Knowledge and Truth

Copyright 2016 Graham Berrisford. One of about 300 papers at Last updated 25/09/2017 19:54


Enterprise architecture is about business system planning.

It can be seen as applying the principles of general system theory – which we’ll get to later.

First, what theory underpins general system theory?

This paper is one of several on theories of information, communication, language, knowledge and description.


Systems are islands of orderly behavior, which are describable.

To apply system theory is to express some knowledge of a system as it is observed now, or envisaged in the future.

What is knowledge? How is it acquired? Must knowledge be true or false, or can it be in between?



Knowledge use and acquisition.. 1

Laws of concrete reality (physics). 3

Laws of abstract description (logic or thought). 4

Testing the truth of knowledge. 5

Conclusions and remarks. 8

Footnote on the laws of logic. 9


Knowledge use and acquisition

You may have come across one or other version of something called a WKID triangle or pyramid.

Most models of this kind are questionable, but here is a version compatible with communication theory.




having enough knowledge (and the ability) to respond effectively to novel information


having information that is accurate enough to be useful


extracting meaningful facts from data, or being able to do that


holding an encoded form of information in a message or memory space


Knowledge is useful information.

Your knowledge of the world is used when you describe it and predict it.


What use is knowledge?

You know you must step off a railway track to avoid be running over by an onrushing train.

That truth of that knowledge is testable by gruesome experimentation.

Knowledge (along with emotions like love and fear) helps you to survive, thrive and pass your genes on.


Knowledge as a descriptive mental model

Suppose you predict the outcome of staying on the track, step off and so live to tell the tale.

This implies the knowledge was somehow encoded in your mental model of train behavior.

The bio-electro-chemical form of that mental model is deeply mysterious.

Perhaps it is a network that connects related images, symbols, sensations and experiences held in memory.

All that matters here is that the presence of that mental model is demonstrable.


Knowledge as a biological phenomenon – inherited

Your innate fear of onrushing objects will surely prompt you to step out of the path of an onrushing train.

Animals inherit mental models that gave their ancestors an evolutionary advantage.

E.g. Kittens innately know the properties of a mouse’s tail, and respond animatedly to any long, thin, wiggly thing.

Experiment has shown that babies fear crawling over the edge of what appears (in a visual illusion) to be a cliff edge.

I have watched a seated cat rise from the road in front of my car, walk to the side of the road and sit down again.


Knowledge as a psychological phenomenon - learnt through experience

Watching a train squash an apple may teach you about the danger of standing on railways tracks.

You acquire some knowledge through perceiving and remembering, through conditioning, through trial and error.

Recognising similarities between (aka classifying or typifying) situations, entities and events is a mark of intelligence and essential to learning.


Knowledge as a social phenomenon - acquired through social communication

You and I have somewhat different mental models of train behavior.

Our mental models of standing on a train track includes the knowledge that it is dangerous.

In other words, it has the quality we encode in verbal language as “dangerous”.

Our knowledge is wrapped up with having a word for that quality.

And our ability to communicate qualities is massively increased by having such rich verbal languages.

Provided we share mental models and languages well enough, you’ll step off the track when I tell you the train is coming,


In short, you acquire knowledge through a mixture of inheritance, experience and communication.

A knowledge triangle


<inherit and acquire>   <describes and predicts>

Knowing actors  <observe and envisage>  Realities


Is knowledge necessarily true? Or is only some of it certainly true?

You may be thinking, surely some knowledge is certain - there are universal and incontrovertible laws?

Laws of concrete reality (physics)

The solar system is a natural system.

Humans have described how the sun and the planets interact by following the laws of gravity and motion.

More generally, people have defined laws that typify the behavior of physical matter and energy in space and time.

This triangle describes the role of physicists in creating and using laws.


Law of space and time

<create and use>               <idealise>

Physicists  <observe and envisage>  The universe


Surely the laws of physics - rules that apply to physical matter and energy – are certain knowledge?

Actually, in history of cosmology one theory has been supplanted by another over several centuries

The sequence is often presented along these lines.


·         Aristotle and Ptolemy thought the sun revolved around the earth.

