Krantz, Luce, Suppes, & Tversky (2006) Foundations of Measurement Volume I: Chapter 1 Introduction

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1 & : ,0-5, Foundatons of Measurement: Chapter Introducton. THREE BASIC PROCEDURES OF FUNDAMENTAL MEASUREMENT [objects] [events] [property][represent] [system] [ fundamental measure]: ) fundamentally [ fundamental measure] fundamental or not a b a b b a b a b b a - -

2 a b a b b a ab 2 [concatenate] a b a b c a b c a b : ) [transtvty] [assocatvty] a b a c b.. Ordnal Measurement a, b φ( a), φ( b) a b φ ( a) > φ( b) ) 2) 2 3) 3 2 : > R( a, b) S( φ( a), φ( b)) R ( a, b) S( φ(a), φ( b))

3 a b b a ab a, b a b bc cd b d φ ( b) = φ( c) φ ( c) = φ( d) ( b) φ( d) φ > perfect copy 2 a, b a b b a perfect copy 2 workng standard copy copy a b b a 5 2 ϕ revealed preference [] - 3 -

4 ..2 Countng of Unts a, a, a a perfect copy a a b b a φ ( a a ) > φ( b) > φ( a) = φ( a ) φ (a a ) = 2φ (a) φ(b) φ(a) 2φ ( a) a a a a b b a a a φ(b) 3φ ( a) 4φ ( a) a 2a = a a 3a = (2a) a 4a 5a a [standard sequence] 000 b na ( n +) a nφ(a) ( n + ) φ( a) φ(a) φ(a) e e ma φ( a ) = / m e m = 000 b e m = 0 φ(b) φ(b) 2 2 a b b a copy. countng 3. satsfactory: - 4 -

5 b c a b na na c φ (b) > φ( na) > φ( c) b c φ (b) > φ(c) 2. : φ ( b c) = φ( b) + φ( c) b a n copy c n a n + n b c 3. b a n copy c n n / n φ(b) / φ( c) [extensvely] Solvng Inequaltes a, a, a : 2, a 5 a5 a3 a4 a a2 a5 a4 a3 a2 a a x = φ( a ), =,, 5 : () x + x 3 5 x + x 4 x 3 x x + x x x x x 4 2 x 5 x x 4 x 3 x x 2 2 x > 0 > 0 > 0 > 0 > 0 > 0 > 0 (2) - 5 -

6 5 7 a, a, a 2, 5 ) < x / x 3/ < 5 () (2) + >. 2. b c (b) φ(c) φ THE PROBLEM OF FOUNDATIONS.2. Qualtatve Assumptons: Axoms. a b a c b a ( n + ) a b b na na ( n +) a n ( n + ) a b ( n + 2) a b b a b [applcable] ()+(3)+(5)+(7) 2-6 -

7 [axomatzaton] [axoms] [procedure].2.2 Homomorphsms of Relatonal Structures: Representaton Theorems..2 2/ φ φ..2 2 φ [homomorphsm] 2 φ [relatonal structure] A A,, c = a b 3 Re, >, + 2 : strong homomorphsm homomorphsm full homomorphsm strong homomorphsm - 7 -

8 φ A Re > + A R Re S A a,b, R φ(a), φ(b), S A Re φ A,, R, Rm Re, S,, Sm R S ( =,, m ) φ n A,, An A An m R,, R m n Re S n R φ φ,,φ φ n ( j A j ) [representaton theorem].2.3 Unqueness Theorems [countng of unts](..2 ) a φ(a) e φ( e) = e φ( a) / φ( e) e [unquely] φ e φ e φ ( a) / φ( e) = φ ( a) / φ ( e) φ( e) = φ (e) = α φ ( a) = αφ( a) φ ( e ) = /α [smlarty transformaton] - 8 -

9 φ αφ = φ, α > 0 φ [permssble transformaton] [rato scale] C = ( 5/ 9)( F 32) 2 [affne transformaton] φ αφ + β, α > 0 [nterval scale] φ ( a) φ ( b) [ αφ( a) + β ] [ αφ( b) + β ] φ( a) φ( b) = = φ ( c) φ ( d) [ αφ( c) + β ] [ αφ( d) + β ] φ( c) φ( d) [power transformaton] β φ αφ, α > 0, β > 0 [log-nterval scale] 0 [monotonc ncreasng transformaton] φ f (φ) f [strctly ncreasng] [ordnal scale] b a copy n 2 n n e 2 2 n n φ φ p.2-9 -

10 φ φ (P) φ φ A R, R S, S,, n Re,, n (H) (H) (P) [unqueness theorem] 2 φ φ φ φ Measurement Axoms as Emprcal Laws Other Aspects of the Problem of Foundatons [formalzaton] - 0 -

11 5 7.3 ILLUSTRATIONS OF MEASUREMENT STRUCTURES.3. Fnte Weak Orders ( [weak order]) A A A, a, b, c A 2. a b b a ( [connectedness]) 4 2. a b b c a c ( ) [antsymmmetry] [smple order] [total order] ( ) A A, a, b A a b φ( a) φ( b) φ Re f a A φ ( a) = f [ φ( a)] A φ φ φ 3 : [pre-order] weak order total preorder. [quas-order] 4 : [totalness]. [reflexvty] - -

