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# Introduction To Logic PHIL 110

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INTRO LOGIC DAY 05 Schedule for Unit 1 Day 1 Intro Day 2 Chapter 1 Day 3 Chapter 2 40 of 8 arguments Day 4 Chapter 3 5 pts each Day5 Chapter4 60 of 12 translations D 6 Ch t 4 ay aper 5ptsea h Day 7 Chapter 4 I Day 8 EXAM 1 xxxx CHAPTER 4 TRANSLATIONS IN SENTENTIAL LOGIC Basic Idea In order to uncover logical forms we paraphrase English sentences so that they contain only standard connectives We also use special abbreviations and special punctuation Abbreviation Scheme atomic sentences are abbreviated by upper case letters of the Roman alphabet standard connectives are abbreviated by special symbols Iogograms compound sentences are abbreviated by algebraic combinations of 1 and 2 Standard Connectives connective symbol and amp or v not if then a NonStandard Connectives a few examples but although or butnotboth neither nor if onlyif ifand only if unless is necessary for is sufficientfor if othenNise unessin which case Simple Transformations Some simple sentences are straightfonNardly equivalent to compound sentences Jay and Kay are students and Jay is a student Kay is a student X90 Kay is a rich student Kay is rich and Kay is a student 19030 Pitfall 1 Kay is a former student Kay is a former and Kay is a student F amp S Pitfall 2 two or more meanings ambi also amphi means both and around otherwords containing ambi and amphi ambivalent ambitious amphitheater ambidextrous ambienoe ambiance amphibian Jayand Kay are married ambiguous First Reading Jay and Kay are married Jay is married to someone J and amp Kay is married to someone K 11 Second Reading Jay and Kay are married Jay and Kay are M marriedtoeachother later 39n red39cate o 39c P 939 Mjk further parts are revealed Pitfall 3 Odd use of and Sometimes and is used as follows 1 keep trying and you will succeed 2 keep it up buster and I will clobber you 3 give him an inch and he will take a mile 4 give me the money and I ll give t 5 you do that and I ll kill ya Pitfall 3 continued What happens if we symbolize keep trying and you will succeed as K amp S K you will keep trying 8 you will succeed K amp 8 you will keep trying and you will succeed 17 appropriate paraphrase 1 if you keeptrying then you will succeed K S 2 if you keep it up buster then I will olobber you K C 3 if you give him an inch then he will take a mile G T 4 if you give me the money then I ll give you the product M P 5 if you do that then I ll kill ya D K 18 Exclusive Or Ram amp Sam will win the election Ram will win or Sam will win but not both at least one of them win R v S also amp both of them will win not R amp S alternatively Ram will win and Sam will not win R amp S or v Sam will win and Ram will not win S amp NR NeitherNor it is neither raining nor sleeting it is not raining and it is not sleeting N R amp N S Jay is not sleeping and Kay is not sleeping neither Jay nor Kay is sleeping JampK Alternative Formulation it is not true that it is raining or it is sleeting it is neither raining nor sleeting it is not raining or sleeting RVS RampS Conditionals standard expression if A then C which is symbolized A is the antecedent C is the consequent Simple Variants of ifthen if A then C all of these are if A C symbolized the C if A same if always introduces the antecedent then always introduces the consequent 23 Other Variants of if C provided A provided A C C in case A in case A C C supposing A supposing A C all symbolized the same way Only If only if is not equivalent to if lwill get an A only if I take all the exams vs lwill get an A if I take all the exams lwill get an A if I get a hundred vs lwill get an A only if I get a hundred How does only work 1 employees only 2 authorized personnel only 3 cars only 4 right turn only only operates as a dualnegative modifier for example 1 means to exclude anyone who is not an employee a more lyrical example of only I only have eyes for youquot The Flamingos 1959 lyrics by Al Dubin music by Harry Warren 1934 only is focussensitive only have eyes for you alternative focus for only only have eyes for you alternative focus for only only have eyes for you How does only modify if only modifies if by introducing two negations ifna B then notA Example lwill get an A only if I take all the exams A ony If E K lwill not get an A if I do not take all the exams 391 A If 391 E if I do not take all the exams then I will not get an A K if not E then not A If And Only If Two approaches to if and only if 1 treat it as a simple connective H 2 treat it as a complex connective We will concentrate on 2 according to which if and only if consists of 3 components 1 if 2 and 3 only if each of which we already know how to paraphrase Example of if and only if Iwill pass if and only if I average 50 P if and only if A P if A and P only if A if A then P not P if not A if not A then not P A gt P amp A gt P 39 Unless The following are all equivalent Iwill pass only if I study P only if S I will not pass unless I study not P unless S I will not pass if I do not study not P if not S only if has two builtin negations unless has one builtin negation if has no builtin negation Example of unless wi not pass unless I study not P unless 8 not P if not 8 not P if not 8 if not 8 then not P 8 gt P THE END INTRO LOGIC DAY 10 Derivations in SL 2 Initial Modes of Reasoning Modus Ponendo Ponens affirming by affirming 54 a C J4 C Modus Tollendo Tollens denying by denying 54 a C C 54 Modus Tollendo Ponens 1 affirming by denying Modus Tollendo Ponens 2 affirming by denying J4 V 13 J4 V 13 54 23 23 54 l aka quot 39 syllogism l Review We demonstrate show that an argument is valid by deriving deducing its conclusion from its premises using a few fundamental modes of reasoning Example 1 review s Res RVT PVT PsQ Q Pr 1write all 1 NS premises 2 R a s Pr 2write show 3 R V NT Pr conclusion 4 NP v T Pr 5 MP N P 3apply rules Q r to available 6 NQ on lines 7 NR 112 MT 8 NT 37 MTP1 43quot 9 MP 48 Mm highfive 10 NO 59 MP Tek gives Manny a highfive Tek gives Rules discussed Today J4 J4 J4ampB J4amp3 3 3 7 7 7 7 3 J4amp3 3ampJ4 J4 J4 Avg 7 7 29 J4 provided on exams available on course web page textbook keep this in front of you when doing homework Rule Sheet Rules of Inference Basic Idea Every connective has 1 a takeout rule also called an eliminationrule OUTrule 2 a putin rule also called an introductionrule lNrule AmpersandOut ampO if you have a conjunction J4 amp B then you are entitled to infer its first conjunct J4 if you have a conjunction J4 amp B then you are entitled to infer its second conjunct B have means have as a whole line rules apply only to whole lines not pieces of lines WedgeOut v0 if you have a disjunction A v B and you have the negation of its 1st disjunct NA then you are entitled to infer its second disjunct B if you have a disjunction A v B and you have the negation of its 2quotd disjunct NB then you are entitled to infer its first disjunct J4 what we earlier called modus tollendo ponens AmpersandIN ampI if you have a formula J4 and you have a formula B then you are entitled to infer their first conjunction J4 amp B if you have a formula J4 and you have a formula B then you are entitled to infer their second conjunction B amp J4 WedgeIN vI if you have a formula J4 then you are entitled to infer its disjunction with any formula to its right J4 v B if you have a formula J4 then you are entitled to infer its disjunction with any formula to its left B v J4 ArrowOut gtO if you have a conditional A a C and you have its antecedent A then you are entitled to infer its consequent C if you have a conditional a C and you have the negation of its consequent NC then you are entitled to infer the negation of its antecedent NA What we earlier called modus ponens and modus tollens DoubleNegation DN if you have a formula A then you are entitled to infer its doublenegative NNA if you have a doublenegative NNA then you are entitled to infer the formula A rules apply only to whole lines not pieces of lines ArrowIntroduction ARROWIN HI 0 V X THERE IS NO SUCH THING AS what we have instead is CONDITIONAL DERIVATION which we examine later Direct Derivation The Original and Fundamental SHOWRule SHe WA ln Direct Derivation DD one directly arrives at the very formula one is trying to show Example 2 NSRaSNRaNTPaTNPaNQINQ 1 Ms Pr 2 R a s Pr 3 NR a NT Pr 4 P a T Pr 5 MP a NQ Pr 6 NQ DD 7 NR 12 so 8 NT 37 so 9 MP 48 so 10 NQ 59 so Example 3 NSRaSRvNT NPVTNPgtNQINQ 1 Ms Pr 2 Rss Pr 3 RV NT Pr 4 Pv T Pr 5 MP NQ Pr 6 NQ DD 7 NR 12 so 8 NT 37 v0 9 MP 48 v0 10 NQ 59 so ArrowOut Strategy If you have a line of the form aC then try to apply arrowout a0 which requires a second