1. 2 3 Exam 1:Sentential LogicTranslations (+) Exam 2:Sentential LogicDerivations Exam 3:Predicate...
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Transcript of 1. 2 3 Exam 1:Sentential LogicTranslations (+) Exam 2:Sentential LogicDerivations Exam 3:Predicate...
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INTRO LOGICINTRO LOGICDAY 15 DAY 15
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UNIT 3UNIT 3TranslationsTranslations
ininPredicate LogicPredicate Logic
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OverviewOverview
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
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Grading PolicyGrading Policy
When computing your final grade,
I count your four four highesthighest scores scores.
(A missedmissed exam counts as a zerozero.)
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Subjects and PredicatesSubjects and Predicates
In predicate logic,
every atomic sentence consists of
one predicatepredicate
and
one or more “subjectssubjects”
including subjects, direct objects, indirect objects, etc.
in mathematics “subjectssubjects” are called “argumentsarguments”(Shakespeare used the term ‘argument’ to mean ‘subject’)
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Example 1Example 1
is a dogis a dogElleElle
is awakeis awakeKayKay
is asleepis asleepJayJay
PredicatePredicateSubjectSubject
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Example 2Example 2
JayJayis taller thanis taller thanElleElle
ElleElleis next tois next toKayKay
KayKayrespectsrespectsJayJay
ObjectObjectPredicatePredicateSubjectSubject
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Example 3Example 3
toto
fromfrom
toto
JayJayElleElleprefersprefersKayKay
JayJayElleElleboughtboughtKayKay
KayKayElleEllesoldsoldJayJay
Indirect Indirect ObjectObject
Direct Direct ObjectObject
PredicatePredicateSubjectSubject
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What is a Predicate?What is a Predicate?
A predicatepredicate is 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.
predicate noun phrase2noun phrase1
sentence
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Compare with ConnectiveCompare with Connective
A connectiveconnective is 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.
connective sentence2sentence1
sentence3
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ExamplesExamples
is tall
is taller than
recommends to
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Symbolization ConventionSymbolization Convention
1. PredicatesPredicates are symbolized by upper case lettersupper case letters.
2. SubjectsSubjects are symbolized by lower case letterslower case letters.
3. PredicatesPredicates are placed firstfirst.
4. SubjectsSubjects are placed secondsecond.
PredPred subsub11 subsub22 … …
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ExamplesExamples
Kay recommended Elle to Jay
Jay recommended Kay to Elle
Kay is taller than Elle
Jay is taller than Kay
Kay is tall
Jay is tall
Rkej
Rjke
Tke
Tjk
Tk
Tj
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Compound Sentences - 1Compound Sentences - 1
neither Jay nor Kay is tall
both Jay and Kay are tall
Jay is not taller than Kay
Jay is not tall
Jay is taller than both Kay and Elle Tjk & Tje
Tj & Tk
Tj & Tk
Tjk
Tj
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Compound Sentences - 2Compound Sentences - 2
JayJay andand KayKay are marriedare married (individually)
=
JayJay is marriedis married, andand KayKay is marriedis married
and are married
MMjj && MMkk
JayJay andand KayKay are married are married (to each other) MMjkjk
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QuantifiersQuantifiers
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.
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Quantifiers – 2Quantifiers – 2
QuantifiersQuantifiers combine common nounscommon nouns and verb phrasesverb phrases
to form sentences.
Examples
everyevery seniorsenior is happyis happy
nono freshmanfreshman is happyis happy
at least oneat least one juniorjunior is happyis happy
fewfew sophomoressophomores are happyare happy
mostmost graduatesgraduates are happyare happy
predicate logic treats both common nouns and verb phrases as predicates
predicate logic treats both common nouns and verb phrases as predicates
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The Two Special Quantifiers The Two Special Quantifiers of Predicate Logicof Predicate Logic
some, at least one
existential quantifier
every, anyuniversal quantifier
symbolEnglish
expressionsofficial name
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Actually, they are both
upside-down.
Names of SymbolsNames of Symbols
backwards ‘E’
A E
upside-down ‘A’
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How Traditional Logic Does QuantifiersHow Traditional Logic Does Quantifiers
Quantifier Phrases are Simply Noun Phrases
Jay is happy
Kay is happy
some one is happy
every one is happy
subject predicate
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How Modern Logic Does QuantifiersHow Modern Logic Does Quantifiers
Quantifier Phrases are
Sentential Adverbs
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Existential QuantifierExistential Quantifier
some one is happy
there is some one who is happy
there is some one such that he/she is happy
there is some x such that x is happy
x Hx
there is an x (such that) H x
pronunciation
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Universal QuantifierUniversal Quantifier
every one is happy
every one is such that he/she is happy
whoever you are you are happy
no matter who you are you are happy
no matter who x is x is happy
x Hx
for any x H x
pronunciation
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Negating QuantifiersNegating 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
notnot one or more
it must be
zero
thus, the
negationnegation of ‘at least oneat least one’
is
‘not not at leastat least oneone’
which is
‘nnoneone’
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Negative-Existential QuantifierNegative-Existential Quantifier
no one is happy
there is no one who is happy
there is no one such that he/she is happy
there is no x such that x is happy
there is not some x such that x is happy
x Hx
there is no x (such that) H x
pronunciation
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Negative-Universal QuantifierNegative-Universal Quantifier
not every one is happy
not every one is such that he/she 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
x Hx
not for any x H x
pronunciation
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Quantifying Negations - 1Quantifying Negations - 1
suppose not everyone is happy
then there is someone
who is
not happy
i.e., there is some x :
x is not happy
xHx
xHx
the converse argument is also valid
=
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Quantifying Negations - 2Quantifying Negations - 2
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
xHx
xHx
the converse argument is also valid
=
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