Symbolic Logic:
Conjunction • , Negation ~,
Disjunction v
Examples
Review
Conjunction = “…and…” = •
-
Negation = “not…” = ~
-
-
Conjunction is only true if both conjuncts are true
Negation of a statement is true if statement is
false
Negation of a statement is false if statement is
true
Disjunction = “…or…” = v
Example: Translate the Following
1.
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
1.
It is not true that evil spirits exist.
1.
~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 • ~O
T=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)
1.
2.
3.
4.
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)



Right disjunct: ~(X • ~Y)





Both A and C are true. This makes (A v C) true.
But (A v C) is negated, so ~(A v C) is false.
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)?