## Logical symbols　 [Mathematical Appendix : Logic]

・connectives:negation ￢～ / disjunction ∨ /conjunction ∧ / implication,conditional⇒ / equivalent ⇔
・quantifiers: universal quantifier ∀ / existential quantifier ∃

cf. symbols in set-theory/φ/Ω,U//=///+///c/( , )/{ , }/×
→[Mathematical Appendix : Logic]
→　Mathematical Appendixes

## negation : ￢P,　～P

・　"P" "P"  means "not P"。

・The truth-table below defines  the truth value of"P".

 P ￢P true false false true

* Whether  "P" is true or false
depends upon
whether  "P" is true or false.

《"P" is true 》 makes "P" false.
《"P" is false 》 makes "P" true.

[in detail]　　→Semantics "￢P "

 [reference] 　・Nakauchi , Logic Workbook,1.2(p.13) 　・Iseki, Sets and Logic , 1.3(p.13); 　・Sugiura , Introduction to Analysis I , 399 　・Kamiya and Urai,Mathematics for Economics 16-7 　・Iritani and Kuga , Introduction to Mathematical Economics,1.1.1(p.2); ・Okada,Mathematics for Economics and Business Administration 附録1(p.245)

## disjunction,logical sum　P∨Q

・"PQ  means "P or Q" .

・The truth-table below defines  the truth value of "PQ".

 P Q P∨Q true true true true false true false true true false false false

* Whether  "PQ" is true or false
depends upon
both
whether  "P" is true or false
and
whether  "Q" is true or false .
《Both of P and Q is false》 makes "PQ" false.
Other situations make "PQ" true.

[in detail]　　→Semantics "PQ"

 [reference] 　・Nakauchi , Logic Workbook ,1.4(p.19); 　・Iseki, Sets and Logic , 1.3(p.14); 　・Sugiura , Introduction to Analysis I , 399　・Kamiya and Urai , Mathematics for Economics , p.17 　・Iritani and Kuga , Introduction to Mathematical Economics, 1.1.1(p.2); ・Okada , Mathematics for Economics and Business Administration,  附録1(p.246)

 * The two expressions below are interchangeable.   ・　 ￢ ( A ∨ B ) 　・　( ￢ A ) ∧ ( ￢ B ) [reference] 　・Sugiura , Introduction to Analysis I , 399; 　・Motohashi『新しい論理序 説(New Introduction to Logic)』6.4問題6(p.121;128) 　・Iritani and Kuga , Introduction to Mathematical Economics,定理1.1(p.4)

## conjunction,logical product :　P∧Q

・"PQ  means "P and Q" "P but Q".

・The truth-table below defines  the truth value of "PQ".

 P Q P∧Q true true true true false false false true false false false false

* Whether  "PQ" is true or false
depends upon
both
whether  "P" is true or false
and
whether  "Q" is true or false .
《Both of P and Q is ture》 makes "PQ" true.
Other situations make "PQ" false.

[in detail]　　→semantics "PQ"

 [reference] 　・Nakauchi , Logic Workbook ,1.3(p.16); 　・Nakatani , Logic , 1.3.A(p.9)合接conjunction・連言・論理積 　・Iseki, Sets and Logic , 1.3(p.14); 　・Sugiura , Introduction to Analysis I , 399　・Kamiya and Urai , Mathematics for Economics , p.17 　・Iritani and Kuga , Introduction to Mathematical Economics, 1.1.1(p.2); ・Okada , Mathematics for Economics and Business Administration,  附録1(p.246)

 * The two expressions below are interchangeable.   ・　 ￢ ( A ∧ B ) 　・　( ￢ A ) ∨ ( ￢ B ) [reference] 　・Sugiura , Introduction to Analysis I , 399; 　・Motohashi『新しい論理序 説(New Introduction to Logic)』6.4問題6(p.121;128) 　・Iritani and Kuga , Introduction to Mathematical Economics,定理1.1(p.4)

 →[logical symbols ：contents] →[Mathematical Appendix : Logic] →[Mathematical Appendixes]

## equivalent:　P⇔Q

・"PQ" means "P if and only if (iff) Q".
 [reference] 　・Nakauchi, Logic Workbook, 1.4(p.22)1.10(p.58); 　・Kamiya and Urai『経 済学のための数学入門』19; 　・Iritani and Kuga , Introduction to Mathematical Economics,1.1.2(p.3); 　・OkadaMathematics for Economics and Business Administration 附録1(p.248);

・The truth-table below defines  the truth value of "PQ".

