Propositional Logic

1
Propositional Logic

• Section 1.1

2
Logical Form

• We use informal logic everyday to express our
  reasoning and conditions for actions using
  connective words like
• Or, and, but, if then, neither nor, etc
• Problems
• We usually dont think through the complications
  of these statements and sometimes other people
  misinterpret our statements
• Natural language doesnt work for technical
  discussions about complex systems where we must
  be precise and maintain a chain of reasoning
• Language sometimes plays tricks on us
• Hard to find simpler ways of stating the same
  thing

3
Logic

• Logic provides a way to deduce the vast amounts
  of knowledge in a discipline from a small number
  of explicitly stated facts
• To make this deduction, we treat logic as a
  language. This allows
• Expression of knowledge concisely and precisely
• A way to reason about the consequences of that
  knowledge rigorously
• Logic addresses the correctness of the form of
  any argument, not its content

4
Logical Form

• To treat logic as a language, we need to create a
  syntax (fundamental concepts)
• Statements (Proposition)
• Declarative sentences that are true or false, but
  not both
• Connectives
• Negation
• Conjunction
• Disjunction
• Implication (Conditional)
• Equivalence
• The truth or falsity of a statement is called the
  truth value and is characterized by assigning the
  statement a value of T or F
• In logic we are particularly interested in
  statements and combinations of statements
  (compound statements) that are either True or
  False

5
Proposition

• A proposition is the smallest building block for
  constructing complex statements
• It is a statement that is either True or False,
  but not both
• Atlanta was the site of the 1996 Summer Olympic
  games
• True - proposition
• 11 2
• True proposition
• 31 5
• False - proposition
• What will my CS1050 grade be?
• That depends not a proposition
• X 10 12
• Alone not a proposition
• For some value of x proposition
• He is ill
• Depends who He is ambiguous not a proposition

6
Propositions (continued)

• Ambiguous words like I, He, Here, Good
  allow the truth of the sentence to vary with the
  speaker, scene or context and must be replaced
  before a declarative sentence can be accepted as
  a statement
• Symbols are used to represent a proposition
  (e.g., letters of the alphabet)
• P John is tall
• Q Susan has brown hair
• R Mammals are warm-blooded
• S 11 2
• Propositions and compound propositions are the
  stepping stones toward the goal of arguments,
  which are sequences of statements that can be
  checked for goodness (next chapter)

7
Connectives

• Connectives allows us to build complicated
  logical expressions (or compound statements) out
  of simple propositions

Order of operations , then ? and ?, ? , and
last is ?
8
Truth Tables

• Since compound propositions can be complex, we
  need rules to work out the truth value
• We do this by knowing the truth value of the
  simpler parts and combining them to find the
  truth value of the compound proposition
• One way to do this is with Truth Tables
• Every connective is defined by one and only one
  Truth Table definition
• Evaluate proposition for every combination of
  truth values

9
Negation of p

• Let p be a proposition. The statement It is not
  the case that p is also a proposition, called
  the negation of p or p (read not p)

10
Conjunction of p and q

• Let p and q be propositions The proposition p
  and q is denoted by p?q
• True when both p and q are true
• False otherwise

Note In a proposition, the phrases both,
but, although, and neither nor mean the
same as and
11
Disjunction of p and q

• Let p and q be propositions. The proposition p
  or q is denoted by p?q
• False when p and q are both false
• True otherwise

Note In a proposition, the phrases either and
unless mean the same as or
12
Exclusive OR of p and q

• Let p and q be propositions.
• The exclusive or of p and q is denoted by p?q
• True when exactly one of p and q is true (but not
  both)
• False otherwise

13
Example Compound Statements

• Define the following propositions
• For a particular integer x,
• p 2 x
• q 2 lt x
• r x lt 6

• Write the compound statement
• 2 ? x p ? q
• 2 lt x lt 6 q ? r
• 2 ? x lt 6 (p ? q) ? r
• 2 ? x ?p

14
Example Truth Tables
(p ? q) (p q)
p q p ? q p q (p q) (p ?
q) (p q) T T T T
F F T F
T F T
T F T T
F T T F
F F F T
F
15
Example Writing Statements and Truth Tables
John is healthy and wealthy and not wise H W
S

• H John is healthy
• W John is wealthy
• S John is wise

H W S H W S H W S T T T
T F F T T F
T T T T F
T F F
F T F F F T
F F T T F F
F F T F F T
F F F T F F
F F F F F
T F
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Logical Equivalence

• Important technique in proofs is to replace a
  statement with another statement that is
  logically equivalent
• Dogs bark and cats meow is logically equivalent
  to Cats meow and dogs bark
• P Dogs bark
• Q Cats Meow
• P Q ? Q P
• Two statements are logically equivalent iff they
  have identical truth values for each possible
  substitution of their statement variables

17
Example
Same truth values, so p?q and q?p are logically
equivalent
18
DeMorgans Law

• The negation of an AND statement is logically
  equivalent to the OR statement in which each
  component is negated
• ?(p ? q) ? ?p ? ?q
• The negation of an OR statement is logically
  equivalent to the AND statement in which each
  component is negated
• ?(p ? q) ? ?p ? ?q
• Proof

19
Tautology and Contradiction

• Tautology compound proposition that is always
  true regardless of the truth values of the
  propositions in it.
• Contradiction Compound proposition that is
  always false regardless of the truth values of
  the propositions in it.

20
List of Logical Equivalences
p?T ? p p?F ? p Identity Laws p?T ? T
p?F ? F Domination Laws p?p ? p p?p ? p
Idempotent Laws ?(?p) ? p Double Negation
Law p?q ? q?p p?q ? q?p Commutative
Laws (p?q)?r?p?(q?r) (p?q)?r?p?(q?r)
Associative Laws
21
List of Equivalences (continued)
p?(q?r) ? (p?q)?(p?r) Distribution Laws p?(q?r)
? (p?q)?(p?r) ?(p?q)?(?p ? ?q) De Morgans
Laws ?(p?q)?(?p ? ?q) p ? ?p ? T Misc.,
Theorem 1.1.1 p ? ?p ? F
(page 14) (p?q) ? (?p ? q)
22
Example Distribution
Prove that p ? (q ? r) ? (p ? q) ? (p ? r)
Same Truth Values ?Logically Equivalent
23
Proving Logical Equivalence

• In addition to Truth Tables, we can also use
  Logical Laws and Logical Equivalences
• Prove (p q) v q ? p v q
• (p q) v q Logical Equivalence
• q v (p q) Commutative
• (q v p) (q v q) Distributive
• (q v p) t Negation
• (q v p) Identity
• p v q Commutative