Introduction To Logic
Introduction To Logic PHIL 110
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This 27 page Class Notes was uploaded by Ms. Jada Ernser on Friday October 30, 2015. The Class Notes belongs to PHIL 110 at University of Massachusetts taught by Barak Krakauer in Fall. Since its upload, it has received 8 views. For similar materials see /class/232329/phil-110-university-of-massachusetts in PHIL-Philosophy at University of Massachusetts.
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Date Created: 10/30/15
Exam 2 Format I N I l 6 argument forms 15 points each plus 10 free points Symbolic argument forms no translations 1 2 For each one you will be asked to construct a derivation of the conclusion from the premises The rule sheet will be provided 39 I I 1 problem from SetD Derivations In SL 2 problem from M 4 2 problems from SetF 1 problem from SetG9196 Schedule Inference Rules so far Day 09 Introductory Material 8 0 g fig 8 1 a a V Day 10 Direct Derivation DD 3 amp B B amp J Ba 11 CondItIonal DerIvatIon CD y Negation Derivation ND V0 N B NBV 3 VI q Indirect Derivation B B Day 12 show atomic 13 show disjunction so a 1 g a 13 DN Ne N1 Day 13 show conjunction NN Day 14 EXAM 2 3 w 2 SHeW Rules so far DirectDerivation Strategy MA In Direct Derivation DD one directly arrives at the very formula one is trying to show DD SHGW DD CD ND SHGW aC CD SHGW N ID As As SHGW C SHGW X Affiliated Rules Assumption Rule CD Ifone has a line of the form SHOW aC then one is entitled to write down the formula on the very next line as an 39 Assumption Rule ND Ifone has a line of the form SHOW N then one is entitled to write down the formula on the very next line as an 39 ContradictionIn XI if you have a formula and you have its negation then you are entitled to infer a contradiction absurdity ShowConditional Strategy SHOW aC ShowNegation Strategy Indirect Derivation SHCW J4 SHOW NA N As J4 As SHOW X SHGW X X X This is exactly parallel to ND and is another version of the traditional mode of reasoning known as REDUCTIO AD ABSURDUM Can we show the following 1 P a Q Pr We are stuck 2 NPa Q Pr 3 SHOW Q we have PaQ we also have NPaQ so to apply a0 so to apply a0 we must find P we must find NP orfind NQ or find NQ Using ID The difference between ID and ND is that ND applies only to negations whereas ID applies in principle to all formulas it is a generic rule like directderivation Although ID can in principle be used on any formula it is best used on two types of formulas 1 atomic formulas P Q R etc 2 disjunctions AVE ShowAtomic Strategy A is atomic PQR etc Example 2 Example 1 ShowDisjunction Strategy SHGW JAVZ 54VZ3 SHGW X o o X As Affiliated InferenceRule TildeWedgeOut v0 v 3 21 13 v B Example 4 Example 3 Example 5 v amp amp v NPamp v NPamp INTROLLKHC DAY 04 CHAPTER3 VALIDITY IN SENTENTIAL LOGIC Schedule for Unit 1 Day1 Intro Day 2 Chapter 1 Day3 Chapter2 Day 4 Chapter 3 2310 Day 5 Chapter 4 60 of Day 6 Chapter 4 Exam 1 Day 7 Chapter 4 Day 8 EXAM 1 Validity in General an argument is valid if and only if it is Mpossible for the conclusion to be false while the premises are true an argument is invaicl if and only if it is possible for the conclusion to be false while the premises are true Validity in Sentential Logic Exampe1 an argument is valid an argument is analid If an argument form has 2 atomic sentences if and only if if and only if then there are 4 cases 4 22 there is there is m case at least one case R S in which in which the premises are true the premises are true case 1 T T and and case 2 T F the conclusion is the conclusion is false false case 3 F T case 4 F F 5 What is a Case