and Cube(b) ; and Cube(b), and so on. Truh Table or (2) Cube(b) ( Cube(b)) Anoher Example Seps 2 & 3 Truh Table or (2) Cube(b) ( Cube(b)) Nex, he innermos connecive : I will lip each value o Cube(b) Now, he nex innermos : akes wors value o he pair Finally, he main connecive : lips he value o he conjuncion we jus compued William Sarr The Logic o Boolean Connecives (Phil ) Rugers Universiy 17/33 William Sarr The Logic o Boolean Connecives (Phil ) Rugers Universiy 18/33 Building Reerence Columns A Helpul Rouine You need o lis each possibiliy exacly once when illing in reerence columns Here s a helpul rouine or his: 1 In he innermos re. column, you alernae s and s 2 In he nex innermos column, you double ha alernaion, and so on or any more rows Truh Table or (2) Cube(b) ( ) Le s pu his rouine o work William Sarr The Logic o Boolean Connecives (Phil ) Rugers Universiy 19/33 Ye Anoher Example Sep 1 We ll consruc a able or: (3) ( Cube(b)) Cube(c) Le A =, B = Cube(b), C = Cube(c) Table or (3) A B C (A B) C Firs, he 8 rows o he reerence columns: 1 Alernae on innermos column 2 Double his alernaion on he nex 3 Double again William Sarr The Logic o Boolean Connecives (Phil ) Rugers Universiy 20/33

5 Ye Anoher Example Seps 2 & 3 You Try I In Class Exercise Table or (3) A B C (A B) C Nex, rows or innermos connecives ( ) B: lips value o B C: lips value o C Now, he row or he nex innermos connecive ( ) akes lowes value Break ino groups o 4-6 and consruc a ruh able or: (4) (B C) A Finally, he row or he main connecive ( ): akes highes value William Sarr The Logic o Boolean Connecives (Phil ) Rugers Universiy 21/33 William Sarr The Logic o Boolean Connecives (Phil ) Rugers Universiy 22/33 Boole An Inroducion Truh Tables LPL conains a program called Boole, which is or consrucing ruh ables Now ha you ve done a one by hand, you can appreciae how nice o a ool his is! Le s run hrough he basics o Boole by using i o consruc a able or (3) Okay, we ve learned how o draw hese prey ables, bu how do hey help us do logic? 1 Truh ables are a powerul way o exploring logical possibiliy 2 This concep underlies several imporan conceps o logic: Logical consequence Logical equivalence Logical ruh We will now look a hese conceps wih he help o ruh ables William Sarr The Logic o Boolean Connecives (Phil ) Rugers Universiy 24/33 William Sarr The Logic o Boolean Connecives (Phil ) Rugers Universiy 25/33

6 Logical Truh Logical Truh P is a logical ruh i and only i i is logically necessary. Tha is, i is no possible or he laws o logic o hold while P is alse Logical ruhs are hose senences which are guaraneed by logic alone o be rue Logical possibiliy is dieren rom physical & oher kinds o possibiliy vs. Te(a) Dodec(a) Traveling he speed o ligh vs. being a round square This sounds vague, can we do any beer? Yes, i we use ruh ables Tauologies Tauology P is a auology i and only i he ruh able or P has only s in he column under P s main connecive Truh Table or (1) So, (1) is a auology Inuiively, is (1) a logical ruh? Yes! So, i looks like he idea o a auology is a way o making he idea o a logical ruh a bi more precise This is because ruh ables are a precise way o hinking abou logical possibiliy William Sarr The Logic o Boolean Connecives (Phil ) Rugers Universiy 27/33 William Sarr The Logic o Boolean Connecives (Phil ) Rugers Universiy 28/33 Tauologies Some Examples Tauologies vs. Logical Truhs Cube(c) Cube(c) is a auology (Te(a) Te(a)) is a auology Te(a) is no a auology Recall ha logical ruhs are senences which are guaraneed o be rue by he laws o logic alone All auologies are logical ruhs Bu are all logical ruhs auologies? In oher words, can we jus replace he idea o a logical ruh wih ha o a auology? No! Seeing his acually akes a lile creaiviy William Sarr The Logic o Boolean Connecives (Phil ) Rugers Universiy 29/33 William Sarr The Logic o Boolean Connecives (Phil ) Rugers Universiy 30/33

7 A Curious Logical Truh Which isn a Tauolgy Surely i is a logical ruh ha Jay is Jay and ha Kay is Kay: (5) j = j k = k Bu consider he ruh able or (5): Truh Table or (5) j = j k = k j = j k = k Firs, reerence columns Second, main connecive Bu wai, here are s in ha column! When you build reerence columns, you jus lis all he combinaions Bu, some combinaions don make sense! Tauologies & Logical Truhs Remember 1 P is a auology i and only i every row o he ruh able assigns o P 2 I P is a auology, hen P is a logical ruh 3 Bu some logical ruhs are no auologies 4 P is -possible i and only i a leas one row o is ruh able assigns o P Le s do exercise 4.5 William Sarr The Logic o Boolean Connecives (Phil ) Rugers Universiy 31/33 William Sarr The Logic o Boolean Connecives (Phil ) Rugers Universiy 32/33 Wha we did oday We learned how o consruc ruh ables, by hand & wih Boole We hen applied ruh ables o he problem o precisely deining: 1 Logical ruh We came up wih a similar concep: 1 Tauology We saw ha all Tauologies are logical ruhs, bu some logical ruhs are no auologies William Sarr The Logic o Boolean Connecives (Phil ) Rugers Universiy 33/33