Download p q

Propositional Logic
Proposition
A proposition is a statement that is either
true or false, but not both.
• Atlanta was the site of the 1996
Summer Olympic games.
• 1+1 = 2
• 3+1 = 5
• Who will win the elections in Israel?
Definition 1. 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”)
p = The sky is blue.
p = It is not the case that
the sky is blue.
p = The sky is not blue.
Table 1.
The Truth Table for the
Negation of a Proposition
p
¬p
T
F
F
T
Definition 2. Conjunction of p
and q
Let p and q be
propositions. The
proposition “p and q,”
denoted by pq is true
when both p and q are
true and is false
otherwise. This is
called the conjunction
of p and q.
Table 2. The Truth Table for
the Conjunction of two
propositions
p
q
pq
T
T
F
F
T
F
T
F
T
F
F
F
Definition 3. Disjunction of p
and q
Table 3. The Truth Table for
the Disjunction of two
propositions
p
q
pq
T
T
F
F
T
F
T
F
T
T
T
F
Let p and q be
propositions. The
proposition “p or q,”
denoted by pq, is the
proposition that is false
when p and q are both
false and true otherwise.
Definition 5. Implication pq
Let p and q be propositions.
The implication pq is the
proposition that is false when
p is true and q is false, and
true otherwise. In this
implication p is called the
hypothesis (or antecedent or
premise) and q is called the
conclusion (or
consequence).
Table 5. The Truth Table for
the Implication of pq.
p
q
pq
T
T
F
F
T
F
T
F
T
F
T
T
Implications
How can both p and q be false, and pq be true?
•Think of p as a “contract” and q as its “obligation” that is
only carried out if the contract is valid.
•Example: “If you make more than $25,000, then you must
file a tax return.” This says nothing about someone who
makes less than $25,000. So the implication is true no
matter what someone making less than $25,000 does.
•Another example:
p: Bill Gates is poor.
q: Pigs can fly.
pq is always true because Bill Gates is not poor. Another
way of saying the implication is
“Pigs can fly whenever Bill Gates is poor” which is true
since neither p nor q is true.
Definition 6. Biconditional
Table 6. The Truth Table for
the biconditional pq.
p
q
pq
T
T
F
F
T
F
T
F
T
F
F
T
Let p and q be
propositions. The
biconditional pq is the
proposition that is true
when p and q have the
same truth values and is
false otherwise. “p if and
only if q, p is necessary
and sufficient for q”
Practice
p: You learn the simple things well.
q: The difficult things become easy.
• You do not learn the
simple things well. p
• If you learn the simple
things well then the
difficult things become
easy.
pq
• If you do not learn the
simple things well, then
the difficult things will
not become easy.
p  q
• The difficult things
become easy but you
did not learn the simple
things well. q  p
• You learn the simple
things well but the
difficult things did not
become easy.
p  q
Truth Table Puzzle
Steve would like to determine the relative
salaries of three coworkers using two facts
(all salaries are distinct):
• If Fred is not the highest paid of the three,
then Janice is.
• If Janice is not the lowest paid, then Maggie
is paid the most.
Who is paid the most and who is paid the least?
p : Janice is paid the most.
q: Maggie is paid the most.
r: Fred is paid the most.
s: Janice is paid the least.
p
T
F
F
F
F
q
F
T
F
T
F
r
F
F
T
F
T
s
F
T
T
F
F
Fred, Maggie, Janice
rp
T
F
T
F
T
•If Fred is not the highest paid
of the three, then Janice is.
•If Janice is not the lowest paid,
then Maggie is paid the most.
s q (rp) (sq)
F
F
T
F
T
T
T
F
F
F
p : Janice is paid the most.
q: Maggie is paid the most.
r: Fred is paid the most.
s: Janice is paid the least.
p
T
F
F
F
F
q
F
T
F
T
F
r
F
F
T
F
T
s
F
T
T
F
F
rp
T
F
T
F
T
•If Fred is not the highest paid
of the three, then Janice is.
•If Janice is the lowest paid,
then Maggie is paid the most.
s q
T
T
F
T
T
(rp) (sq)
T
F
F
F
T
Fred, Janice, Maggie or Janice, Maggie, Fred
or Janice, Fred, Maggie
Logical Equivalence
• An important technique in proofs is to
replace a statement with another
statement that is “logically equivalent.”
• 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.
Logically Equivalent
• Compound propositions P and Q are
logically equivalent if PQ is a
tautology. In other words, P and Q
have the same truth values for all
combinations of truth values of simple
propositions.
• This is denoted: PQ (or by P  Q)
Example: DeMorgans
• Prove that (pq)  (p  q)
(pq) p q
(p  q)
p q
(pq)
TT
TF
FT
FF
T
F
F
F
F
T
F
F
T
F
T
F
F
T
T
T
F
T
F
T
Illustration of De Morgan’s Law
(pq)
p
q
Illustration of De Morgan’s Law
p
p
Illustration of De Morgan’s Law
q
q
Illustration of De Morgan’s Law
p  q
p
q
Example: Distribution
Prove that: p  (q  r)  (p  q)  (p  r)
p
T
T
T
T
F
F
F
F
q
T
T
F
F
T
T
F
F
r
T
F
T
F
T
F
T
F
qr p(qr) pq pr
T
T
T
T
F
T
T
T
F
T
T
T
F
T
T
T
T
T
T
T
F
F
T
F
F
F
F
T
F
F
F
F
(pq)(pr)
T
T
T
T
T
F
F
F
Prove: pq(pq)  (qp)
pq
TT
TF
FT
FF
pq
T
F
F
T
pq qp
T
T
F
T
T
F
T
T
(pq)(qp)
T
F
F
T
We call this biconditional equivalence.
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
List of Equivalences
p(qr)  (pq)(pr)
p(qr)  (pq)(pr)
Distribution Laws
(pq)(p  q)
(pq)(p  q)
De Morgan’s Laws
p  p  T
p  p  F
(pq)  (p  q)
Miscellaneous
Or Tautology
And Contradiction
Implication Equivalence
pq(pq)  (qp)
Biconditional Equivalence
Logic
• Logic is a language for reasoning.
• It is a collection of rules that we use
when doing logical reasoning.
• Human reasoning has been observed
over centuries from at least the times of
Greeks, and patterns appearing in
reasoning have been extracted,
abstracted, and streamlined.
Propositional Logic
• Propositional logic is a logic about truth and
falsity of sentences.
• The smallest unit of propositional logic is thus
a sentence.
• No analysis will be done to the components
of a sentence.
• We are only interested in true or false
sentences, but not both.
• Sentences that are either true or false are
called propositions (or statements).
Propositions
• If a proposition is true, then we say it has a truth
value of "true";
• if a proposition is false, its truth value is "false".
E.g.: 1. Ten is less than seven
2. 10 > 5
3. Open the door.
4. how are you?
5. She is very talented
6. There are life forms on other planets
7. x is great than 3
(1) and (2) are propositions (or statements). (1) is false
and (2) is true. (3) – (7) are not propositions
Identifying logical forms
•
1.
argument 1 and 2 have the same form.
If Jane is a math major or Jane is a computer major,
then Jane will take Math 150.
Jane is a computer science major
Therefore Jane will take Math 150
2. If logic is easy or I will study hard , then I will get a A in
this course .
I will study hard
Therefore, I will get a A in this course
Logic form: if P or Q, then R
Q
Therefore, R
Logic Connectives
• Simple sentences which are true or false are basic
propositions.
• Larger and more complex sentences are constructed
from basic propositions by combining them using
connectives.
• Thus, propositions and connectives are the basic
elements of propositional logic.
• English word
Connective
Symbol
Not
And
Or
If then
if and only if
Negation
Conjunction
Disjunction
Implication
Equivalence
 ()




