Introduction To Logic
Introduction To Logic PHIL 110
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This 42 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 Gary Hardegree in Fall. Since its upload, it has received 13 views. For similar materials see /class/232334/phil-110-university-of-massachusetts in PHIL-Philosophy at University of Massachusetts.
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Date Created: 10/30/15
INTRO LOGIC DAY 04 Schedule for Unit 1 Day1 Intro Day2 Chapter1 Day 3 Chapter 2 40 of Day4 Chapter3 Day 5 Chapter 4 60 of Day6 Chapter4 Day 7 Chapter 4 Day 8 EXAM 1 CHAPTER 3 VALIDITY IN SENTENTIAL LOGIC Validity in General an argument is valid an argument is invaid if and only if if and only if it is Mpossible for it is possible for the conclusion to be the conclusion to be false false while while the premises are true the premises are true What are Possibilities in Sentential Logic at least a possibility is a case which is a possible combination of truthvalues assigned to the atomic sentences Example 1 If an argument form has 2 atomic sentences then there are 4 cases R S case 1 T T case 2 T F case 3 F T case 4 F F Example 2 If an argument form has 3 atomic sentences then there are 8 cases Q R 8 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 case 8 F F F In General If an argument form has n atomic sentences then there are 2quot cases Example 1 Modus Tollens premise2 HI conclusion premise1 ithhenS notS HI notR R gtS S HI R 9 TruthTable RSS F R T T F F T F T Is there a case in which the premises are all true N0 but the conclusion Is false Is the argument form valid or invalid VALID 10 Example 2 Evil Twin of Modus Tollens premise1 premise2 HI conclusion ithhenS notR HI notS R gt S R HI S Counterexample if R then S not R I not S if I am in Boston I am not in I I am not in then I am in Mass Boston Mass T T F TruthTable case R S 1 mmaam 2 3 4 mamam F T T Is there a case in which the premises are all true but the conclusion is false YEs Is the argument form valid or invalid l INVALID l 13 Example 3 Modus Ponens premise1 premise2 HI conclusion lithhens R HI 8 l Rae R HI 3 TruthTable case R S 1 mmaam 2 3 4 mamam Is there a case in which the premises are all true but the conclusion is false l l Is the argument form valid or invalid l VALID l 15 Example 4 Evil Twin of Modus Ponens premise1 premise2 HI conclusion l ithhenS 8 HI R l Rae 3 HI R Counterexample if R then S S IR if I am in Boston I am in I I am in then I am in Mass Mass Boston T T F 17 TruthTable R S R T T F F Is there a case in which the premises are all true but the conclusion is false YES l Is the argument form valid or invalid l INVALID l 18 Example 5 Modus Tollendo Ponens disjunctive syllogism premise1 premise2 HI conclusion RorS notR HI 8 l RVs R M s TruthTable R S R T T F F Is there a case in which the premises are all true but the conclusion is false l l Is the argument form valid or invalid VALID 20 Example 6 Evil Twin of MTP premise HI conclusion premise R or S R HI not S RVs R h s 21 TruthTable R S R T T F F Is there a case in which the premises are all true but the conclusion is false YEs Is the argument form valid or invalid INVALID 22 Example 7 notR notRandS R RampS 23 TruthTable R R amp s F T F T 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 Is the argument form valid or invalid VALID 24 Example 8 notRandS notR RampS l R 25 TruthTable R amp S R F T 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 but the conclusion Is false Is the argument form valid or invalid INVALID 26 Logical Equivalence Two formulas are logically equivalent if and only if they have the same truthvalue