The shape of deduction in Spinoza's Ethics.

Lucian Wischik. M.Phil. essay. March 1997.


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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      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-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
    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!