Construction of Complex Propositions
• Let X and Y represent arbitrary propositions.
Then
(X), (X  Y), (X  Y), (X  Y), and (X  Y),
are propositions.
• E.g., (A)  (B  C) is a proposition.
– It is obtained by first constructing
(A) by applying (X),
(B V C) by applying (X  Y) to propositions B and C,
and then by applying (X  Y) to the two propositions
(A)  (B  C) considering them as X and Y,
respectively.
• A well-formed formula (wff): A legitimate string
yes: (A)  (B  C)
no: ((A  BC((
Logical Reasoning
• Logical reasoning is the process of drawing
conclusions from premises using rules of
inference
• These inference rules are results of observations
of human reasoning over centuries.
• They have contributed significantly to the scientific
and engineering progress of the mankind.
• Today they are universally accepted as the rules
of logical reasoning and they should be followed
in our reasoning.
Valid and invalid arguments
• An argument is a sequence of statements. All
statements but the final one are called
premises (assumptions or hypotheses). The
final statement is called the conclusion. The
symbol , read “therefore” is normally placed
just before the conclusion.
• “An argument form is valid” means that no
matter what statements are substituted for the
statement variables in its premises, if the
resulting premises are all true, then the
conclusion is also true. A valid argument is
called a proof.
Reasoning with Propositions
• The basic inference rule is modus ponens. It
states that if both P  Q and P hold, then Q
can be concluded, and it is written as
P
PQ
--------Q
• The lines above the dotted line are premises
and the line below is the conclusion drawn
from the premises.
Some more
• modus tollens
Q
PQ
-------- P
• Conjunctive Simplification
PQ
-------P
• Conjunctive addition
P
Q
------------P Q
• Rule of contradiction
P  c, where c is a contradiction
--------P
Yet some more
•
•
•
•
Disjunctive Addition
P
------------PQ
Disjunctive syllogism
PQ
Q
--------P
Hypothetical syllogism
PQ
QR
-------PR
Dilemma: proof by division into cases
PQ
PR
QR
-------R
Proof
• A proof is a sequence of sentences, where
each sentence is either a premise or a
sentence derived from earlier sentences in
the proof by one of the rules of inference.
• The last sentence is the theorem (also called
goal or query) that we want to prove.
A complex example
1. If my glasses are on the kitchen table, then I saw
them at breakfast.
2. I was reading the newspaper in the living room or I
was reading the newspaper in the kitchen.
3. If I was reading the newspaper in the living room,
then my glasses are on the coffee table.
4. I did not see my glasses at breakfast.
5. If I was reading my book in bed, then my glasses are
on the bed table.
6. If I was reading the newspaper in the kitchen, then
my glasses are on the kitchen table.
Where are the glasses?
Translate them into symbols
• P = my glasses are on the kitchen table,
• Q = I saw my glasses at breakfast.
• R = I was reading the newspaper in the living room
• S = I was reading the newspaper in the kitchen.
• T = my glasses are on the coffee table.
• U = I was reading my book in bed.
• V= my glasses are on the bed table.
Statements in the previous slide are translated as follows:
1. P  Q
2. R  S
3. R  T
4. Q
5. U  V
6. S  P
• Example: weather problem
If it is humid, it is hot. If it’s hot & humid, it’s raining.
It is humid.
• Example for the “weather problem” given
above.
1 Hu
Premise
“It is humid”
2 HuHo
Premise
“If it is humid, it is hot”
3 Ho
Modus Ponens(1,2)
“It is hot”
4 (HoHu)R
Premise
“If it’s hot &
humid, it’s raining”
5 HoHu
And Introduction(1,2) “It is hot and humid”
6R
Modus Ponens(4,5)
“It is raining”
Entailment
• Entailment: KB |= Q
– Q is entailed by KB (a set of premises or
assumptions) if and only if there is no
logically possible world in which Q is false
while all the premises in KB are true.
– Or, stated positively, Q is entailed by KB if
and only if the conclusion is true in every
logically possible world in which all the
premises in KB are true.
derivation
• Derivation: KB |- Q
– We can derive Q from KB if there is a proof
consisting of a sequence of valid inference
steps starting from the premises in KB and
resulting in Q
Two important properties for
inference
1. Soundness
2. Completeness
Soundness: If KB |- Q then KB |= Q
– If Q is derived from a set of sentences KB
using a given set of rules of inference, then
Q is entailed by KB.
– Hence, inference produces only real
entailments, or any sentence that follows
deductively from the premises is valid.
Completeness: If KB |= Q then KB
|- Q
– If Q is entailed by a set of sentences KB,
then Q can be derived from KB using the
rules of inference.
– Hence, inference produces all entailments,
or all valid sentences can be proved from
the premises.
Logics
Logics are formal languages for representing
information such that conclusions can be
drawn
• Syntax: defines the sentences in the language
• Semantics: define the “meaning” of
sentences: i.e., define true of a sentence in a
world
• Example: arithmetic
Worlds in Propositional Logic
• Assignment of a truth value – true or false –
to every atomic sentence
• Examples:
– Let A, B, C, and D be the propositional symbols
– is m = {A=true, B=false, C=false, D=true} a world?
– is m’ = {A=true, B=false, C=false} a world?
• How many worlds can be defined over n
propositional symbols?
Models
• A world is a possible truth assignment to the
propositional symbols.
• Given a world m, we say m is a model of a
sentence  if  is true in m
• M() is the set of all models of 
• Then KB  if and only if M(KB)  M()
M()
M(KB)
Inference
• KB |-i  : sentence  can be derived from KB
by procedure i
• Soundness: i is sound if
whenever KB |-i  it is also true that KB

