Logic: Learning Objectives

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Logic Learning Objectives

• Learn about statements (propositions)
• Learn how to use logical connectives to combine
  statements
• Explore how to draw conclusions using various
  argument forms
• Become familiar with quantifiers and predicates
• CS
• Boolean data type
• If statement
• Impact of negations
• Implementation of quantifiers

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Mathematical Logic

• Definition Methods of reasoning, provides rules
  and techniques to determine whether an argument
  is valid
• Theorem a statement that can be shown to be true
  (under certain conditions)
• Example If x is an even integer, then x 1 is
  an odd integer
• This statement is true under the condition that x
  is an integer is true

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Mathematical Logic

• A statement, or a proposition, is a declarative
  sentence that is either true or false, but not
  both
• Lowercase letters denote propositions
• Examples
• p 2 is an even number (true)
• q 3 is an odd number (true)
• r A is a consonant (false)
• The following are not propositions
• p My cat is beautiful
• q Are you in charge?

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Mathematical Logic

• Truth value
• One of the values truth or falsity assigned
  to a statement
• True is abbreviated to T or 1
• False is abbreviated to F or 0
• Negation
• The negation of p, written p, is the statement
  obtained by negating statement p
• Truth values of p and p are opposite
• Symbol is called not p is read as as not
  p
• Example
• p A is a consonant
• p it is the case that A is not a consonant
• q Are you in charge?

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Mathematical Logic

• Truth Table
• Conjunction
• Let p and q be statements.The conjunction of p
  and q, written p q , is the statement formed by
  joining statements p and q using the word and
• The statement p?q is true if both p and q are
  true otherwise p?q is false

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Mathematical Logic

• Conjunction
• Truth Table for
• Conjunction

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Mathematical Logic

• Disjunction
• Let p and q be statements. The disjunction of p
  and q, written p v q , is the statement formed by
  joining statements p and q using the word or
• The statement p v q is true if at least one of
  the statements p and q is true otherwise p v q
  is false
• The symbol v is read or

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Mathematical Logic

• Disjunction
• Truth Table for Disjunction

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Mathematical Logic

• Implication
• Let p and q be statements.The statement if p
  then q is called an implication or condition.
• The implication if p then q is written p ? q
• p ? q is read
• If p, then q
• p is sufficient for q
• q if p
• q whenever p

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Mathematical Logic

• Implication
• Truth Table for Implication
• p is called the hypothesis, q is called the
  conclusion

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Mathematical Logic

• Implication
• Let p Today is Sunday and q I will wash the
  car. The conjunction p ? q is the statement
• p ? q If today is Sunday, then I will wash the
  car
• The converse of this implication is written q ? p
• If I wash the car, then today is Sunday
• The inverse of this implication is p ? q
• If today is not Sunday, then I will not wash the
  car
• The contrapositive of this implication is q ? p
• If I do not wash the car, then today is not
  Sunday

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Mathematical Logic

• Biimplication
• Let p and q be statements. The statement p if
  and only if q is called the biimplication or
  biconditional of p and q
• The biconditional p if and only if q is written
  p ? q
• p ? q is read
• p if and only if q
• p is necessary and sufficient for q
• q if and only if p
• q when and only when p

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Mathematical Logic

• Biconditional
• Truth Table for the Biconditional

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Mathematical Logic

• Statement Formulas
• Definitions
• Symbols p ,q ,r ,...,called statement variables
• Symbols , , v, ?,and ? are called logical
  connectives
• A statement variable is a statement formula
• If A and B are statement formulas, then the
  expressions (A ), (A B) , (A v B ), (A ? B )
  and (A ? B ) are statement formulas
• Expressions are statement formulas that are
  constructed only by using 1) and 2) above

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Mathematical Logic

• Precedence of logical connectives is
• highest
• second highest
• v third highest
• ? fourth highest
• ? fifth highest

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Mathematical Logic

• Example
• Let A be the statement formula ((p v q )) ? (q
  p )
• Truth Table for A is

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Mathematical Logic

• Tautology
• A statement formula A is said to be a tautology
  if the truth value of A is T for any assignment
  of the truth values T and F to the statement
  variables occurring in A
• Contradiction
• A statement formula A is said to be a
  contradiction if the truth value of A is F for
  any assignment of the truth values T and F to the
  statement variables occurring in A

