Copyright 1995 by the CREATION RESEARCH SOCIETY (CRS), Inc.
                  by Robert A. Herrmann, Ph.D.
          Mathematics Department, U. S. Naval Academy,
           572 Holloway Rd., Annapolis, Md 21402-5002.
          Received 22 April, 1993; Revised 18 June 1993
          Creation Research Society Quarterly 30(4):218
                           March, 1994
<Abstract> This article gives a general overview of recent 
results in mathematical logic that should have a profound effect 
not only upon the foundations of creation-science but the 
foundations of all religious experiences and thought that either 
assume or logically require the existence of a supernatural 
higher intelligence. In particular, it is shown that the concept 
of the existence of a higher intelligence exterior to the 
material universe can be modeled rationally by means of the 
science of mathematics. It is established that human religious 
experiences and scientific models associated with either an 
assumption of or an implied requirement that a supernatural 
higher intelligence exists are not somehow or other irrational in 
character as it is claimed by many secular scientists and 
philosophers. Indeed, if such experiences or creation-science 
models directly correlate to certain customary Bible 
interpretations, then the assumption of irrationality is 
scientifically proved to be false.
When concepts relative to the DNA molecule are modeled by means 
of information theory, one aspect of the obtained theoretical 
conclusions seems to defy human comprehension unless a very 
special postulate is assumed. As Wilder-Smith (1993) states: 
  "We are forced to come back to basics and assume that there 
  must have been in the beginning -- at the act of creation -- an 
  organ of the kind that makes the human brain tick (but    
  infinitely more powerful, of course) to generate the concepts 
  of biology on a much larger scale than the human brain can ever 
Under the further assumption that all life throughout the 
universe is associated with DNA type molecules and that such 
natural processes are amenable to human thought, then such a 
higher intelligence could not be assigned to biological entities 
within the universe itself. Using the term natural to refer to 
entities, processes, and the like that are within our universe, 
under these assumptions, information theory leads to the 
conclusion that the acceptance of a supernatural higher 
intelligence would be needed in order to properly comprehend the 
model. Unfortunately, the assumption that a supernatural higher 
intelligence exists has been rejected by secular scientists and 
atheistic philosophers as not being consistent with scientific 
logic. Indeed, one of the greatest onslaughts against such an 
assumption and all of the human (religious) experiences that are 
modeled by using such an assumption began in earnest with the 
introduction of the philosophy of ``rationalism.'' This 
philosophy claims that explanations for religious experiences or 
perceived phenomena that include supernatural entities external 
to the natural world are irrational in character. The concept of 
irrational refers to what is considered to be contrary to certain 
established human thought patterns.
Rationalism implies that if you cannot rationally justify the 
existence of such a higher intelligence, especially as such an 
intelligence relates to experiences within the natural world, 
then it is necessary to replace hypotheses stated in specified 
supernatural terms with hypotheses stated in natural terms. 
Feuerbach (1967, p. 110) stated this claim as follows: 
  ". . . there is no way of explaining the thousands and 
  thousands of contradictions, perplexities, difficulties, and 
  inconsistencies in which religious belief involves us, unless 
  we acknowledge that the original God was a being abstracted 
  from nature . . . ." 
Feuerbach (1967, p. 248) also states: "Moreover, religious ideals 
have always involved all manner of irrational and even 
superstitious conceptions." He even  attacks the rationalists as 
being incomplete rationalists. 
  ". . . the rationalists take great pains to point out the 
  obvious fallacies of religion; but these are secondary, 
  subordinate fallacies; as for the fundamental fallacies, which 
  have all others as consequences, the same rationalists let them 
  stand, for they are sacred and inviolable. Consequently, when a
  rationalist asks an atheist what atheism is, the proper answer 
  is: Rationalism is a half-baked, incomplete atheism; atheism is 
  a complete and thoroughgoing rationalism." (Feuerbach, 1967, pp. 
