[GEMATMW] Module 1: Logic

Language

a systematic means of communicating by the use of sound or conventional symbols

Language

the code we all use to express ourselves and communicate to others

Proposition

a declarative statement which is either TRUE (T or 1) or FALSE (F or 0)

Propositional Variable

represents a proposition with an undetermined value

logical operators

propositions + propositional variables

Negation

generally formed by introducing the word “not”.

Negation

denoted by ¬𝑷 𝒐𝒓 ~𝑷 and is read as “not 𝑷”.

Conjunction

The proposition “𝑷 and 𝑸”, denoted 𝑷∧𝑸

Conjunction

true only when both propositions are true.

Disjunction

The proposition “𝑷 or 𝑸”, denoted by 𝑷∨𝑸

Disjunction

false only when both propositions are false

Conditional

If 𝑷, then 𝑸

hypothesis

If 𝑷, then 𝑸; Proposition 𝑷 is the __________

conclusion

If 𝑷, then 𝑸; proposition Q is the _________

Biconditional

𝑷 if and only if 𝑸

Biconditional

(𝑷⇒𝑸)∧(𝑸⇒𝑷)

Biconditional

𝑷⇔𝑸

Converse

𝑷⇒𝑸 → 𝑸⇒𝑷

Contrapositive

𝑷⇒𝑸 → ¬𝑸⇒¬𝑷

Inverse

𝑷⇒𝑸 → ¬𝑷⇒¬𝑸

𝟐^𝒏

in constructing a truth table, the number of rows is equal to ___ where 𝒏 is the number of distinct propositional variables

Tautology

a propositional form that is true under all circumstances

Contradiction

A propositional form that is false under all circumstances

Contingency

A propositional form that is neither a tautology nor a contradiction

⇔𝑷⇒𝑸¬𝑸⇒¬𝑷𝑷∧𝑸𝑷∨𝑸

Idempotence

𝑷 ⇔ (𝑷∨𝑷)

Idempotence

𝑷 ⇔ (𝑷∧𝑷)

Commutativity

(𝑷∧𝑸) ⇔ (𝑸∧𝑷)

Commutativity

(𝑷∨𝑸) ⇔ (𝑸∨𝑷)

Associativity

[(𝑷∨𝑸)∨R] ⇔ [𝑷∨(𝑸∨R)]

Associativity

[(𝑷∧𝑸)∧R] ⇔ [𝑷∧(𝑸∧R)]

De Morgan’s Laws

¬(𝑷∨𝑸) ⇔ (¬𝑷 ∧ ¬𝑸)

De Morgan’s Laws

¬(𝑷∧𝑸) ⇔ (¬𝑷 ∨ ¬𝑸)

Distributivity

[𝑷∨(𝑸∧R)] ⇔ [(𝑷∨𝑸) ∧ (𝑷∨R)]

Distributivity

[𝑷∧(𝑸∨R)] ⇔ [(𝑷∧𝑸) ∨ (𝑷∧R)]

Material Equivalence

(𝑷⇔𝑸) ⇔ [(𝑷⇒𝑸) ∧ (𝑸⇒𝑷)]

Material Equivalence

(𝑷⇔𝑸) ⇔ [(𝑷∧𝑸) ∨ (¬𝑷 ∧ ¬𝑸)]

Involution

𝑷 ⇔ ¬¬𝑷

Material Implication

(𝑷⇒𝑸) ⇔ (¬𝑷 ∨ 𝑸)

Exportation

[(𝑷∧𝑸)⇒R] ⇔ [(𝑷⇒(𝑸⇒R)]

Absurdity

[(𝑷⇒𝑸) ∧ (𝑷⇒¬𝑸)] ⇔ ¬𝑷

Contrapositive

(𝑷⇒𝑸) ⇔ (¬𝑸⇒¬𝑷)

Deductive argument

a collection of propositions where it is claimed that one of the propositions, is called the conclusion, follows from the other propositions, called the premises of the argument.

tautology

The deductive argument is valid if and only if the propositional form is a _________

Addition

P
∴ P∨Q

Simplification

𝑷∧𝑸
∴𝑷

Conjunction

𝑷
𝑸
∴𝑷∧𝑸

Modus Ponens

P⇒Q
P
∴Q

Modus Tollens

𝑷⇒𝑸
¬𝑸
∴¬𝑷

Disjunctive Syllogism

𝑷∨𝑸
¬𝑷
∴𝑸

Hypothetical Syllogism

P⇒Q
Q⇒R
∴P⇒R