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Mathematical Logic

• Logically Implies
• A statement formula A is said to logically imply
  a statement formula B if the statement formula A
  ? B is a tautology. If A logically implies B,
  then symbolically we write A ? B
• Logically Equivalent
• A statement formula A is said to be logically
  equivalent to a statement formula B if the
  statement formula A ? B is a tautology. If A is
  logically equivalent to B , then symbolically we
  write A B

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Mathematical Logic
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Mathematical Logic

• Proof of (p q ) ? ((q ?p ))

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Mathematical Logic

• Proof of (p q ) ? ((q ?p )) continued

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Validity of Arguments

• Proof an argument or a proof of a theorem
  consists of a finite sequence of statements
  ending in a conclusion
• Argument a finite sequence
• of statements.
• The final statement, , is the conclusion,
  and the statements
  are the premises of the argument.
• An argument is logically valid if the statement
  formula
• is a
  tautology.

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Validity of Arguments - Example
P Q R Premises Valid
T T T T T T T T
T T F T F F F T
T F T F T F T T
T F F F T F F T
F T T T T T T T
F T F T F F T T
F F T T T T T T
F F F T T T T T
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Validity of Arguments

• Valid Argument Forms
• Modus Ponens (Method of Affirming)

P Q Premises Conclusion Q Valid
T T T T T T
T F F F F T
F T T F T T
F F T F F T
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Validity of Arguments

• Valid Argument Forms
• Modus Tollens (Method of Denying)

P Q Premises Conclusion Valid
T T T F F F T
T F F T F F T
F T T F F T T
F F T T T T T
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Validity of Arguments

• Valid Argument Forms
• Disjunctive Syllogisms
• Disjunctive Syllogisms

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Validity of Arguments

• Valid Argument Forms
• Hypothetical Syllogism (proven earlier)
• Dilemma

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Validity of Arguments

• Valid Argument Forms
• Conjunctive Simplification
• Conjunctive Simplification

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Validity of Arguments

• Valid Argument Forms
• Disjunctive Addition
• Disjunctive Addition

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Validity of Arguments

• Valid Argument Forms
• Conjunctive Addition

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Validity of Arguments Formal Derivation

• Prove
• Formal Derivation Rule
  Comment1. P ? Q
  Premise2. Q ?R
  Premise3. P
  Assumption Assume P4. Q
  1,3, MP5. R
  2,4, MP R is now proved6. P ?R
  DT
  Discharge P, ie, P is no
  longer to
  be used, and
  conclude that P ? R
• Uses Deduction Theorem (DT)

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Quantifiers and First Order Logic

• Have dealt with Propositional Logic (Calculus) so
  far
• Propositional variables, constants, expressions
• Dealt with truth or falsity of expressions as a
  whole
• Consider1. All cats have tails2. Tom is a
  cat3. Tom has a tail
• Cannot conclude 3, given 1 and 2 using
  propositional logic
• Predicate Calculus allows us to identify
  individuals such as Tom together with properties
  and predicates.

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Quantifiers and First Order Logic

• Predicate or Propositional Function
• Let x be a variable and D be a set P(x) is a
  sentence
• Then P(x) is called a predicate or propositional
  function with respect to the set D if for each
  value of x in D, P(x) is a statement i.e., P(x)
  is true or false
• Moreover, D is called the domain of the discourse
  and x is called the free variable

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Quantifiers and First Order LogicPropositional
function example 1

• Let P(x) be the statement x is an odd integer
• Let D be the set of all positive integers.
• Then P is a propositional function with domain of
  discourse D.
• For each x in D , P(x) is a proposition, i.e. a
  sentence which is either true or false.
• P(1) 1 is an odd integer True
• P(14) 14 is an odd integer - False

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Quantifiers and First Order LogicPropositional
function example 2

• Let P(x) be the statement the baseball player
  hit over .300 in 2003
• Let D be the set of all baseball players.
• Then P is a propositional function with domain of
  discourse D.
• For each x in D , P(x) is a proposition, i.e. a
  sentence which is either true or false.
• P(Barry Bonds) Barry Bonds hit over .300 in 2003
  - True
• P(Alex Rodriguez) Alex Rodriguez hit over .300
  in 2003 - False

