Propositional Logic. Propositions Any statement that is either True (T) or False (F) is a...
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Transcript of Propositional Logic. Propositions Any statement that is either True (T) or False (F) is a...
Propositions
• Any statement that is either True (T) or False (F) is a proposition
• Propositional variables: a variable that can assume a value of T or F
• Propositional constants: T or F• Atomic proposition: A proposition consisting of
only a single propositional variable or constant• Logical connectives: logical operators
Truth Table
• Gives the values of a proposition under all possible assignments of its variables
• Used to define connectives
P P
T F
F T
FF
TF
FT
TT
QP
T
F
F
T
P Q
Biconditional
“equivalent”
T
T
F
T
P Q
Conditional
“implies”
F
T
T
T
P Q
Disjunction
“or”
F
F
F
T
P Q
Conjunction
“and”
Connectives
P implies Q
if P then Q
Q if P (e.g., Q :- P)
P only if Q
P is sufficient for Q
Q is necessary for P
P is equivalent to Q
P if and only if Q
P is necessary and sufficient for Q
P iff Q
Compound Propositions
• Also called logical expressions, formulas, and well-formed formulas (wffs)
• Well-formed formulas are defined inductively:– Basis:
T and F are wffs (these are the constants)
P, Q, … are wffs (these are the variables)
– Induction: if A and B are wffs, then so are:(A)
(A B), (A B), (A B), (A B)
Parentheses• Well-formed formulas are fully parenthesized:
((((P Q)) ((P) Q)) R)• We can remove some parentheses:– Outside parentheses can be removed– Use precedence:
– Use associativity always left associative
T
T
T
T
T
T
F
T
6
F
F
T
T
F
F
F
F
4
T
T
T
T
F
F
F
F
2
T
T
T
T
T
T
F
F
5
F
F
F
F
F
F
T
T
1
T
T
T
T
T
T
F
F
3
FFF
TFF
FTF
TTF
FFT
TFT
FTT
TTT
R QP) PQ(RQP
(P Q) P Q R
Evaluating Logical Expressions
• In general…– To evaluate expressions using truth tables with k variables
and n operations is O(2kn).– If we have one operator (n = 1) and if we can substitute in
T or F and evaluate in 1 sec, then• k time• 30 20 minutes• 40 14 days• 50 40 years• 60 40,000 years!
• Not practical to use truth tables for “large” k!
Evaluating with Truth Tables
Logical Expressions
• Tautology – a logical expression that is true for all variable assignments– The symbol |= (read “entails”) denotes that what follows
holds or is true, so long as what precedes it is true– Since tautologies are always true, we sometimes write “|=
B” to denote that B is a tautology independent of what precedes it
• Contradiction – a logical expression that is false for all variable assignments
• Contingent – a logical expression that is neither a tautology nor a contradiction
Sound Reasoning• A logical argument has the form:
A1 A2 … An B
and is sound if when Ai = T for all i, B = T
(i.e., if the premises are all true, then the conclusion is also true)
• This happens when A1 A2 … An B is a tautology
Logical Arguments
• Consider the following statements
1. if you study then you succeed2. you study3. you succeed
• These three statements create a logical argument– Lines 1 and 2 are the premises– Line 3 is the conclusion
• This logical argument is sound… 1. If P then Q2. P------------------3. Q
( ( P Q ) P ) Q
Modus Ponens
T
T
T
T
F
F
F
T
T
T
F
T
FF
TF
FT
TT
((A B) A) B(A B) A (A B)BA
A B A BT ? T
Hence, modus ponens is sound!
A BA
B
Important!
• We are dealing with the validity of an argument, NOT with the validity of the result!