no matter what in every case Examples 7 and 8 ZOMBIE REASONING notR and S notR and notS notRorS notRornotS IT JUSTLKEMATH X2y2 xy 28 xy2 vxy Examples 7 and 8 cont TnotRandST i T notRandnotS T TnotRorST i T notRornotS T TruthTable for 7 RampSll R amp 3 F T T T F T F F T T T F F F T F T F T F F T T F F F T T F F F T F T T F T Do the formulas match in truth value T N0 T T Are the two formulas logically equivalent T N0 T 30 TruthTable for 8 R v S N R v N S F T T T F T F F T F T T F F T T T F F F T T T F T F T T F F F T F T T F Do the formulas match in truth value N0 Are the two formulas logically equivalent N0 Valid Equivalence 1 SI Do the formulas match in truth value YES Are the two formulas logically equivalent YES 32 Valid Equivalence 2 ll Do the formulas match in truth value lYEsl Are the two formulas logically equivalent YEsl 33 INTRO LOGIC DAY 09 UNIT 2 DERIVATIONS IN SENTENTIAL LOGIC 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 that a given argument form is valid by deriving deducing its conclusion from its premises using a few fundamental modes of reasoning Example 1 Modus Ponens MP A a C if A then C A A C C a derivative 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 S Example 1 continued P PeQ QR RS S UP l Q MP R MP S lt Example 2 Modus Tollens MT A a C if A then C C not C A not A a derivative argument form S R a S we can employ Q a R modus tolens MT P a Q to derive the conclusion from the premises P Example 2 continued NS RS QR PQ P mT l NR MT Q MT P4 Example 3 using both MP and MT aC NC derivative argument form of MP and MT to derive the conclusion from the premises we can employ a combination S RAS RaT PAT PaQ Q Example 3 continued S RS RaT PaTPaQ Q MT NR MP NT MT NP MP NQ 4 Derivations How to Start argument P P gtQ Q gtR R gtSS 1 write down premises 2 write down SHOW conclusion 1 P Premise 2 P gt Q Pr 3 Q gt R Pr 4 R gt S Pr 5 SHOW 8 the goal Derivations How to Continue 3 apply rules as appropriate to available lines until goal is reached 1 P Pr 2 P gtQ Pr 3 Q gtR Pr 4 R gtS Pr 5 SHOWS goal 6 Q 12 MP 7 R 36 MP 8 s 47 MP l follows from lines 1 and 2 by modus ponens 11 Derivations How to Finish 4 Box and Cancel 1 P Pr 2 P gt Q Pr 3 Q gt R Pr 4 R gt S Pr 5 8 DD 6 Q 12 MP 7 R 36 MP 8 8 47 MP DD Direct Derivation 17L dW iW CD dW 00 iW LO q Y l AAAAAAAAAO vvvvvvvvvv ONIONedeiFdfiNFHNfse f I Z 9 adwexa 9L iW iW iW G LO q Y vvvvvvvv dfoedfaebfseafs z adwexa Initial Inference Rules Modus Ponens Modus Tollens 54 a C 54 a C 54 NC C 54 Modus Tollendo Ponens 1 Modus Tollendo Ponens 2 54 v B 54 v 3 54 NTB B 54 Examples of Modus Ponens it a C PampQ a RvS NP a Q it PampQ P C RVS N Q Examples of Modus Tollens it a C PampQ a RvS NP a Q C RVS Q l PampQ P P a Q valid argument E BUT P NOT an instance of MT 17 Form versus Content I Value are the dime and ten pennies the same NO are they the same in value YES these particular ten pennies are 1943 copper pennies and are worth to collectors 2000000 Examples of MTP1 A v B PampQ v RVS P v Q A PampQ P B RVs Q Examples of MTP2 A v B PampQ v RVS P v Q 13 RVS Q A PampQ P 20 THE END INTRO LOGIC DAY 23 Derivations in PL 2 Overview Exam 1 Sentential Logic Translations Exam 2 Sentential Logic Derivations Exam 3 Predicate Logic Translations Exam 4 Predicate Logic Derivations 6 derivations 15 points 10 free points Exam 5 very similar to Exam 3 Exam 6 very similar to Exam 4 Exams 5 and 6 will be given on Friday Dec 19 130330 400 Mahar Auditorium provided on exams available on course web page textbook keep this in front of you when doing homework Rule Sheet x v z x v1 don t make UP your own rules Sentential Logic Rules DD ID CD D ampD eto ampl amp0 VO e0 