• Completeness: i is complete if
whenever KB  it is also true that KB |-i 
Examples of Logics
• Propositional calculus
A  B  C
• First-order predicate calculus
( x)( y) Mother(y,x)
• Logic of Belief
B(John,Father(Zeus,Cronus))
Semantics of PL
• It specifies how to determine the truth value
of any sentence in a world m
• The truth value of True is True
• The truth value of False is False
• The truth value of each atomic sentence is
given by m
• The truth value of every other sentence is
obtained recursively by using truth tables
Truth Tables
A
True
B
True
A
False
AB
True
AB
True
AB
True
True
False
False
False
True
False
False
False
True
False
False
True
False
True
True
False
True
True
Satisfiability of a KB
A KB is satisfiable iff it admits at least
one model; otherwise it is unsatisfiable
KB1 = {P, QR} is satisfiable
KB2 = {PP} is satisifiable
KB3 = {P, P} is unsatisfiable
valid sentence
or tautology
Proof Methods
• Applications of inference rules
– Legitimate (sound) generation of new sentences
from old
– Proof = a sequence of inference rule applications
• Model checking
– Truth table enumeration (exponential in n)
– Improved backtracking
– Heuristic search in model space (sound but
incomplete)
e.g. min-conflicts like hill-climbing algorithms
Inference Rule: Modus Ponens

