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邏輯的世界
Logic
陳鍾誠 2006 年於金門
大綱
Outline
• Boolean Logic (布林邏輯)
• First-Order Logic (一階邏輯)
何謂邏輯?
What is Logic ?
• A world of true and false.
• Data
–T , F
• Function
–And (&)
–Or (|)
–Not (-)
邏輯系統的種類
Types of Logic
• Propositional logic (命題邏輯)
– Also known as Boolean Logic (§1.1-1.2):
– Basic definitions. (§1.1)
– Equivalence rules & derivations. (§1.2)
• Predicate logic (述詞邏輯 或 謂詞邏輯)
– Also known as First-Order Logic (§1.3-1.4)
– Predicates.
– Quantified predicate expressions.
– Equivalences & derivations.
述詞邏輯
Propositional Logic
Propositional Logic is the logic of
compound statements built from simpler
statements
using so-called Boolean connectives.
Some applications in computer science:
• Design of digital electronic circuits.
• Expressing conditions in programs.
• Queries to databases & search engines.
George Boole
(1815-1864)
述詞邏輯的基本運算
Propositional Logic – Operators
Formal Name
Nickname Arity
Symbol
Negation operator
NOT
Unary
¬
Conjunction operator
AND
Binary

Disjunction operator
OR
Binary

Exclusive-OR operator
XOR
Binary

Implication operator
IMPLIES
Binary

Biconditional operator
IFF
Binary
↔
運算一：反相
Negation operator - NOT
p p
T F
F T
運算二：及
Conjunction operator – AND
p
F
F
T
T
q
F
T
F
T
pq
F
F
F
T
運算三：或
Disjunction operator – OR
p
F
F
T
T
q
F
T
F
T
pq
F
T
T
T
述詞邏輯的表示式
Expressions of Propositional Logic
• Use parentheses to group sub-expressions
– Example :
• m : male
• c : child
• m  c : male child (boy)
– Example :
• f  (g  s)
運算四：互斥或
Exclusive-Or – XOR
p
F
F
T
T
q pq
F F
T T
F T
T F
自然語言的語義不清楚
Natural Language is Ambiguous
Note that English “or” can be
ambiguous regarding the “both” case!
“Pat is a singer or
Pat is a writer.” “Pat is a man or
Pat is a woman.” Need context to disambiguate the
meaning!
For this class, assume “or” means
inclusive.
p
F
F
T
T
q p "or" q
F
F
T
T
F
T
T
?
運算五：導致
The Implication Operator
• p  q means p implies q.
– If p is true, then q is true;
– but if p is not true, then q could be either true or false.
• Example :
– E.g., let p = “You study hard.”
q = “You will get a good grade.”
– p  q = “If you study hard, then you will get a good
grade.” (else, it could go either way)
pq 的真值表
Implication Truth Table
• p  q is false only when
p is true but q is not true.
• p  q does not say
that p causes q!
• p  q does not require
that p or q are ever true!
• E.g. “(1=0)  pigs can fly” is
TRUE!
p
F
F
T
T
q pq
F
T
T T
F
F
T T
Imply 運算  的範例
Examples of Implications
• “If this lecture ends, then the sun will rise
tomorrow.” True or False?
• “If Tuesday is a day of the week, then I am a
penguin.” True or False?
• “If 1+1=6, then Bush is president.”
True or False?
• “If the moon is made of green cheese, then I am
richer than Bill Gates.” True or False?
三種反運算
Converse, Inverse, Contrapositive
Some terminology, for an implication p  q:
• Its converse is: q  p.
• Its inverse is:
¬p  ¬q.
• Its contrapositive: ¬q  ¬ p.
• One of these three has the same meaning (same truth
table) as p  q. Can you figure out which?
p  q = -q  -p 兩者相等
How do we know for sure?
Proving the equivalence of p  q and its
contrapositive using truth tables:
p
F
F
T
T
q q
F T
T F
F T
T F
p
T
T
F
F
pq q p
T
T
T
T
F
F
T
T
利用真值表來證明
Proving Equivalence via Truth Tables
• -(-p-q) = pq
Ex.
that
(
q).
p qProve
pq 
p pq
q 
p (p
q 
p  q)
T T
T
F
FF F
T F
F
T
FT T
TF T
F T
F
T
TT T
F F
F
T
• Demorgan’s Theorem : (笛摩根定律)
p q = p + q = p+q
p  q 的真值表
Biconditional Truth Table
• p  q means that p and q
have the same truth value.
• Note this truth table is the
exact opposite of ’s!
– p  q means ¬(p  q)
• p  q does not imply
p and q are true, or cause
each other.
p
F
F
T
T
q pq
F
T
T
F
F
F
T
T
布林運算總整理
Boolean Operations Summary
• We have seen 1 unary operator (out of the 4
possible) and 5 binary operators (out of the 16
possible). Their truth tables are below.
p
F
F
T
T
q
F
T
F
T
p
T
T
F
F
pq pq pq pq pq
F
F
F
T
T
F
T
T
T
F
F
T
T
F
F
T
T
F
T
T
布林運算的幾種符號表達法
Some Alternative Notations
Name:
Propositional logic:
Boolean algebra:
C/C++/Java (wordwise):
C/C++/Java (bitwise):
Logic gates:
not and or
  