formula as input in particular either J4 or NC have AaC find A NC or deduce C 21 WedgeOut Strategy If you have a line of the form AVE then try to apply wedgeout v0 which requires a second formula as input in particular either NA or NZ have AvB nd 21 13 or deduce B A Example 4 PaQvRPQvRNR PaQaQaRINQ NR VR v NR NQ Pr Pr Pr DD 1 1 Example 5 PvQSRvSTPT V V P as aT Pr Pr Pr DD WedgeIn Strategy If you need AVE then look for either disjunct find A OR find B then apply vI to get AvB Example6 PaQRvQaSRaTNTampPIQampS 1 PaQ Pr 2 RVQ S Pr 3 RaT Pr 4 NTampP Pr 5 QampS DD 6 NT 7 P 4ampo 8 Q 17 0 9 NR 36 0 10 Qas 29 v0 11 s 810 o 12 QampS 811ampI AmpersandOut Strategy If you have a line of the form A83 then apply ampersandout 80 which can be applied immediately to produce A and B AmpersandIn Strategy If you are trying to find or show A83 then look for both conjuncts nd A AND nd 3 then apply 81 to get A83 INTRO LOGIC DAY 15 UNIT 3 Translations in Predicate Logic Overview lt Exam 1 Exam 2 Exam 3 Exam 4 Exam 5 Exam 6 Sentential Logic Translations Sentential Logic Derivations Predicate Logic Translations Predicate Logic Derivations finals week finals week very similar to Exam 3 very similar to Exam 4 Exams 5 amp 6 Friday Dec 19 130 330 pm Mahar Auditorium Grading Policy When computing your final grade I count your four highest scores A missed exam counts as a zero Subjects and Predicates In predicate logic every atomic sentence consists of one predicate and one or more subjects including subjects direct objects indirect objects etc in mathematics subjects are called arguments Shakespeare used the term argument to mean subject Examples 1 Subject Predicate Jay is asleep Kay is awake Elle is a dog Examples 2 Subject Predicate Object Jay respects Kay Kay is next to Elle Elle is taller than Jay Examples 3 Direct Indirect Subject Predicate Object Object Jay sold Elle to Kay Kay bought Elle from Jay Kay prefers Elle to Jay What is a Predicate A predicate is an quotincompletequot expression ie an expression with one or more blanks such that whenever the blanks are filled by noun phrases the resulting expression is a sentence noun phrase1 I predicate noun phrase2 Compare with Connective A connective is an quotincompletequot expression ie an expression with one or more blanks such that whenever the blanks are filled by sentences the resulting expression is a sentence sentence1 I connective sentence2 j V sentence3 Examples l I is taller than I W Irecommendsl Symbolization Convention 1 Predicates are symbolized by upper case letters 2 Subjects are symbolized by lower case letters 3 Predicates are placed first 4 Subjects are placed second PRED sub1 sub2 Examples Jay is tall Tj Kay is tall Tk Jay is taller than Kay Tjk Kay is taller than Elle Tke Jay recommended Kay to Elle Rjke Kay recommended Elle to Jay Rkej Compound Sentences 1 Jay is not ta Tj Jay is not taller than Kay NTjk both Jay and Kay are ta Tj amp Tk neither Jay nor Kay is tall NTj amp Tk Jay is tallerthan both Kay and Elle Tjk amp Tje Compound Sentences 2 Jay and Kay are married individualy Jay is married and Kay is married Mj amp Mk I Jay and Kay are married to each other I Mk I and are married to each other 15 Quantifiers quantifiers are linguistic expressions that convey quantity Examples every all any each both either some most many several few no neither at least one at least two at most one at most two exactly one exactly two Quantifiers 2 quantifiers combine commonnouns and verbphrases to form sentences Examples every seni no freshman is happy or is happy predicate logic treats both commonnouns and verbphrases as predicates at least onejunior is happy few sophomores are happy most graduates are happy The Two Special Quantifiers of Predicate Logic official name 5 9quot symbol expressions universal every V quan er any existential some El quantifier at least one Names of Symbols V upsidedown A El backwards E Actually they are both upsidedown How Traditional Logic Does Quantifiers Quantifier Phrases are Simply NounPhrases 20 How Modern Logic Does Quantifiers quantifier phrases are gerrymandered into sentential modifiers aka connectives 21 Existential Quantifier some one is happy there is some one who is happy there is some one such that heshe is happy there is some X such that X is happy Elx Hx pronunciation of symbols i there is an x such that H x 22 Universal Quantifier every one is happy every one is such that heshe is happy whoever you are you are happy no matter who you are you are happy no matterwho X is X is happy Vx Hx pronunciation of symbols for any x H x 23 Negating Quantifiers modern logic takes EI to mean at least one which means one or more which means one ortwo orthree or if a counting number is not oneormore it must be zero thus the negation of at least one is not at least one which is equivalent to none 24 NegativeExistential Quantifier no one is happy there is no one who is happy there is no one such that heshe is happy there is no X such that X is happy there isn t some X such that X is happy EIX HX pronunciation of symbols i there is no X such that H X 25 NegativeUniversal Quantifier not every one is happy heshe is H it is not true that whoever you are you are H it is not true that no matter who you are you are H it is not true that no matter who X is X is H VX HX not every one is such that pronunciation of symbols not for any X H X 26 Quantifying Negations 1 suppose then noteveryoneis happy thaeissomeone vvhois nothappy themissomex xisnothappy mewmmmewwmm bmwvmm VXHX E 27 Quantifying Negations 2 suppose then nooneishappy no matter who you are you are nothappy no matter who X is xisnothappy mewmmmemwmm bmwvwd 3XHX E 28 INTRO LOGIC DAY 11 Derivations in SL 3 provided on exams available on course web page online textbook keep this in front of you when doing homework Rule Sheet m don t make UP your own rules Review We demonstrate show that an argument is valid by deriving deducing its conclusion from its premises using a few fundamental modes of reasoning Inference Rules excerpt w ampo 8593 8593 93 B 7 3 8593 385 w vo vB vB N N vB Bv 93 gtI gtO J4gtC J4gtC NC C N DirectDerivation The Original and Fundamental SHeWRule Conditional Derivation CD SHGVWJA In Direct Derivation DD one directly arrives at the veryformula one is trying to show short for assumption depends upon formula Affiliated Assumption Rule if one has a line of the form then one is entitled to write the formula on the very nextline as an assumption SHOW J4C Can we show the following using DD 1 PaQ 2 QaR Pr We are stuck Pr 3 SHOW P a R 7 we have so to apply we must or nd PaQ we also have 0 so to apply nd P Q or nd we must find Example 1 1 P a Q Pr 2 Q a R Pr 3 P a R CD 4 P As 5 W R DD 6 Q 1 4 o 7 R 2 6 o ArrowOut Strategy Example 2 1 P a Q Pr 2 PampQ a R Pr 3 P a R CD 4 P As 5 R DD 6 Q 14 0 7 PampQ 46 ampI 8 R 27 0 Can we show the following using DD 1 P a Q Pr 2 Q P Pr 3 SHOW P 7 We are stuck Example 3 Pa a a a PeQ PgtR we have PaQ we also have QaP so to apply 0 so to apply 0 we must find P we must find Q or nd Q or find P 11 Negation Derivation D SHOW 54 A As snow x x xx Historically this method of reasoning is called EDUCTIO AD ABSURDUM Latin for reducti on to absurdity In symbolic logic an absurdity is a selfcontradiction both asserting and denying the same proposition N Affiliated Rules ContradictionIn XI if you have a formula J4 and you have its negation N then you are entitled to infer a contradiction absurdity Assumption Rule If one has a line ofthe form SHOW NA then one is entitled to write the formula on the very next line as an assumption Example 5 Example 4 INTRO LOGIC DAY 11 Derivations in SL 3 Review We demonstrate show that an argument is valid by deriving deducing its conclusion from its premises using a few fundamental modes of reasoning provided on exams available on course web page make a 7 copy and I keep it in front of you when doing homework Rule Sheet m x v don make Exam 246 uP your own Exam 46 r39ules Inference Rules excerpt w E A A AampB AampB B B A B AampB BampA Q m A A AVB AvB MA MB AvB BvA B A a1 0 AAC AAC A NC C NA DirectDerivation The Original and Fundamental SHeWRule SHGVVZA o o 54 in Direct Derivation DD one directly arrives at the very formula one is trying to show Can we show the following using DD 1 P gtQ Pr 2 Q gtR Pr 3 SHOWP gtR we are stuck we have P gtQ so to apply gtO we must find P or find Q we also have Q gtR so to apply gtO we must find Q or find R ArrowO ut Strategy Conditional Derivation CD w ag conditional derivation 54 As assumption SHOW C depends on formula 0 o o Affiliated AssumptionRule ifone has a line of the form SHOW AaC then one is entitled to write the formula 54 on the very next line as an assumption Example 1 