From Feuerbach's viewpoint, the hypothesis of the nonexistence of 
a higher intelligence exterior to the natural world, of God, is 
the ultimately correct hypothesis from which to begin a complete 
rationalization for all religious experiences and perceived 
phenomena. Since Feuerbach's lectures, these ideas have been 
championed by numerous influential philosophers, scientists and 
social reformers. Marx (1960, p. 24), using logical terminology, 
states it by writing: "Christianity, . . ., cannot agree with 
reason because `worldly' and `religious' reason contradict each 
other." Santayana (1905, p. 159), utilizing a destructive term 
taken from the language of logic, writes: 
  ". . . the grand contradiction is the idea that the same God 
  who is the ideal of human aspiration is also the creator of 
  the universe and the only primary substance."
In this age of scientism, influential humanists, scientists, 
journalists and the like continue to parrot these claims of 
Feuerbach with the added proviso that the assumption of the 
existence of a supernatural higher intelligence will contradict 
absolutely the logical procedures accepted by the scientific 
community. One quotation will suffice as an example of this world-
view. H. J. Eysenck (1973, pp. 89-90) writes: 
  "Thus the first part of my definition of humanism would involve
  a stress on the use of reason in dealing with inanimate nature 
  and with other human beings . . . . This inevitably involves the 
  rejection of revealed religion . . . . All humanists are agreed 
  that religion is not based on reason . . . . To me, the word 
  reason in this respect implies science. Science is the 
  embodiment of the rational attempts to solve problems posed by 
  nature or human beings . . . . Reason, to me, marks out the 
  method to be used by all humanists."
Individuals who have either had personal religious experiences or 
argue for the scientific acceptance of such a higher intelligence 
certainly do not consider their contributions as irrational. As 
exemplified by the above quotations, many in the philosophic and 
scientific world do consider as irrational the assumption that 
such an higher intelligence needs to be supernatural in character 
and this has inspired their attempts at rationalizing religious 
experiences, or ignoring creation-science models and evidence for 
the acceptance of such models.
If it could be demonstrated scientifically that assuming the 
existence of a supernatural higher intelligence is rational  in 
character, then this would destroy, utterly and completely, the 
philosophical foundations for the philosophy of rationalism as it 
is applied to religious experiences and thought. It would 
eliminate the basic philosophical argument against the existence 
of a supernatural deity. Atheism would have lost its most 
profound intellectual foundation. Further, the necessary 
conclusions of information theory applied to the DNA molecule 
would be upheld and, indeed, the basic foundation of creation-
science could no longer be rejected on scientific grounds. But 
what would constitute a scientific demonstration that it is 
rational to postulate the existence of a supernatural higher 
Timothy Ferris (1979, p. 157) writes: "Scientific theories must 
be logical. They must be expressible in terms of mathematics, the 
most rigorous logical system known." Ferris overstates his 
conclusion when he writes that this ``must'' be the case. 
Actually, the modern scientific approach to theory is rather more 
vague on the subject of rationality. What can be said is that if 
a theory can be closely associated with a mathematical structure, 
then it would follow the most rigorous logical system known.
*Human Intelligence*
No attempt will be made in this paper to give a nearly complete 
definition of human intelligence. But one of the crowning 
achievements of humanity has been the construction of a symbolic 
language as a substitute for oral expressions. Modern computer 
technology also allows for visual or audio impressions that are 
captured by mechanical devices to be translated into a symbolic 
language that can later reproduce, with great clarity, the 
original visual or audio content. Thus, for our purposes, human 
intelligence will include the ability to express thoughts and 
perceptions in a symbolic language comprehensible by others and, 
further, to present written arguments that follow patterns that 
correspond logically to procedures accepted by the majority of 
Throughout this discussion, it will only be assumed that a 
symbolic language corresponds to a portion of human oral 
expression, human perception and mental impression. A symbolic 
language L is constructed intuitively from two or more symbols by 
juxtaposition and yields geometric configurations called symbol 
strings (i.e. strings of symbols). For every natural number n, 
there theoretically exists more than n distinct symbol strings by 
this process. Similar symbol strings are recognized by human 
perception to be equivalent.