NVO eto Predicate Logic Rules Universal Derivation UD day 2 UniversalOut VO day 1 TildeUniversaIOut NVO day 3 ExistentialIn 3 day 1 ExistentialOut 30 day 2 TildeExistentiaIOut 30 day 3 Rules Already Introduced Day 1 VvFv H0 Fo EvFv o is an OLD name more about this later Rules to be Introduced Today Universal Derivation UD ExistentialOut ii oi Example 1 I every F is H everyone is F everyone is H 1 VXFX a HX Pr 2 VXFX Pr 3 W VXHX what 3 W Ha amp Hb amp Hc amp ampampampD is a W Ha ultimately l involved b M Hb in C2 l showing 0 W Ho 7 a 7 7 7 universal Example 1a VXFX a Hx VXFX SHOW VXHX Ha Fa a Ha v0 Fa V0 Ha a0 one down a zillion to go Example 1b VXFX a Hx VXFX SHOW VXHX Hb Fb a Hb VO Fb VO Hb a0 two down a zillion to go Example 1c 1 VXFXHX Pr 2 VXFX Pr 3 SHOW VXHX 4 Hc DD 5 Fcch 1 VO 6 Fc 2 VO 7 Ho 56 a0 three down a zillion to go But wait the derivations all look alike DD VO 0 DD VO 0 DD The UniversalDerivation Strategy All we need to do is do one derivation with one name say a and then argue that all the other derivations will look the same To ensure this we must ensure that the name is general which we can do by making sure the name we select is NEW a name counts as NEW precisely if it occurs nowhere in the derivation unboxed or uncancelled The UniversalDerivation Rule UD SHOW V1 lv v is any variable Fv is any official formula SHOW Fn n replaces v n must be a NEW name 0 Le one that is occurs nowhere in the derivation unboxed or uncancelled Comparison with UniversalOut l3 VvFv F0 SHOW VvFv SHOW Fn E a name counts as OLD precisely if it occurs somewhere in the derivation unboxed and uncancelled a name counts as NEW precisely if it occurs nowhere in the derivation unboxed or uncancelled Example 2 I every F is H if everyone is F then everyone is H 1 VXFX a Hx Pr 2 VXFX a VXHX CD 3 VXFX As 4 VXHX UD a new 5 Ha DD 6 FaaHa 1 vo aoldl 7 Fa 3 vo a old l 8 Ha 67 a0 Example 3 everyFisGeveryGisHeveryFisH I 1 VXFX a GX Pr 2 VXGX a Hx Pr 3 VXFX a Hx UD a new 4 Fa a Ha CD 5 Fa As 6 Ha DD 7 Faaea 1 vo aold 8 Ga a Ha 2 vo a old 9 Ga 57 a0 10 Ha 89 a0 17 Example 4 I everyone R s everyone everyone is R ed by everyone 1 Pr UD DD 1 VO bold 5 VO aold ExistentialOut 30 any variable 2 y X w any formula l gt EvFv Fn l n replaces v any NEW name a b c d Comparison with UniversalOut VvFv F0 EvFv Fn a name counts as OLD precisely if it occurs somewhere unboxed and uncancelled a name counts as NEW precisely if it occurs nowhere unboxed or uncancelled 20 Example 5 every F is unH no F is H 1 VXFX a HX Pr 2 EIXFX amp Hx ND 3 EXFx amp Hx As 4 X DD 5 Fa amp Ha 3 30 new 6 Fa Ha 1 vo old i 7 Fa 5 ampO 8 Ha 9 Ha 67 so 10 X 89 XI 21 Example 6 some F is not H not every F is H 1 EXFx amp HX Pr 2 VXFX a Hx D 3 VXFX a Hx As 4 X DD 5 Fa amp Ha 1 30 new 6 Fa Ha 3 v0 old i 7 Fa 5 ampO 8 Ha 9 Ha 67 so 10 X 89 XI 22 Example 7 every F is G some F is Hsome G is H 1 VXFX a GX Pr 2 EXFx amp Hx Pr 3 S EXGX amp Hx DD 4 Fa amp Ha 2 30 new 5 Fasea 1 vo old i 6 Fa 4 ampO 7 Ha 8 Ga 56 a0 9 Ga amp Ha 78 ampI 10 3XGx amp Hx 9 BI 23 Example 8 ifanyone is F then everyone is H if someone is F then everyone is H 1 VXFX a VyHy Pr 2 EIXFX a VXHX CD 3 EIXFX As 4 VXHX UD new 5 S Ha DD 6 Fb 3 30 new 7 FbaVyHy 1 vo old i 8 VyHy 67 a0 9 Ha 8 vo old i 24 Example 9 if someone is F then everyone is H if anyone is F then everyone is H EIXFX a VXHX Pr VXFX a VyHy UD new Fa a VyHy CD Fa As 8 VyHy UD new 8 Hb DD EIXFX 4 EII VXHX 17 0 Hb 8 VO old Example 10 a fragment someone R s someone missing premises everyone R s everyone EIXEInyy Pr 7 Pr 8 VXVnyy UD new VyRay UD new Rab ElyRcy 1 EIO new Rod 6 iao newi THE END
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