The Logic of
Propositions
Albert R Meyer
February 15, 2013
propositional logic.1
IMPLIES
The value of (P IMPLIES Q) is F iff
P is T and Q is F.
Truth Table for IMPLIES
Albert R Meyer
P
Q
P IMPLIES Q
T
T
T
T
F
F
F
T
T
F
F
T
February 15, 2013
propositional logic.2
A True Implication
(1=-1) IMPLIES (I am Pope)
We reasoned correctly to
reach the false conclusion
Albert R Meyer
February 15, 2013
propositional logic.3
A True Implication
(1=-1) IMPLIES (I am Pope)
We reasoned correctly to
reach the false conclusion
Albert R Meyer
February 15, 2013
propositional logic.4
A True Implication
(1=-1) IMPLIES (I am Pope)
We reasoned correctly to
reach the false conclusion
from the false hypothesis.
Albert R Meyer
February 15, 2013
propositional logic.5
A True Implication
(1=-1) IMPLIES (I am Pope)
We reasoned correctly to
reach the false conclusion
from the false hypothesis.
Albert R Meyer
February 15, 2013
propositional logic.6
A True Implication
(1=-1) IMPLIES (I am Pope)
The whole implication is true,
even though both conclusion
& hypothesis are false.
Albert R Meyer
February 15, 2013
propositional logic.7
Proving Validity
Instead of truth tables,
can try to prove valid
formulas symbolically using
axioms and deduction rules
Albert R Meyer
February 15, 2013
propositional logic.8
modus ponens rule
antece de nts
P,
P IMPLIES Q
Q
conclusion
Albert R Meyer
February 15, 2013
propositional logic..9
Soundness
A sound rule preserves truth:
if all the antecedents are
true is some environment,
then so is the conclusion.
Albert R Meyer
February 15, 2013
propositional logic..10
Soundness
modus ponens is sound:
if P is true,
and P IMPLIES Q is true,
then Q must be true,
―by truth table.
Albert R Meyer
February 15, 2013
propositional logic..11
Soundness & Validity
Lemma: A rule is sound
AND{Antecedents}
IMPLIES
is valid.
Albert R Meyer
iff
Conclusion
February 15, 2013
propositional logic.12
Lukasiewicz’ Proof System
3 Axiom patterns:
1) (¬P  P)  P
2) P (¬P  Q)
3) (P  Q)  ((Q  R)  (P  R))
1 Rule: modus ponens
Albert R Meyer
February 15, 2013
propositional logic.13
Lukasiewicz’ Proof System
3 Axiom forms:
One Rule: modus ponens
Albert R Meyer
February 15, 2013
propositional logic.14
Lukasiewicz’ Proof System
Prove formulas by starting with
axioms and repeatedly applying
the inference rule.
For example, to prove:
P→P
Albert R Meyer
February 15, 2013
propositional logic.15
A Lukasiewicz’ Proof
rd
3
axiom:
replace R by P
Albert R Meyer
February 15, 2013
propositional logic.16
A Lukasiewicz’ Proof
rd
3
axiom:
replace Q by
Albert R Meyer
February 15, 2013
propositional logic.17
A Lukasiewicz’ Proof
rd
3
axiom:
Axiom 2)
Albert R Meyer
February 15, 2013
propositional logic.18
A Lukasiewicz’ Proof
so apply modus ponens:
Axiom 2)
Albert R Meyer
February 15, 2013
propositional logic.19
A Lukasiewicz’ Proof
so apply modus ponens:
Axiom 1)
Albert R Meyer
February 15, 2013
propositional logic.20
A Lukasiewicz’ Proof
so apply modus ponens:
Albert R Meyer
February 15, 2013
propositional logic.21
System
is Sound
Lukasiewicz’ Proof
System
The 3 Axioms are all valid
(verify by truth table).
We know modus ponens is
sound. So every provable
formula is also valid.
Albert R Meyer
February 15, 2013
propositional logic.22
Lukasiewicz’ System is Complete
Conversely, every valid
(NOT,→)-formula is
provable.
System is “complete”.
Not hard to verify but would
take
Albert R Meyer
February 15, 2013
propositional logic.23
validity checking still inefficient
Deduction proofs in general
no better than truth tables.
No efficient method for
verifying validity is known.
Albert R Meyer
February 15, 2013
propositional logic.24
Other Applications
Java Logical Expressions:
OR
AND
if ((x>0) || (x <= 0 && y>100))
(more code)
Albert R Meyer
February 15, 2013
propositional logic.25
Digital Logic
1 ::= T
0 ::= F
i ::= AND
+ ::= OR
x ::= NOT (x)
Albert R Meyer
February 15, 2013
propositional logic.26
Application: Digital Logic
s ::= A XOR B
c ::= A AND B
half adder
from http://en.wikipedia.org/wiki/Adder_(electronics)
Albert R Meyer
February 15, 2013
propositional logic.27
Digital Logic
d ::= c in XOR s
c out ::= (c in AND s) OR c
A
B
d
cin
cout
s
c
full adder
Albert R Meyer
February 15, 2013
propositional logic.28