PROPOSITIONAL LOGIC

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PROPOSITIONAL LOGIC

• SUMMARY

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DEFINITIONS

• A declarative sentence is a sentence that
  declares a fact or facts.
• A proposition is a declarative sentence to which
  we can assign a truth value of either true or
  false, but not both.
• Logical Variables are letters such as p, q, r,
  s, or A, B, C, D, that are used to represent
  propositions

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MORE DEFINITIONS

• Connectives are symbols or words that enable you
  to combine simple propositions to form compound
  propositions.
• Propositional form is an expression involving
  logical variables and connectives such that, if
  all variables are replaced by propositions then
  the form becomes a proposition.
• Truth tables assign truth-values to propositional
  forms when truth-values have been assigned to the
  variables in the form.

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EXAMPLES OF A DECLARATIVE SENTENCE

• The earth is round.
• 5 2 6 1
• 1 0
• This sentence is false.

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EXAMPLES OF A PROPOSITION

• The earth is round. IS TRUE
• 5 2 6 1 IS TRUE
• 1 0 IS FALSE
  
• This sentence is false. IS NOT A
  PROPOSITION

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TRUTH TABLES

• Truth tables are constructs that show the
  resulting value of two propositions and a logical
  operator being joined together to form a more
  complex proposition.

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TRUTH TABLES

• Truth tables are constructs that show the
  resulting value of two propositions and a logical
  operator being joined together to form a more
  complex proposition.
• Each proposition listed in a truth table will
  have a value of either true or false, which will
  allow for a possibility of four combinations.
  When combined with logical operators, or
  connectives, the resulting value of the
  combination changes

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CONNECTIVES

• AND (Ù)
• The and connective joins propositions together in
  such a way that the result is true ONLY if the
  value of both propositions is true.

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CONNECTIVES

• OR (Ú)
• The or connective joins propositions together in
  such a way that the result is true if either
  proposition has a true value.

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CONNECTIVES

• NOT (Ø)
• The not operator is a little different. It only
  works on one proposition at a time, whether that
  proposition is a single entity, or the result of
  another operation. The not operator "negates" the
  given result.

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CONNECTIVES

• EXAMPLE OF A CONJUNCTION
• P This book is boring.
• Q I am hungry.
• And
• PQ This book is boring and I am hungry.

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CONNECTIVES

• EXAMPLE OF A DISJUNCTION
• P Linda is a secretary.
• Q Jim is lazy.
• v Or
• PvQ Linda is a secretary or Jim is lazy.

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CONNECTIVES

• EXAMPLE OF IMPLICATION
• P It is Monday.
• Q We are in Italy.
• gt Implies
• PgtQ If it is Monday then we are in Italy.

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CONNECTIVES

• EXAMPLE OF EQUIVALENCE
• P The world will blow up.
• Q CHAOS rules the world.
• ltgt Equivalence
• PltgtQ The world will blow up if an only if
• CHAOS rules the world.

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NEGATION

• EXAMPLE OF NEGATION
• P I have two kittens.
• P I do not have two kittens.

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LAW OF DETACHMENT

• (MODUS PONENS)
• P gt Q
• P
• _______________
• therefore Q

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LAW OF THE CONTRAPOSITIVE

• P gt Q
  ____________
• therefore Q gt P

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LAW OF MODUS TOLLENS

• P gt Q
• Q
• _______________
• therefore P

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LAW OF SYLLOGISM

• (HYPOTHETICAL SYLLOGISM)
• P gt Q
• Q gt R
• _______________
• therefore P gt R

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LAW OF DISJUNCTIVE INFERENCE

• (DISJUNCTIVE SYLLOGISM)
• P v Q
• P
• __________
• Q
• P v Q
• Q
• ________
• therefore P

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LAW OF THE DOUBLE NEGATION

• (P)
  ____________
• therefore P

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DeMORGANS LAW

• (P Q) (P v Q)
  
  ___________
  ___________
• therefore P v Q therefore P Q

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LAW OF CONJUNCTION

• P
• Q
  ____________
• therefore P Q

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LAW OF DISJUNCTIVE ADDITION

• P
  ____________
• therefore P v Q

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LAW OF CONJUNCTIVE ARGUMENT

• (P Q) (P Q)
• P
  Q
  ___________
  ___________
• therefore Q
  therefore P

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REASONING WITH PROPOSITIONAL LOGIC
If theres a fire there will be smoke. If there
is smoke there will be an alarm. Can we count on
hearing an alarm if there is a fire?
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TAUTOLOGICAL FORM

• (MODUS PONENS)
• P (P gt Q) gt Q

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TAUTOLOGICAL FORM

• (MODUS TOLLENS)
• Q (P gt Q) gt P

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TAUTOLOGICAL FORM

• HYPOTHETICAL SYLLOGISM
• (P gt Q) (Q gt R) gt P gt R

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TAUTOLOGICAL FORM

• DISJUNCTIVE SYLLOGISM
• (P V Q) P gt Q