WPC) 2B)W Z #|x0pt)Courier 10cpi (10pt)RomanMS Serif 16pt GU@Citizen 200GXCITI200G.WRSx  @XhX@2 CS [#|xCourier 10cpi (10pt)RomanEPFX80.WRSSx  @,, X@"^<<dx<PP<P<Pxxxxxxxxxx<<xdxPPPxPxxP<<<dd<d<dY<PdPPxxxxxxxd<d<d<d<x <<Nxx<xxxPPx<x<<2 [[[["*0u^6EQqg3HHnw:w6=nnnnnnnnnn=:wwwagrlrlgu{8Nr\urrrrgb{g{glgJHJrn3glblbCbr88l8rglgWWNr\{gb\j+hwl3gggt\8Ww;քwwCqHHw{sjwsyu_[|lw"^11A>JM++1PPD>>>>>>>>>>JPJ7S>DADA>FJ"/D7MFDDDD>;J>J>A>+++D>>A;A;(;D""A"fD>A>44/D7J>;7++JYD"4>D>>>>>>>>>>>\A;A;A;A;A;""""""""FDD>D>D>D>JDJDJDJDA;>>D>D>A;D>AA N</A7>AA>>>>;>>>>>>v>"^CX JXXCXCJXXh XJXXvvXJXJvhXth<hYA.X_kkX vvvvvhJhJhJhJ XXN{XOXOO  O"*0u^)5?WO'88T[,[)/TTTTTTTTTT/,[[[JkOXSXSOZ^+:"^PZxZnnPdPPPP"nJnPnndPZPnPZYPnnnn"nPnPnPnP PPNZZZPPP"^BY KYYBYBKYYi YKYYwwYKYKwiYwi;iYB,YYkmY wwwwwiKiKiKiK YYNzYPYPP  P"^11A>JM++1PPD>>>>>>>>>>JPJ7S>DADA>FJ"/D7MFDDDD>;J>J>A>+++D>>A;A;(;D""A"fD>A>44/D7J>;7++JYD"5>D>>>>>>>>>>>\A;A;A;A;A;""""""""FDD>D>D>D>JDJDJDJDA;>>D>D>A;D>AA N</A7>AA>>>>;>>>>>>v>2:|\s:V"^--<9DG((-JJ?9999999999DJD3M9?Zi44d4i_d_PPHiUq_ZUb(and/___t{U2Pn7xzznnCzxiCAnqkbnkpkvxW{Uufn{x"^)3==dY((=Q("==========((QQQ=oIIPVIIVV(5PCdPVIVI=CVIcIDC/"32=*==5=5"==""5"V====//"=5P55/0!0AY($01(==I=I=I=I=I=kPP5I5I5I5I5("("("("P=V=V=V=V=V=V=V=V=D5I=V=V=C5V=I; ((N7?=/<==YPx   P*B'U PP2IrYBx)4N r4  pU 3_#;+1x;\  P@UP.s4FFFx@F6X@ [U@49WH6xUbW*f9 xUX;/xC<xnXx  P*R'UXP <W!6(-x="h6\  P@UhP H=7uC2x|oXu\  PAXPEX 0(xUyh0*f9 xUhXP7VC2x)XV4  pUXWXHnnnnnnnaaaaaH=H=H=H=nnnnnnnnn{annnyanj 2) jK    G#x  P*V'U5P##   P*B'U P# On the three equations of magick from Liber Kaos #x  P*V'U5P#  P,-My r #Xx  P*R'UXP# r by 0}Dave Caplan TP  x In this article I have undertaken to analyze the three equations of magick as  P- xlisted by Peter J. CarrollsNh s4I<ԍPeter J. Carroll, Liber Kaos, Samuel Weiser, Maine 1992, pp. 40 51.s and have provided some hermeneutics; I will present  x_philosophic interpretations based on algebraic facts on an intuitive level. Only an  xnelementary familiarity with algebra and with concepts of probability is required to follow the arguments in general.  P] -1. The First Equation  PF - The first equation is " P - xn"!#xs ddddddd  xXM = GL(1 - A)(1 - R) ,x6X@ GU@x6X@ GU@x6X@ GU@.M.GL.A.Rs.g.[..(.1/.).(.1#.).,X$(#(# (#(#!!'#$where M denotes "magick factor" which, I presume would refer to the parameter  P- xdetermined by will as opposed to random occurrence. G denotes 'gnosis', L for  P- x'magickal link', A for 'conscious awareness' and R for 'subconscious resistance'.  xIt is crucial to note that these parameters are probabilities; they can be written, for  P`- xstrictly mathematical generality, as functions over a variable x denoting event or  PI- x`outcome; but insofar as Carroll is treating the parameters on a qualitative scale  P2- xalso, the functional notation would bring about confusion over what 'x' would then denote. " P- x" Now A and R are parameters which interfere with the magickal telos. Hence,  P- xPthe closer they approach to 0, the higher the "purity" of magick. There is no way  x}to empirically measure the factors involved, however, and Carroll does take note  P- xof this; the equations are understood to be intuitive.Dh s4N<ԍ Cf. p.42, Ibid.D Nonetheless it is possible r,   x}with a little algebraic manipulation r,  to assess the scope of the parameters involved.  xROf course we're assumingalong with Carrollthat the parameters are all  Pz- x@evaluated probabilistically, i.e., they take on values over the real unit interval.8zMh s4x"ԍ[0,1]  .8 The  xcloser we look at the equations, however, the more it becomes evident that the  xtopology of the interval over which the values of the parameters range, is open.  xHence r,  under allowable circumstances, letting a variable take on the value 0 (or 1)  P - xpis strictly ad hoc, i.e. for purposes of argument  r ( r and sometimes in terms of a  P!-reductio ad absurdum r ) r .  P!