Chapter 2 Fundamentals of Logic

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Chapter 2 Fundamentals of Logic

• Dept of Information management
• National Central University
• Yen-Liang Chen

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2.1 Basic connectives and truth table

• Assertions, called statements or propositions,
  are declarative sentences that are either true or
  false
• New statements can be obtained from existing ones
  in two ways.
• Transform a given statement p into the statement
  ?p
• Combine two or more statements into a compound
  statement

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Forming a compound statement
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Ex 2.1

• s Phyllis goes out for a walk
• t The moon is out
• u It is snowing
• (t??u)?s
• t?(?u?s)
• ?(s?(u?t))

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Ex 2.2

• If I weigh more than 120 pounds, then I shall
  enroll in an exercise class
• p I weigh more than 120 pounds
• q I shall enroll in an exercise class
• the four cases of p?q

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A word of caution

• In our everyday language, we often find
  situations where an implications is used when the
  intention actually calls for a biconditional.
• If you do your homework, then you will get to
  watch the baseball game.

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p?(q?r)?(p?q)?r
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p?(p?q), p?(?p?q)
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Key ideas

• A compound statement is a tautology if it is true
  for all truth value assignments and a
  contradiction if it is false for all truth value
  assignments
• To show (p1?p2??pn)?q a valid argument, we need
  to show this statement is a tautology. If any pi
  is not true, then no matter what q is the
  statement is true. Thus, we only need to show
  that q follows from (p1?p2??pn), when all of
  them are true.
• Premises and conclusion

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2.2 Logic equivalence the laws of logic

• Ex 2.7, p?q is equivalent to ?p?q
• Definition 2.2. Two statements are said to be
  logically equivalent, s1?s2, when the statement
  s1 is true if and only if the statement s2 is true

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(p?q)?(p?q)?(q?p)
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(p?q)?(p?q)?? (p?q)
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DeMorgans law

• ? (p?q) ??p??q ?(p?q)??p??q

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The distributive law

• p?(q?r) ?(p?q) ? (p?r)
• p? (q?r) ?(p?q) ? (p?r)

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Observations

• When s1?s2, then s1?s2 is a tautology when
  ?s1??s2 then ?s1??s2 is a tautology
• When s1?s2 and s2?s3, then s1?s3

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The laws of logic

• ??p?p
• ?(p?q)??p??q
• ?(p?q)??p??q
• p?q?q?p
• p?q?q?p
• p?(q?r) ? (p?q)?r
• p?(q?r) ? (p?q) ?r
• p?(q?r) ? (p?q) ?(p?r)
• p?(q?r) ? (p?q) ? (p?r)

• p?p?p
• p?p?p
• p?F?p
• p?T?p
• p??p?T
• p??p?F
• p?T?T
• p?F?F
• p?(p?q) ? p
• p?(p?q) ? p

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Observation

• Definition 2.3, sd, the dual of s, is obtained by
  replacing ? with ?, ? with ?, T with F and F with
  T.
• Theorem 2.1. The principle of duality. Let s and
  t be statements that contain no logical
  connectives other than ? and ?. If s?t, then
  sd?td.

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Two substitution rules

• Suppose that the compound statement P is a
  tautology. If p is a primitive statement that
  appears in P and we replace each occurrence of p
  by the same statement q, then the resulting
  compound statement P1 is also a tautology.
• Let P be a compound statement where p is an
  arbitrary statement that appears in P, and let q
  be such a statement such that p?q. Suppose that
  in P we replace one or more occurrences of p by
  q. Then this replacement yields the compound
  statement P1. Under these circumstances P? P1.

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Ex 2.10

• P ?(p?q)?(?p??q) is a tautology
• P1 ?((r?s)?q)?(? (r?s)??q)
• P2 ?((r?s)? (t?u))?(? (r?s)?? (t?u))

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Ex 2.11

• Let P (p?q)?r be a compound statement.
• Because (p?q)??p?q, if P1 (?p?q)?r, then P1?P.
• Let P p?(p?q) be a compound statement.
• Because ??p?p, if P1 ??p ?(??p ?q), then P1?P.