Example 2 A case is a If an argument form has 3 atomic sentences 3 possible combination of truthvalues the there are 8 cases 8 39 2 139 assigned to the atomic formulas Q R 3 case 1 T T T case 2 T T F case 3 T F T case 4 T F F case 5 F T T case 6 F T F case 7 F F T 6 case 8 F F F Example 1 Modus Tollens premise premise HI conclusion ithhenS notS HI notR Res Ms HI NR Example 2 Evil Twin of Modus Tollens premise TruthTable case R S R a S MS I NR 1 T T T F F 2 T F F T F 3 F T T F T 4 F F T T T Is there a case in which the premises are all true No butthe conclusion is false Is the argument form valid or invalid VALID premise HI conclusion ithhenS notR HI notS i R a s NR HI Ms 11 Counterexample ithhen S notR InotS ifl live in Boston I don39t live I don39t live then I live in Mass in Boston in Mass T T F 1E TruthTable TruthTable case R 8 Res NR INS case R 8 Res R IS 1 T T T F F 1 T T T T T 2 T F F F T 2 T F F T F 3 F T T T F 3 F T T F T 4 F F T T T 4 F F T F F Is there a case in which the premises are all true YES Is there a case in which the premises are all true No butthe conclusion is false butthe conclusion is false l Is the argument form valid or invalid l INVALID l l Is the argument form valid or invalid l VALID l 13 1339 Example 3 Example 4 Modus Ponens Evil Twin of Modus Ponens premise premise HI conclusion premise premise HI conclusion lithhens R HI 3 l lithhens 3 HI R l l R a s R HI 3 l l R a s 3 HI R l Counterexample if R then S 8 IR ifl live in Boston I live in I live in then I live in Mass Mass Boston T T F TruthTable case R S R a S S I R 1 T T T T T 2 T F F F T 3 F T T T F 4 F F T F F Is there a case in which the premises are all true YES butthe conclusion Is false Is the argument form valid or invalid INVALID Example 5 Modus Tollendo Ponens disjunctive syllogism premise premise HI conclusion l RorS notR HI 3 l l RvS NR iv 3 l TruthTable case R S R v S NR I S 1 T T T F T 2 T F T F F 3 F T T T T 4 F F F T F Is there a case in which the premises are all true No but the conclusion is false Isthe argument form valid or invalid VALID 20 Example 6 Evil Twin of MTP premise HI conclusion premise R or S R HI not 8 R v S R I NS 21 TruthTable case R S R v S R I NS 1 T T T T F 2 T F T T T 3 F T T F F 4 F F F F T Is there a case in which the premises are all true YES butthe conclusion is false Is the argument form valid or invalid INVALID 21 Example 7 23 TruthTable R I R amp s F T F T T F T T T F F T F T F F T T F T F F F Is there a case in which the premises are all true No but the conclusion is false Isthe argument form valid or invalid VALID 24 Example 8 m Logical Equivalence Two formulas are logically equivalent if and only if they have the same truthvalue no matter what in every case TruthTable R amp s I R F T T F T T T F F F T T F F T T F T F F F T F Is there a case in which the premises are all true YES butthe conclusion is false Is the argument form valid or invalid INVALID 23 Examples 7 and 8 ZOMBIE REASONING notRandS notRandnots notRorS notRornotS l ltxygt2 ll x2v2 i l xy ll xy l 28 TruthTable for 7 Valid Equivalence 1 Rg SH NRg NS RampSIIRVS F T T T F T F F T F T T T T T F F T T T F F F T F T F T T F F F T T T F T F F T T F F F T T F F T T F T F T T F F F T F T T F T F F F T F T T F T Do the formulas match in truth value T NO T T Do the formulas match in truth value T YES T T Are the two formulas logically equivalent T Are the two formulas logically equivalent T YES T 2 2 TruthTable for 8 Valid Equivalence 2 NRVS NRVNS RvSIIRampS F T T T F T F F T F T T T F T F F T F T T F F T T T F F T T F F T F T F F F T T T F T F T F F T T T F F F T T F F F T F T T F T F F F T F T T F