Chapter 2: Propositional Calculus: Formulas, Models, Tableaux
Chapter 2: Propositional Calculus: Formulas, Models, Tableaux
August 22, 2008
Outline
1 2.1 Boolean Operators 2 2.2 Propositional Formulas 3 2.3 Interpretations 4 2.4 Equivalence and Substitution 5 2.5 Satisfiability, Validity, and Consequence 6 2.6 Semantic Tableaux 7 2.7 Soundness and Completeness
2.1 Boolean Operators
? Boolean type: T (true), F (false) ? Boolean operator: a function on the set {T,F}.
These operators can be unary, binary, etc. ? Question: How many n-ary Boolean operators are there
on {T,F}? 22n
? We single out the following five operators:
? (unary)
, , , (binary) ? There are other binary operators that are sometimes used
e.g. in the theory of Boolean circuits:
(XOR, exclusive OR) (NAND) (NOR, Sheffer's stroke)
p q pq TT F TF T FT T FF F
pq TT TF FT FF
pq F T T T
p q pq TT F TF F FT F FF T
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