 P Q P⇔Q true true true true false false false true false false false true

* The two expressions below are interchangeable.
・　 PQ
・　(PQ)(PQ)
 →[logical symbols ：contents] →[Mathematical Appendix : Logic] →[Mathematical Appendixes]

## implication:,conditional:　A⇒B

・"PQ" means "if P , thenQ" "Q,if P ","P,only if Q" ,"when P , Q"。  [→詳細]

・The truth-table below defines  the truth value of  "AB".

 P Q P⇔Q true true true true false false false true true false false true

[→詳細]

* The four expressions below are always interchangeable.
・ A B
・ (B)　　(A)
・ ( A ) ∨ B
・ ( A (B) )

 [reference] 　　　　　・Shimizu『記号論理学』§1.1(p.8)。条件法conditonal, 「もし-ならば- if-,then-」「-であるとき- when-,-」記号⊃、§1.2.2)表1.2真理値表 　　　　　・Todayama『論理学をつくる』2.1.3条件法conditional「もし～ならば」記号→「質料含意material implicationとも言う」(p.20); 2.2.5 定義(6)前件antecedent,後件consequent (p.34);3.1.2(2)(p.38)真理値表;3.1.2(2)(pp.39-40)日本語のならばの語感との違い。 　　　　　　　　　　　　　　　3.2.2(pp.42-43)A→B"A,only if B",B→A"A,if B"　3.10日本語の「ならば」と論理学の「→」(p.81) 　　　　　・Nakatani『論理』 2.1 条件文　(pp.29-32) 　　　　　・Sugiura , Introduction to Analysis I , 400 　　　　　・Nakauchi, Logic Workbook,4.8(pp.43-47);「条件命題」 　　　　　・Motohashi『新 しい論理序説』44-56; 　　　　　・Kamiya and Urai『経 済学のための数学入門』18-19; 　　　　　・Iritani and Kuga , Introduction to Mathematical Economics,1.1.2(p.3) 　　　　　・OkadaMathematics for Economics and Business Administration,附録1(p.247); ・Chiang, Fundamental Methods of Mathematical Economics 759. 　　　　　・Iseki『集合と論 理』1.3(p.15);;
 * The two expressions below are always interchangeable. 　　・ ￢( A ⇒ B ) 　　　　　・ A ∧ ￢ ( B ) [reference] 　・Sugiura , Introduction to Analysis I , 400; 　・Iritani and Kuga『数 理経済学入門』定理1.1(7)(p.4)

## universal quantifier　∀

【1】　"x 　P(x) "  means  "for all x , x has property P" , that is, "everything has property P".  [→ in detail ...]

【1'】　"xS　P(x) "  means  "for all x in X , x has property P" , that is, "everything in X has property P".  [→ in detail ...][→abbreviation]

【2】　" x 　P(x,y)"  means  "for all x the pair (x,y) has property P" .  [→ in detail ...]

【2'】　"xS 　P(x,y) "  means  "for all  x in S , the pair (x,y) has property P."   [→ in detail ...][→abbreviation]

n　" xi 　P(x1, x2, …, xn)"  means  "for all xi ,  (x1, x2, …, xn) has property P" .  [→ in detail ...]

n'】　"xi S 　P(x1, x2, …, xn) "  means  "for all  xi in S ,  (x1, x2, …, xn) has property P."   [→ in detail ...][→abbreviation]

* nesting

* Where is these expressions  used?   definition : limit of a sequence/definition: limit of a function "lim f(x)" /definition : continuity of a function

 * The two expressions below are always interchangeable. 　　　　　 ∀x  ( P(x) ∧ Q(x) )　 　　　　　( ∀x P(x) ) ∧ ( ∀x Q(x) )  * The two expressions below are always interchangeable. 　　　　 ∀x∈S  (  P(x) ∧ Q(x) ) 　 　　　　　( ∀x∈S  P(x) ) ∧ ( ∀x∈S  Q(x) ) [reference] 　 　・Nakauchi, Logic Workbook,定理2.2.10(p.87)：∀(∨) is also mentioned. 　・Nakatani, Logic, 4.3-C-(4・5)(p.90)：∀(∨) is also mentioned. 　・Saitoh , from Japanese to Symbolic Logic, 2 章§7付録(pp.94-95)

 * The two expressions below are always interchangeable. 　　　　　∀x ∀y  P(x,y) 　 　　　　　∀y ∀x  P(x,y)  * The two expressions below are always interchangeable. 　　　　∀x∈S  ∀y∈T P(x,y) 　 　　　　∀y∈T  ∀x∈S  P(x,y)

 →[logical symbols ：contents] →[Mathematical Appendix : Logic] →[Mathematical Appendixes]

## existential quantifier 　∃

【1】　" x  P(x) "  means  "There exists an x such that  x has property P ", "For some x, P(x)","Something is  P "..  [→ in detail ...]

