##
John O. Campbell

This is an excerpt from the book: Einstein's Enlightenment.

## Logic is the branch of philosophy which studies valid reasoning. When Aristotle introduced the subject of logic to philosophy, one of his most significant contributions was the logical syllogism where the truth of some statements is obvious merely due to the form of the statements. For example:

1) All men are mortal

2) Socrates is a man

3) Therefore, Socrates is mortal

It is
obvious, given the first two statements, that the third must be true.

Later
three basic logical operators were identified: ‘and’, ‘or’ and ‘not’. These operators
could be used to describe relationships between logical variables which may
have the value of either ‘true’ or ‘false’. All possible relationships between
logical variables may be described using these operators leading to the field
of Boolean Algebra. In turn logical relationships cast in terms of Boolean
Algebra serve as the foundations from which all mathematics may be derived.
This astonishing understanding shows that all mathematics is but the
implications held within binary true/false relationships.

A more
concrete example of the power of logic is that the three logical operators may
be cast in the form of electronic circuits and used to construct a computer.
General purpose computers or universal Turing machines are capable of computing
all mathematically computable functions.

Therefore,
logic is a central discipline at the basis of mathematics, computation and much
of science. Charles Sanders Peirce, the
great American philosopher, made a number of fundamental advances in these fields and was able to envision the central role played by logic
within philosophy. He understood logic primarily as the rules by which the
‘world mind’ composing the universe evolves, an observation very much in the
spirit of Einstein’s Enlightenment. A first step towards this conclusion is
Peirce’s contention that logic forms the rules of thought or mind (1) :

*The term ”logic” … in its broader sense, it is the science of the necessary laws of thought*

Second, Peirce viewed ‘thought’ and ‘mind’ as not necessarily associated with brains but rather as general properties of
nature (1) :

*Thought is not necessarily connected with a brain. It appears in the work of bees, of crystals, and throughout the purely physical world; and one can no more deny that it is really there, than that the colors, the shapes, etc., of objects are really there. … Not only is thought in the organic world, but it develops there*

Peirce
was a first class logician, mathematician and scientist but he understood these
within a context of mind or thought; logic, mathematics and science merely
expressed the rules for clear thought. He portrayed mind or thought as a
property of nature and thus ventured beyond the usual materialistic view of
science.

To a
surprising extent Peirce demonstrated many of these basic relationships between
logic and mind a half century earlier than the pioneers of our current
information and computer ‘revolutions’. Sadly, his work was ignored and he was left
largely without credited.

Peirce
was the first to prove that all logical propositions, including Boolean algebra
and mathematics, could be constructed from relations between two primitive
fundamental logical quantities: ‘not-and’ and ‘not-or’. He used this insight to
understand that these two logical quantities could be implemented in electronic
circuits and thus was the first to suggest that all logical statements could be
cast in the form of electronic circuitry. He and his colleagues actually developed
a wiring diagram of a ‘mechanical logic machine’ (2) .

In this
he very much foretold the basis of our current understanding of computation.
Given his vision that the processes of logic operate within a universal
property of thought or mind he also presaged the current view among physicist
that the universe is primarily engaged in the computation of an evolving logic (3) .

Peirce’s
great understanding of the relationship between logic and thought or mind was
ignored by researchers largely because of his meager status on the peripheries
of academia but also because he was so far ahead of his time. It was not until
1943 that Warren McCulloch and Walter Pitts
rediscovered the same fundamental relationships underlying logic and thought (4) :

*Many years ago one of us, by considerations impertinent to this argument, was led to conceive of the response of any neuron as factually equivalent to a proposition which proposed its adequate stimulus. He therefore attempted to record the behavior of complicated nets in the notation of the symbolic logic of propositions. The "all-or-none" law of nervous activity is sufficient to insure that the activity of any neuron may be represented as a proposition. Physiological relations existing among nervous activities correspond, of course, to relations among the propositions; and the utility of the representation depends upon the identity of these relations with those of the logic of propositions.*

This seminal understanding of logical propositions is fundamental
to both the fields of artificial intelligence and cognitive psychology.

The philosophy developed by Peirce puts
logic at its core, it insists that the world acts logically. This logic is
played out through semiotic, the relationship among object, sign, and an
interpretant or receiver of the sign.

