The shape of deduction in Spinoza's "Ethics"The shape of deduction in Spinoza's 
Ethics.Lucian Wischik. M.Phil. essay. March 1997. 
Abstract. One of the most remarkable features of Spinoza's Ethics is its 
axiomatic form. Spinoza sets out at the start a small number of definitions and 
axioms that are assuredly true, and proceeds to deduce from these the rest of 
his philosophy. In this respect, the work is an attempt to use a theory of 
philosophy that is modelled upon Euclid's Elements. The Ethics is significant in 
that it is "the only major philosophical work of the 17th century rationalists 
which undertakes, and at least in form achieves, the frequently expressed goal 
of extending the mathematical 'method' [to philosophy]".
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.
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Also see A Dedication to Spinoza's Insights in eclectic hypertext 
The Spinoza Study 
Spinoza's Proof for the Existence of God  as expressed in a logical notation by 
C. Jarrett in The Logical Structure of Spinoza's Ethics, Part I, Synthese 37, 
1978, 15-65, reproduced here for interest.
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-existsFollowing 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=>NYHere 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 == PxyHere 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,A2These 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,D9And there we have it! 
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