Example 2 Example 3 a OAR AQ a AR PAQ PAR Can we show the following using DD 1 P gt Q Pr We are stuck 2 Q gt P Pr 3 SHOW P we have P gtQ we also have Q gtP so to apply gtO so to apply gtO we must find P we must find Q or find Q or find P ArrowO ut Strategy x x Negation Derivation D SHOW 54 54 As SHOW X X XI special symbol for absurdity always done by DD see later for details to show that 54 is false 54 one assumes that 54 is true and shows that this leads to absurdity x 2 Fancy Name Historically this method of reasoning is called RE UCTIO AD ABSURDUM Latin for reduction to absurdity to demonstrate that a proposition is false he assumes that it is true d demonstrates that this assumption leads to absurdity in symbolic logic an absurdity is a selfcontradiction both asserting and denying the same proposition Affiliated Rules ContradictionIn XI if you have a formula and you have its negation then you are entitled to infer 595 a contradiction absurdity X Assumption Rule If one has a line ofthe form SHOW 54 then one is entitled to write the formula 54 on the very next line as an assumption Example 4 Example 5 THE END INTRO LOGIC DAY 05 CHAPTER 4 TRANSLATIONS IN SENTENTIAL LOGIC Schedule for Unit 1 Day 1 Intro Day 2 Chapter 1 Day 3 Chapter 2 40 of 10 arguments Day 4 Chapter 3 4 pts each Day 5 Chapter 4 60 of 12 translations D 6 Ch t 4 ay ap er 5 pts each Day 7 Chapter 4 Day 8 EXAM 1 Basic Idea In order to uncover logical forms we paraphrase English sentences so that they contain only standard connectives We also use special abbreviations and special punctuation Abbreviation Scheme atomic sentences are abbreviated by upper case letters of the Roman alphabet 2 standard connectives are abbreviated by special symbols Iogograms 3 compound sentences are abbreviated by algebraic combinations of 1 and 2 NonStandard Connectives a few examples but ifandonyif although unless or butnotboth neithernor if ifotherwise is necessary for is sufficient for unlessin which case onlyif Standard Connectives connective symbol and amp or v not if then Simple Transformations Some simple sentences are straightforwardly equivalent to compound sentences Jay and Kay are students Jay is a student J and amp Kay is a student K Kay is a rich student Kay is rich R and amp Kay is a student 8 Pitfall 1 Kay is a former student Kay is former and amp Kay is a student I n D Pitfall 2 Jay and Kay are married ambi also amphi means both and around ambiguous other words containing ambi and amphi ambivalent ambitious amphitheater ambidextrous ambience ambiance amphibian First Reading Jay and Kay are married Jay is married to someone J and amp K Kay is married to someone Second Reading Jay and Kay are married Jay and Kay are M marriedtoeachother later in predicate logic Mk further parts are revealed l Pitfall 3 Odd use of and Sometimes and is used as follows 1 keep trying and you will succeed 2 keep it up buster and I will clobber you 3 give him an inch and he will take a mile 4 give me the money and I ll give y pruct 5 you do that and I ll kill ya Pitfall 3 continued What happens if we symbolize keep trying and you will succeed as K amp S K you will keep trying 8 you will succeed K amp S yo u will keep trying and you will succeed 15 other T ll ll 339 liar quot39 was appropriate paraphrase 1 iii uyiievm g eed K a S lili 53 l 3 we 3 Eyeiuhgeivinritg39kinail re G gt T 4 Eli i ivg39ii ih39Z Dprli ua 39V39 n P 5 if you do that D K then I ll kill ya 16 Exclusive Or Ram M Sam will win the election Ram will win or Sam will win but not both at least one of them win R v S also amp both of them will winnot N R amp S alternatively Ram will win and Sam will not win R amp MS or v Sam will win and Ram will not win S amp NR Alternative Formulation it is neither raining nor sleeting it is not raining or sleeting it is not true that it is raining or it is sleeting N R v S RampS 17 NeitherNor it is neither raining nor sleeting it is not raining and it is not sleeting RampS neither Jay nor Kay is sleeping Jay is not sleeping and Kay is not sleeping JampK Conditionals standard expression if A then C which is symbolized A is the antecedent C is the consequent Simple Variants of ifthen if A then C all of these are if A C symbolized the C ifA same if always introduces the antecedent then always introduces the consequent 21 Only If only if is notequivalent to if I will get an A only ifl take all the exams vs I will get an A if I take all the exams I will get an A if I get a hundred vs I will get an A only if I get a hundred Other Variants of if C provided A C in case A C supposing A provided A C in case A C supposing A C all symbolized the same way How does only work 1 employees only 2 authorized personnel only 3 cars only 4 right turn only only operates as a dualnegative modifier for example 1 means to exclude anyone who is not an employee a more lyrical example of only I only have eyes for you39 The Flamingos 1959 lyrics by Al Dubin music by Harry Warren 1934 only is focussensitive I only have eyes for you alternative focus for only I only have eyes for you alternative focus for only I only have eyes for you How does only modify if only modifies if by introducing two negations A only if B A noA if otB Example lwill getanAonly if I take all the exams A 0ny If E I 39II t t A39f W39 quot 9e an 39 notAIfnotE I do not take all the exams ifl do not take all the exams then I will not get an A I if not E then not A If And Only If Two approaches to if and only if 1 treat it as a simple connective lt gt 2 treat it as a complex connective We will concentrate on 2 according to which if and only if consists of 3 components 1 if 2 and 3 only if each of which we already know how to paraphrase Example of if and only if I will pass if and only if I average 50 P if and only if A P ifA and P only ifA if A then P not P if not A if not A then not P A a P amp A a N P 3339 Unless The following are all equivalent I will pass only ifl study P only ifS I will not pass unless I study not P unless S I will not pass ifl do not study not P if not 8 only if has two unless has one if has no builtin negations builtin negation builtin negation Example of unless I will not pass unless I study not P unless 8 not P if not 8 not P if not 8 if not 8 then not P 8 gt P INTRO LOGIC DAY 10 DERIVATIONS IN SENTENTIAL LOGIC 2nd day Review from its premises using a few We demonstrate show that an argument is valid by deriving deducing its conclusion fundamental modes of reasoning Initial Modes of Reasoning Modus Ponendo Ponens affirming by affirming AaC 54 C Modus Tollendo Tollens denying by denying AAC C 54 Modus Tollendo Ponens 1 affirming by denying 52lva 54 TB Modus Tollendo Ponens 2 af rming by denying 52lva NTB 54 l aka dis39unctive sylloqism l Example 1 s Rss RVT PvT PsQI Q 1write all 1 NS Pr premises 2 R 8 Pr 2write show 3 R v NT pr conclusion 4 N P V T Pr 5 N P N P 3apply rules Q r to available 6 NQ on lines 7 NR 12 MT 8 NT 37 MTP1 4 box and 9 NP 4 8 MTPZ cancel highfive 10 NO 59 MP Tek gives Manny a highfive Tek gives ARod a highfive provided on exams available on course web page make a copy and keep it in front of you when doing homework m x v Rule Sheet don make Exam 246 uP your own Exam 46 r39ules Rules discussed Today w ampB ampB B 73 3 ampB 3amp m vB vB N NB vB Bv 73 a1 seeCD gtO ac ac NC C N M M NN NN Rules of Inference Basic Idea every connective has an 0 OUTrule how to breakdown a formula with this connective how to buildup a formula with this connective 0 lNrule AmpersandOut ampO if you have a conjunction 54 amp B then you are entitled to infer its first conjunct if you have a conjunction 54 amp B then you are entitled to infer its second conjunct TB have means have as a whole line rules apply only to whole lines not pieces of lines AmpersandIN ampI if you have a formula 54 and you have a formula B then you are entitled to infer their first conjunction 54 amp B if you have a formula 54 and you have a formula B then you are entitled to infer their second conjunction TB amp 54 11 WedgeOut v0 if you have a disjunction J4va and you have the negation of its 1St disjunct 34 then you are entitled to infer its second disjunct if you have a disjunction and you have the negation of its 2nd disjunct fB then you are entitled to infer its first disjunct what we earlier called modus tollendo ponens WedgeIN vI if you have a formula A then you are entitled to infer its disjunction with any formula to its right A V B if you have a formula A then you are entitled to infer its disjunction with any formula to its left B V A 13 ArrowOut gtO if you have a conditional A a C and you have its antecedent A then you are entitled to infer its consequent C if you have a conditional A gt C and you have the negation