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A Logical Proof of the
Existence of God
By
William S. Hatcher
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William S. Hatcher
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I. The Historical Context.
There is a continuous history
of roofs of the existence of
2
God starting with Aristotles
well known proof of the
existence of an uncausedcause (hisprime mover).
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Aristotles proof is based on an
3
uses attributional logic.
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Attributional logic is the logic that
deals exclusively with propertiesof objects. Example:
------ is reen attributes or
4
assigns the property of greenness
to any object whose name is
substituted for the blank.
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With the appearance ofNajat
(salvation) by the great Muslim
philosopher Avicenna (980-1037)comes the first use ofrelational
5
logic as a basis of a proof ofGods existence. Avicenna thereby
avoids any appeal to Aristotlesinfinite regression principle.
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Relational logic includes
attributional logic but goesbeyond the latter by treating
6
two existents. Example:
----- is a brother of ____
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Avicennas proof was a smallpart of an ambitious
philosophical program ofreconciling revelation (i.e., the
7
oran w t sc ence(essentially Greek
philosophy, especiallyAristotle).
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The immediate successor of
Avicenna was the Arabic-speaking
Jewish Rabbi Maimonides (1134-1204). In his work Guide to the
8
, .reformulation of Avicennas proof but
reverts to attributional logic and an
appeal to Aristotles infiniteregression principle.
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Taking Avicennas work as a
model, Maimonides conceived
9
reconciling the Torah with
Aristotelian philosophy.
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The famed Catholic philosopherand theologian Thomas Aquinas
(1225-1274) followed on theheels of Maimonides, and
10
represents the latters attempt to
reconcile Greek philosophy with
the New Testament of
Christianity.Copyright 2008, The Estate of
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The Summa contains three
ways of knowing (proving)
God. Thomas third way is hisformulation of the Avicenna
11
proof and, like M.s, reverts toattributional logic and appeal to
the principle of infiniteregression.
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The treatment of Godsexistence by later
philosophers such asDescartes, Leibniz, and Kant
12
use attr ut ona og c anappealed to the infinite
regression principle, as wellas relying on modal logic.
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The logic of modalities involvessuch notions as necessary
existence or contingentexistence, instead of simply
-
13
.
These modal notions are so
vague that there is, even today,
no universally agreed upon
system of modal logic.Copyright 2008, The Estate of
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Thus, none of Avicennassuccessors used or understood
his method. Even Avicenna didnot see his method as
14
par c pa ng n a new og c uonly as a novel way he had
found to treat the specificquestion of Gods existence.
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II. The Modern Period: theadvent of relational logic.
The first systematic treatment
15
of relational logic was inBegriffschrift(1879), by G.
Frege. Begriffschrift meansconcept writing.
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Freges basic idea was that written
language was twice removed from
its content, being a transcription of
the phonemes of speech, which in
16
turn, represent ideas. Fregeoriginated the notion of a formallanguage in which each symbol
represents exactly one logical idea.
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Such formal languages now
constitute both the theoreticalfoundations (architecture)
17
and the practical foundations(programming languages) of
computer science.
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The successors to Frege
were B. Russell, E. Zermelo,
and finally J. von Neumann in
18
,
which, in the opinion of many,
carried relational logic to itsmost refined form.
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It was under von Neumann thatthe first electronic computer, the
Eniac, was conceived and builtat Princeton (1938-1947). This
19
an a su sequen compu ersare based on relational logic and
could not exist had relationallogic not been conceived.
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III. The Power of RelationalLogic.
There are several ways of
20
assessing and understandingthe increased power of
relational logic overattributional logic.
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AL is decidable: there existsa computer algorithm A( )
such that, given anystatement in the language of
21
AL, the algorithm willterminate in a finite time and
yield 1 if the statement is atruth of AL, and 0 if it is not.
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RL is semidecidable. This meansthat there exists a computer
algorithm S( ) with the followingproperty: if it terminates when
22
language of relational logic, then
that statement is a truth of relational
logic. In case of nontermination wecan draw no conclusion.
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Furthermore, it is known (and
proved by Church in 1936)that there does not exist and
23
cannot exist a decisionalgorithm for RL. RL is thus
essentially undecidable.