12 2 () A a b a b b a a b a b ( b a) A, A [equvalence relaton] [asymmetry] 4 a = { b A ba} a [equvalence class] b a a b a = b A A / a b a b 5 A / 4 2 a b ( a) ( b) A /, A φ φ( a) = ( a) φ (a) = f [ φ( a)] ( a) = f [ ( a)] a A/ (a ) a b b a b c( b c a c) (b ) c (a) c (a ) (b ) a b b a b a (b) (a ) c (b) > (a) 7 5 : [quotent set] - 2 -

13 φ (.. 6) 2. a b, b c, c a a b b a ab.3.2 Fnte, Equally Spaced, Addtve Conjont Structures A, A 2 A A = A 2 A2 a A a A2 a2 a = a, a ), = b, ) ( 2 b ( b 2 A A 2 A A 2 2 a b c2 A2 ( a, c2) ( b, c2) a 2 2 b 2 c A ( c, ) a2 ( c, b2 ) 2 A a b a,b a b a b b a a b b c a c 2 a b c d a 2 b 2 a a, b ) ( b, ) 2 [addtvty] a b b 2 2 ( 2 a2-3 -

14 A 2 3 ( [ndependent conjont structure]) A A2 A = A A2 A A, a, b, c A a, b, c, d, =, a b b a 2. a b b c a c 3. ( a, c2) ( b, c2) ( a, d2) ( b, d2) (, ) (, ) 4. ( c, a2) c b2 d a2 ( d, b2 ) A 2 4 () A A 2, A A2 2 a b ( a, c2) ( b, c2 ) c2 A2 a 2 2 b 2 ( c, a2) ( c, b2 ) c A 4 A, A A j 5 ( [equally spaced, addtve conjont structure]) A J ( =, 2 ) aj b c A c a b c A A 2, 4 a, b A, =, 2-4 -

15 5. a J b J a a, a ) ( b, ) b ( 2 b2 J a b a A A 2 2 J b 2 ( ) A A2 A A 2, a = ( a, a2), b = ( b, b2 ) A A2 a b φ ( a ) + φ ( a ) φ ( ) + φ ) 2 2 b A 2 ( b 2, =, 2 φ A / 2 φ,φ 2 φ,φ 2 φ = αφ + β, =,2 α > 0, β, β ( φ, φ2) A A 2, Re Re, ( x, y) ( u, v) x + y u + v φ α β, β [ndependence laws]

16 [addtve conjont measurement] [polynomal conjont measurement] 6 7 [utlty measurement] 8 [multdmensonal proxmty measurement] 3 5 a a 2 A b A2 b 2 ( a, b2 ) ( b, a2) A c ( c ), a2 ( b, b 2 ) c2 ( b, b2 ) ( a, c2) ( b, c2) ( c, b2 ) ( a, b2 ) ( b, a2) ( c, a2) ( b, b2 ) ( a, c2) ( c, b2 ) ( b, c2) CHOOSING AN AXIOM SYSTEM 8.4. Necessary Axoms [representaton] A A, Re, φ / [necessary axom] [necessary] - 6 -

17 Nonnecessary Axoms [nonnecessary axoms] [structural] [fnte] [countable]

18 3 [solvablty] 2 A a b a b c ab c A c a, b A a b a b a, b a b = c a, b A a b = c c A a b a, b A A 2,.3.2 a, a2, b ( a, b2 ) ( b, a2) b2 ( a, b2 ) ( b, a2) b ,4,5,6,

19 .4.3 Necessary and Suffcent Axom Systems [necessary and suffcent] satsfactory 9 8 satsfactory unsatsfactory satsfactory.4.4 Archmedean Axoms 3 [Archmedean] x y nx y n 2 y > nx n a,a a = 2a,3a, b b na n - 9 -

20 [strctly bounded] [needed] A A 2, decsve 2 A A 2 A 2 A A A Consstency, Completeness, and Independence [consstent] [not categorcal]

21 .5 EMPIRICAL TESTING OF A THEORY OF MEASUREMENT.5. Error of Measurement

22 4.4.4 a c b c a b [semorder] Selecton of Objects n Tests of Axoms b a, b 2a,, b ma na b n > m n [contnuty] [connectedness n the order topology]

23 2 a, b, c a, b, c a, b, c..3 9 satsfactory

24 .6 ROLES OF THEORIES OF MEASUREMENT IN THE SCIENCES

25 Coombs(964), Guttman(944, 968), Shepard(966) 20.7 PLAN OF THE BOOK

26 ( ) EXERCISES (ry

漸化式のすべてのパターンを解説しましたー高校数学の達人・河見賢司のサイト

漸化式のすべてのパターンを解説しましたー高校数学の達人・河見賢司のサイト https://www.hmg-gen.com/tuusin.html https://www.hmg-gen.com/tuusin1.html 1 2 OK 3 4 {a n } (1) a 1 = 1, a n+1 a n = 2 (2) a 1 = 3, a n+1 a n = 2n a n a n+1 a n = ( ) a n+1 a n = ( ) a n+1 a n {a n } 1,