Propositional and Predicate Logic

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Propositional and Predicate Logic

• 2003. 9.
• ??????? ???????
• ? ? ?

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Logic

• A set of rules for inferencing
• A method applying to any set of concepts
• Axioms (??)
• Statements that are true for a class of objects
  (assumptions or properties)
• Theorem (??)
• Statements that are derived from axioms by a
  sequence of logical steps

Axioms
proof
Theorem

• Statements assumed
• Properties

A sequence of logical steps
proved under the axtioms
Axiomatic System
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Proposition

• A statement has a truth value
• Truth value
• true or false
• True statements
• 100 gt 99
• Harry Truman was a U.S. President.
• The moon is not made of green cheese.
• False statements
• 7 gt 42
• There are 1200 members of the U.S. Congress.

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Paradoxes

• Sentences do not have a truth value
• Impossible to assign a truth value
• This sentence is false.
• If the sentence is true, false
• Otherwise, true
• ????? ??? ?? ??? ????.
• Someone believe it true.
• But others do not think so.

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Logical operations

• Propositions can be built from other propositions
  through the use of logical operators.
• Logical connectives
• Connect two propositions (binary operators)
• and, or, and if-then
• Logical operator
• Unary operator
• Not
• With these operators, the result is always a
  compound proposition with a value of true or
  false.

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not operator

• Negation
• Reverse the truth value
• P 7 gt 5
• (Harry Truman was a U.S. President.)
• not P 7 5
• (Harry Truman was not a U.S. President.)

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Logically equivalent

• If two propositions have the same truth value,
  they are logically equivalent.

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and operator

• P 7 gt 5
• Q 8 is even
• P and Q 7 gt 5 and 8 is even.

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Example
and
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or operator

• In natural language,
• Exclusive or
• either one or the other, but not both
• Inclusive or
• Logical or operator
• disjunction

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Example

• Verify that
• R not ( P and Q) and
• S (not P) or (not Q)
• are logically equivalent.

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Implication (if-then) operator

• if P then Q
• P Þ Q
• P implies Q
• P The moon is made of green cheese.
• Q Ludwig van Beethoven was the King of France.

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Example

• If it is sunny Saturday and my homework is done,
  I am going cycling.
• If it is sunny Saturday, then, if my homework is
  done, I am going cycling.
• P It is sunny Saturday.
• Q My homework is done.
• R I am going cycling.

( P and Q ) Þ R
( P Þ ( Q Þ R )
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Example (Verify)
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DeMorgans Laws

• not ( P and Q) º (not P) or (not Q)
• not ( P or Q) º (not P) and (not Q)

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Predicate Logic

• There exists an integer n such that n2 25.
• Property n2 25 ? Predicate
• P(x) property of the object x
• Quantification
• Existential quantifier
• There exits an object such that
• x P(x)
• Universal quantifier
• For all objects,
• "x P(x)

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Example

• for all students there exists one plate
• one plate " students
• There exits one plate for all students
• " students one plate
• Assume x is an integer.
• P(x) x x2
• " x P(x) for all x, x x2
• x P(x)
• there exists x such that x x2

false
true
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Some Characteristics of quantifiers

• " x P(x)
• not ( " x P(x) ) º x ( not P(x))
• If a predicate P(x) is not true for all x, there
  exists some x such that P(x) is false.
• not ( x P(x) ) º " x ( not P(x))

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HW 2

• Exercise 2.1
• 2, 6
• Exercise 2.2
• 1, 2