In 1930, Tarski characterized and abstracted mathematically those 
general procedures that correspond to the most significant human 
mental processes that, for finite collects of such symbol 
strings, yield deductive conclusions. The mathematical operator 
so obtained is termed a consequence operator. In modern 
mathematical logic, there are two types of such logic operators. 
The most basic is the finitary consequence operator of Tarski 
(1930). However, there is a similar operator that is more general 
in character and is often termed simply as a consequence 
The small amount of set-theoretic language that is employed in 
this paper is taken from a standard high-school algebra course 
and, in some cases, is only considered as an abbreviation. 
Indeed, each abbreviation is specifically defined. No actual 
mathematics appears in this paper. The formal mathematics can be 
found in Herrmann (1987, 1991). The symbol used to represent the 
finitary consequence operator is the symbol Cn. The more general 
consequence operator is often denoted simply by C. Informally, 
such operators take any subset A of L (i.e. A subset L) and yield 
all those members of L that can be deduced from A (i.e. Cn(A)). A 
basic requirement is that the assumed premises can always be 
deduced logically (i.e. A subset Cn(A)). Once a human being has 
deduced all of the consequences, then no more consequences can be 
deduced from the same set of premises (i.e. Cn(Cn(A)) =Cn(A)). 
For C, if one set of premises B is a subset of another such set A 
(i.e B subset A subset L), then deductions from B form a subset 
of those deductions from A (i.e. C(B) subset C(A)). For a 
finitary consequence operator, the human argument of using only 
finitely many symbol strings from a set of premises A to obtain a 
deduction is modeled by the  additional requirement that if x is  
deduced from  A (i.e. x in Cn(A) or x is a member of Cn(A)), then 
there is a finite set of premises  F subset A such that x can 
also be deduced from F. One can show that this last requirement 
also implies the last property listed for the general consequence 
operator C. Consequence operators that correspond to  specific 
deductive processes such as those defined for propositional, 
predicate, and higher-order formal languages (i.e. those logical 
processes used in modern scientific discourse) can be further 
characterized so that each can be differentiated one from another.
What Tarski did was to take a concrete everyday experience and 
mathematically abstract its most basic properties. From this 
abstraction, mathematical arguments establish other properties. 
These other properties  may then be interpreted with respect to 
the original linguistic terms that generate the Tarski 
abstraction. Thus new insight is gained into what constitutes 
human thought patterns. As will be discussed later, the same type 
of formal abstraction is possible for certain dialectic logics.
In 1978 (Herrmann, 1981), Tarski's consequence operator theory 
was investigated through application of the new mathematical 
discipline called Nonstandard Analysis for the specific purpose 
of finding a nonnumerical model for the concept of subliminal 
perception. Nonstandard does not mean that different mathematical 
procedures are employed. This is a technical term relative to 
abstract model theory. After many years of refinement, the basic 
properties of nonstandard consequence operators appeared in 
mathematical journal form (Herrmann, 1987) and book form 
(Herrmann, 1991). Cosmological interpretations of these results 
have been reported upon numerously many times within other 
scientific and philosophic journals as well. However, also of 
significance is a linguistic interpretation of these fundamental 
results. Generating the mathematical structure is not extremely 
difficult. But interpreting it linguistically has been arduous.
*A Special Linguistic Interpretation*
In order to interpret a formal mathematical structure relative to 
different disciplines, a correspondence is created between terms 
in one discipline and the abstract entities of the structure. 
This actually yields a many-to-one correspondence since numerous 
disciplines can be correlated to the same mathematical structure. 
Each time this is done, a mathematical model is constructed. Our 
interest in this paper is a specific correspondence between some 
terms relative to intelligence, linguistic, and similar human 
activities associated with a physical world and the mathematical 
structure. With respect to nonstandard structures, however, many 
new objects emerge that are not present within the standard 
structure. Although these new objects have all of the properties 
of the original entities and thus the same properties as the 
nonabstract objects from which they were originally abstracted, 
they also have many additional properties not shared by any of 
the original entities. What one does, in this case, is to created 
new terms that have a similar linguistic-like character as the 
original linguistic terms and assign these new terms to 
appropriate unassigned entities within the nonstandard structure. 
