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INTRO LOGIC

This overview provides an introduction to predicate logic, focusing on translations and derivations. Learn about subjects, predicates, symbolization conventions, compound sentences, quantifiers, and negating quantifiers.

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INTRO LOGIC

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  1. INTRO LOGIC DAY 15

  2. UNIT 3TranslationsinPredicate Logic

  3. Overview •  Exam 1: Sentential Logic Translations (+) •  Exam 2: Sentential Logic Derivations • Exam 3: Predicate Logic Translations • Exam 4: Predicate Logic Derivations • Exam 5: (finals) very similar to Exam 3 • Exam 6: (finals) very similar to Exam 4

  4. Grading Policy • When computing your final grade, • I count your four highest scores. • (A missed exam counts as a zero.)

  5. 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’)

  6. Example 1 Subject Predicate Jay is asleep Kay is awake Elle is a dog

  7. Example 2 Subject Predicate Object Jay respects Kay Kay is next to Elle Elle is taller than Jay

  8. Example 3 Subject Predicate Direct Object Indirect Object Jay sold Elle to Kay Kay bought Elle from Jay Kay prefers Elle to Jay

  9. What is a Predicate? • A predicateis an "incomplete" expression – • i.e., 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 noun phrase2 sentence

  10. Compare with Connective • A connectiveis an "incomplete" expression – • i.e., an expression with one or more blanks – • such that, • whenever the blanks are filled by sentences, • the resulting expression is a sentence. sentence1 sentence2 sentence3

  11. Examples

  12. 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. Predsub1sub2 …

  13. Examples Jayis tall Tj Kayis tall Tk Jayis taller thanKay Tjk Kayis taller thanElle Tke JayrecommendedKaytoElle Rjke KayrecommendedElletoJay Rkej

  14. Compound Sentences - 1 Jayisnottall Tj Jayisnottaller thanKay Tjk bothJayandKayare tall Tj&Tk neitherJaynorKayis tall Tj&Tk Jayis taller thanbothKayandElle Tjk&Tje

  15. Compound Sentences - 2 Mj&Mk • JayandKayare married (individually) • = • Jayis married, andKayis married JayandKayare married (to each other) Mjk

  16. Quantifiers are linguistic expressions denoting quantity. Examples every, all, any, each, both, either some, most, many, several, few no, neither at least one, at least two, etc. at most one, at most two, etc. exactly one, exactly two, etc. Quantifiers

  17. Quantifiers combine common nouns and verb phrases to form sentences. Examples everysenioris happy nofreshmanis happy at least onejunioris happy fewsophomoresare happy mostgraduatesare happy Quantifiers – 2 predicate logic treats both common nouns and verb phrases as predicates

  18. The Two Special Quantifiers of Predicate Logic official name English expressions symbol universal quantifier every, any  existential quantifier some, at least one 

  19. Names of Symbols  backwards ‘E’  upside-down ‘A’ Actually, they are both upside-down. A E

  20. How Traditional Logic Does Quantifiers • Quantifier Phrases are Simply Noun Phrases

  21. How Modern Logic Does Quantifiers • Quantifier Phrases are • Sentential Adverbs

  22. Existential Quantifier • some oneis happy • there is some onewhois happy • there is some one such that he/sheis happy • there is somex such that xis happy • xHx pronunciation there is anx (such that) Hx

  23. Universal Quantifier • every oneis happy • every one is such that he/sheis happy • whoever you areyouare happy • no matter who you areyouare happy • no matter who x isxis happy • xHx pronunciation for anyxHx

  24. Negating Quantifiers • modern logic takes ‘’ to mean at least one • which means one or more • which means one, or two, or three, or … if a (counting) number is not one or more it must be zero thus, the negation of ‘at least one’ is ‘not at leastone’ which is ‘none’

  25. Negative-Existential Quantifier • no oneis happy • there is no onewhois happy • there is no one such that he/sheis happy • there is nox such that xis happy • there is not somex such that xis happy • xHx pronunciation there is nox (such that) Hx

  26. Negative-Universal Quantifier • notevery oneis happy • notevery one is such that he/sheis H • it is not true thatwhoever you areyouare H • it is not true thatno matter who you areyouare H • it is not true thatno matter who x isxis H • xHx pronunciation not for anyxHx

  27. Quantifying Negations - 1 xHx • suppose not everyone is happy • then there is someone • who is • not happy • i.e., there is some x : • x is not happy = xHx the converse argument is also valid

  28. Quantifying Negations - 2 xHx • suppose no one is happy • then no matter who you are • you are • not happy • i.e. no matter who x is • x is not happy = xHx the converse argument is also valid

  29. A E A THE END E A E A E

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