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Ex 2.12, Ex 2.13

• ?(p?q)?r?
• ??(p?q)? r?
• ??(p?q)?? r?
• (p?q)?? r

• ?(p?q)?
• ?(?p?q)?
• ??p??q?
• p??q

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Definitions

• Implication p?q
• contrapositive, ?q?? p
• converse, q? p
• inverse ? p ?? q

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Ex 2.16, Ex 2.17

• (p?q)??(?p?q)?
• (p?q)?(??p??q)?
• (p?q)?(p??q)?
• p?(q??q)?
• p?F?p

• ??(p?q)?r??q ?
• ??(p?q)?r???q ?
• (p?q)?r?q ?
• (p?q)?(q?r) ?
• (p?q)?q?r ?
• q?r

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Simplifying the switch network

• (p?q?r)?(p?t??q)?(p??t?r)?p?r?(t??q)

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2.3 Logic implication rules of inference

• (p1?p2??pn)?q is a valid argument, if the
  premises are true, then the conclusion is also
  true.
• If any one of p1, p2,, pn is false, the
  implication is automatically true.
• To establish the validity of a given argument is
  to show that the statement (p1?p2??pn)?q is a
  tautology.
• The conclusion is deduced or inferred from the
  truth of premises.

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Ex 2.19

• (p?r)?(?q?p)??r?q

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Ex 2.20

• p?((p?r)?s)?(r?s)

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Key concepts

• Definition 2.4. If p and q are arbitrary
  statements such that p?q is a tautology, then we
  say that p is logically implies q and we write
  p?q to denote this situation.
• When p?q, we refer to p?q as a logical
  implication.
• If p?q, then p?q is a tautology, and we have p?q
  and q? p. Conversely, suppose that p?q and q? p,
  then we have p?q.

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The rule of inferences

• The rule of Modus Ponens
• (method of affirming), the rule of detachment
• p?( p?q)?q
• (r?s)?(r?s)?(?t?u)? (?t?u)
• The rule of syllogism
• ( p?q)?( q?r)? ( p?r)

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Ex 2.24

• (p)? (p??q) ? (?q??r) ? ?r

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the rule of Modus Tollens

• (method of denying), ?q?( p?q)? ?p
• Ex 2.25
• (p?r)? (r?s)? (t??s)? (?t?u)? (?u) ? ?p

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Some notes

• Some arguments look similar in appearance but are
  indeed invalid.
• q?( p?q)?p
• ?p?( p?q)??q
• the rule of conjunction, (p)?(q)?(p?q)
• the rule of disjunctive syllogism, (
  p?q)?(?p)? q

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the rule of contradiction

• (?p)?(F)?(p)

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The rule of contradiction

• When we want to establish the validity of the
  argument (p1?p2??pn)?q, we can establish the
  validity of the logically equivalent argument
  (p1?p2??pn??q)?F

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Ex 2.30
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Ex 2.31
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Ex 2.32

• (?p?q)?(q?r)??r??p?F

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Another inference rule

• ( p)?(q?r) ?( p?q) ?r
• ((p1?p2??pn)?(q?r)) ?(p1?p2??pn?q) ?r
• This result tells us that if we want to establish
  the validity of the first argument, we may be
  able to do so by establishing the validity of the
  corresponding argument.

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Ex 2.33
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2.4 The use of Quantifiers

• Definition 2.5. A declarative sentence is an open
  statement if
• (1) it contains one or more variables, and
• (2) it is not a statement, but
• (3) it becomes a statement when the variables in
  it are replaced by certain allowable choices.
• These allowable choices constitute what is called
  the universe or universe of discourse. The
  universe comprises the choices we wish to
  consider or allow for the variables in the open
  statement.

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definitions

• Existential quantifier (?) and universal
  quantifier (?) are used to quantify the open
  statements.
• In an open statement p(x) the variable x is
  called a free variable. In the statement ?x p(x)
  the variable x is called a bound variableit is
  bound by the existential quantifier ?. Similarly,
  in the statement ?x p(x) the variable x is bound
  by the universal quantifier ?.

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Ex 2.36

• p(x) x?0
• r(x) x2-3x-40
• q(x) x2 ?0
• s(x) x2-3gt0

• ?x p(x)?r(x)
• ?x p(x)?q(x)
• ?x q(x)?s(x)
• ?x r(x)?s(x)
• ?x r(x)?p(x)

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Ex 2.37

• p(x) x is a rational number, q(x) x is a real
  number
• ?x p(x)?q(x)
• e(t) triangle t is equilateral, a(t) triangle t
  has three angles of 60?
• ?t e(t)?a(t)
• ?x sin2xcos2x1
• ?m?n 41m2n2

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Ex 2.39

• For n1 to 20 do Ann?n-n
• ?n (An?0)
• ?n (An12An)
• ?n (1?n?19)?(AnltAn1)
• ?m ?n (m?n)?(Am?An)