But can you assign a concrete dictionary meaning to these new 
A dictionary meaning to these new terms will not carry the 
appropriate content. One reasonable method to obtain an in-depth 
comprehension is to have a strong understanding of the workings 
of the mathematical structure and to reflect upon the relations 
between these new linguistic-like terms themselves, as well as 
between the new terms and the standard linguistic expressions. 
What this means is that you must study the written statements 
depicting these relationships. The model that this creates forms 
a portion of the deductive world model or, simply, the D-world 
model. There is, however, a new method that has been devised that 
renders these new concepts comprehensible without the necessity 
of an in-depth study. The method is termed negative comparison.
Negative comparison is a description as to how these new concepts 
negatively compare with the original standard concepts. Certain 
aspects of such linguistic type interpretations have been 
discussed elsewhere (Herrmann, 1991) but not as it directly 
relates to the concept of a higher intelligence. Further, this 
present interpretation uses a few special terms not previously 
introduced. The linguistic-like terms that correspond to new 
abstract entities that, at least, have similar properties as the 
original have the prefix ``ultra-'' attached. It is always to be 
understood that prior to each statement one should insert an 
expression such as ``It is rational to assume that . . .'' where 
the term ``rational'' means the logical  processes science uses 
to develop its most cherished theories. To be as simplistic as 
possible within this section, only one of many distinct logical 
processes will be compared. What can be said about this one 
process will hold for all similar processes that can be 
characterized by the consequence operator. Note that logical 
processes are also termed mental processes.
The use of the ``ultra-'' prefix does not remove the term from 
being only a defined mathematical abstraction. Within a 
description, additional phrases that correlate such terms to a 
specific discipline are either inserted or, at least, understood 
by the reader. Relative to a supernatural higher intelligence, 
one basic correlating phrase is ``entity within the universe.'' 
This signifies any corporeal entity of which the human mind can 
conceive and which makes its home within the material universe. 
The insertion of this phrase is the basic change in the 
interpretation from those previously used. Other obvious 
correlating terms will appear when relationships between the 
ultra-objects and the concrete linguistic entities from which the 
model was generated are discussed.
There exists an ultra-language, denoted by *P, that at least has 
all of the properties of the most simplistic of human languages, 
the propositional language P. The language P is a subset of *P. A 
simple informal propositional language P can be constructed from 
but two primitive words such as ``house'' and ``door'' and the 
usual additional symbol strings such as ``or'' ``and'' ``not'' 
and ``implication.'' In this case, all of the expressions in P 
are meaningful in the sense that they impress on the human mind 
various images. Assume that all of the members of P are 
meaningful in this sense. There are many members of the ultra-
language *P that cannot be used for any purposes by, and have no 
specific meaning to, any entity within the universe. However, all 
members of *P are ultra-meaningful. The mathematical model would 
require ``ultra-meaningful'' to correspond to a statement such as 
``they ultra-impress on an ultra-mind various ultra-images.'' 
Remember that deep understanding of what these new terms might 
signify requires an investigation of the relationships between 
such terms as expressed by hundreds of such statements. Suppose S 
denotes the consequence operator that characterizes the simple 
human mental process called propositional (sentential) deduction. 
Then S is a finitary consequence operator and all of the 
consequences S(B) that can be deduced from a set of premises B 
subset P are obtained by deduction from the finite subsets of B. 
Now there exists an ultra-logical process, denoted by D, defined 
on subsets of the ultra-language *P, where D has, at least, the 
same properties as those of the logical process S when D 
operators on finite subsets of the humanly comprehensible 
language P (Note 1).
What happens when the ultra-mental process D is applied to any 
finite subset F of the humanly comprehensible language P. The set 
of consequences D(F) contains all of these consequences S(F) 
comprehensible by entities within the universe (i.e. S(F) subset 
D(F)) and many that are not comprehensible by entities within the 
universe. Using consequence operator terminology, when this 
occurs, the ultra-mental process being modeled by the consequence 
D is said to be stronger than the mental process modeled by S. It 
is this and other, yet to be described, properties that led to 
the selection of the term  ``ultra'' as a prefix. Further, no 
entity within the universe can duplicate the ultra-mental process 
D, and this process also has numerous properties that are not 
comprehensible by any entity within the universe (Note 2).