of its consequent C then you are entitled to infer the negation of its antecedent A what we earlier called modus ponens and modus tollens ArrowIntroduction THERE IS NO SUCH THING AS ARROWIN Hi 0 V X what we have instead is CONDITIONAL DERIVATION CD which we examine later DoubleNegation DN if you have a formula A then you are entitled to infer its doublenegative J4 if you have a doublenegative J4 then you are entitled to infer the formula J4 rules apply only to whole lines not pieces of lines Direct Derivation The Original and Fundamental SHeWRule SHGWJA o o o 54 In Direct Derivation DD one directly arrives at the very formula one is trying to show Example 2 SRaSRaTPaTNPANQINQ 1 S Pr RAS Pr NR NT Pr P T Pr Pa Q Pr Q DD R 1 T 7 P Q ArrowOut Strategy If you have a line ofthe form AAC then try to apply arrowout a0 which requires a second formula as input in particular either 54 or C have A gtC find A C or deduce C 21 19 Example 3 8RaSRvTPvTPaQQ 1 S Pr RAS Pr Rv T Pr NPVT Pr Pa Q Pr Q DD R 1 T 7 P Q 20 WedgeOut Strategy If you have a line of the form Ava then try to apply wedgeout vO which requires a second formula as input in particular either 54 or TB have AVE find NA 13 or deduce B A 21 Example 4 PaQvRPaQRANR PaQaQaRQ 1 P a Q v R Pr 2 PaQv Ra NR Pr 3 P a Q a Q a R Pr 4 Q DD 5 NR 12 gt0 6 P gt Q 15 v0 7 Q gt R 36 gt0 8 Q 57 gto 22 Example 5 PvQ gtSRvS gtTPIT anS Pr vSaT Pr Pr DD 15 23 WedgeIn Strategy if you need v B then look for either disjunct find A OR find 23 then apply vI to get AVE 24 Example6 l PaQRvQaSRaTTampPIQampS l 1 P a Q Pr 2 R v Q a S Pr 3 R a T Pr 4 T amp P Pr 5 Q amp S DD 6 T 7 P 4ampO 8 Q 17 gt0 9 R 36 gtO 10 Q gt S 29 v0 11 S 810 gtO 12 QampS 811ampI 25 AmpersandOut Strategy if you have a line of the form A843 then apply ampersandout ampO which can be applied immediately to produce A and B 26 AmpersandIn Strategy if you are trying to find or show A843 then look for both conjuncts find A AND find 23 then apply ampI to get 1843 27 THE END 28 154 Hardegree Symbolic Logic 6 THE OFFICIAL INFERENCE RULES So far we have discussed only four inference rules modus ponens modus tollens and the two forms of modus tollendo ponens In the present section we add quite a few more inference rules to our list Since the new rules will be given more pictorial nonLatin names we are going to rename our original four rules in order to maintain consistency Also we are going to consolidate our original four rules into two rules In constructing the full set of inference rules we would like to pursue the following overall plan For each of the ve connectives we want two rules on the one hand we want a rule for quotintroducingquot the connective on the other hand we want a rule for quoteliminatingquot the connective An introductionrule is also called an I39mrule an eliminationrule is called an outrule Also it would be nice if the name of each rule is suggestive of what the rule does In particular the name should consist of two parts 1 reference to the spe cific connective involved and 2 indication whether the rule is an introduction in rule or an elimination out rule Thus if we were to follow the overall plan we would have a total of ten rules listed as follows AmpersandIn ampl AmpersandOut ampO WedgeIn vl WedgeOut vO DoubleArrowln lt gtl DoubleArrowOut lt gtO gtl Arrowln gtl ArrowOut gtO Tildeln NI gt TildeOut NO However for reasons of simplicity of presentation the general plan is not followed completely In particular there are three points of difference which are marked by an asterisk What we adopt instead in the derivation system SL are the following inference rules Chapter 5 Derivations in Sentential Logic 155 INFERENCE RULES INITIAL SET AmpersandIn amp1 J4 J4 B B JZI amp B B amp JZI AmpersandOut amp0 J4 amp B JZI amp B JZI B WedgeIn v1 J4 J4 14 v B B v 14 WedgeOut v0 JZI v B JZI v B NJZI NB B JZI DoubleArrowIn lt gtI JZI gt B JZI gt B B gt A B gt J4 J4 lt gt B B lt gt JZI DoubleArrowOut lt gtO JZI lt gt B JZI lt gt B JZI gt B B gt JZI ArrowOut gtO JZI gt B JZI gt B JZI NB B NJZI Double Negation DN JZI NNJZI N NJZI JZI 156 Hardegree Symbolic Logic A few notes may help clarify the above inference rules Notes 1 2 3 4 5 6 7 Arrowout gtO the rule for decomposing conditional formulas re places both modus ponens and modus tollens Wedgeout v0 the rule for decomposing disjunctions replaces both forms of modus tollendo ponens Double negation DN stands in place of both the tildein and the tilde out rule There is no arrowin rule The rule for introducing arrow is not an in ference rule but rather a showrule which is a different kind of rule to be discussed later In each of the rules J4 and B are arbitrary formulas of sentential logic Each rule is short for infinitely many substitution instances In each of the rules the order of the premises is completely irrelevant In the wedgein vl rule the formula 3 is any formula whatsoever it does not even have to be anywhere near the derivation in question There is one point that is extremely important given as follows which will be repeated as the need arises Inference rules apply to whole lines not to pieces of lines In other words what are given above are not actually the inference rules themselves but only pictures suggestive of the rules The actual rules are more properly written as follows 0 INFERENCE RULES OFFICIAL FORMULATION AmpersandIn ampl If one has available lines A and B then one is entitled to write down their conjunction in one order 1484 or the other order BampJZL AmpersandOut ampO If one has available a line of the form 1484 then one is entitled to write down either conjunct A or conjunct B Chapter 5 Derivations in Sentential Logic 157 WedgeIn vl If one has available a line A then one is entitled to write down the disjunction of A with any formula B in one order JZlvB or the other order BvJZL WedgeOut v0 If one has available a line of the form JZlvB and if one additionally has available a line which is the negation of the first disjunct NJZl then one is entitled to write down the second disjunct B Likewise if one has available a line of the form JZlvB and if one additionally has available a line which is the negation of the second disjunct NB then one is enti tled to write down the first disjunct JZL DoubleArrowln lt gtl If one has available a line that is a conditional JZl gtB and one additionally has available a line that is the converse B gtJZl then one is entitled to write down either the biconditional JZllt gtB or the biconditional Blt gtJZL DoubleArrowOut lt gtO If one has available a line of the form JZllt gtB then one is entitled to write down both the conditional JZl gtB and its converse B gtJZL ArrowOut gtO If one has available a line of the form JZl gtB and if one additionally has available a line which is the antecedent JZl then one is entitled to write down the consequent B Likewise if one has available a line of the form JZl gtB and if one additionally has available a line which is the negation of the consequent NB then one is entitled to write down the negation of the antecedent NJZL Double Negation DN If one has available a line JZl then one is entitled to write down the doublenegation NNJZL Similarly if one has available a line of the form NNJZl then one is entitled to write down the formula JZL The word available is used in a technical sense that will be explained in a later section 158 Hardegree Symbolic Logic To this list we will add a few further inference rules in a later section They are not crucial to the derivation system they merely make doing derivations more convenient 7 SHOWLINES AND SHOWRULES DIRECT DERIVATION Having discussed simple derivations we now begin the official presentation of the derivation system SL ln constructing system SL we lay down a set of system rules the rules of SL It39s a bit confusing we have inference rules already presented now we have system rules as well System rules are simply the official rules for constructing derivations and include among other things all the inference rules For example we have already seen two system rules in effect They are the two principles of simple derivation which are now officially formulated as system rules System Rule 1 The Premise Rule At any point in a derivation prior to the first showline any premise may be written down The annotation is Pr System Rule 2 The InferenceRule Rule At any point in a derivation a formula may be written down if it follows from previous available lines by an inference rule The annotation cites the line numbers and the inference rule in that order System Rule 2 is actually shorthand for the list of all the inference rules as formulated at the end of Section 6 The next thing we do in elaborating system SL is to enhance the notion of