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Quantifiers and First Order Logic

• Predicate or Propositional Function
• Example
• Q(x,y) x gt y, where the Domain is the set of
  integers
• Q is a 2-place predicate
• Q is T for Q(4,3) and Q is F for Q (3,4)

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Quantifiers and First Order Logic

• Universal Quantifier
• Let P(x) be a predicate and let D be the domain
  of the discourse. The universal quantification of
  P(x) is the statement
• For all x, P(x) or
• For every x, P(x)
• The symbol is read as for all and every
• Two-place predicate

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Quantifiers and First Order Logic

• Universal Quantifier Examples
• Consider the statement
• It is true if P(x) is true for every x in D
• It is false if P(x) is false for at least one x
  in D
• Consider with D being the set of all real
  numbers.
• The statement is true because for every real
  number x, it is true that the square of x is
  positive or zero.
• Consider that with D being the
  set of real numbers is false. Why?

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Quantifiers and First Order Logic

• Existential Quantifier
• Let P(x) be a predicate and let D be the domain
  of the discourse. The existential quantification
  of P(x) is the statement
• There exists x, P(x)
• The symbol is read as there exists
• Bound Variable
• The variable appearing in
  or

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Quantifiers and First Order Logic

• Existential Quantifier Example
• Consider
• It is true since there is at least one real
  number x for which the proposition is true. Try
  x2
• Suppose that P is a propositional function whose
  domain of discourse consists of the elements
  d1,,dn. The following pseudocode determines
  whether is true.

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Quantifiers and First Order Logic

• Negation of Predicates (DeMorgans Laws)
• Example
• If P(x) is the statement x has won a race
  where the domain of discourse is all runners,
  then the universal quantification of P(x) is
  , i.e., every runner has won a
  race. The negation of this statement is it is
  not the case that every runner has won a race.
  Therefore there exists at least one runner who
  has not won a race. Therefore
• and so,

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Quantifiers and First Order Logic

• Negation of Predicates (DeMorgans Laws)

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Quantifiers and First Order Logic

• Formulas in Predicate Logic
• All statement formulas are considered formulas
• Each n, n 1,2,...,n-place predicate P(
  ) containing the variables
  is a formula.
• If A and B are formulas, then the expressions A,
  (A?B), (A?B) , A ?B and A?B are statement
  formulas, where , ?, ?, ? and ? are logical
  connectives
• If A is a formula and x is a variable, then ?x
  A(x) and ?x A(x) are formulas
• All formulas constructed using only above rules
  are considered formulas in predicate logic

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Quantifiers and First Order Logic

• Additional Rules of Inference
• If the statement ?x P(x) is assumed to be true,
  then P(a) is also true,where a is an arbitrary
  member of the domain of the discourse. This rule
  is called the universal specification (US)
• If P(a) is true, where a is an arbitrary member
  of the domain of the discourse, then ?x P(x) is
  true. This rule is called the universal
  generalization (UG)
• If the statement ?x P (x) is true, then P(a) is
  true, for some member of the domain of the
  discourse. This rule is called the existential
  specification (ES)
• If P(a) is true for some member a of the domain
  of the discourse, then ?x P(x) is also true. This
  rule is called the existential generalization (EG)

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Quantifiers and First Order Logic

• Counterexample
• An argument has the form ?x (P(x ) ? Q(x )),
  where the domain of discourse is D
• To show that this implication is not true in the
  domain D, it must be shown that there exists some
  x in D such that (P(x ) ? Q(x )) is not
  true
• This means that there exists some x in D such
  that P(x) is true but Q(x) is not true. Such an x
  is called a counterexample of the above
  implication
• To show that ?x (P(x) ? Q(x)) is false by finding
  an x in D such that P(x) ? Q(x) is false is
  called the disproof of the given statement by
  counterexample

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Logic and CS

• Logic is basis of ALU
• Logic is crucial to IF statements
• AND
• OR
• NOT
• Implementation of quantifiers
• Looping
• Database Query Languages
• Relational Algebra
• Relational Calculus
• SQL