Chapter three

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Chapter three

• By Mohamed Al Sukaiti

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Sentential Logic, Symbols Truth Tables

• It is a branch of formal logic in which sentences
  are basic units.
• It helps in development of analytical and
  communication skills effectively.

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Conjunction

• Two simple sentences connected by and()
• Example
• Ahmed is wise and tarique is playful.

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Truth TableAhmed is wise WTarique is
playful pW and Pare two simple sentences
called conjuncts and joined together WP is
called conjunction
W P W P
T T T
T F F
F T F
F F F
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Cont.

• The conjunction is true if and only if both the
  conjuncts are true and is false otherwise.
• Example
• Both Tom and Jerry are naughty
• It means Tom is naughty and Jerry is
  naughty.
• 2. Ali loves Maths but hates physics
• But means Ali loves Maths and he hates
  physics

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Cont.

• 3. The students came today, however, the exam is
  tomorrow.
• It means The students came today and The
  exam is tomorrow
• 4. Although Sara worked hard, she failed
• It means Sara worked hard and Sara
  Failed.

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Disjunction

• Tow simple sentences (disjuncts) connected by
  or(v) to form a disjunction.
• Ex Farid is football player OR Tarique is a
  banker.
• Farid is a football player F
• Tarique is banker B
• A disjunction is true if and only one of
  disjuncts is true.

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Truth Table
F B F or B(FVB)
T T T
T F T
F T T
F F F
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Cont.

• At least one of the two boys, Ali and Said, will
  pass the exam.
• It means Either Ali will pass the exam or
  said will pass the exam( or both will pass)
• 2. Tom is acientist or philosopher
• It means Tom is scientist or Tom is a
  philosopher(or he is both)
• This kind of or is called inclusive or as
  it means any one or both

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Exclusive or

• Which means either this or that but not both.
• Examples
• You will get an ice cream or sweet free with
  this meal
• Tom will pass or fail in this exam

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Negation

• Consider the following sentence
• Muscat is the capital of Oman
• If This sentence is true , then the negation of
  it would be Muscat is not the capital of Oman,
  or, It is not the case that Muscat is the
  capital of Oman.
• This sentence is true if its component simple
  sentence Muscat is the capital of Oman is false.

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• It is false if the component sentence is true.
• In this case Muscat is the capital of Oman is
  true , so the negation is false.
• If Muscat is the capital of Oman is denoted as
  C, Then It is not the case
• that Muscat is the capital of Oman is denoted
  by C(negation of C)

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Truth Table
C C
T F
F T
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Cont.

• Not all sailors are good swimmers can be
  modefied as it is not the case that all sailors
  are good swimmers.
• All Sailors are good swimmers S
• It is not the case that all sailors are
  good swimmers S
• No doctor is rich is true if and only if some
  doctors are rich is false
• So sentence might be rephrased as
• It is not the case that some doctors are
  rich.

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Cont.

• Water is not solid can be rephrased as
• It is not the Case that water is soliod.
• Water is solid W

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Material Conditional (If then)

• Ex 1 If Ali got the job, then he applied for it.
• Ali got the job G
• Ali applied for it A
• This casn be expressed as G ? A
• G The sentence to left of ? is the antecedent.
• A The sentence to the right of ? is the
  consequent.

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Cont.

• It can also be expressed as
• Either it is not the case that ali got the job
  or ali applied for the job. G V A

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Cont.

• If ali is in Muscat then ali is in Oman
• If Ali is in Muscat M
• Then Ali is in Oman O
• M ? O
• Or
• Either it is not the case that ali is in
  Muscat or ali is in Oman M V O

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Truth Table
A E AE
T T T
T F F
F T F
F F T
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Combination of Sentential Connectives

• The connectives such as conjunctions,
• disjunctions, and negations can be used
• in combinations.
• Examples A and B will not both become president
  of USA.
• It means both A and B together cannt become
  president , but separately they can be or it is
  possible that neither will be the president.

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Cont.

• It is not the casae that A willl become president
  and B will become president.
• (A and B)
• (A A will become president
• B will become president)
• Ex 2 Ali and Salim cannt both become
  captin of the football team. (AS)

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7. Neither-nor

• Neither Ali nor said is fond of sweets.
• Can be expressed as It is not the case that
  Ali is fond of sweets and it is not the case that
  said is fond of sweets.
• Ali is fond of sweets A
• Said is fond of sweets S
• We can express the sentence as
• A and S (A S)

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Cont.

• The previous serntence can be expressed as
• It is not the case that either Ali is fond of
  sweets or said is fond of sweets
• (A V S)

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A S (A V S) Or (A S)
T T F
T F F
F T F
F F T
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Class practice( Logical Symbols)

• Write the following in logical symbols. Use the
  following abbrviations
• A Amy jobs regularly
• B Bob Jobs regurarly
• C Carl jogs regurarly

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Cont.

1. Bob and Carl jog regularly
2. Bob does not jog regularly but Carl does
3. Either Bob Jogs regularly or Carl does
4. Neither Bob nor Carl jogs regularly

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Cont.

1. It is not the case that Carl or Bob jogs
   regularly, moreover Amy doesnt either.
2. Bob doesnt jog regularly unless Carl does
3. Either Amy jogs regularly or he doesnt
4. Either Bob or Amy jogs regularly but they dont
   jog regularly

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Exercise Page 9

• A Amy jogs regularly
• B Bob jogs regularly
• C Carl jogs regularly
• L Bob is Lazy
• M Carl is a marathon runner
• H Amy is healthy

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Cont.

• a) If Bob jogs regurarly he is not lazy.
• b) If Bob is not lazy he jogs regurarly
• c)Bob jogs regurarly if and only if he is not
  lazy
• d)Carl is marathon runner if and only if she jogs
  regurarly

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Cont.

• e) If both Carl and Bob jog regularly, then Amy
  does too
• f) If either Carl and Bob doesnt jog regularly
  then Amy doesnt either.
• g) If Amy is healthy and Babo is not Lazy then
  both jog regularly.
• h) If either Bob nor Amy jogs then Carl jogs if
  and only if he is a marathon runner.

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