Is Spinoa's approach a form (synthetic, referring to the manner in which knowledge is laid out), or a method (analytic, referring to the manner in which knowledge is gained)? Is there a kernel of truth hidden inside an unpalatable shell, or is the form an essential part of the work? Did Spinoza use the method to compel readers through logic, or style? All commentators have disagreed on these points.
We present the results of a computer-based quantitative analysis of the structure of the whole of Spinoza's Ethics. An understanding of the shape of deduction helps answer some of the above questions.
The predicates used are Ax x is an attribute Bx x is free Dx x is (an instance of) desire Ex x is eternal Fx x is finite Gx x is a god Jx x is (an instance of) love Kx x is an idea Mx x is a mode Nx x is necessary Sx x is a substance Tx x is true Ux x is an intellect Wx x is a will Axy x is an attribute of y Cxy x is conceived through y Ixy x is in y Kxy x is a cause of y Lxy x limits y Mxy x is a mode of y Oxy x is an object of y Pxy x is the power of y Rxy x has more reality than y Vxy x has more attributes than y Cxyz x is common to y and z Dxyz x is divisible into y and z *A for-all *E there-exists
Following are just a normal way for logic to work. US: If *Ax.X appears on an earlier line, *Eg.(g=b)=>X[g/b] may be entered on a new line, with same premise-numbers as on earlier UG: If *Eg.g=b=>X[a/b] appears on a line, then *Aa.X may be entered on a line, if b occurs neither in X nor in any premise on the earlier line. The premise-numbers of the new line are those of the earlier line. Ia: *Aa.a=a may be entered on a line, with premises 0 PA: If X[a/b] appears on an earlier line, *Eg.g=b may be entered on a new line, if X is atomic and undefined. Premise numbers are same. EX: If *Aa.X appears on an earlier line, *Ea.X may be entered on new line, sharing premise numbers.
Two modal operators L and N, with meaning as follows: R1: LX => NX R2: NX => X R3: L(X=>Y) => LX => LY R4: MX => LMX R5: X => LX only when premises of X are 0 R6: N(X=>Y)=>NX=>NY
Here are the definitions of the terms. (== is a definition) D1 *Ax. Kxx &-(*Ey.y<>x & Kyx) == L(*Ey.y=x) D2 *Ax. Fx == *Ey. ((y<>x & Lyx) & *Az.Azx==Azy) D3 *Ax. Sx == Ixx & Cxx D4a *Ax. Ax == *Ey. (((Sy & Ixy) & Cxy) & Iyx ) & Cyx D4b *Ax,y. Axy == Ax & CYx D5a *Ax,y. Mxy == (x<>y & Ixy) & Cxy) D5b *Ax. Mx == *Ey.(Sy & Mxy) D6 *Ax. Gx == (Sx & *Ay.Ay=>Ayx) <--- defn. of God! D7a *Ax. Bx == (Kxx & -*Ey.(y<>x & Kyx)) D7b *Ax. Nx == *Ey.y<>x & Kyx D8 *Ax. Ex == L(*Ey.y=x)
These are Spinoza's axioms A1 *Ax. Ixx v *Ey.(y<>x & Ixy) A2 *Ax. -(*Ey.(y<>x & Cxy)) == Cxx A3 *Ax,y. Kyx => N( *Eu.u=y == *Eu.u=x ) A4 *Ax,y. Kxy == Cyx A5 *Ax,y. -(*Ez.Czxy) == -Cxy & -Cyx A6 *Ax. Kx => (Tx == *Ey.Oyx & Hxy) A7 *Ax. M - (*Ey.y=x) == - L(*Ey.y=x)
Some extra necessary axioms, ommitted by Spinoza A8 *Ax,y. Ixy -> Cxy A9 *Ax. *Ey.Ayx A10 *Axyz. Dxyz -> M -(*Ew.w=x) A11 *Ax,y. (Sx & Lyx) -> Sy A12 *Ax. (*Ey. Mxy->Mx) A13 M(*Ex.Gx) A14 *Ax. N(*Ey.y=x) == -Fx) A15 *Ax. ( -Fx -> [(-L(*Ey.y=x) & N(*Ey.y=x)) == *At.(*Ey.y.y=x at t)] A16 *Ax,y. (*Ez.Azx & Azy) -> *Ez.Czxy A17a *Ax. Ux -> -Ax A17b *Ax. Wx -> -Ax A17c *Ax. Dx -> -Ax A17d *Ax. Jx -> -Ax A18 *Ax,y. (Sx & Sy) -> (Rxy -> Vxy) A19 *Ax,y. ((Ixy & Cxy)&Iyx)&Cyx == Pxy
Here are a few derivations DP1 *Ax.Sx==Ixx from D3,A8 DP2 *Ax.Cxx->Ixx from A1,A2,A8 DP3 *Ax.Sx->Ax from D3,D4a DP4 *Ax.Sx=Cxx from D3,DP2 DP5 *Ax.Sx v Mx from A1,A8,A12,D5a,DP1 DP6 *Ax.-(Sx & Mx) from D3,D5a,D5b,A2 DP7 *Ax,y. (Axy & Sy) -> x=y) from D3,D4b,A2
These are the first 11 propositions in Spinoza's Ethics
P1 *Ax,y. Mxy & Sy -> Ixy & Iyy from D5a,D3
P2 *Ax,y. (Sx & Sy) & x<>y -> -(*Ez.Czxy) from D3,A2,A5
P3 *Ax,y. -(*Ez.Czxy) -> -Kxy & -Kyx from A4,A5
P4 *Ax,y. x<>y -> *Ez,z'. [((((Azx&Az'x)&z<>z') v ((Azx & z=x)&My))
v ((Az'y & z'=y)&Mx)) v (Mx & My) from A9,Dp5,DP6,DP7
P5 *Ax,y. (Sx & Sy) & x<>y -> -(*Ew.Awx & Awy) from DP7
P6 *Ax,y. (Sx & Sy) & x<>y -> -(Kxy & -Kyx) from P2,P3
P6c *Ax. Sx -> -(*Ey. y<>x & Kyx) from D3,A2,A4
P7 *Ax. Sx -> L(*Ey.y=x) from D3,P6c,A4,D1
P8 *Ax. Sx -> -Fx from D2,A9,A11,P5
P9 *Ax,y. (Sx & Sy) -> (Rxy -> Vxy) from A18
P10 *Ax. Ax -> Cxx from D3,D4a and A2
P11 L (*Ex.Gx) <------ that God exists!
from D1,D3,D4a,D4b,D6,A1,A2,A4,A8,D9
And there we have it!