In Peirce’s philosophy, it is the shunting
of signs between objects and interpretants - which themselves may act as signs
- that causes the universe to learn and evolve. Our minds are described as
subsumed within this greater world mind. According to his philosophy,
information bearing signs are perhaps the most fundamental entity in the
universe (5) :

*all this universe is perfused with signs, if it is not composed exclusively of signs*

Figure 4: Peircean semiotics describes logic and
information transfer as a triadic relationship among objects, signs which
signify the objects, and interpretants which draw conclusions when receiving
signs. In the case pictured, the fire is the object, the smoke is a sign
signifying the fire and a human who sees the smoke is the interpretant who
interprets the smoke as signifying a fire. Peirce consider semiotics as the
basic rule governing the relationship of entities.

This view is remarkably similar to that of
the current ‘information revolution’; the understanding, across many branches
of science, that information is perhaps the most fundamental entity in our
universe. How could Peirce have scooped current researchers by over fifty years? Why doesn’t our information age rhetoric acknowledge the
contributions made by Peirce’s philosophy? The answer is that Peirce’s
philosophy was ignored, in fact it was nearly lost.

Upon his death, Peirce’s widow sold his
largely unpublished papers to Harvard’s library. There they moldered for many
years and many still remain unpublished. Although currently gaining in
recognition Peirce’s work is still largely unknown. Perhaps Peirce was simply
too far ahead of his times to be understood.

Instead it was the mathematical description
of information, developed by Claude Shannon (6) in 1948, which would
form the foundation of information theory. The business of Bell Labs, where
Shannon was a researcher, was to understand the nature of information and
Shannon’s work is widely regarded as the foundation of our modern understanding
of information and its context.

In his description, information is related
to probability; information is a function of probability, if a high probability
is assigned to a possible outcome of an event and if it happens to be the
actual outcome then there is not much information gain but if a low probability is
assigned to an actual outcome then a lot of information is received. A complete
set of possible outcomes may be assigned probabilities and the resulting
distribution constitutes a probabilistic model of the event.

The information contained in evidence
regarding the actual outcome implies a recalculation of all the probabilities
making up the model. The model is updated in accord with Bayes’ theorem. In
this understanding information does not exist in isolation, something must be
informed and that something is a model. This view of information’s reliance on
a model is analogous to Peirce's view of a sign’s reliance on an interpretant.

Only now is an important implication of Shannon’s theory becoming apparent, that although information may be a
fundamental component of the universe it is far from simple. Information does
not exist outside of the complex context of a probabilistic model (7) . Long before
Shannon, Peircean semiotic captured that notion, that information or signs do not
have an existence outside of a context of objects which generate the sign and
interpretants which interpret them.

One of
the many technical contributions which Peirce made to logic was his development
of existential graphs. This system of logical notation serves to greatly
simplify complicated logical expressions and his work initiated a program
within the study of logic towards notations which would reveal the underlying
simplicity of logical expressions. After all, in the end, once all the values
of variables are stated, every logical expression is simply either true or
false.

The program for the simplification of logical notation which Peirce initiated was concluded in 1969 with the publication of
George Spencer-Brown’s Laws of Form (8) . Spencer-Brown
developed an ultimate simplification of logical notation which uses only one
symbol. As Louis Kauffmann explains (9) :

Using this symbol two types of logical
operations may be described, the first a type of repetition with two marks in
succession, the second adding no new information. We may think of the mark as
the crossing of a distinction; from that view the first operation merely states
that to twice state a position in regard to a distinction is the same as only
stating it once. The second logical operation is a type of negation with one
mark inside another which cancel each other to nothing. From the view of distinction,
we can interpret this to say that if we cross a distinction twice it is the
same as not crossing it at all.

The mark may be conceived of as a type of
negation and our two operations as two types of double negatives. The first is
redundant as in ‘I cannot, I cannot go to sleep’, the second ‘cannot’ is
redundant and provides no further meaning than does ‘I cannot go to sleep’. The
second type of double negative is cancellation as in ‘I cannot not go to sleep’.
Here the negatives cancel and the meaning is ‘I will go to sleep’.