There is a delicate analysis that can reveal the composition for 
some of the ultra-words in *P, where w in the ultra-language *P 
is an ultra-world if it is not a member of P. What this analysis 
details is often quite startling. For example, there are ultra-
hypotheses, a single one of which is denoted by w, that cannot be 
comprehended by entities within the universe and that, when the 
ultra-mental process D is applied to w, yields a consequence that 
can be comprehended by entities within the universe. These ultra-
hypotheses exist in subsets of *P that, at least, have the same 
characterizing properties as sets that describe human behavior, 
natural laws and the like. For example, if a sentence x in P 
describes a certain human behavior trait, then, although there 
may not appear to be a hypothesis h in P from which x can be 
deduced by the human mind, there does exist in *P an ultra-
hypothesis w such that the ultra-mind process D when applied to w 
yields the conclusion x.
There are other mental processes that seem to correspond to 
intelligence. One of these is choosing from a list of statements, 
that is potentially infinite, a specific finite set that is 
meaningful for a particular application. Embedding this finite 
choice process into the deductive-world model yields the same 
type of conclusions as those for the ultra-logic D. This ultra-
mind process cannot be duplicated by any entity within the 
universe, it is stronger than all such mental processes and has 
properties that in all cases improve upon the mental process of 
finite choice (Herrmann, 1991).
Another human reasoning process is the dialectic. Basic 
characterizing expressions can be listed for many such dialectics 
(Gagnon, 1980). Such dialectics can be applied to any language E 
constructed from two or more symbols. The basic ingredients are a 
set of theses T, a set of antitheses A, and an operator Sy, among 
others, which yields a synthesis z for any t in T and some a in 
A. For all the dialectics listed by Gagnon (1980), it is not 
difficult to show that there exist sets of symbol strings T and A 
and operators such as Sy that when embedded into the deductive-
world model become sets of ultra-theses, ultra-antitheses and, an 
ultra-mental process, the ultra-synthesis operator *Sy (Herrmann, 
1992). Once again, the same type of conclusions hold for these 
ultra-dialectics as holds for the ultra-logic D.
It appears that all forms of such mental-like processes are 
improved upon, to an extreme degree, by their corresponding ultra-
mental processes. When the collection UM of ultra-mental 
processes is compared, as a whole, with the corresponding set M 
of mental processes that are displayed by humanity, then it 
appears reasonable to characterize the collection UM as 
representing a higher intelligence. The logical existence of UM 
is obtained by use of the most fundamental tool of modern science 
and establishes that the acceptance of the existence of a 
supernatural higher intelligence is scientifically rational and 
verifies the conclusions discussed in the introduction to this 
paper. Moreover, any properly stated model MH that either 
specifically utilizes such a postulate or logically implies the 
existence of a supernatural higher intelligence cannot be 
rejected as somehow or other not being scientific in character. 
Indeed, if such a model MH explains past natural events or human 
experiences, and predicts other events as they are observed 
today, then the scientific method explicitly states that such 
models are to be considered as good as or even better than other 
Although this discussion could be concluded at this point, one 
interesting question is suggested. Has such a higher intelligence 
been previously described using terms and concepts that parallel 
those for the above ultra-mental processes?
*Significance of Results*
Although a comparison with the doctrine of all of the major 
religious belief-systems has not been made, there does exist a 
strong correlation between these results and statements that 
appear in the Jewish and Christian Bibles. The Bible, when 
literally interpreted, often describes God's attributes in terms 
of a linguistic or a mental model. This is especially the case 
when the mind of God is compared to the mind of man. In every 
single case, the ``mind of God'' Scriptural statements are 
modeled by the above special deductive-world interpretations. 
This is a startling fact since the deductive-world model was not 
created originally for application to theological concepts.