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Definitions

• p(x) and q(x) are called logically equivalent,
  written as ?x p(x)?q(x), when p(a) ? q(a) is
  true for each replacement a from the universe.
  We say that p(x) logically implies q(x), written
  as ?x p(x)?q(x), when p(a)?q(a) is true for
  each replacement a from the universe.
• ?x p(x)?q(x) if and only if ?x p(x)?q(x) and
  ?x q(x)?p(x)
• ?x p?q contrapositive, ?x ?q?? p converse, ?x
  q? p inverse ?x ? p ?? q

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Examples

• Ex 2.40.
• s(x) x is a square e(x) x is a equilateral
• ?x s(x)?e(x) contrapositive, converse, inverse
• Ex 2.41.
• p(x) ?x?gt3 q(x) xgt3
• ?x p(x)?q(x) contrapositive, converse, inverse
• Ex 2.42. r(x) 2x15 s(x) x29
• ?x r(x)?s(x) ?x r(x) ? ?x s(x)
• but we have ?x r(x)?s(x) ? ?x r(x) ? ?x
  s(x)

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Table 2.22

• ?x r(x)?s(x) ? ?x r(x) ? ?x s(x)
• ?x r(x)? s(x) ? ?x r(x) ??x s(x)
• ?x r(x)?s(x) ? ?x r(x) ? ?x s(x)
• ?x r(x)?s(x) ??x r(x) ? ?x s(x)

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Ex 2.43

• ?x p(x)?(q(x)?r(x)) ? ?x (p(x)?q(x))?r(x)
• ?x p(x)?q(x)??x(?p(x)?q(x))
• ?x ??p(x)??x p(x)
• ?x ?p(x)?q(x)??x ?p(x)??q(x)
• ?x ?p(x)?q(x)??x ?p(x)??q(x)

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Rules for negation

• ??x p(x) ? ?x ? p(x)
• ??x p(x) ? ?x ? p(x)
• ??x ? p(x) ? ?x?? p(x) ? ?x p(x)
• ??x ? p(x) ? ?x?? p(x) ? ?x p(x)

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Ex 2.44

• p(x) x is odd, q(x) x2-1 is even
• ?x (p(x)?q(x)). If x is odd, x2-1 is even.
• ? ?x (p(x)?q(x))
• ??x ?(p(x)?q(x))
• ??x ?(?p(x)?q(x))
• ??x ??p(x)??q(x))
• ??x p(x)??q(x)
• There exists an integer x such that x is odd and
  x2-1 is odd.

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examples

• Ex 2.45
• ?x ?y p(x, y) ? ?y ?x p(x, y)
• Ex 2.46
• ?x ?y ?z p(x, y, z) can be written as ?x, y, z
  p(x, y, z)
• Ex 2.47
• ?x ?y p(x, y) ? ?y ?x p(x, y)

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Ex 2.48

• when a statement involves both existential and
  universal quantifiers, we must be careful about
  the order in which the quantifiers are written.
• p(x, y) xy17
• ?x ?y p(x, y) is different from ?y ?x p(x, y)

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Ex 2.49

• What is the negation of ?x?y(p(x,y)?q(x,y))
  ?r(x,y)

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Ex 2.50
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2.5 Quantifiers, definitions and the proofs of
theorems

• Ex 2.52.
• For all n in 2, 4, 6,, 26, we can write n as the
  sum of at most three perfect squares.
• Table 2.4 shows this by the method of exhaustion.
• The method is reasonable when we dealing with a
  fairly small universe.
• When the universe is very large, it is impossible
  to use the method of exhaustion.

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The rule of universal specification.

• If p(x) is an open statement for a given
  universe, and if ?x p(x) is true, then p(a) is
  true for each a in the universe.
• Note that this a is a specific but arbitrarily
  chosen member from the prescribed universe.

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Ex 2.53 (b)(c)
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The rule of universal generalization.

• If an open statement p(x) is proved to be true
  when x is replaced by a specific but arbitrarily
  chosen element c from our universe, then the
  universally quantified statement ?x p(x) is true.
• Furthermore, the rule extends beyond a single
  variable. That is, the same holds for ?x ?y p(x,
  y), ?x ?y ?z p(x, y, z) or more variables.

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Ex 2.54
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Ex 2.56
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Theorems Proving

• The rule of universal specification and the rule
  of universal generalization can be applied to
  prove theorems.
• Theorem 2.2. If k and l are both odd, then kl is
  even.
• Theorem 2.3. If k and l are both odd, then k?l is
  also odd.

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Theorem 2.4

• If m is an even integer, the m7 is odd.
• Theorem 2.4 uses three different ways to prove
  the theorem.
• (1) p?q, if m is even then m7 is even
• (2) ?q??p, if m7 is even then m is odd
• (3)p??q?F, if m and m7 are both even, then it is
  a contradiction.