Spencer-Brown used two relationships
between the marked symbols to indicate these two types of negation, the first a
repetition and the second a cancellation:

Using this deceptively simple notation it
is possible to almost immediately see the solution to some very complicated
logical puzzles. For example, Spencer-Brown demonstrates the solution to a
famous puzzle posed by Lewis Carrol (10) :

This
amazing ability of Spencer-Brown’s system to simplify complicated logical
problems is just one aspect of its power. It has been demonstrated that it can also
be used to derive the axioms of Boolean Algebra and hence all of mathematics. It
is an amazing fact that all of mathematics may be derived merely from the
nature of a binary distinction (10) .

The
great mathematician and physicist Louis Kauffman has made extensive
investigations into Spencer-Brown’s research and has concluded (8) :

*It remained for Spencer-Brown (some fifty years after Peirce and Nicod) to see the relevance of an arithmetic of forms underlying his notation and thus putting the final touch on a development that, from a broad perspective, looks like the world mind doing its best to remember the significant patterns that join logic, speech and mathematics.*

Indeed,
the image of a ‘world mind’ at the basis of the universe appears almost
inescapable in the philosophical systems of both Peirce and Spencer-Brown. This
may be disturbing to some as it denies a long-held scientific assumption that
materialism is the basic property of the universe but this would seem
unavoidable once one concedes that information is also a basic constituent. After all mind may only be a euphemism for information processing
which, it now seems clear, is a basic natural operation.

We
should remember that Spencer-Brown’s powerful system is based upon only two
simple observations concerning the nature of distinctions. We should also
remember that the bit, the fundamental unit of information, is merely the unit
of information which records a binary distinction.

Figure
5: George Spencer-Brown

Some researchers
have marveled at ‘the unreasonable effectiveness of mathematics in the natural
sciences’ (11) ; why are so much of
the natural sciences so well described by the patterns of mathematics? We may
perceive the outlines of an answer:

1) Information is perhaps the most fundamental component of reality.

2) Information is composed of yes/no binary distinctions.

3) Any effective description of reality, such as mathematics, must be
derivable from the nature of binary distinctions.

Inherent to Spencer-Browns conception of
‘distinction’ is both a ‘mark’ to mark the distinction and an ‘observer’ to
interpret it. The concluding sentence of Laws of Form, is:

*We see now that the first distinction, the mark and the observer are not only interchangeable, but, in the form, identical.*

It has been noted that Spencer-Brown’s
paradigm meshes well with Peircean semiotics (8) if we equate
distinction with object, mark with sign and observer with interpretant. The
choice of the word ‘observer’ should be understood here in the Einsteinian
manner as a generalized entity able to react to measurements or data or some
other instantiation of a sign. I suggest the word ‘model’ be used to guard
against anthropocentric interpretations which may be attached to ‘observer’.

Spencer-Brown’s logic should be understood
as an extension of Peirce’s philosophical program, one which presages
Einstein’s Enlightenment. Although the model of logic and semiotics developed
by Peirce and Spencer-Brown may be suggestive of a world mind capable of
processing and building knowledge it lacks details. Clearly it offers some
insight into the relationship between models (interpretants) and that which is
modeled (objects), a relationship which might be considered to be knowledge,
however details of the mechanisms which relates object to sign and sign to
model are lacking.

If semiotics is to serve as a useful
paradigm for knowledge it should be expected to encompass science. Indeed,
science, understood as a process of Bayesian inference, may be cast in a triad
form compatible with semiotics: phenomena, data, and model. The inclusion of
Bayesian inference brings a particular strength to the semiotic paradigm.
Bayesian experimental design provides a mechanism for extracting data (signs)
from the phenomena (object) so as to maximize the expected knowledge gain of
the model (interpretant). It also provides the mechanism for using data to most
effectively update models in the form of the Bayesian update.

Probability distributions employed as
models within Bayesian inference have the property of information entropy; the
amount of information which separates the model from certainty. If the log of
the probabilities used in the inference is to base 2 then this information is
measured in bits. Bits are the basic unit of distinction. The model’s entropy
is the number of binary distinctions which separate it from certainty. While models treated within Bayesian inference may be separated from certainty by
any finite number of distinctions, the special case of one bit separations from
certainty is the case where Bayesian inference becomes equivalent to classical
logic. In Spencer-Brown’s terms single bit models are ‘observers’ of the first
distinction.