As examples, every time the Scriptures state that God ``speaks'' 
to a prophet, or a Jew or Christian then the above special 
interpretation is verified. Indeed, all statements that compare 
God's wisdom, intelligence and the like with that of humanity are 
satisfied by this special interpretation as are numerous 
statements relative to the supernatural means that God employs to 
communicate with an individual.
Here is a partial list of such statements. Genesis 1:26; Numbers 
23:19; Deuteronomy 33:26; 1 Kings 8:23, 27; 2 Chronicles 2:5; Job 
9:4, 10, 11:7, 8, 12:13, 15:8, 28:12--13, 20--24, 32:8, 33:12, 
14, 37:23, 38:33, 36; Psalm 35:10, 53:2, 77:13, 86:5, 93:5, 
94:11, 119:27, 99, 100, 139:2, 6, 17--18, 147:5; Proverbs 2:6: 
Ecclesiastes 2:26, 3:11, 8:17; Isaiah 55:8--9; Jeremiah 10:10--
13, 17:10, 31:10; Daniel 2:21--22, 46; Matthew 10:20; Mark 13:12, 
13; Luke 6:8, 10:21, 22, 21:15, 24:45: John 8:47, 10:16, 27, 
12:40, 14:26; Romans 11:33--34; 1 Corinthians 1:10, 19--20; 2:10, 
13, 16; 2 Corinthians 10:4; Ephesians 1:17; Colossians 2:3, 4; 2 
Timothy 2:7; James 1:5.
Even if not specifically related to doctrinal statements, the 
logical existence of a supernatural higher intelligence is 
obviously significant for any supernaturally related belief-
system and modern creation-science. It is no long advisable to  
categorize human religious experiences and scientific models that 
are associated with a supernatural higher intelligence as being 
somehow or other irrational in character. Indeed, if such 
experiences or creation-science models directly correlate to a 
literal Bible interpretation, then the assumption of 
irrationality can be scientifically proved to be false. Finally, 
since application of the basic tool used for modern scientific 
research has established that it is  scientific to assume the 
existence of a supernatural higher intelligence, a properly 
constituted creation-science model that relies upon this 
assumption is not ``pseudoscience'' as has been claimed. Note 
once again that if such a model increases our capacity to 
understand the workings of the natural realm, then the scientific 
method specifically states that such a model is the preferred 
1. Mathematically the purely subtle consequence operator C1 on 
all of the internal subsets of *P is D. See Herrmann (1987), 
Theorem 4.5.
2. This comes from the fact that the formalized first-order 
theory of the propositional calculus is an infinite set and as 
such when embedded into the D-world model this metatheory 
generates infinitely many incomprehensible statements that behave 
like logical rules for the ultra-logic D.
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definitions of humanism. ed. P. Kurtz,  Prometheus Books. 
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Feuerbach, L. 1967.  Lectures on the essence of religion. 
Translated by R. Manheim. Harper & Row. New York.
Gagnon, L. S. 1980. Three theories of dialectic. Notre Dame 
Journal of Formal Logic. XXI(2):316--318.
Herrmann, R. A. 1981. Mathematical philosophy. Abstracts American 
Mathematical Society. 2(6):527.
Herrmann, R. A. 1987. Nonstandard consequence operators. Kobe 
Journal of Mathematics. 4(1):1--14.
Herrmann, R. A. 1991. Ultralogics and more. Institute for 
Mathematics and Philosophy, P. O. Box 3268, Annapolis, MD 21403-
Herrmann, R. A. 1992. Ultra-dialectics. Institute for Mathematics 
and Philosophy, P. O. Box 3268, Annapolis, MD 21403-0268.
Marx, K.  and F. Engles. 1960. On religion. Translated by the the 
Institute of Marxism-Leninism. Foreign Languages Publishing 
House. Moscow.
Santayana, G. 1905. Reason in religion. Charles Scribner's & 
Sons. New York.
Tarski, A. 1930. "Uber einige fundamentale begriffe der 
metamathematik. Comptes Rendus de seaces de la Spciete des 
Sciences et des Lettres de Varsovie. 23 cl. iii:22--29.
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living systems. Impact # 236. Institute for Creation Science, El 
Cajon, CA.
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