Symbolic Logic: Conjunction

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Symbolic LogicConjunction , Negation ,
Disjunction v

• Examples

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Review

• Conjunction and
• Conjunction is only true if both conjuncts are
  true
• Negation not
• Negation of a statement is true if statement is
  falseNegation of a statement is false if
  statement is true
• Disjunction or v
• Disjunction is true if either disjunct is true

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Example Translate the Following

• It is not true that evil spirits exist.
• First step Make a dictionary (define statements)
• Second step Look at the sentence, symbolize
  statements correctly (using , , or v)
• (Third step Determine truth values)

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Solution

• It is not true that evil spirits exist.
• E
• EEvil spirits exist.
• If evil spirits do exist (E is True), then E is
  false.
• If evil spirits do not exist (E is False), then
  E is true.

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Example

• 2. There were three people involved in the
  accident, and no one was injured.
• Note When symbolizing statements, always make
  the statement a positive one. If you have a
  negative statement in the sentence, put its
  positive in the dictionary then when you
  translate, simply negate that sentence.

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Solution

• 2. There were three people involved in the
  accident, and no one was injured.
• T OTThree people were involved in the
  accident.OSomeone was injured.

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Example

• 3. You cannot be a sailor and a marine both.

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Solution

• 3. You cannot be a sailor and a marine both.
• (S M)
• SYou can be a sailor.MYou can be a marine.

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Example

• 4. More educators, more administrators, and more
  students are turning to philosophy to provide
  them with the skills of reasoning.

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Solution

• 4. More educators, more administrators, and more
  students are turning to philosophy to provide
  them with the skills of reasoning.
• (E A) S or E (A S)
• EMore educators are turning to philosophy to
  provide them with the skills of reasoning.
• AMore administrators are turning
• SMore students are turning

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Example

• 5. Either you are male or female but not both.

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Solution

• 5. Either you are male or female but not both.
• (M v F) (M F)
• MYou are male.
• FYou are female.

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Example Determine Truth Values

• Given A, B, and C are TRUE statements
• Given X, Y, and Z are FALSE statements
• Is the following true or false?
• Y v C

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Solution
Y v C 1. We know that Y is False 2. Since Y is
false, this makes Y True. 3. We also know that C
is True 4. Therefore, we have two true disjuncts
(C and Y) 5. The main connective here is the
wedge (v) and we know that a disjunction is false
only if both disjuncts are false. 6. Therefore,
Y v C is true.
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Example

• Determine whether the following is true
• (B v C) (Y v Z)
• Given A, B, and C are True X, Y, and
  Z are False

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Solution

• (B v C) (Y v Z)
• Look at one conjunct at a time. We have two
  here (B v C) and (Y v Z)
• (B v C) since we know B and C are both true,
  this makes this disjunction true
• (Y v Z) since we know that Y and Z are both
  false, this makes this disjunction false
• Since we now know the whole left conjunct (B v C)
  is true, and that the right conjunct (Y v Z) is
  false, the conjunction of the two must be false
  (for a conjunction to be true, both conjuncts
  must be true)

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Example

• Determine whether the following is true
• (A v C) v (X Y)
• Given A, B, and C are True X, Y, and Z
  are False

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Solution

• (A v C) v (X Y)
• The main connective the middle wedge (v)
  (disjunction)
• Therefore we have two disjuncts
• Left disjunct (A v C)
• Right disjunct (X Y)
• Strategy determine truth values of each
  disjunct, then we know if at least one disjunct
  is true, this will make the whole statement true

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Solution (continued)

• (A v C) v (X Y)
• Left disjunct (A v C)
• Both A and C are true. This makes (A v C) true.
• But (A v C) is negated, so (A v C) is false.
• Right disjunct (X Y)
• X is false.
• Y is false, so this means Y is true.
• This makes the inner conjunction false (to be
  true, both conjuncts (X and Y) must both be
  true)
• Because the whole statement (X Y) is false,
  this makes its negated form (X Y) true
• Since the left disjunct is false, and the right
  disjunct is true, this means (A v C) v (X Y)
  is true (since at least one disjunct is true)

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Questions?Any problems you want to see worked
out (if time permits)?