- x The first matter at hand is to explore the parameters G and L within the contextX!0*(("3 '#! Mdd^X of the first equation. Solving for them we find that   A#x'ddddd}_dd d xnG = M over {L(1 - R)(1 - A)}x6X@ GU@x6X@ GU@x6X@ GU@GM.L.R.A@.4.x.(.1.)l.(.1.)n$(#(#(#(#!A'#$and " P- x"a#x' ddddd}_dd  xnL = M over {G(1 - R)(1 - A)}x6X@ GU@x6X@ GU@x6X@ GU@LM.G.R.A@.4.x.(.1.)l.(.1.)n$(#(#(#(#!a'#$which yield the following respective corollaries: G = 0 iff YHh s4 ԍ'iff' denotes the biconditional 'if and only if'.Y M = 0, L c 0 and R, A  P1- xc 1; L = 0 iff M = 0, G c 0 and R, A c 1. If G is defined then L can never equal 0,  P - xand R, A, can never equal 1. If L is defined then G can never equal 0, R, A, can never equal 1.  P - x Let's look at the first solution for G, above. The solution tells us that r,  if the  P - xvaluation of gnosis is such that it is a nonevent or absent, then correspondingly  xPthere will be no magick involved. In such a case, however, if the assessment is to  xbe meaningful (pardon the relative equivocation here, i.e., in terms of mathematical  P- x%syntax and philosophic "meaning" in general) the magickal link, L must be  Py- x`(measurably?) presenti.e., it is essential that L > 0and the opposing factors  xcannot have probability of 1. While it makes direct intuitive sense for the  PK- xopposition factors, the reason why L would beNK7h s43<ԍI say 'would be' viz. because the equation for G would  s4otherwise be undefined, i.e., it would be meaningless.ĵ  r larger than 0 is not as  P4- xhermeneutically clear; we only know that it is algebraically necessary that L > 0 for the solution to be defined.  r   r I'll return to it below.  P- x Now the situation for L in the second solution above parallels that of G in the  x`first. If there is no magickal link then correspondingly there will be no magick  xoinvolvedwhich admittedly is quite obvious. However it is not as obvious that  P- xthere must be some measurable quantity of G, gnosis, for the statement to be  P- xQmeaningful. Hence, it becomes evident that G and L are related to the magick  P- xfactor in complementary ways: hermeneutically, if there is no gnosis then there will  xbe no magick (and vice versa), which in turn implies that r   r in order for the  xcorrespondence to hold, there must be some magickal link; and if there is no  xRmagickal link, there is no magick, which in turn implies that r   r in order for the  P7- x2correspondence to hold, there must be some underlying gnosis. Admittedly I am  xtaking great philosophic liberty here in my hermeneusis; however, the very  P - xBintuitiveness of the parameters at hand invites a wide range of possibilities for  P-understanding them.  P- xC Solving for G and L in the first equation of magick affords us the   84hCopyright  1994 by Dave Caplan8insight  P - xregarding their complementary relation to the magick factor, M. The first equation  P!- xalone only tells us that M  would equal 0 if either G or L is 0, or both, or if  A  P"- xand/or R are equal to 1 which are all intuitively and immediately obvious. Let's see  P#-what it would take for G  and L, respectively, to equal 1. #M0*((r$e'# AdMdd^ '#6 a j#Ԍ P- x G = 1 iff M = L(1 R)(1 A), M c 0, L c 0 and R, A c 1 and likewise, L= 1 iff  P- xM = G(1 R)(1 A), M c 0, G c 0 and R, A c 1. To interpret philosophically, this  P- x#means that, in order for perfect gnosis to exist,n s4KԍIt is important to note that here, we're not speaking of  s4<gnosis in general but highly specific gnosis corresponding to the  s4<means of some specific magickal telos.  the magick factorwhich cannot  xbe absentmust be perfectly balanced with the magickal link (corresponding to  P- x#the telos of whatever magick factor in question) in relation to all factors resisting  xmagick. For the magickal link to be 100% certain, the magick factor must be  Pv-perfectly balanced with gnosis in relation to all factors resisting magick.  