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Symbolic Logic : Conjunction • , Negation ~, Disjunction v

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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

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

E=Evil 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 • ~OT=Three people were involved in the accident.O=Someone 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)

S=You can be a sailor.M=You 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)

E=More educators are turning to philosophy to provide them with the skills of reasoning.

A=More administrators are turning…

S=More 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)

M=You are male.

F=You 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)

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