Chapter 2 Fundamentals of Logic

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Chapter 2 Fundamentals of Logic

• Yung-Ling Lai

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Propositions
A proposition??or statement ?? is a declarative
sentence that may be assigned a true or false
value. This value is the truth value of the
proposition.
Propositions 224, The earth is flat, Not
Propositions Is 224?, Dont lie! Also
Propositions Green is a beautiful
color,Jesus is Gods son.
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Example

• We use the lowercase letters of the alphabet
  (such as p, q, and r) to represent these
  statements.
• p Discrete math is a required course for
  freshmen.
• q Margaret Mitchell wrote Gone with the Wind.
• r 2 3 5.

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Example

• We do not regard sentences such as the
  exclamation(??)
• What a beautiful evening!
• or the command
• Get up and do your exercises. as statements.
• The preceding statements represented by p, q, and
  r are considered to be primitive statements ????,
  for there is really no way to break them down
  into anything simpler.

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True or False, Thats All
We are not going to worry how to define if a
proposition is true or not. The only thing
that matters is the fact that a proposition is
True or False. Sometimes we allow ourselves
to use 0 for False and 1 for True. Things become
interesting if we combine propositions
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• New statements can be obtained from primitive
  statements in two ways.
• Transform a given statement p into the statement
  ?p, which denotes its negation ??and is read
  Not p. ?p (Negation statements)
• Combine two or more statements into a compound
  statement, using logical connectives. (Compound
  statements????)

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Compound Propositions

• If p and q are propositions, then so isthe
  conjunction p?q (read p and q) and the
  disjunction p?q (read p or q).
• The negation of p is denoted by ?p (or also p).
• Given the truth values of p and q, we can
  determine the truth values of these other
  propositions.

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Implication or Conditional

• Given the propositions p and q, we can make
  theimplication or conditional p?q (If p, then
  q, or p implies q). Do not use p ? q.
• Note 3 out of 4 are Trueif p is false, p?q is
  always true
• p?q is equivalent with (?p ? q)
• Do not think in terms of cause and effectIf
  224 then fish live in water is True.

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Biconditionals

• Given the propositions p and q, we can make
  thebiconditional p?q (p if and only if
  q)????.Equivalent with (p?q)?(q?p)

Note that it is fine to talk aboutp?p. This
is just False. Do not write p ? q when you
mean p ? q
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Truth Tables
Using the short hand T and F for True and False,
we have the following truth tables
Note the difference between or and exclusive
or (?).
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Truth Tables in Action
Truth tables summarize what should be obviousIf
John is 6 feet is true and 112 is true,
then John is 6 feet and 112 is also true.
Using parenthesis, we can make longer and more
complicated propositions, like p?(q??(p))
whichcould stand for John is 6 feet, or 112
and Johnis not 6 feet. Observe that the last
proposition is equivalent withJohn is 6 feet or
112. But how to prove this?
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Proving with Truth Tables

• To find the Truth Table of a compound statement,
  build it up from its more elementary statements

Example 2.4 The truth table of q ?
(?r?p) Write down all8 combinationsfor the
truth values of p,q,r.
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Example 2.5
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Equivalence

• Two propositions are (logically) equivalent if
  and only if for each case their truth values are
  the same (?).

Examples p ? (p?p) 112 ? 112 or 112
??p ? p He did not not do it ? He did
it ?(p?q) ? ?p??q If p then not p ? not p
Proving that two propositions are equivalent
canbe done by comparing the two truth tables.
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Example of Equivalence

• How to prove q ? (?r?p) ? (r?q)?(p?q)?
• Earlier we saw

while for the other proposition
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Def. 2.1 Tautology / Contradiction

• Tautology A proposition that is equivalent with
  True.Also called logically true. Example
  p??p.Contradiction A proposition equivalent
  with False. Also called logically false.
  Example p??p.
• Pay attention to the phrase logically2
  31 is not a tautology, but 21 or 2?1
  is113 is not a contradiction, but 11 and
  1?1 is.As with equivalence look at the truth
  tables.

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Writing Down Truth Tables

• True, False is preferred (less ambiguous) than
  1, 0
• List the elementary truth values in a consistent
  way(from FF to TT or from TT to FF).
• They get long with n variables, the length is
  2n.
• In the end we want to be able to reason about
  logic without having to write down such tables.

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Def.2.2 Logical Equivalence

• Two propositions are (logically) equivalent if
  and only if for each case their truth values are
  the same (?).

