Class Note for PHIL 110 at UMass(8)
Class Note for PHIL 110 at UMass(8)
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This 14 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Massachusetts taught by a professor in Fall. Since its upload, it has received 21 views.
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Date Created: 02/06/15
INTRO LOGIC DAY 15 UNIT 3 Translations in Predicate Logic Overview 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 phrase2 noun phrase1 predicate V sentence 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 connec ve sentence1 sentence2 V sentence3 10 Examples is tall is taller than recommends to 11 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 12 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 13 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 14 Compound Sentences 2 Jay and Kay are married individualy Jay is married and Kay is married Mj amp Mk Jay and Kay are married to each other Mjk and are married to each other 15 Quantifiers that convey quantity quantifiers are linguistic expressions Examples no neither every all any each both either some most many several few at least one at least two at most one at most two exactly one exactly two 16 Quantifiers 2 quantifiers combine commonnouns and verbphrases to form sentences predicate logic treats Examples both commonnouns every senior is happy no freshman is happy at least onejunior is happy few sophomores are happy most graduates are happy and verbphrases as predicates 17 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 E A 19 How Traditional Logic Does Quantifiers QuantifierPhrases are Simply NounPhrases every one is happy some one is happy Jay is happy Kay is happy subject predicate 20 1O 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 there is an x such that H x 22 11 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 12 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 EX HX pronunciation of symbols there is no X such that H X 25 NegativeUniversal Quantifier not every one is happy not every one is such that 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 pronunciation of symbols not for any X H X 26 13 Quantifying Negations 1 suppose then noteveryoneis happy thaeissomeone vvhois nothappy themissomex xisnothappy VXHX Exerx mecmwemeamunmn sabovmw 27 Quantifying Negations 2 suppose then nooneishappy no matter who you are you are nothappy no matter who X is xisnothappy 3XHX erwa thecmwemeamumeMisabovmm 28 14
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