x Of course, to weaken the interpretation in the context of an open topology, this  PH- xameans that the higher the gnosis factor, the closer the relation between the  x#magickal link to factors resisting magick, and the higher the quantity (quality?) of  P - xthe magickal link, the closer the relation between gnosis and the factors of magick resistance.  xo As for the resistance factors, they are quite intuitive, even more so than the factors pertaining to magick. Solving for them individually,   #x'ddddd_dd , x?A = 1 - {M over {GL(1 - R)}}x6X@ GU@x6X@ GU@x6X@ GU@AM.GL.Rs;`.1.(.1(.)?߼$(#(# (#(#!'#$and  x_#x'ddddd_dd  x@R = 1 - {M over {GL(1 - A)}}x6X@ GU@x6X@ GU@x6X@ GU@RM.GL.As;`.1.(.1(.)@߽$(#(#b(#(#!'#$in which we notice that they too are related in a complementary way; they cannot  xboth equal 1 at the same time, and in order for one of them to be 1 (i.e., 100%  xcertain in terms of conscious and unconscious magick resistance), the magick factor must be completely absent. This, to repeat, is intuitive and straightforward.  P- r   P-  r   P|-2. The Second Equation  Pe-  PN- xB As for the second equation, I will depart from Carroll's notationNNn s4 ԍHe uses 'P' for nonmagickal probability and 'Pm' for magickal probability (p.45 f.). slightly by  xadopting the general form for probability as a function over stochastic events or  P - xover discrete or continuous random variables. I'll simply denote it by P(x) in order r   P i x#to avoid a possible conflation with 'Pm(x)' denoting the magickal probability over  xothe same event. r"   r"  Anyway the second equation is for computing the probability  PiPm(x)0*&&aa ,e'#,Mdd^'#Mj#Ԍ  P- of an event brought about via magickal means. The equation isy#xJddddddd  xy " P- xA"$(#(#(#(#$'#$#x ddddd| dd X x&%P_m (x) = P(x) + (1 - P(x))M^{1/P(x)}x6X@ GU@x6X@ GU@x6X@ GU@NPoow$mNxJNPNxjNP2Nx^NMoo`PooxN(N)N(vN)>N(N1N(N)N)oo1oo/oo(oo) )NNN&ߣ$(#(#(#(#!'#$where M denotes the magick factor, as in the first equation. It is important to note  P_- xAthat P(x) can never equal 0 for the equation to be defined. Hence, for all x, P(x)  PH- x> 0. This takes into account random factors involved in magickal telos, implying  x#also that when magickal outcomes are assessed probabilistically, we have to take  xinto account the certain existence of random perturbations to which magickal  P -energy is subject.   P i x Interestingly, Pm(x) > 0 for all events over which it operates, even if P(x) = 1  P - xor M = 0. This means that the probability of a magickal event occurring is still  xQhigher than 0 even if the magickal factor is completely absent r$ . P r$ hilosophically  P - x"speaking, randomness alone can bring about the magickal telos in question. Hence,  Pi xif M = 0 then Pm(x) = P(x). Furthermore, if P(x) is 100% probable then so is Pm(x).  x4The converse also holds true. Interpreting philosophically, the higher the  Pb- xprobability of an event x occurring randomly with or without any magick factor, the  PK- x}higher the probability of accomplishing the magickal telos for actualizing event x.  P4- xThe lower the probability of an event x occurring randomly without any magickal  xPfactor, the higher the dependence of magickal probability upon the surety of the  x2magick factor alone. The lesser both factors on the right side of the equation are  x(i.e., the closer they approach 0), the lesser the probability of having the magickal  P-event x come to pass.  P- x Now solving for M under the context of the second equation makes for a significant hermeneusis. Solving, " P- x#"#xddddd dd equ.1 x7&(1 - P(x))M^{1/P(x)} = P_m (x) - P(x),x6X@ GU@x6X@ GU@x6X@ GU@N(sN1N(gN)N)oo1oo/oot(oo)LN(N)@N( N)l N,N=NxN;NPNx/NMoo1PooxNPoo $mNxNPNx7ߴ$(#(#(#(#!'#$!