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More importantly for philosophy,
AL and RL lead us to ask quite
different kinds of questions ofreality. In AL, we seek to know
24
an object by asking what are itsintrinsic properties. In RL, we
want to know how the objectrelates to other objects.
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It turns out that the relational
approach often yields more
useful information while
25
clichs as fire burns
because it is the nature of fire
to burn.
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But why logic?
Because logical deduction
26
unobvious from the obvious
through a series of
individually obvious steps.
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IV. The Proof Itself, part 1:the causality relation
Our proof depends on exactly
27
,
extralogical principle and three
logical principles. As we present
each principle, we will see that itis empirically grounded.
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We say that a metaphysical
principle is empirically grounded
if the restriction of the principleto physical reality yields a known
28
truth of empirical science. Itthem becomes a metaphysical
generalization of an empiricallaw.
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This way of doing metaphysicsis part of a general
philosophical method, calledMinimalism. Our articulation of
29
our logical proof of Godsexistence is, in general, an
illustration of Minimalism atwork.
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We begin with our one
extralogical assumption:
30
P.0. Something exists (there
is not nothing).
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P.0 is obviously empirically groundedand is, in fact, obviously true.
However, the spirit of Minimalism isthat we make all assumptions
31
,
obviousness. Our assumption of the
extralogical P.0 makes our proof a
cosmologicalproof rather than anontologicalproof.
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We now define realityas the
totality of actual existence =everything there is (or was or
32
will be).Aphenomenon is some
nonempty portion of reality.
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Example: Let V symbolize reality, the latter
being conceived as the interior of the largercircle.
V Every subdomain of V ofany
33
phenomenon.
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We now consider a binary
relationship called causality
which may hold between anytwo phenomena A and B. If the
34
relationship AB does indeedhold, then we say that A causes
B. This means B exists by virtueof A.
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Generically, causality is a
logical relation, but this
relation has an empirical
35
world: If AB holds, then it
can never occur that A holds
without B holding.
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Never A without B is thus anecessary (but not sufficient)
condition for AB to hold. This islike semidecidability. If ever we
36
,
then we know certainly that A does
not cause B. But in the absence of
such a clear counterexample, wecan draw no conclusion either way.
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However, the empiricalrequirement that never A
without B is clearly enough toground empirically the causality
37
relationship. The point is thatcausal links are inferred
(logically) and not observed, asHume already indicated.
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Causality is thus a legitimate
principle of minimalisticmetaphysics. We now
38
proceed with certaindefinitions related to the
causality relationship.
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D.0.A phenomenon B is withouta cause if, for no A, does AB
hold.D.1. B is caused (other-caused)
39
if for some AB, AB holds andBB does not hold (i.e., BB).
D.2.B is uncaused (self-caused)if BB and never AB for AB.
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We can now articulate the first of
our three logical principles, the
principle of sufficient reason.
40
P.1. (POSR) Every phenomenon
B is either caused or uncaused
(and never both).
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P.1 implies that no phenomenonB can exist without a cause, be
that cause either wholly within Bor (partly or wholly) outside of B.
41
,
described by D.0 cannot occur.
Either D.1 or D.2 must occur, for
any given phenomenon B
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P.1 says that the why question isalways meaningful (even if we
never find the answer). If we askwhy B? the answer there is no
42
,
not acceptable. POSR is thus the
fundament and basis of (scientific)
rationality. It is the essential logicalprecondition for all of science.
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Example: why is grass green?Answer: Because white light
contains the full spectrum of all thecolors and because the structure of
43
(pigment) which absorbs all except
the green portion of the light
spectrum, and reflects the rest (thegreen portion).
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Note: This is a case of other-
causation, because self-
44
whole cause be within the
phenomenon itself.
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Part V. A Last Look at Aristotle.
We are now in a position to givea modern version of Aristotles
45
-
logic. This is useful not only for
understanding our proof but also
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POSR was used in Aristotles proof,but without being explicitly
identified, only tacitly assumed. Theprinciple is clearly named and
46
.
principle needed for Aristotles
proof is Transitivity. We posit this
as a temporary principle (we will notneed it for our proof).
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T.1. (Transitivity) If AB and BC,then also AC.
P.1 and T.1 are all that is needed toprove the following Lemma in
47
.
Lemma 1. There cannot be any
circular causal chain among distinctphenomena.