Models within Bayesian inference are moved
towards certainty by incorporating the implications of data (sign) into the
model (interpretant); through the process of Bayesian updating. Mathematically
this is the unique method of moving a model towards certainty (12) ;
that is of increasing knowledge.

Bayesians have frequently committed the error of assuming that knowledge is a human property and that Bayesian inference is solely descriptive of human activities. Spencer-Brown did not indulge in this misconception; he maintained a focus on the world mind (10) :

Bayesians have frequently committed the error of assuming that knowledge is a human property and that Bayesian inference is solely descriptive of human activities. Spencer-Brown did not indulge in this misconception; he maintained a focus on the world mind

*Thus we cannot escape the fact that the world we know is constructed in order (and thus in such a way as to be able) to see itself.*

*This is indeed amazing.*

*Not so much in view of what it sees, although this may appear fantastic enough, but in respect of the fact that it can see at all.*

*But in order to do so, evidently it must first cut itself up into at least one state which sees, and at least one state which is seen. In this severed and mutilated condition, whatever it sees is only partially itself.*

Like Peirce, Spencer-Brown did not have a
large impact on the scientific community. His work is provocative and its
reception was varied but muted. Some extreme claims have been made for its mathematical importance.
The book jacket of some editions of Laws of Form contains an endorsement from
Bertram Russell “Not since Euclid’s Elements have we seen anything like it’. The
book jacket of my edition contains excerpts from reviews in Nature and the
British Journal of the Philosophy of Science; both describe it as ‘a work of genius’. Others however have dismissed it as a work of mysticism with little
mathematical content. That view has not been mitigated by Spencer-Brown’s
subsequent claim to be the reincarnation of the Buddha and not just any old
Buddha but rather

__the__Buddha who comes only once every 2500 years. Perhaps as a result his work has been largely shunned within the mathematical community.
A ‘world mind’ may at first seem more akin
to a supernatural rather than a scientific concept but it is clearly resolved
within a scientific context if we accept that:

1)
Information is a fundamental
component of the universe.

2)
The universe evolves through
numerous nested Darwinian processes (universal Darwinism).

3)
Darwinian processes are
isomorphic to Bayesian inference. The process of evolution accumulates
information into knowledge through an evidence-based processes.

This scientific context describes a
continual universal process of evolution which processes information into
knowledge. This is a process, aptly described as that of a ‘world mind’, in
which the world comes to better know itself.

This view, initiated by Peirce, places
logic or the ‘necessary laws of thought’ guiding the world mind at the
foundation of metaphysics or the study of the nature of reality.

#
Bibliography

1.

**Peirce, Charles S.***Collected Papers of Charles Sanders Peirce.*1906. p. 101. Vol. 6.
2.

**Wikipedia.**Charles Sanders Peirce bibliography.*Wikipedia.*[Online] [Cited: 1 10, 2016.] https://en.wikipedia.org/wiki/Charles_Sanders_Peirce_bibliography.
3.

**Lloyd, Seth.***Programming the Universe.*s.l. : Vintage; Reprint edition, 2007.
4.

*A Logical calculus of the ideas immanent in nervous activity.***McCulloch, Warren and Pitts, Walter.**1943, Journal of Mathematical Biophysics, Vol. 5.
5.

**Peirce, C.S.***Collected papers of Charles Sanders Peirce: The basis of pragmatism.*1906. Vol. Voume 5.
6.

*A mathematical theory of communications.***Shannon, Claude.**1948, Bell System Technical Journal.
7.

**Campbell, John O.***Darwin does physics.*s.l. : CreateSpace, 2015.
8.

*The Mathematics of Charles Sanders Peirce.***Kauffman, Louis H.**2001, Cybernetics & Human Knowing, Vol.8, no.1–2, pp. 79–110.
9.

**Kauffman, Louis.***Laws of Form - An exploration in mathematics and foundations.*s.l. : Unpublished rough draft.
10.

**Spencer-Brown, G.***The Laws of Form.*New York : E.P. Dutton, 1979.
11.

*The Unreasonable Effectiveness of Mathematics in the Natural Sciences.***Wigner, Eugene.**1960, Communications on Pure and Applied Mathematics 13, pp. 1–14.
12.

**Jaynes, Edwin T.***Probability Theory: the logic of science.*Cambridge : Cambridge University Press, 2003.