5'1'1 PROPOSITIONAL CALCULUS

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5.1.1 PROPOSITIONAL CALCULUS
An atom is an elemental proposition. A
well-formed proposition results from combining
atoms, using the symbols ? ? ? ? ? using
enough parentheses to make clear the order in
which the logical operations are applied. Thus
p?q?r is not a WFP because it could mean either
(p?q)?r or p?(q?r) .
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EXAMPLES OF WFP's

• EXAMPLE 5.1.1
• EXAMPLE 5.1.2

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EXAMPLES OF TRUTH TABLES

• EXAMPLE 5.1.2 TRUTH VALUES
• EXAMPLE 5.1.3 TAUTOLOGY/THEOREM

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EXCLUDED MIDDLE
The excluded middle is the statement that every
WFP is either true or false ( x ??x) . This
is a tautology.
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? ?
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? ?
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IMPLICATION ?
To say that p ? q is true does not mean q is
true because p is true, but only that if p is
true, q will be true. Thus if p is false, we are
not assured that q is false.
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MODUS PONENS
Modus ponens is a form of argument that exploits
the implication. ( p ? (p ? q) ) ?q
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CONTRAPOSITIVE
( (p?q) ? (?q ? ?p) )
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5.1.1 PROPOSITIONAL CALCULUS

• ATOM
• WELL-FORMED PROPOSITION
• NEGATION
• TRUTH TABLES
• MODUS PONENS
• ARGUMENT
• EASY IDENTITIES
• HARD IDENTITIES

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5.1.1 EASY IDENTITIES

• Associativity (2)
• Commutativity (2)
• Weak Absorption (2)
• Involution
• Idempotency (2)
• Complementarity (2)
• Identity (2)

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5.1.1 HARD IDENTITIES

• Distributivity (2)
• De Morgan (2)
• Strong Absorption (2)
• Redundancy (2)
• Concensus (2)

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REDUNDANCY
Redundancy is one of the statements ( p ? (?p ?
q) ) ? (p ? q) ( p ? (?p ? q) ) ? (p ? q)
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MATHEMATICAL SYMBOLS

• ? ? ? ? ? ? ? ? ?
• ? ? ? ? ? ?
• ? ? ? ? ? ? ?
• ? ? ? ? ?