Examples p ? (p?p) 112 ? 112 or 112
??p ? p He did not not do it ? He did
it ?(p?q) ? ?p??q If p then not p ? not p
Proving that two propositions are equivalent
canbe done by comparing their two truth tables.
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Biconditional vs. Equivalence
Dont confuse the equivalence ? with the
biconditional ? (only the biconditional has a
truth table). For examplep ? p is a
proposition/tautology, a statement within logic,
p ? p is mathematically correct, about
logic. p ? ?p is a contradiction (False), p ? ?p
is incorrect Hence p?p ? ?(p??p), and so on.
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Proving Things in Logic

• The standard approach is to use truth tables.
• If we deal with n simple propositions p1,,pn,
  our truth table will have size at least 2n.
• This becomes a substantial disadvantage if n is
  big.
• Sometimes there is a much more efficient way to
  prove equivalences there is more to
  propositional logic than truth tables.
• First, look at some very simple equivalences

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The Laws of Logic 1

• Double negation law ??p ? p
• De Morgans laws ?(p?q) ? ?p??q and
  ?(p?q) ? ?p??q
• Commutative laws p?q ? q?p and p?q ? q?p
• Associative laws p?(q?r) ? (p?q)?r and
  p?(q?r) ? (p?q)?r
• Distributive laws p?(q?r) ? (p?q)?(p?r)
  and p?(q?r) ? (p?q)?(p?r)

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The Laws of Logic 2

• Idempotent laws p?p ? p and p?p ? p
• Identity laws p?False ? p and p?True ? p
• Inverse laws p??p ? True and p??p ? False
• Domination laws p?True ? True and p?False ?
  False
• Absorption laws p?(p?q) ? p and p?(p?q) ? p

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Definition 2.3

• Let s be a statement. If s contains no logical
  connectives other than ? and ?, then the dual of
  s, denoted sd , is the statement obtained from s
  by replacing each occurrence of ? and ? by ? and
  ?, respectively, and each occurrence of T0 and F0
  by F0 and T0, respectively.

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• Given the primitive statements p, q, r and the
  compound statement
• s (p ??q) ? (r ? T0),
• we find that the dual of s is
• sd (p ??q) ? (r ? F0).
• (Note that ?q is unchanged as we go from s to sd
  .)

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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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Substitution Rules

• 1) Rule of substitution If q is a (compound)
  statement and an equivalence f(p) ? ?(p) holds
  with p elementary, then the equivalence f(q) ?
  ?(q) holds as well.
• 2) Rule of substitution If p and q are
  (compound) statement and the equivalence p ? q
  holds, then if we replace some p by q to get ?
  from f,then f ? ? holds as well.

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Example 2.10

• From the first of DeMorgans Laws
• ?(p ? q)?(?p ??q)
• from the first substitution rule
• ?(r ? s) ? q ? ?(r ? s)??q
• replace each occurrence of q by t ?u
• ?(r?s)?(t ?u) ? ?(r?s)??(t ?u)

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Combining Rules

• Additional equivalence rule for implication p?q
  ? ?p?q.
• Using the previous (simple) equivalences in
  combination with the rules of substitution allows
  us to prove all true equivalences in
  propositional logic.

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Proving Equivalences

• Here is how to prove our q ? (?r?p) ?
  (r?q)?(p?q)
• q ? (?r?p) ? q?(??r?p) p?q ? ?p?q rule
• ? q?(r?p) double negation
• ? (q?r)?(q?p) distributive law
• ? (r?q)?(q?p) commutative law
• ? (r?q)?(p?q) commutative law

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Associativity

• By repeatedly using the Associative laws we see
  that the parentheses in ((p1 ? p2) ? ? pn) do
  not matter. Same for ((p1 ? p2) ? ? pn).
• When deriving equivalences you are allowed to
  apply the rule multiple associative laws.
• Even more informal we can drop the parentheses
  altogether.

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Commutativity

• By repeatedly using the Commutative laws we see
  that the order of the pj in ((p3 ? p7) ? ? p2
  do not matter.
• Same for ((p3 ? p7) ? ? p2.
• When deriving equivalences you are allowed to
  apply the rule multiple commutative laws.

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Rules of Inference

• In real life when proving mathematical statements
  we do not establish an equivalence but a
  consequence.
• Typically, an argument uses hypotheses or
  premises to reach a conclusion.
• How to do this is described by the rules of
  inference.

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Rules of Inference

• Theorems are true/correct mathematical
  statements.
• Axioms are self-evident theorems.
• Using the rules of inference we can make a
  (valid) argument to derive other theorems from
  the axioms.
• An argument is valid if and only if the validity
  of the hypotheses implies the validity of the
  conclusion.