·         Copernicus placed the sun at the centre of the universe.

·         Galileo championed Copernicanism, and made discoveries in the kinematics of motion and astronomy

·         Kepler’s laws of planetary motion provided a foundation for Newton's theory of universal gravitation.

·         Newton defined the theory of universal gravitation and three laws of motion.

·         Einstein showed Newton’s laws only approximately describe the motion of things – in our part of the universe.


Newton's laws of motion remain true enough at the scales of space, time and energy humans deal with.

Einstein’s laws are not the whole truth either.

“As Einstein would have happily admitted [his] new physics was not a definitive answer, nor did it negate the importance of Newton’s contribution.

It was not “right” or “true”, but simply a more accurate explanation that Newton’s”, which was perfectly good for its time.

As a pragmatist would say, it was a valid explanation” Marcus Weeks


In retrospect, Newton’s and Einstein’s laws presume three basic axioms about the universe.

1.      An object cannot both exist and not exist at a given point in space and time.

2.     An object must either exist or not exist at a given point in space and time.

3.     An object cannot exist at different points in space at the same time.


These seem intuitively obvious and certainly true statements.

Yet even these are undermined by quantum mechanics.

Heisenberg’s uncertainty principle says an object’s position and speed cannot both be described exactly, at the same time.

And to Einstein’s great discomfort, a bizarre property of quantum mechanics is that an electron can be two places at the same time.


The modern scientific view is that we never know the universe as it really is, only as it is described in mental/documented/other models.

Laws of abstract description (logic or thought)

OK, some laws of physics are generalisations that are imperfect but accurate enough to be useful.

What about the laws of logic?


The three basic laws of logic are premises that describe mathematical expressions and verbal propositions.

Interestingly, they can be seen as reflecting the presumed laws of concrete reality above.

1.      The law of contradiction: A proposition is not both true and false.

2.      The law of excluded middle (or third): A proposition is either true or false.

3.      The principle of identity: A proposition true of x must be true of x; OR x is identical with x.


I guess many mathematicians see those as intuitively obvious, certainly true statements.

Indeed, they are true in most mathematical descriptions of the universe.

Yet mathematicians have debated these laws, and now allow that a proposition might be neither true nor false.


Suppose we try to recast the laws of logic as laws of description.

1.      The law of contradiction: A description is not both true and false.

2.      The law of excluded middle (or third): A description is true or false.

3.      The principle of identity: A description true of x must be true of x; OR x is identical with x.


Intuition rejects at least the first two laws.

We all know a natural language description might be entirely true, entirely false, a mix of both, or uncertain.

Testing the truth of knowledge

Social communication spreads exhortations, suppositions, babble and nonsense as well as objective knowledge.

Our survival depends in part on recognising when assertion is not truth, fake news is not news, correlation is not causation, and pseudo science is not science.


How do we know knowledge is true?

A J Ayer said knowledge is “justified true belief”.

The justification must be verification by one of two means:

·         Logical analysis or manipulation of descriptive elements according to agreed rules (e.g. 2 + 2 = 4).

·         Empirical testing of propositions about real-world entities and events.

Logical analysis

Mathematicians use logical analysis to prove conclusions drawn from axioms (which they can’t prove).

Logical analysis has been wildly successful in mathematics.

By this means, mathematicians may conclude their models are perfect.

Logical analysis is not so in successful in investigations of nature.

Empirical testing

Science means knowledge in Middle English (via Old French from the Latin scientia and scire “know”).

Outside of mathematics, the scientific method is how we seek to prove the truth of knowledge.



It is easy to deceive ourselves.

We continually validate our knowledge by inspecting it and discussing it with others.

We engage in a kind of collective mental modelling, to hone the accuracy of our knowledge.

However, the people we converse with may be equally deluded.



Better, we can seek to verify a proposition or model by realising and testing it.

We judge the truth of knowledge by how well it predicts what happens in reality.

Still, no amount of successful tests can prove a model is true, since it may fail the next test.



Karl Popper famously proposed the test of a good theory is that it can be falsified.