p pq +
! && ||
~ & |
xor implies



!=
^
iff

==
Boolean Algebra
- 把布林邏輯轉換為代數
陳鍾誠 2006 年於金門
位元運算
Bits and Bit Operations
• A bit is a binary (base 2) digit: 0 or 1.
John Tukey
(1915-2000)
• Bits may be used to represent truth values.
• By convention:
0 represents “false”; 1 represents “true”.
• Boolean algebra is like ordinary algebra except that
variables stand for bits, + means “or”, and multiplication
means “and”.
– See chapter 10 for more details.
布林代數的基本運算
Boolean Operator
• Boolean Complement (-)
1=0
• Boolean Sum (+)
1+1=1
1+0=1
0+1=1
0+0=0
0=1
• Boolean Product (‧)
1‧1=1
1‧0=0
0‧1=0
0‧0=0
布林集合
Boolean Set
• Boolean Set : B
– B = {0,1}
– Example : 0
• n tuple boolean set
– Bn = {(x1, x2, …, xn)| xi B for 1  i  n}
– Example :
•
•
•
•
(0,0)
(0,1)
(1,0)
(1,1)
布林變數
Boolean Variable
• The variable x is called a boolean variable
if it assume values only from B, that is , if
its only possible value are 0 and 1.
布林代數式
Boolean Expression
• Boolean expression
– 0,1,x1,x2,…,xn
– If E1 and E2 are Boolean expression, then the
following expressions are boolean expressions
• E1
• (E1E2)
• (E1+E2)
• Example :
–
–
–
–
0
x
(1+x)
(x+y)(zy)
Counter example (反例）
xy+)(+z
布林函數
Boolean Function (1)
• Boolean function of degree n
– A function mapping from Bn to B
• Example :
– f(x,y) = xy
布林函數
Boolean Function (2)
• Drawing the truth table of
– f(x,y,z) = xy + z
Some Applications of
Boolean Logic
陳鍾誠 2006 年於金門
布林邏輯的應用
Applications of Boolean Logic
• Circuit Design :
– Building Truth Table
– Minimizing the number of circuit
• By Karnaugh Maps (卡諾圖)
• Database Querying
• Logical Inference in Artificial Intelligence
– An extension of Boolean Logic called
First-Order Logic
• Example : Prolog,
電路設計中的布林代數
Boolean Algebra in Circuit Design
• Truth Table (真值表)
• Karnaugh Maps (卡諾圖)
卡諾圖化簡方法
Karnaugh Maps
B0
B2 B1
0
1
00
01
1
11
10
1
1
ＱＭ布林化簡法
Quine-McCluskey Method
• Minimizing the boolean expression with
any number of variable
(10.4, pp. 727).
Circuit Design - Adder
邏輯推論
Logical Reasoning
陳鍾誠 2006 年於金門
問題
Question
• 要幾條邏輯推論規則才能推出所有真理呢 ?
pTp
pFp
pTT
pFF
ppp
ppp
p  p
pqqp pqqp
(p  q)  r  p  (q  r)
(p  q)  r  p  (q  r)
p(qr)  (pq)(pr)
p(qr)  (pq)(pr)
(pq)  p  q
(pq)  p  q
….
邏輯推論的歷史
History of Proof System
• Aristotle
• Boole
• Hilbert
• Gentzen
• Robinson
亞里斯多德
Aristotle
• 三段論
– http://zh.wikipedia.org/wiki/%E9%A1%B9%E9
%80%BB%E8%BE%91
經典範例1
這個例子是亞里士多德給出的經典的 "Barbara" 三段論:
如果所有人(M)都是必死的(P), (大前提)
並且所有希臘人(S)都是人(M), (小前提)
那麼所有希臘人(S)都是必死的(P). (結論)
經典範例2
還有，
所有人都是必死的. (普遍原理)
蘇格拉底是人. (特殊陳述)
蘇格拉底是必死的. (把特殊(小)代換入一般(大))
型式 1
PQ
QR
PR
型式 2
P
PQ
Q
Mr. 洪
Mr. Horn
• Horn Clause
P1  …  Pk Q
P1
…
Pk
Q
布爾
George Boole
• 真值表
• 從真值表可推出三段
論嗎 ?
P
PQ
Q
Hilbert
• Modus Ponens
Hilbert’s Axiom
Gentzen
• Axioms and Rules of Inference
Gentzen’s Rules (1)
• Disjunction rules
• Conjunction rules
