1 Propositions A proposition is a declarative sentence that is either true or false Examples of propositions The Moon is made of green cheese Trenton is the capital of New Jersey Toronto is the capital of Canada ID: 260809
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Slide1
Propositional Logic
1Slide2
Propositions
A
proposition
is a declarative sentence that is either true or false.
Examples of propositions:The Moon is made of green cheese.Trenton is the capital of New Jersey.Toronto is the capital of Canada.1 + 0 = 10 + 0 = 2Examples that are not propositions.Sit down!What time is it?x + 1 = 2x + y = z
2Slide3
Propositional Logic (or Calculus)
Constructing Propositions
Propositional
Variables
: p, q, r, s, …The proposition that is always true is denoted by T and the proposition that is always false is denoted by F.Compound Propositions: constructed from other propositions using logical connectivesNegation ¬Conjunction ∧Disjunction ∨Implication →Biconditional ↔
3Slide4
Compound Propositions: Negation
The
negation
of a proposition
p is denoted by ¬p and has this truth table:Example: If p denotes “The earth is round” then ¬p denotes “It is not the case that the earth is round,” or more simply “The earth is not round.” p
¬
p
T
F
F
T
4Slide5
Conjunction
The
conjunction
of propositions
p and q is denoted by p ∧ q and has this truth table:Example: If p denotes “I am at home” and q denotes “It is raining” then p ∧q denotes “I am at home and it is raining.”
p
q
p
∧
q
T
T
T
T
F
FFTFFFF
5Slide6
Disjunction
The
disjunction
of propositions
p and q is denoted by p ∨q and has this truth table:Example: If p denotes “I am at home” and q denotes “It is raining” then p ∨q denotes “I am at home or it is raining.”
p
q
p
∨
q
T
T
T
T
FTFTTFF
F
6Slide7
The Connective Or in English
In English “or” has two distinct meanings.
Inclusive
Or: For p ∨q to be T, either p or q or both must be T Example: “CS202 or Math120 may be taken as a prerequisite.”
Meaning: take either one or both
Exclusive
Or
(
Xor
). In p⊕q ,
either
p
or q but not both must be TExample: “Soup or salad comes with this entrée.” Meaning: do not expect to get both soup and salad
p
q
p
⊕
q
T
T
FTFTFTTFFF
7Slide8
Implication
If
p
and
q are propositions, then p →q is a conditional statement or implication, which is read as “if p, then q ”p is the hypothesis (antecedent or premise) and q is the
conclusion
(or
consequence
).
Example
: If
p
denotes “It
is
raining” and
q denotes “The streets are wet” then p →q denotes “If it is raining then streets are wet.”p
q
p
→
q
T
T
TTFFFTTFFT8Slide9
Understanding Implication
In
p
→
q there does not need to be any connection between p and q. The meaning of p →q depends only on the truth values of p and q. Examples of
valid
but
counterintuitive
implications:
“If the moon is made of green cheese, then you have more money than Bill Gates” --
True
“If Juan has a smartphone, then 2 + 3 = 6” -- False if Juan does have a smartphone, True if he does NOT
9Slide10
Understanding Implication
View logical conditional as an
obligation
or
contract:“If I am elected, then I will lower taxes”“If you get 100% on the final, then you will earn an A”If the politician is elected and does not lower taxes, then he or she has broken the campaign pledge. Similarly for the professor. This corresponds to the case where p is T and q is F. 10Slide11
Different Ways of Expressing
p
→
q
if p, then qif p, qp implies q q if p
q
when
p
q
whenever pNote
:
(
p
only if q) ≡ (q if p) ≢ (q only if p)It is raining
→
streets
are wet:
T
It is raining
only if
streets are wet: T
Streets are wet if it is raining: TStreets are wet only if it’s raining: F11
q
follows from
p
q
unless
¬p
p
only if
q
p
is sufficient for
q
q
is necessary for
pSlide12
Sufficient versus Necessary
Example 1:
p
→
q : get 100% on final → earn A in classp is sufficient for q: getting 100% on final is sufficient for earning A
q
is
necessary
for
p
:
earning A is necessary for getting 100% on final
Counterintuitive
in English:
“Necessary” suggests a precondition
Example is sequentialExample 2:Nobel laureate → is intelligentBeing intelligent is necessary for being Nobel laureateBeing a Nobel laureate is sufficient for being intelligent (better if expressed as: implies)
12Slide13
Converse, Inverse, and Contrapositive
From
p
→
q we can form new conditional statements .q →p is the converse of p →q ¬ p → ¬ q is the inverse of p →
q
¬
q
→ ¬
p
is the
contrapositive of p →
q
How are these statements related
to the original?
How are they related to each other?Are any of them equivalent?13Slide14
Converse, Inverse, and Contrapositive
Example
:
it’s raining
→ streets are wetconverse: streets are wet → it’s raininginverse: it’s not raining → streets are not wetcontrapositive: streets are not wet → it’s not rainingonly 3 is equivalent to original statement1
and
2
are not equivalent to original statement: streets could be wet for other reasons
1
and
2 are equivalent to
each other
14Slide15
Biconditional
If
p
and
q are propositions, then we can form the biconditional proposition p ↔q, read as “p if and only if q ”Example: If p denotes “You can take a flight” and q denotes “You buy a ticket” then
p
↔
q
denotes “You can take a
flight if and only if you buy a ticket”
True only if you do both or neitherDoing only one or the other makes the proposition false
p
q
p
↔q TTT
T
F
F
F
T
F
F
FT15Slide16
Expressing the Biconditional
Alternative ways to say “
p
if and only if
q”:p is necessary and sufficient for qif p then q, and converselyp iff q16Slide17
Compound Propositions
conjunction, disjunction, negation, conditionals, and
biconditionals
can be combined into arbitrarily complex
compound propositionAny proposition can become a term inside another propositionPropositions can be nested arbitrarilyExample: p, q, r, t are simple propositionsp ∨q , r ∧t , r →t are compound propositions using logical connectives(p ∨
q)
∧
t
and
(p
∨
q)
→
t are compound propositions formed by nesting17Slide18
Precedence of Logical Operators
Operator
Precedence
1 23 45
18
With multiple operators we need to know the
order
To reduce number of parentheses use precedence rules
Example
:
p
∨
q
→ ¬
r
is equivalent to
(
p
∨
q
) → ¬ r
If the intended meaning is
p
∨(
q
→ ¬
r
)
then parentheses must be usedSlide19
Truth Tables of Compound Propositions
Construction of a truth table:
Rows
One for every possible combination of values of all propositional variables
ColumnsOne for the compound proposition (usually at far right)One for each expression in the compound proposition as it is built up by nesting19Slide20
Example Truth Table
Construct a truth table for
p
q
rp qrp q →
r
T
T
T
T
F
F
T
T
F
TTTT FTTFFT
F
F
T
T
T
F
T
TTFFFTFTTTFFTFFTFFF
F
T
T
20