Monday, February 20, 2012

Logic: Propositional Logic

I. Propositional Logic

Deals with propositions involving "not", "and", "or", and "if..then".

II. Propositional Terms

P (or any letter) represents a proposition, such as "Cats are mammals." This can be negated by putting "not" in front of it, so not-P means "It is not the case that cats are mammals." The negative can be symbolized: ~P

(P and Q) means two propositions are true. "Cats are mammals and lizards are reptiles." This can be symbolized: P • Q

(P or Q) means that at least one of the propositions is true. They can both be true, but at least one of the must be true. "Cats are mammals or leprechauns are green." This can be symbolized: P ∨ Q

(If P then Q) means that P is true and Q is true. This statement can only be false if P is true and Q is false. The P is called the antecedent, and the Q is called the consequent. In natural language, there is expected to be a relevance between the two. "If there are puddles outside, then it rained." This can be symbolized: P ⊃ Q

(P if and only if Q) means that both P and Q must be true.This can be symbolized: P ≡ Q
    III. Rules of Inference

    To create propositional syllogisms, use the four main logical rules of inference, all of which are logically valid.
    1. Modus Ponens: (if P then Q), P, therefore Q
    2. Modus Tollens: (if P then Q), not-Q, therefore not-P
    3. Conjunction: not(P and Q), P, therefore not-Q
    4. Disjunction: (P or Q), not-P, therefore Q
    IV. Rules of Simplification

    Propositions can be broken down according to the following rules:
    1. AND - (P and Q) can be simplified to: therefore P, therefore Q
    2. NOR - not(P or Q) can be simplified to: therefore not-P, therefore not-Q
    3. NIF - not(if P then Q) can be simplified to: therefore P, therefore not-Q
    4. IFF - (P if and only if Q) can be simplified to: therefore (if P then Q), therefore (if Q then P)
    5. NIFF - not(P if and only if Q) can be simplified to: therefore (P or Q), therefore not(P and Q)
    V. Equivalency

    There are several statements that are logically equivalent, and these rules can also be used to reverse the order of a proposition:

    1. Commutation: (P and Q) is equivalent to (Q and P)
    2. Association:  (P and (Q and R) is equivalent to ((P and Q) and R)
    3. Contraposition: (If P then Q) is equivalent to (If not-Q then not-P)
    4. De Morgan:  Not(P and Q) is equivalent to (not-P or not-Q) and  not(P or Q) is equivalent to (not-P and not-Q)

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