{  ,  }

Example: Modus Ponens

{  ,  }

Battery-OK  Bulbs-OK  Headlights-Work
Battery-OK  Starter-OK  Empty-Gas-Tank  Engine-Starts
Engine-Starts  Flat-Tire  Car-OK
Battery-OK  Bulbs-OK
Example: Modus Ponens

{  ,  }

Battery-OK  Bulbs-OK  Headlights-Work
Battery-OK  Starter-OK  Empty-Gas-Tank  Engine-Starts
Engine-Starts  Flat-Tire  Car-OK
Battery-OK  Bulbs-OK
Example: Modus Ponens

{  ,  }

Battery-OK  Bulbs-OK  Headlights-Work
Battery-OK  Starter-OK  Empty-Gas-Tank  Engine-Starts
Engine-Starts  Flat-Tire  Car-OK
Battery-OK  Bulbs-OK
Example: Modus Ponens

{  ,  }

Battery-OK  Bulbs-OK  Headlights-Work
Battery-OK  Starter-OK  Empty-Gas-Tank  Engine-Starts
Engine-Starts  Flat-Tire  Car-OK
Battery-OK  Bulbs-OK
Headlights-Work
Review: 2 Important Properties
• #1: If KB |- Q then KB |= Q
– If Q is derived from a set of sentences KB using a
given set of rules of inference, then Q is entailed
by KB.
– Hence, inference produces only real entailments,
or any sentence that follows deductively from the
premises is valid.
• #2: If KB |= Q then KB |- Q
– If Q is entailed by a set of sentences KB, then Q
can be derived from KB using the rules of
inference.
– Hence, inference produces all entailments, or all
valid sentences can be proved from the premises.
Summary
• Basic concepts of logic:
– Syntax: formal structure of sentences
– Semantics: truth of sentences wrt worlds
– Entailment: necessary truth of one
sentence given another
– Inference: deriving sentences from other
sentences
– Soundness: derivations produce only
entailed sentences
– Completeness: derivations can produce
all entailed sentences