simple derivation to obtain the notion of a direct derivation This enhancement is quite simple it even seems redundant at the moment But as we further elaborate system SL this enhancement will become increasingly crucial Specifically we add the following additional system rule which concerns a new kind of line called a showline which may be introduced at any point in a derivation System Rule 3 The ShowLine Rule At any point in a derivation one is entitled to write down the expression SHOW JZl for any formula JZl whatsoever Chapter 5 Derivations in Sentential Logic 159 In writing down the line SHOW JZl all one is saying is I will now attempt to show the formula J21 What the rule amounts to then is that at any point one is entitled to attempt to show anything one pleases This is very much like saying that any citizen over a certain age is entitled to run for president But rights are not guarantees you can try but you may not succeed Allowing showlines changes the derivation system quite a bit at least in the long run However at the current stage of development of system SL there is generally only one reasonable kind of showline Speci cally one writes down SHOW C where C is the conclusion of the argument one is trying to prove valid Later we will see other uses of showlines All derivations start pretty much the same way one writes down all the premises as permitted by System Rule 1 then one writes down SHOW C where C is the conclusion which is permitted by System Rule 3 Consider the following example which is the beginning of a derivation Example 1 1 P v Q gt NR Pr 2 P amp T Pr 3 RV s Pr 4 U gt s Pr 5 SHOW NU 7 These five lines may be regarded as simply stating the problem we want to show one formula given four others I write 777 in the annotation column because this still needs explaining more about this later Given the problem we can construct what is very similar to a simple deriva tion as follows 1 P v Q gt NR Pr 2 P amp T Pr 3 R v NS Pr 4 U gt S Pr 5 SHOW U 6 P 2ampO 7 P v Q 6vl 8 NR 17 gtO 9 NS 38vO 10 NU 49 gtO Notice that if we deleted the showline 5 the result is a simple derivation We are allowed to try to show anything But how do we know when we have succeeded In order to decide when a formula has in fact been shown we need additional system rules which we call quotshowrulesquot The first showrule is so simple it barely requires mentioning Nevertheless in order to make system SL completely clear and precise we must make this rule explicit 160 Hardegree Symbolic Logic The first showrule may be intuitively formulated as follows Direct Derivation Intuitive Formulation If one is trying to show formula J4 and one actually obtains A as a later line then one has succeeded The intuitive formulation is unfortunately not sufficiently precise for the purposes to which it will ultimately be put So we formulate the following official system rule of derivation System Rule 4 a showrule Direct Derivation DD If one has a showline SHOW 14 and one obtains A as a later available line and there are no intervening uncancelled showlines then one is entitled to box and cancel SHOW 14 The annotation is DD As it is officially written direct derivation is a very complicated rule Don39t worry about it now The subtleties of the rule don39t come into play until later For the moment however we do need to understand the idea of cancelling a showline and boxing off the associated subderivation Cancelling a showline simply amounts to striking through the word SHOW to obtain SHGW This indi cates that the formula has in fact been shown Now the formula JZl can be used The tradeoff is that one must box off the associated derivation No line inside a box can be further used One in effect trades the derivation for the formula shown More about this restriction later The intuitive content of direct derivation is pictorially presented as follows Direct Derivation DD SHGW le JZl The box is of little importance right now but later it becomes very important in helping organize very complex derivations ones that involve several showlines For the moment simply think of the box as a decoration a ourish if you like to celebrate having shown the formula Chapter 5 Derivations in Sentential Logic 161 Let us return to our original derivation problem Completing it according to the strict rules yields the following l P v Q gt NR Pr 2 P amp T Pr 3 R v NS Pr 4 U gt S Pr 5 SHGW NU DD 6 P 2ampO 7 P v Q 6vl 8 NR l7 gtO 9 NS 38VO 10 NU 49 gtO Note that SHOW has been struck through resulting in SHGW Note the annotation for line 5 DD indicates that the showline has been cancelled in accordance with the showrule Direct Derivation Finally note that every formula below the showline has been boxed off Later we will have other more complicated showrules For the moment however we just have direct derivation 8 EXAMPLES OF DIRECT DERIVATIONS In the present section we look at several examples of direct derivations Example 1 1 MP gt Q v R Pr 2 P gt Q Pr 3 Q Pr 4 SHGW R DD 5 NP 23 gtO 6 Q v R 15 gto 7 R 36vO Example 2 1 P amp Q Pr 2 SHGW P amp Q DD 3 P 1ampo 4 Q 1ampO 5 P 3DN 6 Q 4DN 7 P amp Q 56ampI 162 Hardegree Symbolic Logic Example 3 1 2 3 4 5 6 7 8 PampQ QvR gtS SHGW PampS P 2 QvR S PampS ExmnMe4 1 2 3 4 5 6 7 8 9 AampB AvE gtC D gtC SHGWD A AVE C NNC ND ExmnMeS 1 2 3 4 5 6 7 8 9 10 11 12 AampB vaam ampEHD WMAampC A NB A gtD D D gtCampE CampE C AampC Pr Pr 1ampO 1ampO 5 VI 26 gtO 47ampI Pr Pr Pr 1ampO 5VI 26 38 Pr Pr Pr 1ampO 1ampO 26VO 57 gtO 3lt gtO 89 gtO 10ampO 511ampI Chapter 5 Derivations in Sentential Logic 163 Example 6 1 A gt B 2 A gtB gtB gtA 3 A lt gt B gt A 4 SHGW A amp B 5 B gt A 6 A lt gt B 7 A 8 B 9 A amp B Example 7 1 NA amp B 2 CvB gt ND gtA 3 ND lt gt E 4 SHGW NE 5 NA 6 B 7 C v B 8 MD gt A 9 D 10 E gt ND 11 NE 58 gtO 3lt gtO 910 gtO NOTE From now on for the sake of typographical neatness we will draw boxes in a purely skeletal fashion In particular we will only draw the left side of each box the remaining sides of each box should be mentally lled in using skeletal boxes the last two derivations are written as follows Example 6 rewritten 1 2 3 4 5 6 7 8 9 A gtB A gtB gtB gtA Alt gtB gtA SHGWAampB B gtA Alt gtB A B AampB Pr Pr Pr DD 12 gtO 15lt gtI 36 gtO 17 gtO 78ampI For example 164 Hardegree Symbolic Logic Example 7 rewritten 1 NA amp B Pr 2 C v B gt ND gt A Pr 3 ND lt gt E Pr 4 SHGW NE DD 5 NA 1ampO 6 B 1ampO 7 C v B 6vl 8 ND gt A 27 gtO 9 D 58 gtO 10 E gt ND 3lt gtO 11 NE 910 gtO NOTE In your own derivations you can draw as much or as little of a box as you like so long as you include at a minimum its left side For example you can use any of the following schemes WVWL Finally we end this section by rewriting the Direct Derivation Picture in accor dance with our minimal boxing scheme SHGW Direct Derivation DD SHGW JZl DD 0 O O O O O O INTRO LOGIC DAY 17 Translations in PL 3 Existential Quantifier someone is happy there is someone who is happy there is some x x is happy EIX Hx REVIEW of DAY 1 and DAY 2 ExistentialNegative Quantifier someone is unhappy there is someone who is not happy there is some x x is not happy EIX Hx D Universal Quantifier everyone is happy no matter who you are you are happy no matter who x is x is happy Vx Hx UniversalNegative Quantifier eve ryo ne is unhappy no matter who you are you are not happy no matter who x is x is not happy Vx Hx NegativeExistentialQuantifier 3 no one is happy there is no one who is happy there is no x x is happy 3x Hx 7 D NegativeUniversal Quantifier not everyone is happy not no matter who you are you are happy not no matter who x is x is happy Vx Hx Quantifier Specification some I I some Freshman is Happy I there is someone who is F and who is H there is some X X is F and X is H EIXFX amp HX QuantifierSpecification every I I every Freshman is Happy I no matter who you are IF you are F THEN you are H no matter who X is IF X is F THEN X is H VX FX a HX D Quantifier Specification no I no Freshman is Happy I there is no one who is F and who is H thereisnox XisF and XisH EXFX amp HX new material for day 3 Multiple Quantification l sentences with more than one quantifier GENERAL STRATEGY 1 Count the number of quantifiers in original sentence 2 3 4 5 Count the number of quantifiers in final formula 6 Compare 5 with 1 Determine the overall structure ofthe sentence Work on constituents separately Substitute constituents back into overall formula Example 2 every CAT is a PET but not every PET is a CAT every C is P but not every P is C VXCXgtPX amp VXPXgtCX VXCXgtPX VXPXgtCX Example 1 everyone is FRIENDLY but not everyone is HAPPY everyone is F but not everyone is H Vx Fx amp Vx Hx Vx Fx Vx Hx Example 3 if everyone is FRIENDLY then everyone is HAPPY i everyone is F then everyone is H