T Dome formUIas matCh intrUth value T No T T Do the formulas match in truth value T YES T Are the two formUIas logically equwalent T No T T Are the two formulas logically equivalent T YES T 31 3339 INTRO LOGIC DAY 09 Basic Idea We start with a few argument forms which we presume are valid and we use these to demonstrate that other argument forms are valid We demonstrate show that a given argument form is valid by deriving deducing its conclusion from its premises using a few fundamental modes of reasoning UNIT 2 DERIVATIONS IN SENTENTIAL LOGIC Example 1 Modus Ponens MP gtC if lthenC A A C C Example Argument Form P P a Q we can employ Q a R modus ponens MP R a S to derive the conclusion from the premises 8 Example 1 continued Example 2 continued P PgtQ Q R RaS IS S R S Q R PgtQ P Q R MP MT R Q VP MT S P4 5 Example 2 Modus Tollens MT Example 3 using both MP and MT A gtC if thenC A gtC A gtC C not C A C 21 not A C 21 Example Argument Form NS Example Argument Form NS R a S R a S we can employ QaR I b ER modus tollens MT P a Q men g egg39939 a com Inatlon to derive the conclusion from f d anth I N NQ the premises NP 0 erlve e conc usnon from the premises NQ e Example 3 continued RaS wRaNT PeTNPaNQ Q MT NR MP NT VT IMP Derivations How to Continue Derivations How to Start argument P PaQ QaR Res 8 write down premises write down SHOWquot conclusion P Pr premise P a Q Pr Q a R Pr R a S Pr SHOW 8 what one is trying to derive 3 Apply rules as applicable to available lines until goal is reached 1 P Pr premise 2 P a Q Pr premise 3 Q a R Pr premise 4 R a S Pr premise 5 SHOW 8 goal 6 Q 12MP follows from 7 R 36MP lt lines 4 and 7 8 S 47MP by modus ponens 11 Derivations How to Finish 4 Box and Cancel 1 P Pr 2 P a Q Pr 3 Q a R Pr 4 R a S Pr 5 SHOW 8 Direct Derivation 6 Q 12MP 7 R 36MP 8 S 47MP Derivation Example 2 argument NS RaS QaR PaRP NS Pr R a S Pr Q a R Pr P a Q Pr SHOW NP W R 12MT Q 36MT P 47MT Derivation Example 3 argument NS R s NR NTPgtT NPaNQ NQ 1 2 3 4 5 6 7 8 9 1 0 S Res NRaNT PaT PaQ SHOVVQ Pr Pr Pr R T P Q Initial Inference Rules Modus Ponens aC C Modus Tollens a C NC N Modus Tollendo Ponens 1 vB N 3 Modus Tollendo Ponens 2 vB Examples of Modus Ponens A gt C PampQ gt RVS P gt Q A PampQ NP C RVS NQ Examples of Modus Tollens A gt C PampQ gt RvS P gt Q C RvS Q l PampQ P P gt Q valid argument BUT P NOT an example of MT 17 Form versus ContentNalue are the dime and ten pennies the same NO are they the same in value YES actually in a very important sense they are not the same in value 2000000 Examples of MTP1 A v B PampQ v RVS NP v Q NA PampQ P E E Q 19 Examples of MTPZ A v B PampQ v RVS NP v Q B RVS Q E p ao T INTRO LOGIC DAY 19 Translations in PL 5 Review 2 Quantifiers 1 Predicate everyone respects everyone VX Vy RXy someone respects someone EIX Ely RXy no one respects everyone NEIX Vy RXy no one is respected by everyone NEIX Vy RyX everyone respects someone or other VX Ely RXy there is someone Whom everyone R s EIX Vy RyX there is someone who respects no one Elx Ey ny there is someone Whom no one respects Elx Ey Ryx Review 1 Quantifier 1 Predicate Jay respects someone EIX RjX Jay respects everyone VX RjX Jay respects no one N EIX RjX some one respects Kay EIX RXK everyone respects Kay VX RXK noone respects Kay EX RXK New Material 1 quantifier 2 predicates 2 quantifiers 2 predicates 2 quantifiers 3 predicates Example 1 3 Example 3 3 every student respects Kay Jay respects some at least one student IF W are a student there is someone Who is a S whomj respects no matter who you are THyEN you respeCt ay who is a S and whomj R39s no ma erwho X is IF x is a student there is someone x x is a S