Logical Operations

Negation or Logical Complement

Notation:  ¬

AND

Notation:  ∧

NAND

Notation: ⊼

OR

Notation:  ∨

NOR

Notation: ⊽

XNOR

Notation:  ⇔

EQUIVLANT

Notation:  ≡

XOR

Notation:  ⊻

NONEQUIVALENT

Notation:  ≢

IMPLIES

Notation:  ⇒

THEREFORE

Notation:  ∴

BECAUSE

Notation:  ∵

ABSOLUTE FALSEHOOD

Notation:  ⊥

Logical Rules

A Modus Ponens is that if p implies q and p is true then q is true.

Notation:  p , p ⇒ q ∴q

A Modus Tollens is if not q is true and p implies q, then not p is true.

Notation: ¬q , p ⇒ q ∴ ¬p

A Hypothetical Syllogism is if p implies q and q implies r then p implies r.

Notation:  (p ⇒ q), (q ⇒ r) ∴ (p ⇒ r)

A Disjunctive Syllogism is if not p is true and p or q is true, then q is true.

Notation:  ¬p, (p ∨ q) ∴ q

An Addition is if p is true then p or q is true.

Notation:  p ∴ (p ∨ q)

A Simplification is if p and q are true, then p is true.

Notation:  (p∧q ) ∴ p

A Conjunction is is if p is true and q is true, then p and q are true.

Notation:  p , q ∴ (p ∧q)

A Comment on my Experiences:

Suppose somebody postulates p, and the response is “I am the opposite.”  The first person is trying to be logically reasonable, but the second person is only performing a logical operation. The second person is being logical, but the first person is being reasonable.  So, Logic is easier than Reason.