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Proof Let a circular causal chain
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Proof. Let a circular causal chain
A1A2. . .AnA1 of length n begiven. We claim that necessarily
A1=A2=. . .=An. Indeed, by repeateduse of transitivity we have the
48
.
A2
A3A4
An
. ..
A1
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Thus, A1AnA1. Invokingtransitivity yet again, we have
A1
A1. Thus, by P.1, A1 isuncaused and thus cannot be
49
- . n 1. ,
An is not other, i.e., An=A1.
Indeed, by transitivity, every
AiA1 and thus every Ai=A1 asclaimed.Copyright 2008, The Estate of
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We now posit the secondtemporary principle, Aristotles
principle of infinite regression.T.2.An infinite regression of
50
. ,
we cannot have an infinitely
descending causal chain
. . . An. . . A2A1where theAi are all different.
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We now use the principles P.0,P.1, T.1, our Lemma, and T.2 to
prove Aristotles theorem:AT. There exists at least one
51
u u - uphenomenon.
Proof. The proof is by
contradiction. Suppose that there
is no uncaused phenomenon. ByCopyright 2008, The Estate of
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P.0 we know that at least onephenomenon A1 exists. By
hypothesis, A1 is not uncausedand thus, by P.1, it is other-
52
2 1
,2 1
.
But A2 is also not uncaused (hyp)
and thus other-caused (by P.1)
by some A3A2. Thus,A3A2A1
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Since A2A1, it must also be thatA3A1, because otherwise we
would have a circular causalchain among the distinct entities
53
A1 and A2. More generally, if wehave a causal chain of length n
among distinct phenomenaAn. . .A3A2A1, then our
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Lemma tells us that we can addan An+1An where An+1 is
different not only from An, butfrom all the other A , because if
54
An+1 is equal to any other Ai,then we will have a circular
causal chain among distinctphenomena. Thus, if there is no
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uncaused phenomenon (i.e., if allphenomena are other-caused), then
we can construct an infinitelydescending causal chain of distinct
55
, . .
we assume T.2 true, we concludethat there must be at least one
uncaused (self-sufficient)phenomenon.
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Evaluating Aristotles Proof1. Burden of proof rests on T.2.
2. A. held that an infinite regress
was logically impossible. Modernmathematics shows that this is false.
56
Counterexample, the negativeintegers:
-n
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an infinite regress ofcauses. In this
weaker form, the principle is
defendable but still controversial.4. AT does not deny the possibility
57
of many different uncaused causes--even an infinity of them.
Contradicts monotheism. Each u.c.
is equally a candidate for
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Godhood. Thus, if we areconsistent monotheists, none of
them is a candidate for God(lack of uniqueness).
58
. ,
any u.c. is in fact a universal
cause and thus a candidate for
Creator of all things.
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6. Thus (Avicennas criticism),even if we grant the cogency
of AT, this theorem does notreally do the job of proving
59
phenomenon whoseexistence is proved (at least
one u.c.) does not satisfy theminimal criteria for Godhood.
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VI. Completing Our Proof.
Avicennas criticism of Aristotlesproof raises the question: How will
60
existence of God? We must nowgive a precise logical definition of
God so that we will know when andif we are successful.
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We will shortly give such adefinition, but for the moment the
minimal conditions are that Godmust be a phenomenon G that is
61
, ,
cause, i.e., the ultimate cause of all
phenomena in existence. The
following diagram illustrates theserelationships:
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G A
..
..
.
62
.
C
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Note: It is important to realizethat causality need not be direct.
Suppose that AB, AB, and
that there is some C different
63
ACB. We say that C is aninterpolant cause between A
and B and that the causalitybetween A and B is indirect.Copyright 2008, The Estate of
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If AB and there is nointerpolant cause C between
them, then we say that thecausality of B by A is direct.
64
Thus, to say that G is auniversal cause does not
mean that God has directlycaused every phenomenon.
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It means that every existingphenomenon is the end effect of
a causal chain, of possibly
infinite length, starting with G.
65
controversies result from afailure to understand the
distinction between direct andindirect causation.Copyright 2008, The Estate of
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Following Avicenna, we nowintroduce a second binary
relationship
which may holdbetween two phenomena A and B.
66
component of B. Given B, if ABfor at least one A, then we say that
B is composite. Otherwise, B issimple (noncomposite).
