Knowledge and truth

(Preface to “Scientific Idealism” as a philosophy for system theorists)

Copyright 2016 Graham Berrisford. One of about 300 papers at Last updated 02/04/2017 20:50



What is knowledge?. 1

How certain is knowledge?. 2

How do we know knowledge is true?. 4

Footnotes on fuzziness. 5


What is knowledge?

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

That truth of that knowledge is testable by gruesome experimentation.


Your knowledge is somehow encoded in what might be called a “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 empirically demonstrable.

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


Your mental model of standing on a train track includes the knowledge that it is dangerous.

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

It gives a particular value to the universal concept, quality, property or type that people call “dangerous”.

Your particular situation manifests, exemplifies, exhibits or instantiates that universal.


Is knowledge a linguistic phenomenon?

For communication and discussion, we encode knowledge in a shared language.

But knowledge is not inherently linguistic.

E.g. A dog knows enough to jump out of the path of an onrushing train.

Dogs are famous for their ability to remember and recognise things by smell alone.

Evidently, animals do encode knowledge in other forms than in verbal language.


Is knowledge a biological phenomenon?

Some knowledge of the world is innate or inherited.

Animals inherit some 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 anything that is long, thin and wiggly.

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

So, your innate fear of onrushing objects may prompt you to step out of the path of an onrushing train.


Is knowledge a psychological phenomenon?

Some knowledge of the world is learnt through experience.

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 mark of intelligence and essential to learning.

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


Is knowledge a social phenomenon?

Some knowledge of the world is acquired through social communication.

E.g. Honey bees learn the location of a pollen source by watching the wiggle dance of another bee.

You may be told about the danger of standing on railways tracks before you ever see one.


Is knowledge an evolutionary phenomenon?

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

All of these are explainable as by-products of biological evolution.

Knowledge (along with emotions like love and fear) helps to increase the likelihood you will pass your genes on.


A knowledge triangle


<inherit and acquire>   <describes and predicts>

Knowing actors  <observe and envisage>   Realities

How certain is knowledge?

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

Science proceeds by modelling realities, with a view to predicting future realities.

Scientists strive to refine their theoretical models through intuition, research and testing.


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.

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

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


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.

(Pseudo-scientific papers may survive longer because it is impossible to disprove them.)


Mathematicians use logical analysis to prove conclusions drawn from axioms.

Logical analysis can be peer reviewed a thousand times or more.

The conclusions may be turned into propositions that can be tested in physical reality.


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

The history of cosmology is interesting.

The theories of Aristotle/Ptolemy were supplanted by those of Copernicus, Kepler and Galileo, then Newton and then Einstein.

Physicists know Newton's laws of motion are not true through all space and time, but also know they are reliable in our space and time.

They also know Einstein's cosmological physics is irreconcilable with quantum mechanics, but also know each is fit for its context and purpose.

They recognise the limits of their models and where to apply them.


Models in sociology and economics are far less certain and reliable those in physics.

There is so much pseudo science about 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.”



Hypotheses / Models

<create and use>   <describe and predict>

Scientists  <observe and envisage>   Realities

How do we know knowledge is true?

Social communication spreads exhortations, suppositions, babble, nonsense and “fake news” as well as knowledge.

How do we know knowledge is true?

A J Ayer said a proposition can be discounted if it is not verifiable by:

·         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 has been wildly successful in mathematics.

Some kinds of knowledge are provable as true by logical analysis with respect to initial axioms.

By this means, mathematicians may conclude their models are perfect (but see footnote).


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

Instead, we judge the truth of knowledge by how well it predicts what happens in reality.

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

·         a supposition or hypothesis if it has never been tested

·         knowledge if it has been tested successfully

·         babble or nonsense if it fails to pass our chosen tests.


In practice, these distinctions are fuzzy.

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


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.

But the people we converse with may be equally deluded.

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

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.

But even falsification does not necessarily invalidate knowledge.


“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

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.


In short, 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

Footnotes on fuzziness

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.

Fuzziness in physical measurement

The truth of a proposition about the physical world is determined by measuring physical matter and energy.

How accurate do we need the measurement to be?

Suppose it is proposed the Yankees baseball ground is bigger than Lords’ cricket ground.

However you measure them, your certainty will depend on the accuracy of that measurement.

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.


You can tell or confirm the truth as you see it, but others may have different mental models of the same reality.

Ideally, first, you present a description that (accurately enough) reflects your own mental model of some reality

And then one or more observers confirm that description to be an (accurate enough) description of a reality.


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

We describe particular things by using “universals” to classify or typify them.

A circus ring is “circular”. The earth is “spherical”. A chessboard is “square”.

A play is a performance of “Hamlet”. A plant is a “rose bush”.

These statements may not be 100% true, yet true enough for practical purposes.


A particular thing

instantiates a named “type” when

it embodies property types that define an instance of that type

A circus ring

instantiates the type named “circle” because

its diameter is the same in all directions

A play

instantiates the type named “Hamlet” because

it uses most of the script written by Shakespeare

The rose bush in my garden

instantiates the type named “rosea” because

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


There is fuzziness in defining types and identifying instances.

What margin of error is allowed in measuring the diameter of a circus ring?

How strictly must a play follow the script?

What if a particular rose bush has only two thorns and it grows sturdily to the height of a tree?


Again, you can ask a judge or jury to examine a particular thing against a type definition, and give 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 in a type’s definition and the match of any particular thing to that.

Fuzziness in the laws of logic

The laws of physics are rules that apply to physical matter and energy.

The laws of logic are rules that apply to abstraction descriptions - to propositions.

It can be argued that certain truth and falsehood exists in the logical world of description.

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


"Laws of thought" (after the Editors of Encyclopædia Britannica)

This table lists the symbols of logic.

Logic symbol






formally implies

for every


is the same as


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 thing 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 thing 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 thing 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)


Modern systems go beyond this: certain probability logics have various degrees of truth-value between truth and falsity.

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


[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.



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