#xG!ddddd dd   x20M^{1/P(x)} = {P_m (x) - P(x)} over {1 - P(x)},x6X@ GU@x6X@ GU@x6X@ GU@MooPooxPoo&mxPx@.P.xoo1oo/oo`(oo)i(1)](%)x.1.(l.),).2߯$(#(#N(#(#!!'#$A#xj$ddddd\dd  xL6M = left[{P_m (x) - P(x)} over {1 - P(x)}right]^{P(x)}x6X@ GU@x6X@ GU@x6X@ GU@M,Poom;xgP/x.Pv.xoo@mPoomxsJ.vw}~()().1.(.)oom(oo m)L$(#(# (#(#!A'#$where P(x) c 1. We have to posit hereconsidering the domain of probabilistic  P"i xdiscoursethat for all x, Pm(x)  P(x) in order for the solution to be meaningful for  P#-M.  x Where magick is involved, the random factor is never 100%. If the magickal  PQ%i xbfactor is 0, then necessarily P(x) = Pm(x) as noted above. To interpret, a Q%0*&&aa&,J'#uJMdd^ '#*  Xj#'#!^(d!'#Y$!!  d^j$'#`'Aj$  A   P- xphenomenon is never magickal if its telos, desired effect or outcome is the same  P- xas its probability of occurring randomly by itself. The higher the probability of x  xDoccurring randomly, the lesser the quantity/quality of magick involved.  P- x@Correspondingly, the lesser the probability of x occurring randomly, the higher the  P- x2magick factor. The higher the probability of the magickal event x coming to pass  Pi x(Pm(x)) and the lower the random probability of x, the higher the magick factor.  Pvi xIf random probability and magickal probability (Pm(x)) are both low and P(x)  P_i xapproaches Pm(x), the higher the magick factor. The last interpretative corollary is counterintuitive.  xA As an example, let's say that someone rolls twelve on a pair of fair dice twenty  xtimes in a row. If anyone exclaims that what occurred is "magick," there is  P - xundoubtedly some substance to the predication notwithstanding the person who rolled the dice; the event alone is beyond the pale of ordinary expectation.  P -3. The Third Equation  P -  x" The third equation pertains to the probability of magickally preventing a random  Py- x}event x from occurring. In order to avoid confusion with the magickal probability  x`function of the second equation, I will depart from Carroll's notation and write  PKi!m(x) for the probability just described. The equation is " P4i x"a#xddddddd  x'pi _m (x) = P(x) - P(x)M^{1/{1-P(x)}} ,x6X@ GU@x6X@ GU@x6X@ GU@N!oo}$m$NxPNPNxDNP NxNMoo\PooxN(N)N(|N)N(pN)ooP1oo/oo1oo(oo%)hN,NNooߖ$(#(#4(#(#!a'#$P(x) c 1. If P(x) = 0 then !m(x) = 0, which holds biconditionally. If M = 0 then  Pi!m(x) = P(x).  x The above (temporary) hypotheses lend themselves to the interpretation that  x%if there is no magick factor accountable, then the probability of magickally  xpreventing a random event from occurring is the same as the probability of the  xevent occurring randomly, i.e., by itself. Now there appears to be a minor intuitive  xproblem here, viz. in the thought that if the random event is impossible, then the  x3probability of magickally preventing it from happening should be 1, i.e. 100%  xprobable. But upon closer inspection we see that the magickal factor is annihilated  P - x3by 0 probability of the random event x and hence r,,  there is simply no magickal  P - xprobability to be accounted for for x not occurring. It will be clear below that using  P-values at two ends of the probabilistic spectrum is strictly ad hoc for now.  P- Solving for M in the context of the equation will provide further insights:X0*&&aa)"3'#a  Mdd^XԌ" P- x2"#xaddddddd L x'P(x) M^{1/{1-P(x)}} = P(x) - pi_m(x) , x6X@ GU@x6X@ GU@x6X@ GU@NPNxNMoo'PooxNP_Nxoo$mNxsN(;N)oo1oo^/oo1ooj(oo)N(N)<N(N)hN,oo3N'NN!ߖ$(#(#(#(#!'#$#xG ddddddd  x30M^{1/{1-P(x)}} = {P(x) - pi_m (x)} over {P(x)}, x6X@ GU@x6X@ GU@x6X@ GU@MooPoo)xDP xoomMxe.P-.xoo1oo/oo1oo(ool)(p)().(.).,oo`,8!3߰$(#(#(#(#!'#$#x dddddLdd  x_8M = left[{P(x) - pi_m (x)} over {P(x)} right]^{1-P(x)} ,x6X@ GU@x6X@ GU@x6X@ GU@M,Pxoom5xM.P.xoomPooRmxsoomvw}~(X)().(y.)ooFm1oom(oom), !_$(#(#v(#(#!'