andj R39s x THEN x respects Kay 3x Sx amp RJX Vx Sx gt ka 5 7 Example 2 D Example 4 D no Student respects Jay noone respects every politician there is no one who respects every politician there is no39one Who is a S and Who R39sj there is no x x respects every politician there is no x x is a S and x R39sj NEIX X respects eVery POIitiCian 3x Sx amp ij Example 4b 3 x respects every politician no matter who you are IF you are P THEN X R39s you IF y is P THEN x R s y Vy Py a ny wax no matter who y is VyPyany 3x Vy Py gt ny Example 5b x R39s some one VxSxe there is someone who is R39ed by X whom x R39s there is some y xR39sy 3y ny VxSx gt3nyy Example 5 every Student respects someone or other no matter who you are IF you are S THEN you R someone no matter who x is IF x is 8 THEN x R39s someone Vx Sx x R39s someone Vx Sx a x R39s someone Example 6 there is a politician whom noone respects there is someone who is a P whom noone R s who is P AND whom noone R39s there is some x x is P AND noone R39s x EIX PX amp noone R39s x 3x Px amp noone R39s x Example 6b 3 no one R39s X there is no one who R s X EIX PX amp there is no y y R39s X Ey RyX EIX PX amp 3y Ryx Example 7b 3 every C R39s X no matter who you are if you are C then you R X 3X PX amp no matter who y is ify is C then y R39s X Vy Cy RyX VyCy a RyX EX PX amp VyCy gt Ryx Example 7 D there is a Politician whom every Citizen Respects there is who is a P whom every C R39s someone who is P AND whom every C R39s there is some X X is P AND every C R39s X EIX PX amp every C R39s X EXPX ampeveryCR39sx Example 8 D every Citizen Respects some Politician or other no matter who you are if you are C then you R some P no matter who X is ifX is C then X R39s some P VX CX X R39s some P VXCX gtXR39ssomeP Example 8b x R39s some P there is someone VX OX 4 who is a P R39ed by x who is a P AND who is R39ed byx who is P AND Whom X R39s V there is some y yisP AND xR39sy 3y Py amp ny 3H Pyampny VXCX gt 3yPy amp ny Example 9b xR39s no P there is no one who is a P AND who is R39ed by x 3X CX 8 there is no y y is P AND y is R ed byx yisP AND xR39sy Nay Py amp ny EyPyampny V EXCX amp 3yPy amp ny Example 9 there is a Citizen who Respects no Politician there is someone who is a C who R39s no P who is a C AND who R39s no P there is some x xisaC AND xR39snoP Elx Cx ampxR39snoP 3xCxampxR39snoP INTRO LOGIC Derivations in PL DAY 25 4 Rules Introduced Day 1 VvFv Fo E0 317 a name counts as OLD precisely if it occurs somewhere unboxed and uncancelled Overview Exam 1 Exam 2 Exam 3 Exam 4 6 derivations Exam 5 Exam 6 Sentential Logic Translations Sentential Logic Derivations Predicate Logic Translations Predicate Logic Derivations 15 points 10 free points very similarto Exam 3 very similarto Exam 4 Rules Introduced Day 2 Q ElvFv SHOW VvFv Fn SHOW Fn a name counts as NEW precisely if it occurs nowhere unboxed or uncancelled Rules Introduced Day 3 NVO 30 Vv Elv EIUNQD Vv lt1 is any formula v is any variable ShowV Strategy UD SHGVWVUHU W W o o o n must be a NEW name Strategies main operator showstrategy N a amp v SL strategy V UD El DD orEID Show3 Strategy EID 346va EIvCD 3vc1gt SHOW x o o X Example 3 there is someone Whom everyone R s leveryone R s someone or other 1 HXVyRyx Pr 2 Vx nyy UD 3 W HyRay 3D ID 4 N yRay As 5 W X DD 6 VyRyb 1 30 7 vyw Ray 4 N30 8 Rab 6 V0 9 NRab 7 V0 10 X 89 X someone 4 is F then 1 there is someone who R s noone is disR ed someone or other Pr UD there is someone who R s every F F is Red someone or other Pr UD is some F who R s no one is disR ed some F or other amp Pr amp UD amp amp Fbe NNRba NRba X
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