VXFX a VxHx VXFX VxHx Example 4 if every STUDENT is FRIENDLY then every STUDENT is HAPPY every 8 is F then every Sis H VXSX Fx VXSX Hx VXSX Fx VXSX Hx Any versus Every Basic Principle both any and every are universal quantifiers BUT they are usually not interchangeable Sometimes they are 3 interchangeable any one can Dance VxDx every one can Dance if can Dance then any one can if can Dance then every one can Di VxDx Usually they are not interchangeable is every one here 739 is any one here Jay doesn t respect Jay doesn t respect every one 397 quot any one if any one can fix your car then I can if every one can fix your car then I can no one respects no one respects every one 397quot any one 2ol Difference between every and any I the scope of every is narrow I the scope of any is wide I Example NotEvery Jay doesn t respect everyone not Jay respects everyone Vijx not has wide scope every V has narrow scope Example NotAny Jay doesn t respect anyone does Jay respect a I Rja I no I NRja does Jay respect b I ij I no I Nij does Jay respect c I ch I no I Nch etc no matter who you are Jay does not respect you no matter who x is Jay does not respect x VX ij any V has wide scope not has narrow scope D NotAny None NotSome Jay respects no one there is no one whom Jay respects there is no X Jay respects X EIX ij Recall Elv q Vv N G Elx ij Vx ij Jay respects Jay doesn t no one respect anyone 24 Example FEVERY if everyone fails then satan wins everyone fails then satan wins VX FX a Ws VX FX Ws evew has narrow scope ifthen has wide scope H How do we SHOW such a formula 1 SHOW Vxe a Ws CD 2 Vx Fx 3 SHOW Ws D Example FANY 2 if anyone fails then satan wins if a fails then satan wins if b fails then satan wins if c fails then satan wins I etc I if anyone fails then satan wins 27 In Other Words D no matter who you are if you fail then satan Wins no matter who X is if X fails then satan Wins Vx Fx Ws any has wide scope ifthen has narrow scope How do we SHOW such a formula 1 SHOW VxFx aWs UD 2 SHOW Fa a Ws CD 3 Fa As 4 SHOW Ws UD Universal Derivation later Special Note Sometimes but not always if any if some Vx Fx q Elex gt d provided d has no free occurrence of x INTRO LOGIC DAY 17 Translations in PL 3 REVIEW of DAY 1 and DAY 2 Existential Quantifier someone is happy there is someone who is happy there is some X X is happy 3X HX ExistentialNegative Quantifier someone is unhappy there is someone who is not happy there is some X X is not happy 3X HX Universal Quantifier everyone is happy no matterwho you are you are happy no matter who X is X is happy VX HX UniversalNegative Quantifier everyone is unhappy no matter who you are you are not happy no matter who X is X is not happy VX HX NegativeExistential Quantifier no one is happy there is no one who is happy there is no X X is happy 3X HX NegativeUniversal Quantifier not everyone is happy not no matter who you are you are happy not no matter who X is X is happy VX HX QuantifierSpecification some some Freshman is Happy there is someone who is F and who is H there is some X X is F and X is H EIX FX amp HX Quantifier Specification no no Freshman is Happy there is no one who is F and who is H there is no X X is F and X is H EX FX amp HX QuantifierSpecification every I every Freshman is Happy no matter who you are IF you are F THEN you are H no matter who X is F X is F THEN X is H VX FX gt HX new material for day 3 Multiple Quantification sentences with more than one quantifier GENERAL STRATEGY Determine the overall structure of the sentence Work on constituents separately Substitute constituents back into overall formula Count the number of quantifiers in final formula Compare 5 with 1 Count the number of quantifiers in original sentence Example 1 everyone is FRIENDLY but not everyone is HAPPY everyone is F but not everyone is H Vx Fx amp Vx Hx Vx Fx Vx Hx Example 2 every CAT is a PET but not every PET is a CAT everyCisP but not everyPisC VxCx gtPx amp VxPx gtCx VxCx gtPx VxPx gtCx 15 Example 3 if everyone is FRIENDLY then everyone is HAPPY 4 everyone is F then everyone is H VXFX gt VXHX VXFX VXHX Example 4 if every STUDENT is FRIENDLY then every STUDENT is HAPPY every S is F then every S is H VXSX gt Fx gt VXSX gt Hx VXSX gt Fx VxSx gt Hx Any versus Every Basic Principle both any and every are universal quantifiers iBUTi they are usually not interchangeable Some times they are interchangeable any one canDance VXDX every one canDance if I canDance then any one canDance if I canDance then every one canDance Di gt VXDX Usually they are not interchangeable is every one here 7 is any one here Jaydoesn t respect i Jaydoesn t respect every one any one every one any one if every one can fix your i if any one can fix your car then I can car then I can no one respects i no one respects 20 Difference between every and any the scope of every is narrow the scope of any is wide every any 21 Example NotEvery Jay doesn t respect everyone not Jay respects everyone VXRjX not has wide scope every V has narrow scope 22 Example NotAny Jay doesn t respect anyone does Jay respect a Rja no Rja does Jay respect b ij no ij does Jay respect c ch no ch etc no matter who you are Jay does not respect you no matter who X is Jay does not respect X VX RjX any V ha not N has 5 wide scope narrow scope 23 NotAny None NotSome Jay respects no one there is no one whom Jay respects there is no X Jay respects X 3X RjX Recall 3v Vvq3 EIX RjX VX RjX Jay respects no one Jay doesn t respect anyone 24 Example FEVERY I if everyone fails then Satan wins I I if everyone fails then Satan wins Vxe Ws Vxe Ws every has narrow scope ifthen has wide scope 25 How do we SHOW such a formula 1 SHOW VXFX gt Ws CD 2 VXFX As 3 SHOW W3 4 5 26 Example FANY if anyone fails then Satan wins I if a fails then Satan wins if b fails then Satan wins if c fails then Satan wins I etc I Satan wins I if anyone failsl then 27 In Other Words no matter who you are if you fail then Satan wins no matter who X is if X fails then Satan wins Vx Fx gt Ws any has wide scope ifthen has narrow scope 28 How do we SHOW such a formula 1 SHOW VXFX gtWs UD 2 SHOW Fa gt Ws CD 3 Fa As 4 SHOW W3 5 UD Universal Derivation later 29 Special Note Sometimes but not always if any if some VXFX gtCI EIXFX a d provided d has no free occurrence of x 30 THE END INTRO LOGIC DAY 06 TRANSLATIONS IN SENTENTIAL LOGIC 2 Schedule for Unit 1 Day 1 Intro Day 2 Chapter 1 Day 3 Chapter 2 40 of 8 arguments Day 4 Chapter 3 5 pts each Day 5 Chapter 4 60 of 12 translations D 6 Ch t 4 ay ap er 5 pts each Day 7 Chapter 4 I Day 8 EXAM 1 Standard Connectives have special symbols and amp or v not if then a NonStandard Connectives must be paraphrased neithernor ifotherwise if unlessinwhichcase only if necessary ifand only if sufficient NonStandard Connectives 1 A xor B A orB but not A and B AvB amp AampB AviBampAampiB NonStandard Connectives 1b exclusive or alternative A xor B A but not B or B but not A AampB BampA NonStandard Connectives 2 neither A nor 2 not A and not B N A amp N B NonStandard Connectives 2b alternative neither A nor 39339 lnotl either I A I IN or not A or B N A v B A v 3 NonStandard Connectives 2c recall truthtables RvS T R T T T T F F T T NonStandard Connectives 3 ifB ithhen L B gtJL NonStandard Connectives 3 NonStandard Connectives 4 only if B A A not A if not B ifnofZB then not mamw NonStandard Connectives 5 if and only if A if and only if B A if B and A only if B jg Lil amp NB gtN l NonStandard Connectives 6 A unless B A if not B A if not B if not B then A 2 gt A if otherwise wi play tennis if it is sunny otherwise I will play racketball T if S otherwise R this answers two questions what will I do IF it is sunny play tennis what will I do OTHERWISE ie IF it is NOT sunny play racketball 15 if otherwise cont if it s sunny if it s not then I ll play furthermore sunny then I ll tennis play racketball if S then T and if not 8 then R S a T amp S gt R unessin which case I will play tennis unless it rains in which case I will play squash T unless R in which case S this answers two questions what will I do unless it rains ie if it does not rain play tennis what will I do in case it rains ie if it does rain play squash 17 unlessinwhichcase cont if it does not if it does rain rain then I ll furthermore then I ll play play tennis squash if notR then T and if R then S R a T amp R a S Necessary Conditions in orderthat get an A it is necessary that I take 4 exams in order for me to get an A it is necessary for me to take 4 exams in order to get an A it is necessary to take 4 exams taking 4 exams is necessary for getting an A Necessary Conditions cont simplest paraphrase A is necessary for B this amounts to saying if A does not happen then neither does 2 if not A then not B A a 