BR TO “WHITEHEAD'S NOTATION FOR MULTIPLE RELATIONS”, 22 NOV. 1905
BRACERS 53270. ALS. La Chaux-de-Fonds Bib., Suisse. Russell–Couturat 1: #188
Edited by A.-F. Schmid

Whitehead’s notation for multiple relations

22 nov. 1905.

??φ(x, y) relation (in extension defined by φ(x, y)

Whitehead’s notation for multiple relations.

R^;(x, y) . = . xRy

R^;(x, y, z) . = . x, y, z have the relation R⁽¹⁾

R^;(x, y, z, w) . = . x, y, z, w « « « etc.

R^;(;y) = ??{R^;(x, y)} Df

R^;(x;) = ??{R^;(x, y)} Df

R^;(;yz) = ??{R^;(xyz)} Df

R^;(x;z) = ??{   } Df  R^;(xy;) = ??{   } Df

R^;(x;;) = ??{...} Df

R^;(;y;) = ??{...} Df R^;(;;z)= ??{...} Df

R^;(;yzw) = ??{R^;(xyzw) Df etc.

R^;;(uv) = ??{(Ǝx, y) . x ε u . y ε v . s = R^;(x, y)} Df

R^;;(u;) = ??{(Ǝx) . x ε u . q = R^;(x;)} Df

R^;;(;v) = ??{(Ǝy) . y ε v . q = R^;(;y)} Df

R^;;(uvw) = ??{(Ǝx, y, z) . x ε u . y ε v . z ε w . s = R^;(xyz)} Df etc.

R^;;(;vw) = ??{(Ǝy, z) . y ε v . z ε w . q = R^;(;yz)} Df etc.

R^;;(;;w) = ??{(Ǝz) . z ε w . P = R^;(;;z)} Df etc.

R^;;(tuvw) = ??[(Ǝk, l, m, n) . k ε t . l ε u . m ε v . n ε w . s = R^;(klmn)] Df

R^;;(;uvw) = ??[(Ǝx, y, z) . x ε u . y ε v . z ε w . q = R^;(;xyz)] Df etc.

R^;;(;;vw) = ??[(Ǝy, z) . y ε v . z ε w . P = R^;(;;yz)] Df etc.

(;.)‘R = ??[(Ǝy) . R^;(x, y)] (= D‘R) Df

(.;)‘R = ??[(Ǝx)   ] (= $\breve {D}$‘R) Df

(..)‘R = (;.)‘R ∪ (.;)‘R (C‘R) Df

(;..)‘R = ??{(Ǝy, z) . R^;(xyz)} Df

(.;.)‘R = etc.

(..;)‘R = etc.

(...)‘R = (;..)‘R ∪ (.;.)‘R ∪ (..;)‘R Df

(;...)‘R = ??{(Ǝyzu) . R^;(xyzu)} Df etc.

In practice, the following plan (which avoids the introduction of relations in extension having more than two terms, and therefore avoids the introduction of new indefinables) is usually preferable :

φ^;(;y) = ??φ^;(xy) Df φ^;(x;) = ??φ^;(xy) Df

φ^;(.y) . = . (Ǝx) . φ^;(xy) Df

φ^;(x.) . = . (Ǝy) . φ^;(xy) Df

φ^;(:y) . = . (x) . φ^;(xy) Df

φ^;(x:) . = . (y) . φ^;(xy) Df

φ^;(;;) = ??(,,)φ^;(x;) Df φ^;(.;) = ??φ^;(.y) Df

φ^;(;.) = ??φ^;(x.) Df

φ^;(:;) = ??φ^;(:y) Df φ^;(;:) = ??φ^;(x:) Df

φ^;(..) . = . (Ǝx) . φ^;(x.) Df

φ^;(::) . = . (x) . φ^;(x:) Df

...φ^;(;;) = (..)‘φ = φ^;(.;) ∪ φ^;(;.) Df

(::)‘φ^;(;;) = (::)‘φ = φ^;(:;) ∪ φ^;(;:) Df

Three arguments.

φ^;(;yz) = ??φ^;(xyz) etc. φ^;(.yz) . = . (Ǝx) . φ^;(xyz)

etc.

φ^;(x;;) = ??(,,)φ^;(xy;) etc. φ^;(:y) . = . (x) . φ^;(xyz)

etc.

φ^;(x;.) = ??φ^;(xy.) etc. φ^;(x;:) = ??φ^;(xy:)

etc.

φ^;(x..) . = . (Ǝy) . φ^;(xy.) φ^;(x::) . = . (y) . φ^;(xy:)

φ^;(.;;) - ??(,,)φ^;(.y;) φ^;(:;;) = ??(,,)φ^;(:y;)

φ^;(;..) = ??φ^;(x..) φ^;(;::) = ??φ^;(x::)

φ^;(...) . = . (Ǝx) . φ^;(x..) φ^;(:::) . = . (x) . φ^;(x::)

(..)‘φ^;(a;;) = φ^;(a;.) ∪ φ^;(a.;)

(...)‘φ = φ^;(;..) ∪ φ^;(.;.) ∪ φ(..;) etc.^a Similarly for 4 arguments.

Equivalence of position

φ^;(??) = ??[(Ǝx) . x ε (..)‘φ . s = ??{φ^;(xz) . ≡_z . φ(yz) : φ^;(zx) . ≡_z . φ^;(zy)}] Df

φ^;(a??) = ??[(Ǝx) . x ε (..)‘φ^;(a;;) . s = ??{φ^;(axz) . ≡_z . φ^;(ayz) : φ^;(azx) . ≡_z . φ^;(azy)}] Df

φ^;(???) = ??[(Ǝx) . x ε (...)‘φ . s = ??{φ^;(xuv) . ≡_u,v . φ^;(yuv) : φ^;(uxv) . ≡_u,v . φ^;(uyv) : φ^;(uvx) . ≡_u,v . φ^;(uvy)}] Df

These notations are very useful in Geometry and Dynamics. E. g. Whitehead puts

R^;(axyzt) . = . The straight lines x, y, z are intersected by the straight line a at the time t in the order x, y, z. Then

The class of lines in space = R^;(;;;;.)

The class of moments of time = R^;(....;)

Out of the 5-term relation R^;(axyzt) it appears that both geometry and dynamics can be developed, without the necessity of distinguishing between space and matter.

Notes

^aRussell a sans aucun doute oublié le « ^; » après le dernier φ.