2. SYMBOLIC LOGIC

Negation

**Symbol:** \~ (tilde)

**Translation:**


1. Not
2. It is not the case that…
3. It is false that…

The world’s first oil refinery was not founded in **E**ngland.

E = 0

\~E = 1

Negation Scenarios

* If a value is \~1, then it is false or 0.
* If a value is 1, then it is true.
* If a value is \~0, then it is true or 1.
* If a value is 0, then it is false.

Conjunction

**Symbol:** . (dot)

**Translation:**


1. And
2. Yet
3. Moreover
4. Nevertheless
5. Also
6. Still
7. While
8. However
9. But
10. While
11. Although

**P**oland declared its independence from Russia, while **G**reece restored its monarchy.

P . G

Conjunction Scenarios

Instances where ==P . Q is false== is when ==either P or Q is 0==, or ==both P and Q are 0.==

Disjunction

Symbol: ∨ (wedge)

Translation:


1. Either…or
2. Unless
3. If not

Either **P**hilippines increases its labor capital or its **e**conomy continues to go down.

P ∨ E

Disjunction Scenarios

The only scenario where P v Q is false is when Both P and Q are false.

Material Implication (Conditional)

Symbol: ⊃ (horseshoe) OR → (right-pointing arrow)

Translation:


1. If…then
2. in case
3. sufficient for
4. only if
5. given that
6. necessary for
7. If
8. implies
9. provided that

Note: You cannot interchange the statements unlike the other operations. Pay attention to the sentence.

If P, then Q.

If P → then Q

P → Q

P only if Q

P ⊃ Q

P is a sufficient condition for Q

P ⊃ Q

P is a necessary condition for Q

Q ⊃ P

If **P**hilippines raises its minimum wage, then so does **Q**atar.

Philippines = P

Qatar = Q

\
P ⊃ Q

Material Implication Scenarios

**1st Scenario:** If P happens, then Q happens. Therefore, it is true.

**2nd Scenario:** If P happens, then Q doesn’t happen. It is not true.

**3rd Scenario:** If P doesn’t happen, then Q happens.

**4th Scenario:** If P doesn’t happen, then Q doesn’t happen. It is true.

\
These scenarios are reliant on the “then statements” or the “Q.” If Q cannot happen if P does, then the whole statement is invalid.

Material Equivalence (Bi-Conditional)

**Symbol:** ≡ (triple bar)

**Translation:**


1. If and only if
2. Equivalent to
3. Is a sufficient and necessary condition for

**W**orld War I would end if and only if **G**ermany would sign an armistice agreement.

W ≡ G

Material Equivalence Scenarios

Instances where P ≡ Q is false is when either P or Q is true.

* P and Q must have the same value in material equivalence for P ≡ Q to equal 1, regardless of whether both P and Q are 1 or 0.

Truth Table

* Gives the truth value of a compound proposition for every possible truth value of its simple components.
* Each line in the truth table represents one such possible arrangement of truth values
* Tests the validity of arguments in propositional logic.

Formula for making a truth table

 L = 2ⁿ

\-   L = no. of rows

\-   N = no. of propositions