13 Example taking 4 exams E is necessary for getting an A A if E does not happen then neither does A i if not E then not A i E gt A Sufficient Conditions in orderthat get an A it is sufficient that I get a hundred in order for me to get an A it is sufficient for me to get a hundred in order to get an A it is sufficient to get a hundred getting a hundred is sufficient for getting an A Sufficient Conditions cont simplest paraphrase A is sufficient for 2 this amounts to saying if i does happen then so does 2 if i then 13 A a B Example getting a hundred H is sufficient for getting an A A if H does happen then so does A i if H then A i i H a A Negation of Necessity getting a hundred H is NOT necessary for getting an A A H is nec for A if not H then not A lt H a A gt1 H gt A it is not you don t you won t true that If get a H then get an A 25 Negation of Sufficiency taking all the exams E is NOT sufficient for getting an A A E is suf for A if E then A lt E a A ll you E gt A merely it is not take the you will true that If exams then get an A 26 Basic Statements A is necessary for B A is sufficient for B A is not necessary for B A is not sufficient for B Combinations A is both necessary and sufficient for B A is necessary but not sufficient for B A is sufficient but not necessary for B A is neither necessary nor sufficient for B Example 1 averaging at least fifty A is both necessary and sufficient for passing P A is necessary for P A P and amp A is sufficient for P A P 29 Example 2 taking four exams E is necessary but not sufficient for getting an A A E is necessary forA E A but amp E is not sufficient for A E gtA Example 3 getting a hundred H is sufficient but not necessary for getting an A A H is sufficient forA H A but amp E is not necessary forA H NA 31 Example 4 attending class A is neither necessary nor sufficient for passing P A is not necessary for P A P and amp A is not sufficient for P A P Derivations in SL INTRO LOGIC DAY 13 5 Inference Rules so far Exam 2 Format 6 argument forms 15 points each plus 10 free points Symbolic argument forms no translations For each one you will be asked to construct a derivation of the conclusion from the premises The rule sheet will be provided 1 problem 2 problem 2 problems 1 problem from from from from Set D Set E Set F Set G 91 96 ampo A a B A a B amp1 A A Nao NAampB B 3 3 a NB amp B B amp V0 34 3 3 VI 34 NVO N B N N 34 B B 34 N 3 NB 0 3 B NaO N aB NB amp NB 3 N DN NN XI 34 N NN X W Rules so far DD CD SHGW DD SHGW aC CD As 39 SHGWI C ND ID SHew NA ND W A ID As N As SHGWZX SHGWZX SHOWSTRATEGIES There are 6 kinds of formulas in Sentential Logic 1 atomic formulas P Q etc 2 negations N 3 conjunctions amp 4 conditionals a 5 disjunctions v 6 biconditionals lt gt For each of these there is a suggested showstrategy ShowNegation Strategy ShowConditional Strategy SHOW J4gtC J4 As SHOW C o o ShowAtomic Strategy SHOW A NA As SHOW X X A is atomic PQR etc ShowDisjunction Strategy Example 1 m AVE 454m As SHOW X x ShowConjunction Strategy SHOW NEW RULE NEW STRATEGY 10 INTRO LOGIC DAY 07 EXAM 1 Thursday in class Office Hours Wed 1230 230 Thu 1100 1240 363 Bartlett 1 Validity and Invalidity 8 argument forms 5 points each 2 Translations in SL 12 translations 5 points each Exam1page1 INTRO LOGIC EXAM 1 a EWIIIIIIIIIE mm 1 15 pts each inmeruiimngwuaregvensevmiargum funnsufsemmuallu cwhere eprermsesareseparaledby ammecundusmmsmamdbyv heath msemdmlemthebuxwhelhertheargummtfunmsVALlD M urlNVALID m I mthrtables are optional not graded 1 5 T T I T H T T T i F T F T F IT T F F T F you will be given space to do truthtables 39 but these will not be graded 2 D only your final answer will be graded 3 A Iii l 7 T iT T T T T F T F T 37 T T i T T T F F T T T E T F Alma Exam 1 page 2 WRI39I39E on Lv vouR FINAL FORMULA aELaw no ALL OYHER WORK ON scRAP PAPER 2 l 1 only the final symbolization will be graded do intermediate work on scratchpaper byo 7 IE partial credit will be given as appropriate TRANSLATIONS IN SENTENTIAL LOGIC 3 Complex Statements Step by Step Procedure Identify the simple atomic statements abbreviate them by upper case letters Identify all the connectives Which are standard Which are nonstandard Rewrite the sentence replacing the simple statements by their abbreviations Retain internal punctuation Identity the major connective Notice commasl If the major connective is standard symbolize it otherwise paraphrase it and repeat 4 Work on the constituent formulas resulting from 5 go to 4 Substitute constituents back into overall formula Translate the formula back into English and compare it with original sentence Example 1 if neither JAY nor KAY is working then we will go on VACATION 1 simple sentences J Jay is working K Kay is working V we go on vacation 2 connectives ifthen standard neithernor nonstandard Example 1 b 3 first formula if neither J nor K then V 4 main connective if neither J nor K then V 5 standard or nonstandard standard so we symbolize it neither J nor K gt V Example 1 c work on constituents neitherJ nor K gt V not J and not K J amp K substitute constituents J amp K gt V translate formula back into English if Jay is not working and Kay is not working then we will go on vacation Example 2 unless the exam is EASY lwill PASS only ifl STUDY 1 simple sentences E the exam is easy P I pass S I study 2 connectives unless nonstandard only if nonstandard Example 2 b first formula unless E P only if S main connective unless E P only ifS standard or nonstandard nonstandard so we paraphrase it unless if not if not E then P only ifS Example 2 c 4 5 6 if not E then P only if 8 main connective if not E then P only if S standard or nonstandard standard so we symbolize it notE gt PonlyifS work on constituents notE gt PonlyifS E notPifnotS if not S then not P SeP Example 2 d 7 Substitute constituents E gt S gt P 8 Translate formula back into English if the exam is not easy then if I do not study then lwill not pass Example 3 ifl am HAPPY only ifl am DRUNK then I am not HAPPY unless I am DRUNK simple sentences H I am happy D I am drunk 2 connectives ifthen standard only if nonstandard not standard unless nonstandard Example 3 b first formula if H only if D then notH unless D main connective if H only if D then notH unless D standard or nonstandard standard so we symbolize it H only if D gt notH unless D Example 3 c 6 Work on constituents H only if D gt notH unless D notHifnotD if not D then not H D gt H H only if D gt notH unless D notH if not D if notD then not H D gt H Example 3 d Substitute constituents D gtH gt D gtH Translate formula back into English IF if lam not drunk then I am not happy THEN if lam not drunk then I am not happy Example 4 ifl CONCENTRATE well only ifl am ALERT then provided I am WISE lwill not FLY an airplane unless I am SOBER simple sentences C concentrate well A I am alert W I am wise F I fly an airplane S I am sober connectives ifthen standard only if nonstandard provided nonstandard not standard unless nonstandard Example 4 b 3 rst formula if C only if A then provided W not F unless S 4 main connective if C only ifA then provided W not F unless S 5 standard or nonstandard standard so we symbolize it C only ifA gt provided W not F unless S Example 4 c 6a work on first constituent antecedent C only ifA a provided W notF unless S notC ifnotA if not A then not C A gt C 20 Example 4 d 6b work on second constituent consequent C only if A a provided W notF unless S main connective provided W notF unless S paraphrase if W then notF unless S W a not F unless S W a notFifnotS W a if not S then not F W a S a F 21 Example 4 e 7 substitute constituents A 9C gtVV Sa FH 8 Translate formula back into English IF ifl am not alert then I do not concentrate well THEN ifl am wise then ifl am not sober then I will not fly an airplane INTRO LOGIC DAY 12 Derivations in SL 4 Schedule Day 09 Introductory Material Day 10 Direct Derivation DD Conditional Derivation CD Negation Derivation ND Indirect Derivation Day 12 show atomic show disjunction Day 11 Day 13 show conjunction Day 14 EXAM 2 Exam 2 Format 6 argument forms 15 points each plus 10 free points Symbolic argument forms no translations For each one you will be asked to construct a derivation of the conclusion from the premises The rule sheet will be provided 1 problem 2 problem 2 problems 1 problem from from from from Set D Set E Set F Set G 9196 Inference Rules so far ampO A amp B A amp B ampI A A B B A B A amp B B amp A v0 A v B A v 3 VI A A NA N3 A v B B v A B A gtO A gt B A gt 13 DN A A A NB A A 3 NA SHeW Rules so far DD SW A DD A CD D SHeW AAC CD SHGW A ID A As A As SHeWC SHGWX Affiliated Rules Assumption Rule CD If one has a line ofthe form SHOW AAC then one is entitled to write down the formula A on the