Propositional Logic
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Building Blocks
o Rudiments
o Arguments
o Derivation Rules for Propositional Logic
o Verbal Arguments

Rudiments
o Constants – true and false
o Atoms – Propositions
o Connectives
o Semantics
Propositions
o Statements (or Atomic Proposition) – sentence that is either true or false
 Examples:
 Ten is less than seven
 That guy is going to Hawaii
 Are you that guy?
 This statement is false
o Math
 There are statements in math like 10 – 7 = 3 and 1 + 1 = 3
 One is trye and one is false, but they are both propositions
Boolean connectives
o Motivation – to make compound statements from simple ones
o Basic connectives are
 Conjunction (AND) ^
 Disjunction (OR) V
 Negation (NOT) ‘
 Implication (IF) ->
 Equivalence (IF AND ONLY IF) <->
Well-formed formulas
o A simple proposition is a well formed formula (wff)
o If A is a wff so is A’
o If A and B are wffs, then so are (A), AVB, A^B, A->B and A<->B
Resolving ambiguity
o Precedence
 Parentheses
 Negation
 Conjunction, disjunction
 Implication
 Equivalence
Tautologies
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o A wff which is always true is called a tautology, while a wff which is always false is
called a contradiction
o Examples
 A -> A – tautology
o Tautology checking
 Truth tables
Two important symbols
o If A and B are two wffs, and A -> B is a tautology we say that B follows from A,
written as A B
o If A and B are two wffs, and A <-> B is a tautology then we say that A and B are
equivalent wffs written as AB
o Both A B and AB have to be proved whereas A -> B and A<->B are merely
compound sentences
 In order to prove AB it suffices to prove A B and B A
o If you can prove AB then A and B can be substituted for each other
Examples
o Show that A^(A -> B)  B
 Use A and A->B for A^(A -> B), using conjunction truth table
A
T
T
F
F

A
T
T
F
F

B
T
F
T
F
A ->B
T
F
T
F
A^(A -> B)
T
F
F
F
A^(A -> B) -> B
T
T
T
T
Show that (A->B) (B’->A’)
B
T
F
T
F
(A->B)
T
F
T
T
B’
F
T
F
T
A’
F
F
T
T
(B’ -> A’)
T
F
T
T
Common Tautological Equivalences
o DeMogan’s Laws
 (A V B)’  (A’^B’)
(A ^B)’  (A’ V B’)
o Commutativity
 (A V B)  (B V A)
o Associativity
 (A V B) V C  A V (B V C)
o Distributivity
 A V (B ^ C)  (A V B) ^ (A ^ C)
(A->B) <-> (B’->A’)
T
T
T
T
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Arguments
o An argument is a statement of the form:
 (P1 ^ P2 ^ … Pn) -> Q
o Where each of Pis and Q are propositions
o Pis = hypothesis and Q is conclusion
o If al Pi are true, is Q is necessarily true?
o When can Q be logically deduced from P1, P2, …. Pn?
Valid argument
o Definiciton
 The areument (P 1 ^ P2 ^…. Pn) -> Q Is said to be valid if it is a
tautology
o Note
 If we have shown that it is a tautology, we can write (P1 ^ P2 ^…. Pn)
Q
 The validity of an argument is based purely on its intrinsic structure an
not on the specific meanings attached to its constitutent propositions