His idea has proved immensely useful to the progress of science; the best kind of scientific theory

a)      fits all circumstantial evidence,

b)      passes all tests devised to disprove it and

c)      could be disproved by a future test.


Hovering behind a good theory is always the spectre of falsification by a future test.

A theory that could never conceivably be disproved is considered weak – more a declaration of faith or belief.


And yet - even falsification does not necessarily invalidate knowledge.

Newton’s laws of motion remain useful knowledge, they help us deal with the world we live in.

They are propositions or models that describe and predict reality well enough.

Scientists strive to recognise the limits of their models and understand when/where they are best applied.


The limitations of science

Wonderful and powerful as science is, how certain is the knowledge that it gives us?

“The half-life of knowledge is the amount of time that has to elapse before half of the knowledge in a particular area is superseded or shown to be untrue.”

Fritz Machlup (1962) The Production and Distribution of Knowledge in the United States.


There is a spectrum of precision and certainty in science - from hard mathematics and physics to softer economics and sociology.

“The half-life of a physics paper is on average 13.07 years, in Math it’s 9.17 years, and in Psychology it’s 7.15.”

Samuel Arbesman (2012). The Half-life of Facts: Why Everything We Know Has an Expiration Date.


Worse: there is now so much published in the name of “science” that scientific journals have become an unreliable source.

“Most published scientific research papers are wrong, according to a new analysis. …

there is less than a 50% chance that the results of any randomly chosen scientific paper are true.”


And perhaps most depressing of all to a scientist.

Pseudo-scientific papers can survive longer than scientific ones, because it is impossible to disprove them.


Fluid and fuzzy truth

Philosophers have long debated the nature of truth.

In the natural sciences, truth is a fuzzy concept that can be determined with a degree of certainty rather than complete certainty.


Absolute truth

Absolute truth and falsehood appear to exist in the logical world mathematics and description.

You can be certain a square is a rectangle; because the description of those things is entirely in your gift.

A scientific theory may be considered true when experimental results agree with predictions made using theory.

Similarly, a description of a particular thing is considered true when the qualities of that thing match the qualities in the description.

For example, you say the sky is blue, and when tested by observation, the sky is indeed blue.


Fluid truth

Outside of very stable domains of knowledge, the types used to define qualities can be fluid.

For example: Is Pluto a planet? Once, the question was unanswerable, then the answer was true, now it is false.

To begin with, there was an unknown, unnamed and unclassified body in space.

When that body was discovered, it was named “Pluto” and classified as an instance of “planet”.

But the definition of a planet’s qualities has changed over the centuries, and changed recently such that Pluto is no longer an instance of “planet”.

The thing may have remained the same, the definitive description of it has changed.


Fuzzy truth

Remember the law of the excluded middle?

In modern systems of knowledge, some probability logics have degrees of truth-value between truth and falsity.

Especially, or at least, when the proposition declares something will be true in the future.


In the mathematics of fuzzy logic, predicates are the functions of a probability distribution.

This replaces a strict true/false valuation by a quantity interpreted as the degree of truth.

In natural language, the words we use to describe things are fuzzy polythetic types, meaning the described thing need not wholly match the word describing it

So, we convey meaning with probability rather certainty, and interpreting the meaning of a communication is somewhat fuzzy.


Fuzziness in physical measurement

Is the Yankees baseball ground bigger than Lords cricket ground?

Your certainty about the answer depends on how accurately you can measure them.

The truth of a proposition about the physical world is determined by how accurately you measure physical matter and energy.


Fuzziness in social communication

Suppose two honey bees observe a third honey bee dancing to describe where pollen can be found.

The first bee finds the pollen, and regards the first honey as telling the truth.

The second bee fails to find the pollen, and regards the first honey bee as telling a lie.


The truth as you see it depends on your mental model, but others may have different mental models of the same reality.

How to test the truth of a proposition?

We can ask observers (a judge or jury) to examine a proposition, and give us a verdict

Or else devise test cases with predicted results, and compare the predictions with the actual results of running the tests.

Either way, there is room for fuzziness, or a margin of error.