#$and P(x) c 0. When we take the third equation and its domain into account, there  P i xis only one way for the magick factor to be 100% certain, i.e., iff !m(x) = 0. Now  xthis at first appears problematic and completely counterintuitive: if magickal  P - x#prevention of x is impossible, then how could it be that in such a case the magick factor is absolutely certain?  x However, we have to take the domain into consideration and remember that we  Pi x~are working over an open topology; !m(x) = 0 iff P(x) = 0, and we see that this  Pyi xAcannot be the case when we solved for M. So for all x, 1 > P(x) > 0 and 1 > !m(x)  Pbi x> 0.  Also, for all x, P(x)  !m(x). Hence in this case the magick factor can never  PK-be absolutely certain (i.e., M = 1).  P4i x So let P(x) and !m(x), P(x)  !m(x), approach arbitrarily close to 0 and see how  P- xit affects M. Again we encounter the counterintuitive; M approaches 1 as P(x)  Pi xand !m(x), P(x)  !m(x), approach closer to 0. It's a seeming paradox, because  Pi xO!m(x) is the measure of preventing event x from occurring; so notwithstanding !m(x)  P- xEapproaching 0simultaneously with P(x), which, as we have seen is  P- xCmathematically necessarythe magickal telos of preventing x increases in  Piprobability. Thus we see that M  and !m(x) are complementary.  Pi xR Lastly, let !m(x) approach arbitrarily close to 1. Insofar as P(x)  !m(x),  P|i xalways, P(x) must be one step ahead of or identical to !m(x) as it increases toward  Pe- x}1. We see that when this occurs, placing M in the light of its solution, M decreases  xAand approaches 0. Again it appears that there is a paradox involved, inasmuch as  P7i x!m(x) becoming larger would seem to imply that the magick factor is very high.  P - xThis is not the case for the preventative telos insofar as the event x (to be  xmagickally prevented) is increasing in probability of randomly occurring also, thus  xnegating the effectiveness of the preventative magick, notwithstanding its  Pi x4probability approaching 1. As we have seen above, when !m(x) = P(x), M vanishes.  x According to these equations, Carroll rightfully notes that magick is very  xdifficult to come by, and many of us, perhaps Carroll himself, finds the equations  xPto be incongruo r0 us r0  with experience. This can be accounted for by the simple fact#0*&&aa$'a'#oaL  Mdd^ '#    j# '#^  (d  P- xthat magick is of the ethereal domain@n s4y<ԍCf.pp.1829, Ibid.@; its quantification would be hypostatically  xdistinct from objects and events within the "mundane," probabilistic arena. Thus  x2we would have to apply two different types of measurement, and there clearly is  x_no distinctive wayso farto measure _magick_ as probabilistic quantity. In fact,  xolumping magick into the category of quantifiable phenomena would reduce its  xpverity by virtue of its low probability. And of course, if we were to conflate  xmagickwhich is ethereal and psychicwith mundane energies, then it would  xoalso stand to reason that magick is highly improbable insofar as, for example, it  xwould be nearly impossible to instantaneously, magickally make a feather fly off  xQfrom a bird cage while a simple sneeze would probably do the job with almost  xo100% probability. It becomes a matter of comparing apples and oranges; since  xOthere is yet no definite method of measuring magickally related phenomena (except  P - xprobabilistically) by taking into account its domain, it would entail an intrinsic  xcategory mistake to place it among the objects of a mundane ontology. Granted,  x r3 the scope of magick thus considered depends entirely upon its definition.  xNonetheless we've made it clear, hopefully,  r9 the results of treating magick as  x2observational phenomena without recourse to suprarational considerations.  r9 And speaking of suprarational phenomena, how would we, for example, measure art?  X- Bibliography  P - Carroll, Peter J. Liber Kaos, Samuel Weiser, Maine (1992). "#70*&&aa$"Ԍ