Philosophy - Proofs

Logical Operators

1. Negation [NOT]: “It is not the case that…”

· “It is false that…”

· “It is not true that…”

· “...not…”

· EXAMPLES: It is not true that Obama is the current president of the USA.

It’s not raining.

You are not my father.

· FUNCTION: The negation operator asserts the falsity of a statement.

· NOTE: In logic, you may have double negatives (or triple, etc.). For example, “It is not the case that it is not raining” means the same as “It is raining”.

· NOTE: The negation operator is the only unary operator we use in traditional logic. This means that it only attaches to single statements. It cannot be a logical relation between two statements, as the other operators are.

· TRUTH FUNCTION: “NOT p” is true if and only if P is false.

2. Disjunction [OR]: “Either I will drink a coffee, or I will drink a beer.

· EXAMPLES: You may order coffee or milk.

Either you will enjoy completing the module, or you will be annoyed with this example.

· FUNCTION: The disjunction operator asserts the truth of either one statement, or the other, or both.

· NOTE: The disjunction operator is a binary operator. That means it will always be a logical relation between exactly two statements. For example, it relates “you may order coffee” to “you may order milk”.

· NOTE: The two statements related by the OR operator are called disjuncts .

· NOTE: In logic, we always use the inclusive OR operator. This means that disjunctions are still true even when both disjuncts are true.

· TRUTH FUNCTION: “p OR q” is true if and only if either P, or Q are true, or they are both true.

3. Conjunction [AND]: “I will drink a coffee and I will drink a beer.”

· Greg did well on the exam, but Susan aced it.

· Greg did well on the exam; Susan aced it.

· Greg did well on the exam, yet Susan aced it.

· FUNCTION: The AND operator asserts the truth of two statements.

· NOTE: The conjunction operator is a binary operator. That means it will always be a logical relation between exactly two statements.

· NOTE: The two statements related by the AND operator are called conjuncts .

· TRUTH FUNCTION: “P AND Q” is true if and only if both P and Q are true.

4. Conditional [If/then]: “If squirrels are cute, then Carol will want to pet them.”

· If Saitama is bored, he punches monsters.

· Saitama punches monsters if he is bored.

· Saitama is bored only if he punches monsters.

· Given that Kant is hard to read, we will instead read Korsgaard.

· FUNCTION: The If...then operator asserts a conditional relationship between two statements. The antecedent is sufficient for the consequent, and the consequent is necessary for the antecedent.

· NOTE: The conditional operator is a binary operator. That means it will always be a logical relation between exactly two statements.

· NOTE: The two statements related by the If...then operator are called the “antecedent” and the “consequent”. The order does matter!

· EXAMPLE: If {...antecedent} then {consequent}

· NOTE: In traditional logic, we use the material conditional. (In English, conditional statements can be vague -- there are different kinds.)

· TRUTH FUNCTION: “IF P, THEN Q” is true if and only if either Q is true or P is false. (It is false only when P is true and Q is false.)

5. Biconditional [...if and only if]: “Students will receive A range grades in this class if and only if their final score is at least 900/1000 points.”

· Clear liquids are water if and only if they are H2O.

· FUNCTION: The biconditional asserts the logical equivalency of two statements. (Loosely, “their truth value hangs together”)

· NOTE: The biconditional operator is a binary operator. That means it will always be a logical relation between exactly two statements.

· NOTE: The two statements related by the biconditional can be called “conditions” or “conditionals”.

· TRUTH FUNCTION: “P IF AND ONLY IF Q” is true only when both P and Q or true, or when both P and Q are false.