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Propositional Logic

This guide explores fundamental concepts of propositional logic, focusing on negation, conjunction, disjunction, implication, and equivalence. Negation transforms propositions using 'not', while conjunction ('and') is true only if both propositions are true. Disjunction ('or') holds true if at least one proposition is. Implication indicates that 'if p then q' is true except when p is true and q is false. Equivalence signifies that two propositions are equivalent only if they share the same truth value. Rules for associativity, commutativity, and distributivity are also highlighted.

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Propositional Logic

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  1. Propositional Logic

  2. Negation • Given a proposition p, negation of p is the ‘not’ of p

  3. Conjunction • Representing ‘and’ between propositions. • Given two propositions p and q, conjunction of p and q is true only when both of the propositions are true

  4. Disjunction • Representing ‘or’ between proposition • Given two propositions p and q, disjunction of p and q is true when either or both of the propositions are true

  5. Implication • Representing ‘if …then…’ to connect propositions • Given two propositions p and q, we say that “p implies q” which is the implication of q by p , the result is true in all cases except where p is true and q is false

  6. Equivalence • Representing “if and only if” • Two propositions are equivalent if and only if they have the same truth value

  7. Some rules • Disjunction is an associative and a commutative truth function • p  ( q  r )  (p  q )  r  p  q  r • p  q  q  p • Conjunction is a commutative and associative truth function • Distributive • p  ( q  r )  (p  q)  (p  r) • p  ( q  r )  (p  q)  (p  r) • Implication is not commutative p  q is not the same as q  p • is not associative p ( q  r ) is not the same as (p q ) r • Is transitive p q and q  r , p r

  8. Exercise • Let s, t, and u denote the following atomic propositions: s : Sally goes out for a walk. t : The moon is out u : It is windy Write a possible translation for each of the following statements: • If the moon is out and it is not windy, then Sally goes out for a walk • If the moon is not out, then if it is not windy Sally goes out for a walk

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