【1'】　"xS 　P(x) "  means  "there is at least one element x of  S such that P(x) is true" , that is, "Some elements of S is  P".  [→ in detail ...][→abbreviation]

【2】　" x　P(x,y)　"  means  " there exists an x such that the pair (x,y) has property P." "for some x, P(x,y)" .  [→ in detail ...]

【2'】　"xS　P(x,y)"  means  "there exists an element x in S such that the pair (x,y) has property P."   [→ in detail ...][→abbreviation]

n　"xi 　P(x1, x2, …, xn)"  means  "there exists xi such that P(x1, x2, …, xn) " "for some x, P(x1, x2, …, xn)".  [→ in detail ...]

n'】　"xi S 　P(x1, x2, …, xn) "  means  "there exists xi in S ,  (x1, x2, …, xn) has property P."   [→ in detail ...][→abbreviation]

* nesting

* Where is these expressions  used?   definition : limit of a sequence/definition: limit of a function "lim f(x)" /definition : continuity of a function

 * The two expressions below are interchangeable. 　　　　　 ∃x ( P(x) ∨ Q(x) )　　 　　　　　( ∃x P(x) ) ∨ ( ∃x Q(x) )  * The two expressions below are interchangeable. 　　　　 ∃x∈S  (  P(x) ∨ Q(x) ) 　 　　　　　( ∃x∈S  P(x) ) ∨ ( ∃x∈S  Q(x) ) [reference] 　 　・Nakauchi, Logic Workbook,定理2.2.10(p.87)：∃(∧) is also mentioned. 　・Nakatani, Logic, 4.3-C-(4・5)(p.90)：∃(∧)is also mentioned.

 * The two expressions below are always interchangeable. 　　　　　 ∃x ∃y  P(x,y)　　 　　　　　 ∃y ∃x  P(x,y)　　  * The two expressions below are always interchangeable. 　　　　　 ∃x∈S ∃y∈T  P(x,y)　　 　　　　　 ∃y∈T ∃x∈S   P(x,y)

 →[logical symbols ：contents] →[Mathematical Appendix : Logic] →[Mathematical Appendixes]

## partial negation  - two expressions

【1】

The two expressions below are always interchangeable.

・ ( x  P(x) )
・x (P(x) )

【2】

The two expressions below are always interchangeable.
・ ( xS   P(x) )
・xS  (P(x) )

 【文献】 　・De La Fuente, Mathematical Methods and Models for Economists,1.2.a.Properties and Quantifiers(pp.8-9): 　・Nakauchi, Logic Workbook,106-110; 　・Sugiura , Introduction to Analysis I , 401; 　・Iritani and Kuga , Introduction to Mathematical Economics,1.1.3(p.6); 　・Kamiya and Urai『経 済学のため の数学入門』20; ・Okada,Mathematics for Economics and Business Administration,253-4;

## total negation - two expressions

【1】

The two expressions below are always interchangeable.
( x P(x) )
( P(x) )

【2】

The two expressions below are always interchangeable.

( xS P(x) )
xS  ( P(x) )
 【文献】　・De La Fuente, Mathematical Methods and Models for Economists,1.2.a.Properties and Quantifiers(pp.8-9): 　・Nakauchi, Logic Workbook,106-110; 　・Sugiura , Introduction to Analysis I , 401; ・Okada,Mathematics for Economics and Business Administration,253-4;

## universal and existential quantifiers are not interchangeable

【1】
・　" y x P(x,y)　　x y P(x,y) "  is always true.
・However , its  converse   is not always true.
" x y P(x,y)　 y x P(x,y)"   may sometimes be true, sometimes be false.

【2】
・　" yT xS P(x,y)　　xS yT P(x,y) "  is always true.
・However , its  converse   is not always true.
" xS yT P(x,y)　 yT xS P(x,y)"   may sometimes be true, sometimes be false.

 →[logical symbols ：contents] →[Mathematical Appendix : Logic] →[Mathematical Appendixes]