Propositional Logic

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Propositional Logic

• By
• Jeremy Butkovich
• jebutkov_at_nmu.edu

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Propositional Logic

• Propositional logic is a branch of symbolic logic
  dealing with simple propositions as units and
  their compound statements and connectives.
• Example of a compound propositional statement
• It is raining outside and it is Tuesday.

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Propositions

• A proposition is simply a declarative sentence
  that is either true or False.
• The proposition must be either true or false,
  statements without a truth value are not
  propositions.
• Examples
• It is raining outside.
• 124.
• Note the statement 12. is NOT a proposition.

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Truth tables

• Truth tables are used to display the
  relationships between the truth values of
  propositions.
• Example

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Logical Operators

• Logical operators are operators used on
  propositions rather than on numbers.
• Used to connect statements or negate statements.
• Once operators are used on propositions, they
  form a new proposition.

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Logical Operators
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Negation Operator

• The negation operator () reverses the truth
  value of any proposition or compound Proposition.
  
• Example
• If proposition p I am a donkey.
• Then p I am not a donkey.

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Conjunction Operator

• The conjunction operator (?) combines two
  propositions to form their logical conjunction.
• Example
• If p It is raining.
• And q It is Tuesday.
• Then p ? q It is raining and it is Tuesday.

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Disjunction Operator

• The disjunction operator (?) combines two
  propositions to form their logical disjunction.
• Example
• If p It is raining.
• And q It is Tuesday.
• Then p?q It is raining or it is Tuesday.

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Exclusive-Or Operator

• The exclusive-or operator (?) combines two
  propositions to form their logical exclusive or
  disjunction.
• Example
• If p It is raining.
• And q It is Tuesday.
• Then p?q It is raining or it is Tuesday, but
  not both.

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Implication Operator

• The implication operator (?) combines two
  propositions in an if, then implication.
• The only time a proposition using an implication
  operator is false is if the first statement is
  false.
• Example
• If p It is raining.
• And q It is Tuesday.
• Then p?q If it is raining, then it is
  Tuesday.
• The only time this statement would be false,
  would be if the statement It is raining. is
  false.

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Bi-Conditional Operator

• The bi-conditional operator (?) combines two
  propositions where the first statement is true if
  and only if the second statement is true.
• Example
• If p It is raining.
• And q It is Tuesday.
• Then p?q It is raining if and only if it is
  Tuesday.

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Additional Resources

• Click here to see the truth tables for the
  logical operators discussed.
• Click here for to further research Logic.
• Click here to research propositions in more
  depth.
• Click here to research truth tables in more
  depth.