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Shape of an Argument
premises or hypotheses
thereforesymbol
conclusion
The therefore symbol ? is a bit old fashioned.
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Some Small Arguments
inversefallacy
ValidArguments
InvalidArguments
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About Arguments

• Def. 2.4 For propositions p,q, if p?q is a
  tautology, then p logically implies q. This is
  denoted by p?q.
• Arguments are correct or incorrect / valid or
  invalid a conditional is True or False.
• Arguments are to conditionals (?), what
  Equivalences (?) are to biconditionals (?)

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Checking Arguments

• An argument (H1??Hn) ? C is valid if for all
  cases where the hypotheses Hj are True, the
  Conclusion C is True as well.
• We can check arguments with the help of truth
  tables.But just as with equivalences there are
  other ways of proving the validity of an argument.

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Rules of Inference I
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Rules of Inference II
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Rules of Inference III
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Proving Validity of Arguments

• Using basic inference steps and equivalence rules
  one can prove the validity of arguments.
• Example

Yes, according to truth tables.
valid?
And because p??p ? ?p??p ? ?pwe have the
validityproven a second time.
But also,
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Longer Arguments

• Example 2.31 ((?p??q)?(r?s)) ? (r?t) ? (?t) ? p.
• r?t Premise
• ?t Premise
• ?r Steps 1,2 and Modus Tollens
• ?r??s Step 3 and Disjunctive Amplification
• ?(r?s) Step 4 and DeMorgans Law
• (?p??q)?(r?s) Premise
• ?(?p??q) Steps 5,6 and Modus Tollens
• ??p???q Step 7 and DeMorgans Law
• p?q Step 8 and Double Negation
• p Step 9 and Conjunctive Simplification

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General Remarks

• Propositions that only use ?,?,?,(,) are the
  objects in Boolean algebra (without the
  implication ?). Note the Laws of Logic do not
  use ?.
• This is what you typically have in IF THEN
  construction.
• The implication becomes useful when you want to
  connect Boolean algebra with the rules of
  inference.
• False ? p ? True follows from proof by
  contradiction.It holds that (p??p) ? p hence
  (p??p) ? p ? True.Take the two cases p ? True
  and p ? False.

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Terminology 4 Conditionals

• For the propositions p and q and the conditional
  p ? q,we have the three other conditionals
• converse q ? p
• inverse ?p ? ?q
• contrapositive ?q ? ?p

Only one of these is equivalent with p ? q
the contrapositive, hence (p?q) ? (?q??p). We
also have for the other two (q?p) ? (?p??q) but
not (p?q) ? (q?p) or (p?q) ? (?p??q)
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Proving Techniques
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Arguments in Real Life

• Theorem ?2 is not a rational number that is
  there are no integers a,b such that (a/b)22.
• Assume that there are a,b with (a/b)22, with
  gcd(a,b)1.
• It follows that a22b2, and hence that a has to
  be even.With a2c we see that 4c22b2, hence
  2c2b2.
• This shows that b is even as well, contradicting
  thereduction assumptions about a and b.
• Conclusion There are no such a,b ?2 is not
  rational.

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When Writing Real Proofs

• When writing (real) proofs, you should make it
  clear to yourself and your reader why your
  statements are true
• by hypothesis? (a/b)22
• by definition? a is even, hence a2c
• an earlier proposition? a22b2, hence a is
  even
• following directly? 4c22b2, hence 2c2b2
• logically equivalent?
• Dont confuse A is (logically) equivalent with
  B with B follows (logically) from A.Dont
  abuse the word logic/logically.

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Predicate Calculus

• Statements like 3x5, anbncn are not
  propositions because they have free variables
  they are open statements or predicates(??).With
  n free variables, it is an n-place predicate.
• If we specify the variables, they become
  propositions,like 325, 232333 (True and
  False respectively)x2 satisfies the predicate
  P(x) 3x5.
• Central question for predicates Is the predicate
  satisfied for all, some, or none of the possible
  values?

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Definition 2.5

• A declarative sentence is an open statement if
• it contains one or more variables, and
• it is not a statement, but
• it becomes a statement when the variables in it
  are replaced by certain allowable choices.

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Example

• The number x 2 is an even integer is an open
  statement and is denoted by P(x). The allowable
  choices for x is called the universe (set) for
  P(x).
• If x 3, P(3) is a false statement.
• Q(x,y) The numbers y 2, x y, and x 2y
  are even integers., then Q(4,2) is true.
• ?for some x, P(x) (TRUE), for some x, y, Q(x,y)
  (TRUE), or for all x, P(x) (FALSE).

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Quantifiers

• Given a predicate P(x)The universal quantifier
  ? describes for all x, hence ?x P(x) stands
  for for all x, P(x) holds
• The existential quantifier ? describes there is
  an x ?x P(x) stands for there is an x such
  that P(x) holds
• It is crucial to understand what the universe or
  domain (of discourse) is.
• Compare ?x x25, x?Z or x?R?