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Composites are phenomena which
have parts. All known physical
phenomena are compositesexcept, possibly, the elementary
67
particles of quantum mechanics(e.g., quarks or photons). The
question of the simplicity of these
latter particles is still controversial.
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A composite phenomenon willalso be called a system. We use
the componenthood relationship
to define another relation that
68
.
D.3. When every component
EA is also a component EB,
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The following diagram illustratesthe difference between
componenthood and subness.
BThe components of B
and A are the points
69
A
boundaries. Every pointin A is certainly in B so
that AB clearly holds.
But AB since the
components of B arepoints, and A is a circle,
not a point.Copyright 2008, The Estate of
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However, we do use the wordpart to refer indifferently to
components or to subsystems.
D.4. If either AB or AB holds,
70
.
Note: in spite of the distinction
between component and
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both a component and asubsystem.
Example: The digestive system
is both a component of the body
71
-
system since the components(organs) of the digestive system
are also components of thebody.Copyright 2008, The Estate of
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With this terminology
established, we can now state
72
.
For ease of reference, we alsorestate P.1.
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P.1. (POSR) Every phenomenonA is either uncaused or caused,
and never both.P.2. (Potency) If AB and if
73
either EB or EB (i.e., E is apart of B), then AE.
P.3.(Limitation) If E
B, thenBE.
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Comments on definitions:
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1. P.2 asserts that our causalityrelation is complete causality. In
science, causality corresponds to
Aristotles notion of efficient cause.
74
finally breaks the camels back. Thecomplete cause is all of the other
straws which, together with the last
one, have broken the back of the
camel.Copyright 2008, The Estate of
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We can thus make the followingequation: IP + EC = CC,RP. The
initial phenomenon plus the efficientcause equals the complete cause,
75
phenomenon, RP. In science, IP isassumed already given and we are
trying to determine EC in order toobtain CC and thus RP.
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2. P.3 asserts that a whole B cannotbe the cause of one of its own
components E. This is because thewhole does not even exist (to be a
76
components exist.3. Notice that nothing excludes that
a component may be the cause of awhole of which it is a part.
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Clearly, both P.2 and P.3 areempirically grounded. P.2 is
virtually a definition of the notion
of complete cause and P.3 is,
77
,
second law of thermo-dynamics,which negates the possibility of
purely holistic causality, i.e.,the transfer ofCopyright 2008, The Estate of
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order from a whole to a proper part,without any input of organizing
energy from outside the system.Finally, we have:
78
. . , ,
mean a unique, self-caused(uncaused), noncomposite,
universal cause, if such aphenomenon exists.
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We will prove by pure, formalnonmodal logic that P.0&P.1&
&P.2&P.3 imply that G exists. Let
us recall our observation that logic
79
obvious. P.0-P.3 are so obviousthat most people would not even
feel the necessity to assume themexplicitly.
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Yet, the conclusion that G existsis far from obvious. This
illustrates the power logic haswhen it is properly deployed.
80
e ore g v ng e proo , we nee
one further definition, which will
enable us to give a more formal,
precise definition of the universe
V of existence.Copyright 2008, The Estate of
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D.6.A phenomenon B is an entityif
it is a component of at least one
other system A: for some AB, BA.Thus, all components are entities
81
and all entities are components. Weassume that all non-composites are
entities. We thus have the following
tripartite ontology.
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For noncomposites B, ?BA forsome AB. (noncomposites are
always entities). For compositeentities B, EBA for some EB
82
and for some AB. For non-entitycomposites B, EB? for some
EB. We now (re)define the global
phenomenon V in these terms.
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D.7.
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Let V be the phenomenonwhose components are
precisely the (all) entities.
Now, to be a component is to be
83
be a component of V. Thus, tobe a component (of something--
anything) is to be a componentof V. We thus have:Copyright 2008, The Estate of
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Lemma 2. Every phenomenon B isa part of V.
Proof. If B is an entity (composite ornot), then BV by definition. If B is
84
compos e w e er an en y or
not), then every component EB is
an entity (def.) and thus a compo-
nent of V (def.). Hence (def.) BV.In either case, B is a part of V.
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This terminology allows us to
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restate P.0 in a more formal and
elegant manner:
P.0. V is composite.