• In logic, it doesn’t matter if a logical statement makes sense or not
• What does matter is that:– IF the premises are correct– THEN so is the conclusion
1. If P then Q2. P------------------3. Q
• P: I study hardQ: I do well on my exam
• P: cows give milkQ: doors open
• P: I obeyQ: I am happy
• P: I sail past the end of the worldQ: I will fall off
Makes sense
Doesn’t make sense
Makes sense
Doesn’t make sense
Valid Logical Arguments
Disjunctive Syllogism
T
T
F
F
T
T
T
T
F
T
F
F
F
T
T
T
A
FF
TF
FT
TT
(A B) A B(A B) A A BBA
Hence, disjunctive syllogism is sound!
A BAB
Logical Implication• If A and B are two logical expressions and if A
B is a tautology, we say that A logically implies B, and we write A > B
• > is a meta-symbol to say a logical argument is sound
(P Q) P > Q(P Q) P > Q
If A B is a tautology, then A > B
Modus ponens
Disjunctive syllogism
Logical Equivalence• If A and B are two logical expressions and if A and B
always have the same truth value, then A and B are said to be logically equivalent, and we write A B
• is a meta-symbol to say that A B is a tautology.
T
T
F
T
F
T
AA AA Thus, A A A
Which means, you can replace A A with A.
A B if and only if A B is a tautology
Laws of , , and
Excluded middle law
Contradiction law
P P TP P F
NameLaw
Identity lawsP F PP T P
Domination lawsP T TP F F
Idempotent lawsP P PP P P
Double-negation law(P) P
Commutative lawsP Q Q PP Q Q P
NameLaw
Associative laws(P Q) R P (Q R)
(P Q) R P (Q R)
Distributive laws(P Q) (P R) P (Q R)
(P Q) (P R) P (Q R)
De Morgan’s laws(P Q) P Q(P Q) P Q
Absorption lawsP (P Q) PP (P Q) P
Can prove all laws by truth tables…
T
F
T
F
T
T
T
F
T
T
F
F
T
T
T
T
F
F
F
T
T
T
T
F
FF
TF
FT
TT
QP(P Q)QP
De Morgan’s law holds!
Two Other Useful Laws
• Law of implication– P Q P Q
• Law of contrapositive– P Q Q P
• Note that the converse of P Q is Q P, and the two are NOT equivalent
Duals• To create the dual of a logical expression
1) swap propositional constants T and F2) swap connective operators and
P P T Excluded Middle P P F Contradiction
• The dual of a law is always a law!• Thus, most laws come in pairs pairs of duals
Normal Forms
• Normal forms are standard forms, sometimes called canonical or accepted forms
• A logical expression is said to be in disjunctive normal form (DNF) if it is written as a disjunction, in which all terms are conjunctions of literals
• Similarly, a logical expression is said to be in conjunctive normal form (CNF) if it is written as a conjunction of disjunctions of literals
• Disjunctive Normal Form (DNF)( .. .. .. ) ( .. .. .. ) … ( .. .. )
Term Literal, i.e. P or P
• Conjunctive Normal Form (CNF)
( .. .. .. ) ( .. .. .. ) … ( .. .. )
Examples: (P Q) (P Q)
P (Q R)
DNF and CNF
Examples: (P Q) (P Q)
P (Q R)
Converting Expressionsto DNF or CNF
The following procedure converts an expression to DNF or CNF:
1. Remove all and
2. Move inside (use De Morgan’s law)
3. Use distributive laws to get proper form
Simplify as you go (e.g., double-neg., idemp., comm., assoc.)
CNF Conversion Example( .. .. .. ) ( .. .. .. ) … ( .. .. )
((P Q) R (P Q)) ((P Q) R (P Q)) impl. (P Q) R (P Q) deM. (P Q) R (P Q) deM. (P Q) R (P Q) double neg. (P Q) (P Q) R comm. (P Q) R
idemp. (P R) (Q R) distr.
(DNF)
(CNF)
DNF Expression Generation
F
T
F
F
F
T
T
F
FFF
TFF
FTF
TTF
FFT
TFT
FTT
TTT
RQP
(P Q R)
(P Q R)
(P Q R)
¦ (P Q R) (P Q R) (P Q R)
minterms
The only definition of is the truth table