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/( , )/{ , }/×   
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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 PQ

 ・"PQ  means "P or Q" . 

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

 

P
Q
PQ
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)


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

conjunction,logical product : PQ 

 ・"PQ  means "P and Q" "P but Q". 

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

 

P
Q
PQ
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)


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→[Mathematical Appendix : Logic]
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equivalent: PQ  

・"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
PQ
true true true
true false false
false true false
false false true
 

 * The two expressions below are interchangeable.
  ・  PQ
 ・ (PQ)(PQ)

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→[Mathematical Appendix : Logic]
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implication:,conditional: AB 

・"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
PQ
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.
     xS  (  P(x) Q(x) )  
     ( xS  P(x) ) ( xS  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.
    xS  yT P(x,y)  
    yT  xS  P(x,y) 




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→[Mathematical Appendix : Logic]
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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.
     xS  (  P(x) Q(x) )  
     ( xS  P(x) ) ( xS  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.
      xS yT  P(x,y)  
      yT xS   P(x,y)  



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→[Mathematical Appendix : Logic]
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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;



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→[Mathematical Appendixes : Logic]
→[Mathematical Appendixes]

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.
 

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→[Mathematical Appendix : Logic]
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