hatcher, logical proof of the existence of god

7/29/2019 Hatcher, Logical Proof of the Existence of God

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A Logical Proof of the

Existence of God

By

William S. Hatcher

Copyright 2008, The Estate of

William S. Hatcher

Source: The William S. Hatcher Library. Can be used under terms of the

Librarys license found at http://william.hatcher.org/license


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I. The Historical Context.

There is a continuous history

of roofs of the existence of

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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

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uses attributional logic.


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Attributional logic is the logic that

deals exclusively with propertiesof objects. Example:

------ is reen attributes or

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assigns the property of greenness

to any object whose name is

substituted for the blank.


William S. Hatcher




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With the appearance ofNajat

(salvation) by the great Muslim

philosopher Avicenna (980-1037)comes the first use ofrelational

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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

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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

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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

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, .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

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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

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represents the latters attempt to

reconcile Greek philosophy with

the New Testament of

Christianity.Copyright 2008, The Estate of

William S. Hatcher




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The Summa contains three

ways of knowing (proving)

God. Thomas third way is hisformulation of the Avicenna

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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

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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

-

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.

These modal notions are so

vague that there is, even today,

no universally agreed upon

system of modal logic.Copyright 2008, The Estate of

William S. Hatcher




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Thus, none of Avicennassuccessors used or understood

his method. Even Avicenna didnot see his method as

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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

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of relational logic was inBegriffschrift(1879), by G.

Frege. Begriffschrift meansconcept writing.

Copyright 2008, The Estate ofWilliam S. Hatcher

Source: The William S. Hatcher Library. Can be used under terms of theLibrarys license found at http://william.hatcher.org/license


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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

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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)

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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

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,

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

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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

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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

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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

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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

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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

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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

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clichs as fire burns

because it is the nature of fire

to burn.




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But why logic?

Because logical deduction

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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

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,

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

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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

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our logical proof of Godsexistence is, in general, an

illustration of Minimalism atwork.




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We begin with our one

extralogical assumption:

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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

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,

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

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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

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phenomenon.




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We now consider a binary

relationship called causality

which may hold between anytwo phenomena A and B. If the

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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

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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

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,

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

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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

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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)

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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.

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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.

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,

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

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,

not acceptable. POSR is thus the

fundament and basis of (scientific)

rationality. It is the essential logicalprecondition for all of science.

Copyright 2008, The Estate ofWilliam S. Hatcher Source: The William S. Hatcher Library. Can be used under terms of theLibrarys license found at http://william.hatcher.org/license


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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

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(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-

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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

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-

logic. This is useful not only for

understanding our proof but also

as a good exercise in theapplication of relational logic.Copyright 2008, The Estate ofWilliam S. Hatcher Source: The William S. Hatcher Library. Can be used under terms of theLibrarys license found at http://william.hatcher.org/license


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POSR was used in Aristotles proof,but without being explicitly

identified, only tacitly assumed. Theprinciple is clearly named and

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.

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

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.

Lemma 1. There cannot be any

circular causal chain among distinctphenomena.


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

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.

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

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- . 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

William S. Hatcher




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We now posit the secondtemporary principle, Aristotles

principle of infinite regression.T.2.An infinite regression of

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. ,

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

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u u - uphenomenon.

Proof. The proof is by

contradiction. Suppose that there

is no uncaused phenomenon. ByCopyright 2008, The Estate of

William S. Hatcher




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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-

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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

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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

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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

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, . .

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.

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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

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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).

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. ,

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

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phenomenon whoseexistence is proved (at least

one u.c.) does not satisfy theminimal criteria for Godhood.


William S. Hatcher




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VI. Completing Our Proof.

Avicennas criticism of Aristotlesproof raises the question: How will

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existence of God? We must nowgive a precise logical definition of

God so that we will know when andif we are successful.


William S. Hatcher




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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

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, ,

cause, i.e., the ultimate cause of all

phenomena in existence. The

following diagram illustrates theserelationships:


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G A

..

..

.

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.

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

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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

William S. Hatcher




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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.

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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.

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controversies result from afailure to understand the

distinction between direct andindirect causation.Copyright 2008, The Estate of

William S. Hatcher




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Following Avicenna, we nowintroduce a second binary

relationship

which may holdbetween two phenomena A and B.

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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

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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

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.

D.3. When every component

EA is also a component EB,

we write AB and say that A is asubsystem of B.Copyright 2008, The Estate of

William S. Hatcher




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The following diagram illustratesthe difference between

componenthood and subness.

BThe components of B

and A are the points

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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

William S. Hatcher




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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,

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.

Note: in spite of the distinction

between component and

subsystem, it is neverthelesspossible for something to beCopyright 2008, The Estate of

William S. Hatcher




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both a component and asubsystem.

Example: The digestive system

is both a component of the body

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-

system since the components(organs) of the digestive system

are also components of thebody.Copyright 2008, The Estate of

William S. Hatcher




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With this terminology

established, we can now state

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.

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

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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.

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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

William S. Hatcher




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We can thus make the followingequation: IP + EC = CC,RP. The

initial phenomenon plus the efficientcause equals the complete cause,

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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

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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,

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,

second law of thermo-dynamics,which negates the possibility of

purely holistic causality, i.e.,the transfer ofCopyright 2008, The Estate of

William S. Hatcher




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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:

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. . , ,

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

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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.

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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

William S. Hatcher




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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

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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

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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

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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

William S. Hatcher




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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

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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.

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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

William S. Hatcher




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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,

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, - . .

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,

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. .

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

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. 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

William S. Hatcher




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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

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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

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(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.

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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

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. . . .

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

William S. Hatcher



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

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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


William S. Hatcher



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

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ever observed or postulated hassuch a property, but if I insist on

negating P.3, I must believe that

such a thing really exists.


William S. Hatcher



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

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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

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much more reasonablehypothesis than is such a B.

A similar remark holds forP.2.Copyright 2008, The Estate of

William S. Hatcher



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

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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

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existence of any abstract, non-observable entity, contradicts

pure logic itself -- independentlyof any assumptions whatsoever.Copyright 2008, The Estate of

William S. Hatcher



In other words strict materialism


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In other words, strict materialismis logically untenable. Atheism

involves existential commitmentand cannot be consistently

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maintained as the denial of beliefin any nonobservables. At the very

least, our argument definitively

shifts the existential burden of


William S. Hatcher



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

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irrational preference forbelieving in one of the bizarre

phenomena posited by one ormore of P.i.


William S. Hatcher



hatcher, logical proof of the existence of god

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