LOGIC

1
LOGIC

• I refuse to join any club that would have me for
  a member.
• -Grouch Marx

2
WHAT IS LOGIC?

• "I know what you're thinking about," said
  Tweedledum "but it isn't so, no how."
  "Contrariwise," continued Tweedledee, "if it was
  so, it might be and if it were so, it would be
  but as it isn't, it ain't. That's logic."

3
Deductive vs. Inductive Reasoning
DAY ONE

• Deductive If Jimi Hendrix is a great guitarist,
  then I have all his albums.
• Jimi Hendrix is a great guitarist. I have all
  his albums.
• Inductive If Jimi Hendrix is a great guitarist,
  then it is likely I have all his albums.

4
Propositions

• Logic is a system based on propositions.
• A proposition is a statement that is either true
  or false (not both).
• A statement is a sentence that is true or false.
• 1 1 2 1 1
  3
• We say that the truth value of a proposition is
  either true (T) or false (F).

5
Elephants are bigger than mice.
Is this a statement?
yes
Is this a proposition?
yes
What is the truth value of the proposition?
true
6
520
Is this a statement?
yes
Is this a proposition?
yes
What is the truth value of the proposition?
false
7
Today is January 1 and 99
Is this a statement?
yes
Is this a proposition?
yes
What is the truth value of the proposition?
false
8
Please do not fall asleep.
Is this a statement?
no
Its a request.
Is this a proposition?
no
Only statements can be propositions.
9
Propositional Functions

• Propositional function (open sentence)
• statement involving one or more variables,
• x-3 5
• Let us call this propositional function P(x),
  where P is the predicate and x is the variable.

What is the truth value of P(2) ?
false
What is the truth value of P(8) ?
false
What is the truth value of P(9) ?
true
10
Propositional Functions

• Let us consider the propositional function Q(x,
  y, z) defined as
• x y z.
• Here, Q is the predicate and x, y, and z are the
  variables.

What is the truth value of Q(2, 3, 5) ?
true
What is the truth value of Q(0, 1, 2) ?
false
What is the truth value of Q(9, -9, 0) ?
true
11
Universal Quantification

• Let P(x) be a propositional function.
• Universally quantified sentence
• For all x in the universe of discourse P(x) is
  true.
• ? x ? S, P(x)
• x is an element in a set
• S is the set of elements
• ? means belongs to or is a member of

? Means for ALL
12
Universal Quantification

• Examples
• 1. ? x ?1,2,3,4, 2 x is even
• 2. ? x ?s, x2 0

13
Existential Quantification

• Existentially quantified sentence
• There exists an x in the universe of discourse
  for which P(x) is true.
• Using the existential quantifier ?
• ?x ?S, P(x) There is an x such that P(x).
• There is at least one x such that P(x).

? Means there exists
14
Existential Quantification

• Examples
• 3. ?x ? 2,6,8,10,, x 5 11
• 4. ?x ?,-1,0,1,2,, x2 2

15
Symbols represent facts
DAY TWO
Puff Daddy is a rapper is a fact that could be
represented by the symbol P
16
Logical connectives
? and conjunction ? or disjunction ? not negation
? implication if, then conditional ?
equivalence if and only if biconditional
17
Propositional Logic

• P represents the fact Puff Daddy is a rapper
• Q represents the fact Puff Daddy is rich
• P ? Q Puff Daddy is a rapper and Puff Daddy is
  rich
• P ? Q Puff Daddy is a rapper or Puff Daddy is
  rich
• ? Q Puff Daddy is not rich
• Q ? P Puff Daddy is rich therefore Puff Daddy
  is a rapper
• P ? Q Puff Daddy is a rapper therefore Puff
  Daddy is rich and Puff Daddy is rich therefore
  Puff Daddy is a rapper

18
Propositional Logic Semantics

• Logic is made up sentences
• P might be a sentence
• P ? Q is also a sentence
• If we know the truth values of P and Q, we can
  work out the truth value of the sentence
• If P and Q are both true then P ? Q is true,
  otherwise it is false
• Can use truth tables to ascertain the truth of a
  sentence

19
Negation truth table
20
Conjunction truth table
21
Disjunction truth table
22
Conditional truth table
23
Negation of a conditional

• p ? q
• Negation p ? ? q
• or p ? ? q

24
Example
25
Example
26
Logical Equivalence

• Two statements are logically equivalent if their
  truth tables have the same values for all
  substitutions

27
(No Transcript)
28
A tautology is a statement that is always true.
DAY THREE

• Examples
• R?(? R)
• ? (P?Q)?(? P)?(? Q)
• If S?T is a tautology, we write S?T.
• If S?T is a tautology, we write S?T.

29
A contradiction is a statement that is
alwaysfalse

• Examples
• R?(? R)
• ? (? (P?Q)?(? P)?(? Q))
• The negation of any tautology is a contradiction,
  and the negation of any contradiction is a
  tautology.

30
Converse

• If p then q.
• If q then p.

31
Inverse

• Negate the hypothesis and conclusion
• If p then q.
• If not p then not q.

32
Contrapositive

• Reverse and negate both.
• If p then q.
• If not q then not p.

33
A conditional and its contrapositive are
logically equivalent.
34
Modus Ponens
DAY FOUR

• Law of Detachment
• If Jennifer married Marc Anthony, then Ben
  Affleck is lucky. Jennifer married Marc
  Anthony. 
• Therefore, Ben Affleck is lucky.

method of affirming
35
Modus tollens

• Ben Affleck is not lucky. If J-Lo married Marc
  Anthony, then Ben Affleck is lucky.
• Therefore, J-Lo did not marry Marc Anthony.

If q is false, and if p implies q, (p ?   q),
then p is also false
Method of denying
36
If Paris Hilton is rich, then she will travel the
world in an RV.

37
De Morgans Law

• For all statements p and q
• ? (p ? q) ?p ? ?q
• And
• ? (p ? q) ?p ? ?q

38
Example

• I want to watch the Matrix or Harry Potter on dvd.

39
Fallacy

• An error in reasoning.

• No one has ever proven that ghosts don't exist.
  Therefore, they obviously do.

• Our society is filled with violence and there is
  a lot of violence on TV. It is obvious that the
  violence in society is caused by people watching
  television.