Symbolic LogicSymbolic Logic::Conjunction Conjunction •• , ,

Negation ~, Negation ~, Disjunction vDisjunction v

ExamplesExamples

ReviewReview

- Conjunction = “…and…” = Conjunction = “…and…” = ••- Conjunction is only true if both conjuncts are Conjunction is only true if both conjuncts are

truetrue

- Negation = “not…” = ~Negation = “not…” = ~- Negation of a statement is true if statement is Negation of a statement is true if statement is

falsefalseNegation of a statement is false if statement is Negation of a statement is false if statement is truetrue

- Disjunction = “…or…” = vDisjunction = “…or…” = v- Disjunction is true if either disjunct is trueDisjunction is true if either disjunct is true

Example: Translate the Example: Translate the FollowingFollowing

1.1. It is not true that evil spirits exist.It is not true that evil spirits exist.

First stepFirst step: Make a dictionary (define : Make a dictionary (define statements)statements)

Second stepSecond step: Look at the sentence, : Look at the sentence, symbolize statements correctly (using symbolize statements correctly (using •, ~, or v)•, ~, or v)

((Third stepThird step: Determine truth values): Determine truth values)

SolutionSolution

1.1. It is not true that evil spirits exist.It is not true that evil spirits exist.

1.1. ~E ~E E=Evil spirits exist.E=Evil spirits exist.

If evil spirits do exist (E is True), then ~E is If evil spirits do exist (E is True), then ~E is false.false.

If evil spirits do not exist (E is False), then If evil spirits do not exist (E is False), then ~E is true.~E is true.

ExampleExample

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

Note: When symbolizing statements, Note: When symbolizing statements, always make the statement a positive always make the statement a positive one. If you have a negative statement one. If you have a negative statement in the sentence, put its positive in the in the sentence, put its positive in the dictionary – then when you translate, dictionary – then when you translate, simply negate that sentence.simply negate that sentence.

SolutionSolution

2. There were three people involved in 2. There were three people involved in the accident, and no one was the accident, and no one was

injured.injured. T • ~OT • ~O

T=Three people were involved in the T=Three people were involved in the accident.accident.O=Someone was injured.O=Someone was injured.

ExampleExample

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

SolutionSolution

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

~(S • M)~(S • M)S=You can be a sailor.S=You can be a sailor.M=You can be a marine.M=You can be a marine.

ExampleExample

4. More educators, more 4. More educators, more administrators, and more students administrators, and more students

are turning to philosophy to are turning to philosophy to provide them with the skills of provide them with the skills of

reasoning.reasoning.

SolutionSolution

4. More educators, more administrators, and 4. More educators, more administrators, and more students are turning to philosophy more students are turning to philosophy

to provide them with the skills of to provide them with the skills of reasoning.reasoning.

(E • A) • S or E • (A • S)(E • A) • S or E • (A • S)E=More educators are turning to E=More educators are turning to

philosophy to provide them with the skills philosophy to provide them with the skills of reasoning.of reasoning.

A=More administrators are turning…A=More administrators are turning…S=More students are turning…S=More students are turning…

ExampleExample

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

SolutionSolution

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

(M v F) • ~(M • F)(M v F) • ~(M • F)

M=You are male.M=You are male.

F=You are female.F=You are female.

Example: Determine Truth Example: Determine Truth ValuesValues

GivenGiven: A, B, and C are TRUE : A, B, and C are TRUE statementsstatements

GivenGiven: X, Y, and Z are FALSE : X, Y, and Z are FALSE statementsstatements

Is the following true or false?Is the following true or false?

~Y v C~Y v C

SolutionSolution~Y v C~Y v C

1. We know that Y is False1. We know that Y is False

2. Since Y is false, this makes ~Y True.2. Since Y is false, this makes ~Y True.

3. We also know that C is True3. We also know that C is True

4. Therefore, we have two true disjuncts (C 4. Therefore, we have two true disjuncts (C and ~Y)and ~Y)

5. The main connective here is the wedge (v) 5. The main connective here is the wedge (v) and we know that a disjunction is false only and we know that a disjunction is false only if both disjuncts are false.if both disjuncts are false.

6. Therefore, ~Y v C is 6. Therefore, ~Y v C is truetrue..

ExampleExample

Determine whether the following is Determine whether the following is true:true:

(B v C) • (Y v Z)(B v C) • (Y v Z)

GivenGiven: A, B, and C are True: A, B, and C are True X, Y, and Z are False X, Y, and Z are False

SolutionSolution

(B v C) • (Y v Z)(B v C) • (Y v Z)

1.1. Look at one conjunct at a time. We have Look at one conjunct at a time. We have two here: (B v C) and (Y v Z)two here: (B v C) and (Y v Z)

2.2. (B v C): since we know B and C are both (B v C): since we know B and C are both true, this makes this disjunction truetrue, this makes this disjunction true

3.3. (Y v Z): since we know that Y and Z are (Y v Z): since we know that Y and Z are both false, this makes this disjunction both false, this makes this disjunction falsefalse

4.4. Since we now know the whole left Since we now know the whole left conjunct (B v C) is true, and that the right conjunct (B v C) is true, and that the right conjunct (Y v Z) is false, the conjunction of conjunct (Y v Z) is false, the conjunction of the two must be the two must be falsefalse (for a conjunction to (for a conjunction to be true, be true, bothboth conjuncts must be true) conjuncts must be true)

ExampleExample

Determine whether the following is Determine whether the following is true:true:

~(A v C) v ~(X • ~Y)~(A v C) v ~(X • ~Y)

GivenGiven: A, B, and C are True: A, B, and C are True X, Y, and Z are False X, Y, and Z are False

SolutionSolution

~(A v C) v ~(X • ~Y)~(A v C) v ~(X • ~Y) The main connective = the middle The main connective = the middle

wedge (v) (disjunction)wedge (v) (disjunction) Therefore we have two disjunctsTherefore we have two disjuncts

Left disjunct= ~(A v C)Left disjunct= ~(A v C) Right disjunct = ~(X • ~Y)Right disjunct = ~(X • ~Y)

Strategy: determine truth values of Strategy: determine truth values of each disjunct, then we know if at least each disjunct, then we know if at least one disjunct is true, this will make the one disjunct is true, this will make the wholewhole statement true statement true

Solution (continued)Solution (continued)~(A v C) v ~(X • ~Y)~(A v C) v ~(X • ~Y)

Left disjunct: ~(A v C)Left disjunct: ~(A v C) Both A and C are true. This makes (A v C) true.Both A and C are true. This makes (A v C) true. But (A v C) is negated, so ~(A v C) is false.But (A v C) is negated, so ~(A v C) is false.

Right disjunct: ~(X • ~Y)Right disjunct: ~(X • ~Y) X is false.X is false. Y is false, so this means ~Y is true.Y is false, so this means ~Y is true. This makes the inner conjunction false (to be This makes the inner conjunction false (to be

true, both conjuncts (X and ~Y) must both be true, both conjuncts (X and ~Y) must both be true)true)

Because the whole statement (X • ~Y) is false, Because the whole statement (X • ~Y) is false, this makes its negated form ~(X • ~Y) truethis makes its negated form ~(X • ~Y) true

Since the left disjunct is false, and the right Since the left disjunct is false, and the right disjunct is true, this means ~(A v C) v ~(X • disjunct is true, this means ~(A v C) v ~(X • ~Y) is ~Y) is truetrue (since at least one disjunct is (since at least one disjunct is true)true)

Questions?Questions?Any problems you want to Any problems you want to

see worked out (if time see worked out (if time permits)?permits)?