very next line as an assumption Assumption Rule D If one has a line ofthe form SHOW A then one is entitled to write down the formula A on the very next line as an assumption ContradictionIn XI if you have a formula A and you have its negation A then you are entitled to infer a contradiction absurdity X DirectDerivation Strategy SHGVVJA o o o 54 in Direct Derivation DD one directly arrives at the very formula one is trying to show ShowConditional Strategy SHGW AAC 54 As SHOW C o o o ShowNegation Strategy SHOW 54 54 As SHOW X X Can we show the following 1 P a Q Pr We are stuck 2 NP 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 or find or find Q Indirect Derivation SHOW A SHOW 54 54 As 54 As SHOW X SHOW X X X this is exactly parallel to ND and is another version of the traditional mode of reasoning known as REDUCTIO AD ABSURDUM Using ID the difference between ID and ND is that D 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 P Q R etc lva 1 atomic formulas 2 disjunctions ShowAtomic Strategy W A A SHOW x o X As A is atomic PQR etc Example 1 1 P gt Q Pr 2 P gt Q Pr 3 Q ID 4 Q As 5 x DD 6 P 14 gt0 7 Q 26 gt0 8 x 47 XI Example 2 ShowDisjunction Strategy SHeW Jilva NMVB As SHGW X o o X Affiliated InferenceRule TildeWedgeOut v0 L v B L v B NA NB Example 3 Example 4 P gt vR gt vR Q Example 5 v a amp amp v Pamp amp v Pamp X ampQ Pamp vQ P NO X 20 INTRO LOGIC DAY 18 Translations in PL 4 Chapter 7 Polyadic Predicate Logic Examples of Polyadic Predication Jay respects Kay Jay respects someF Jay respects everyF someF respects Kay everyF respects Kay some F respects some G some F respects every G every F respects some G every F respects every G Example 1 Jay respects someone there is someone who is respected by Jay there is some X X is respected by Jay Jay respects X 3X RjX R is always read in active voice Example 1 redone Jay respects someone there is someone whom Jay respects there is someone Jay respects him her there is some X Jay respects X 3X RjX Example 2 everyone respects Kay no matter who you are you respect Kay no matter who X is X respects Kay VX RXk Example 3 noone respects Jay there is no one who respects Jay there is no X X respects Jay EX RXj Example 4 everyone respects everyone no matter who you are you respect everyone no matter who X is X respects everyone VX X respects everyone no matterwho you are X respects you no matter who y is X respects y Vy ny VX Vy ny Example 5 someone respects someone there is someone who respects someone there is some X X respects someone EIX X respects someone there is someone whom X respects there is some y X respects y 3y ny 3X 3y ny Example 6 no one respects everyone there is no one who respects everyone there is no X X respects everyone X respects everyone no matterwho you are X respects you no matter who y is X respects y Vy ny 3X Vy ny Example 7 no one is respected by everyone there is no one who is Red by everyone there is no X X is Red by everyone everyone respects X everyone respects X no matter who you are you respect X no matter who y is y respects X Vy RyX 3X Vy RyX A Scope Ambiguity everyone respects someone is ambiguous everyone respects someone or other vs everyone respects someone the same one there is someone whom everyone respects What is the Difference Situation 1 Jay Kay gt Kay Elle gt Elle Jay l does everyone respect someone l YES l l is there someone whom everyone respects l NO l What is the Difference cont Situation 2 Jay Jay Kay gt Kay Elle gt Elle l does everyone respects someone l YES l l is there someone whom everyone respects l YES l l NOTE the latter implies the former l 14 Example 8 everyone respects someone or other no matter who you are you respect someone no matter who X is X respects someone X respects someone there is someone whom X respects there is some y X respects y 3y ny VX 3y ny Example 9 everyone respects someone the same there is some one whom everyone respects there is some X everyone respects X EIX everyone respects X no matterwho you are you respect X no matter who y is y respects X Vy RyX 3X Vy RyX Example 10 there is some one who respects no one there is some X X respects no one EIX X respects no one there is no one whom X respects there is no y X respects y Nay ny 3X 3y ny Example 11 there is some one whom no one respects there is some X no one respects X EIX no one respects X there is no one who respects X there is no y y respects X Nay RyX 3X 3y RyX Example 12 Jay doesn t respect anyone no matter who you are Jay does not respect you no matter who X is Jay does not respect X Vx Jay does not respect x ij Vx ij THE END 20 INTRO LOGIC DAY 06 TRANSLATIONS IN SENTENTIAL LOGIC Standard Connectives have special symbols and amp or v not if then Day 1 Intro Day 2 Chapter 1 Day 3 Chapter 2 Day 4 Chapter 3 Day 5 Chapter 4 Day 6 Chapter 4 Day 7 Chapter 4 Day 8 EXAM 1 Schedule for Unit 1 40 of 10 arguments Exam 4 pts each 60 of 12 translations Exam 5 pts each NonStandard Connectives must be paraphrased I neithernor I I ifotherwise I I if I I unlessinwhichcase I I only if I I necessary I I if and only if I I suf cient I NonStandard Connectives 1 NonStandard Connectives 3 exclusive or ifB AxorB Aor B but not both andB AVBampAampB ithhenA 3 1 AampBVBampA NonStandard Connectives 4 NonStandard Connectives 2 only if 54 only if B A A neither norB not A and not B NA amp B if nofTB then not l A v B no l if 0th mam NonStandard Connectives 5 if and only if A i if and only if B AifB and AonlyifB BaA amp BaA NonStandard Connectives 6 A unless B A if not B A if not B if not B then A B a A ifotherwise I will play tennis if it is sunny otherwise I will play racketball T if S otherwise R this answers two questions what will I do if it39s sunny what will I do otherwise ie if it is not sunny play tennis play racketball 11 if otherwise cont if it s sunny if it s not then I ll play furthermore sunny then I ll tennis play racketball if S then T and if not 8 then R S T amp S R unessin which case I will play tennis unless it rains in which case I will play squash I T unless R in which case s this answers two questions what will I do unless it rains ie if it does not rain what will I do in case it rains ie if it does rain play tennis play squash 13 unlessinwhichcase cont if it does not if it does rain rain then I ll furthermore then I ll play play tennis squash if R then T and if R then S R T amp R S Necessary Conditions in order that I get an A it is necessary that I I take 4 exams it is necessary for I me to take 4 exams in order I to get an A I in order for I me to get an A it is necessary I to take 4 exams taking 4 exams I is necessary for I getting an A El Necessary Conditions cont simplest paraphrase A is necessary for B This amounts to saying if A does not happen then neither does B if not 54 then not B m Example taking 4 exams E is necessary for getting an A A if E does not happen Sufficient Conditions cont simplest paraphrase A is sufficient for B This amounts to saying if A does happen then so does B if 54 then B JZL gtB Sufficient Conditions get an A get a hundred in order that I it is sufficient that l in order for I me to get an A it is sufficient for I me to get a hundred to get an A to get a hundred in order I it is sufficient getting a hundred is sufficient for I getting an A 18 Example getting a hundred H is suf cient for getting an A A if H does happen then so does A if H then A m Negations Of Necessity Basic Statements getting a hundred H is not necessary for A 398 necessary for 3 getting an A A I A is suf cient for B t H f A l A is not necessary for B no Is nec or if not H then not A A Is not suf crent for B H a A it is not if you don t then you won t true that get a get an A hundred 21 2339 Negations of Sufficiency Combinations t k39 H th E i u ijigf A is both necessary and suf cient for B getting an A A A is necessary but not suf cient for B t E ff A l A is sufficient but not necessary for B no Is su or if E then A A Is neither necessary nor suf crent for B E a A i it is not ifyou then you will true that merely get an A take all the exams 23 24 Example 1 averaging at least fifty A is both necessary and suf cient for passing P A is necessary for P NA gt NP and amp A is suf cient for P A gt P 2339 Example 2 taking four exams E is necessary but not suf cient for getting an A A E is necessary forA NE gt NA but amp E is not sufficient for A N E gt A Example 3 getting a hundred H is suf cient but not necessary for getting an A A H is sufficient for A H gt A but amp E is not necessary for A N NH gt NA 27 Example 4 attending class A is neither necessary nor sufficient for passing P A is not necessary for P N NA gt NP and amp A is not suf cient for P N A gt P

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