Gentzen’ Rules (2)
• Implication rules
• Negation rule
Completeness
• Gentzen’s Rules are complete
– All logical theorem can be proofed from these
rules.
Robinson
• Resolution Proof :
PQ
;
QR
-P  R
Robinson – second form
-P  Q
;
QR
PR
Completeness
• Robinson’s Rule is Complete under the
resolution procedure.
• Refutation
– Proof by contradiction
Example
Question
PQ
QS
-PR
RS
Is S true under these rules ?
Question in 
-P  Q
-Q  S
-
PR
-R  S
Proof
-R  S -Q  S
-S
-P  Q
PR
-Q
-P
R
S
Reference
• Symbolic Logic
– http://en.wikipedia.org/wiki/Symbolic_logic
• 三段論
– http://zh.wikipedia.org/wiki/%E4%B8%89%E6
%AE%B5%E8%AB%96
邏輯式之間的相等關係
述詞邏輯=布林代數的邏輯系統
• 述詞邏輯的表示式
(Expression in Propositional Logic)
– f  (g  s)
• 布林代數的表示式(Boolean Expression)
– f (g + s)
相等關係的推導
Equivalence Laws
• These are similar to the arithmetic
identities you may have learned in algebra,
but for propositional equivalences instead.
• They provide a pattern or template that
can be used to match all or part of a much
more complicated proposition and to find
an equivalence for it.
相等關係的範例 (1)
Equivalence Laws - Examples
• Identity:
pTp
pFp
• Domination:
pTT
pFF
• Idempotent:
ppp
ppp
相等關係的範例 (2)
Equivalence Laws - Examples
• Double negation:
p  p
• Commutative:
pqqp
pqqp
• Associative:
(p  q)  r  p  (q  r)
(p  q)  r  p  (q  r)
更多的相等關係 (1)
More Equivalence Laws
• Distributive:
p(qr)  (pq)(pr)
p(qr)  (pq)(pr)
• De Morgan’s:
(pq)  p  q
(pq)  p  q
Augustus
De Morgan
(1806-1871)
更多的相等關係 (2)
More Equivalence Laws
• Trivial tautology/contradiction:
ppT
ppF
以相等關係做為運算的定義(1)
Defining Operators via Equivalences
Using equivalences, we can define
operators in terms of other operators.
• Exclusive or:
pq(pq)(pq)
pq(pq)(qp)
以相等關係做為運算的定義(2)
Defining Operators via Equivalences
• Implies:
pq pq
• Biconditional:
pq(pq)(qp)
pq(pq)
一個很長推論的範例
An Example Problem
• Check using a symbolic derivation whether
(p  q)  (p  r)  p  q  r.
(p  q)  (p  r) 
[Expand definition of ] (p  q)  (p  r)
[Defn. of ]
 (p  q)  ((p  r)  (p  r))
[DeMorgan’s Law]
 (p  q)  ((p  r)  (p  r))
 [associative law] cont.
繼續推論…
Example Continued...
(p  q)  ((p  r)  (p  r))  [ commutes]
 (q  p)  ((p  r)  (p  r)) [ associative]
 q  (p  ((p  r)  (p  r))) [distrib.  over ]
 q  (((p  (p  r))  (p  (p  r)))
[assoc.]  q  (((p  p)  r)  (p  (p  r)))
[trivail taut.]  q  ((T  r)  (p  (p  r)))
[domination]  q  (T  (p  (p  r)))
[identity]
 q  (p  (p  r))  cont.
終於推導完畢
End of Long Example
q  (p  (p  r))
[DeMorgan’s]  q  (p  (p  r))
[Assoc.]
 q  ((p  p)  r)
[Idempotent]  q  (p  r)
[Assoc.]
 (q  p)  r
[Commut.]
 p  q  r
Q.E.D. (quod erat demonstrandum)
回顧：述詞邏輯
• Atomic propositions: p, q, r, …
• Boolean operators:      
• Compound propositions: s : (p  q)  r
• Equivalences: pq  (p  q)
• Proving equivalences using:
– Truth tables.
– Symbolic derivations. p  q  r …