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Symbolic LogicConjunction , Negation ,

Disjunction v

- Examples

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

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)

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.

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.

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.

Example

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

Solution

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

Example

- 4. More educators, more administrators, and more

students are turning to philosophy to provide

them with the skills of reasoning.

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

Example

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

Solution

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

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

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.

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

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)

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

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

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)

Questions?Any problems you want to see worked

out (if time permits)?

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