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Rules of Predicate Logic

• Relation between universal and existential
  quantifiers ??x P(x) ? ?x ?P(x) hence ??x
  P(x) ? ?x ?P(x)
• Assuming a non-zero domain for x,?x P(x)
  implies ?x P(x)
• Dont confuse ??x P(x) with ?x ?P(x) or
  ??x P(x) with ?x ?P(x)

the power ofcounterexamples
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Table 2.21 summarize and extend some results for
quantifiers.
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Commuting Quantifiers

• Identical quantifiers commute ?x?y P(x,y) ?
  ?y?x P(x,y) and ?x?y P(x,y) ? ?y?x P(x,y)
• But non-identical ones do not, see ?x?y xy
  versus ?y?x xy

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Compound Predicates

• Table 2.22 ?x P(x) ? Q(x) ? ?x P(x) ? ?x
  Q(x) ?x P(x) ? Q(x) ? ?x P(x) ? ?x Q(x)
• ?x P(x) ? Q(x) ? ?x P(x) ? ?x Q(x) ?x P(x) ?
  ?x Q(x) ? ?x P(x) ? Q(x)
• Note Not ?.

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Table 2.23 Rules for negating statements with
one quantifier.
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What is the negation of
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The Rule of Universal Specification

• If an open statement becomes true for all
  replacements by the members in a given universe,
  then that open statement is true for each
  specific individual member in that universe.
• (A bit more symbolically 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.)

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EXAMPLE 2.53 m(x) x is a mathematics professor
c(x) x has studied calculus.

• Now consider the following argument.
• All mathematics professors have studied calculus.
• Leona is a mathematics professor.
• Therefore Leona has studied calculus.
• Steps Reasons
• 1)?x m(x)?c(x) Premise
• 2) m(l) Premise
• 3) m(l)?c(l) Step (1) and the Rule of Universal
  Specification
• 4) ? c(l) Steps (2) and (3) and the Rule of
  Detachment

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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 any arbitrarily chosen
  element c from our universe, then the universally
  quantified statement ?x p(x) is true.
• If an open statement q(x,y) that is proved to be
  true when x and y are replaced by arbitrarily
  chosen elements from the same universe, or their
  own respective universes, then the universally
  quantified statement ?x ?y q(x, y) or, ?x, y
  q(x, y) is true. Similar results hold for the
  cases of three or more variables.

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• Example2.54Let p(x), q(x), and r(x) be open
  statements that are defined for a given
  universe.We show that the argument
• ?x p(x)?q(x)
• ?x q(x)?r(x)
• __________________
• ? ?x p(x)?r(x)
• is valid by considering the following.
• Steps Reasons
• x (p(x)?q(x) Premise
• p(c)?q(c) Step (1) and the Rule of Universal
  Specification
• x q(x)?r(x) Premise
• q(c)?r(c) Step (3) and the Rule of Universal
  Specification
• p(c)?r(c) Steps (2) and (4) and the Law of the
  Syllogism
• ? x p(x)?r(x) Step (5) and the Rule of
  Universal Generalization

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Theorem 2.2 For all integers k and l, if k, l
are both odd, then k l is even.

• Since k and l are odd, we may write k 2a 1 and
  l 2b 1, for some integers a, b. This is due
  to Definition 2.8.
• Then
• k l (2a 1) (2b 1) 2(a b 1),
• by virtue of the Commutative and Associative
  Laws of Addition and the Distributive Law of
  Multiplication over Additionall of which hold
  for integers.
• Since a, b are integers, a b 1 c is an
  integer with k l 2c, it follows from
  Definition 2.8 that k l is even.

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THEOREM 2.3For all integers k and l, if k and l
are both odd, then their product kl is also odd.

• Proof Since k and l are both odd, we may write k
  2a 1 and l 2b 1, for some integers a and
  bbecause of Definition 2.8. Then the product
• kl (2a 1)(2b 1) 4ab 2a 2b 1
• 2(2ab a b) 1,
• where 2ab a b is an integer.
• Therefore, by Definition 2.8 once again, it
  follows that kl is odd.

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THEOREM 2.5 For all positive real numbers x and
y, if the product xy exceeds 25, then x gt5 or y
gt5.

• Proof Consider the negation of the
  conclusionthat is, suppose that 0 lt x 5 and 0
  lty 5. Under these circumstances we find that 0
  0 0 lt x y 5 5 25, so the product xy does
  not exceed 25.
• (This indirect method of proof now establishes
  the given statement, since we know that an
  implication is logically equivalent to its
  contrapositive.)