Fuzziness in classification or type definition

Again, we describe particular things by using universal types to classify or typify them.

A particular thing

instantiates a named “type” when

it gives values to property types of that type

A circus ring

instantiates the type named “circular” when

its diameter is the same in all directions

A play

instantiates the type named “Hamlet” when

it follows the script written by Shakespeare

The rose bush in my garden

instantiates the type named “rosea” when

it exhibits the property types “thorny, flowering, bushy”


But there is fuzziness in the definition of types and in the identification of instances.

So, we often allow a margin of error.

·         This circus ring is “circular”. The earth is “spherical”. This chessboard is “square”. (What margin of error is allowed in measurement?)

·         This play is a “performance of Hamlet”. (How strictly must it follow the script?)

·         This plant is a “rose bush”. (What if it has only two thorns and it grows sturdily to the height of a tree?


It might be that none of above statements is 100% true, yet all are true enough for practical purposes.

Conclusions and remarks

Knowledge is fuzzy, there are degrees of truth.

We can reasonably point to a particular circus ring and call it “circular”.

But on close inspection, no circus ring is perfectly circular; it is only near enough circular to be usefully described thus.


A truth triangle

True-enough propositions

<create and use>    <describe and predict>

Rational actors <observe and envisage> Realities


Given a proposition or model, then we might call it:

·         a supposition, speculation or hypothesis if it has never been tested

·         knowledge if it has been tested successfully

·         babble or nonsense if it fails to pass tests we agree are important.


In practice, these distinctions are fuzzy.

Much knowledge is a proposition or model that describes or predicts a reality well enough.

Generally speaking, the propositions of sociologists and economists are less certain and reliable those of chemists and physicists, and the “true enough” tests are more relaxed.

Footnote on the laws of logic


The law of excluded middle – the debate

The following is lightly edited from:

[A doctrine of traditional logicians was that] the laws of thought are a sufficient foundation for the whole of logic.

[And] all other principles of logic are mere elaborations of them.

It has been shown, however, that these laws do not even comprise a sufficient set of axioms for the most elementary branch of logic (the propositional calculus)...


Aristotle cited the laws of contradiction and of excluded middle as examples of axioms.

He partly exempted future contingents, or statements about unsure future events, from the law of excluded middle.

Holding that it is not (now) either true or false that there will be a naval battle tomorrow.

[Rather] the complex proposition that either there will be a naval battle tomorrow or that there will not is (now) true.

In the epochal Principia Mathematica (1910–13) of A.N. Whitehead and Bertrand Russell, this law occurs as a theorem rather than as an axiom.


The law of excluded middle and certain related laws have been rejected by L.E.J. Brouwer, a Dutch mathematical intuitionist.

His school do not admit their use in mathematical proofs in which all members of an infinite class are involved.

Brouwer would not accept, for example, the disjunction that either there occur ten successive 7’s somewhere in the decimal expansion of π or else not, since no proof is known of either alternative.

But he would accept it if applied, for instance, to the first 10100 digits of the decimal, since these could in principle actually be computed.


In 1920 Jan Łukasiewicz, a leading member of the Polish school of logic, formulated a propositional calculus that had a third truth-value,

neither truth nor falsity, for Aristotle’s future contingents, a calculus in which the laws of contradiction and of excluded middle both failed.

Other systems have gone beyond three-valued to many-valued logics—e.g., certain probability logics having various degrees of truth-value between truth and falsity.


Symbols used in "Laws of thought"

The following is lightly edited from:

The three traditional laws of logic are listed in the table below.






The law of contradiction

For all propositions p, it is impossible for both p and not p to be true

A description is not both true and not true

(p · p)

The law of excluded middle (or third)

Either p or p must be true, there being no third or middle true proposition between them

A description is true or not true

 p p

The principle of identity

If a propositional function F is true of an individual variable x, then F is indeed true of x

A description true of x must be true of x

F(x) F(x)

OR, a thing is identical with itself

For every x, x is the same as x

(x) (x = x)


This table lists the symbols of logic (after the Editors of Encyclopćdia Britannica).

Logic symbol






formally implies

for every


is the same as



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