85
Theorem 1. Assuming P.0-P.3and our various definitions, then
there exists a unique, non-composite, universal cause G.Copyright 2008, The Estate of
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Proof. By P.1, V is either self-caused or other-caused. Suppose
V
V. By P.0, V is composite. Thus,EV for some E. But then, by P.2,
86
, - . .
Thus, VV. Hence, by P.1, GVfor some pheno-menon GV. Like
every pheno-menon, G is a part ofV. Thus,
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b P 2 GG G i th f lf
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by P.2, GG. G is therefore self-caused. But this means that G is
noncomposite, since E
G
G forsome E implies, by P.2, that GE,
87
. .
universal, because every phenomenonB is a part of V (Lemma 2). Thus, by
P.2 and GV, it follows that, for everyphenomenon B, GB. Finally, G
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i th i d h
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is the unique uncaused pheno-menon, for suppose that, for some
phenomenon G1, G1
G1. Now wehave already estab-lished that G is
88
. 1. . , 1
cannot be both self-caused andother-caused. But G is a cause of
G1. Thus, G is not other, i.e.,G=G1 as claimed.
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We have thus proved the
i f i d
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existence of a unique uncaused,
noncomposite, universal cause.
89Copyright 2008, The Estate of
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VII. Comments and Evaluation
The strength of the proof
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1 A i t ll d
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1. Avicenna actually used acomplicated system of modal-
ities, which we have
eliminated entirely. He also
91
assumptions, which we haveshown either to be
unnecessary or else deduciblefrom our assumptions.
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2 T f ilit t di i
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2. To facilitate discussion, wehenceforth assume P.0 to be
given and true (something exists).Our roof thus shows that
92
(P.1&P.2&P.3)G. The logicalcogency of the proof is beyond
question: the proof can be andhas been totally formalized.
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3 The proof is not an abstract
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3. The proof is not an abstractword game. If the three logical
principles P.1-P.3 are valid (true),then our G does in fact exist.
93
4. Anyone who rejects theconclusion G has only one
rational option. That person must
deny one or more of P.1-P.3:
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(P 1&P 2&P 3)
G thus G
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(P.1&P.2&P.3) G, thus, GP.1 or P.2 or P.3. But to deny a
proposition P is to affirm that its
negation P is true. Thus G
94
. . . .
not such a simple affair as itmight seem at first.
5. Indeed, each of the P.i is auniversal statement, i.e., aCopyright 2008, The Estate of
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statement that makes no existence
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statement that makes no existenceassertion. The negation of such a
universal statement is always anexistence statement. Thus, to deny
95
any o e . s o comm onese o
the existence of certain abstractentities. For example, if I deny P.3,
then I must believe that somewhere inPlatos universe of
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forms there is a system B which
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forms there is a system B whichis the cause of one of its own
components. Certainly nophysical system that we have
96
ever observed or postulated hassuch a property, but if I insist on
negating P.3, I must believe that
such a thing really exists.
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Or if I negate P 1 I am com
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Or, if I negate P.1, I am com-mitted to believing that there is
some phenomenon B whichexists without any cause or
97
reason whatever. Such anexception to the POSR would,
itself, but a good candidate for
God, because according to
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the principles of modern
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the principles of modernscience, it could not be any
physical system. Indeed, Godas we have defined Him is a
98
much more reasonablehypothesis than is such a B.
A similar remark holds forP.2.Copyright 2008, The Estate of
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6 Let us sum up Each of the
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6. Let us sum up. Each of theP.i is empirically grounded
and thus far more reasonablethan its ne ation. Moreover
99
the negation of any P.icommits us to belief in an
abstract entity satisfyinghighly unlikely conditions.
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Since the conjunction of the P.i
also imply the existence of the
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also imply the existence of the
abstract entity G, we conclude
that nihilistic atheism, i.e., the
100
existence of any abstract, non-observable entity, contradicts
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In other words strict materialism
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In other words, strict materialismis logically untenable. Atheism
involves existential commitmentand cannot be consistently
101
maintained as the denial of beliefin any nonobservables. At the very
least, our argument definitively
shifts the existential burden of
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proof from the theist (whoh P i d h h
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proof from the theist (whoaccepts the P.i and thus the
existence of G) to the atheist,who must now justify his
102
irrational preference forbelieving in one of the bizarre
phenomena posited by one ormore of P.i.
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