What Is Knowledge?


Raphael’s School At Athens

(Please read On the Nature of Truth and Reality first if you haven’t already.)

What is knowledge, and how do we justify it?

In The Theaetetus, Plato records a dialogue between Socrates and Theaetetus regarding the nature of knowledge. Socrates, just before his death, suggests three ideas:

Knowledge is nothing but perception. This idea, attributed to Protagoras, is easily discounted by the two men (“things are to you as they appear to you, and to me as they appear to me”).

Knowledge is true judgment. This idea is refuted because many people have true knowledge for the wrong reasons or have true knowledge without justification. This gave way to the last idea.

Knowledge is true judgment with an account. With an account means knowledge justified by logos. This idea was also rejected and Plato closes the dialogue without a conclusion. Let’s deal with this last point in just a moment. But before talking about logos, let’s consider why any of this important. There are two main reasons.

First, objective reality and objective truth has come under assault. If truth is only in the eye of the beholder, then intellectual, scientific, and philosophical progress is impossible. If there is no objective truth but only personal truth, then no idea is better than another, and individual experience (you are the author of your own subjective reality) is most important. But some ideas are better than others, and truth does exist. This means that some people are wrong and some people are right about their ideas or views of reality.

Second, if objective reality does exist, then we must strive towards understanding it. This was the Platonic view. This means that we must accept the idea that there is a knowable truth and seek to use philosophical and scientific methods that help us best describe that truth. If, philosophically, you are stuck debating whether truth is even real, then it is unlikely that you are making practical progress towards discovering any truth and you truly are a Sophist. Or perhaps you are lost in a fantasy of solipsism and believe that only your mind truly exists and can be known (and only by you). But for those who live in the real world and would like to make philosophical and scientific progress, let’s accept objective reality and strive to learn best how to know it (even if it is ultimately unknowable).

Today, few people are able to categorically distinguish between justified belief and opinion. Ten minutes on Facebook is sufficient evidence to justify my proposition. A justified belief is what I have called truth – something evidenced sufficiently to afford an adequate degree of probability to serve practical purposes. An opinion is everything else. It is a justified belief that OJ killed Nicole; it is an opinion that Nicole committed suicide. Only in a world of objective reality are these two ideas distinguishable. I have no justified belief about when I first ate peanut butter and, barring new evidence becoming available, I never will.

Much of what is passed off today as “fact” is actually just opinion. I shirk every time I hear someone say, “Science proves…” because that is not the purpose of science. These are the opening words of conmen. Journalists today are unable to distinguish fact from opinion. A brief foray into political “fact checking” shows nothing but fallacy; fact checking is really just opinion checking and any opinion that the reporter disagrees with is a “lie.” This type of error results either from deliberate attempts at rhetorical deception or from a reporter who actually believes that his subjective reality is the reality. So epistemology has more relevance than ever and in every aspect of modern life.

Plato records that Socrates failed to support the idea that knowledge was true belief supported by logos. He gave three definitions of logos and found none acceptable (he considered logos to be either a statement or speech, an understanding of the elements of the belief, or an understanding of the salient and unique characteristics of the belief). I suggest that knowledge is a justified belief (or a true belief supported by logos); the problem really is what do we use as logos? How should we justify our beliefs?

Heraclitus recognized that logos linked rational thought to the rational reality of the world; but he did not define what logos is in a way that made it accessible to us. In fact, he recognized that logos was elusive:

“For this reason it is necessary to follow what is common. But although the logos is common, most people live as if they had their own private understanding” (Diels-Kranz 22B2).

Plato used the word logos in a similar way. But Aristotle expanded the use of the word and the concept and tried to make it useful. For Aristotle, logos was an argument from reason – that is, logic. This was juxtaposed to pathos (an argument by emotional appeal) and ethos (an argument from moral authority). If logos is logic, then what are the rules of logic?

Aristotle, of course, is the father of the traditional system of logic that dominated philosophy for over 2000 years, until being supplanted by more modern predicate logic. Aristotelian logic establishes the “truth” of propositions and allows for inferences to be made about other propositions or conclusions. But how can we know if a proposition is true in the first place? Propositions may be true by definition, much like 2+2=4; but in most cases we are forced to accept the truth of a proposition without considering the probability of its truthfulness or its falsity. Aristotle’s logic did not do well with uncertainty; it prefers absolutes and clear-cut issues. Most real world problems, however, are ripe with uncertainty.

For “hard sciences,” like mathematics, physics, engineering, and astronomy, Aristotle’s logic and the prevailing Greek ideas of “knowledge” led to an unprecedented expanse of technical understanding. Greek geometry, for example, was dominated by axioms, taken as truths, that underpinned all other propositions. Math in general has followed this course.

In any field where a known input results in a known output, Aristotle is king. Our modern, scientific process is predicated on these ideas, and as math and physics and astronomy were dominant fields in science, neither the objective reality of the universe nor our ability to use logic to know and predict that reality were questioned.

But other scientific fields have not been quite as simple. In true Platonic fashion, we can understand why 2+2=4, and we can demonstrate the consistency and lack of contradiction in this truth. But we don’t know why some people who smoke get lung cancer and others don’t. We do not know exactly what the weather is going to be like tomorrow in Tampa. That’s not to say that there might not be a knowable answer, but we don’t fully understand it since we don’t understand all of the variables involved nor do we understand how they are intertwined.

Therefore, we cannot define knowledge related to these types of realities in an Aristotelian way. Yet, we have to deal with lung cancer and the weather as a practical reality, so we still need to describe our knowledge related to these issues. Because Aristotelian logic fails so many aspects of science and philosophy, Socrates would have been the first to point out that the system is fundamentally flawed and that we need a system of epistemology that is generalizable to all fields, not just rhetoric, math, and grammar. Aristotle didn’t deal well with uncertainty, but we need to if we are going to deal with complex issues.

How can we deal with uncertainty?

Of course, we attempt to deal with uncertainty mathematically with statistics and the theory of probability. But the orthodox view of statistics, called Frequentism, grew out of the field of Mendelian genetics. Ronald Fisher, who largely developed the field of Frequentist statistics that has dominated the last century of scientific research, was a biologist who studied Mendelian genetics. This means that he studied discrete genes with predictable and understandable phenotypic effects. His statistics did not need to describe any uncertainty that might exist in the model, only the variation seen as a result of chance when the model (discreet inheritance patterns) was executed.

Let’s consider three sorts of scientific problems.

First, there are problems where a known input always produces a known output. For example, an electrical circuit, once all the variables are understood, always produces the same output. It is understandably deterministic and the variables involved are understood so that the whole mechanism is understood; even if the details of the mechanism are not fully elucidated, observation can still yield a predictable output for a given input. We don’t need probability theory to understand the outcome. Aristotle and scientists love this type of certainty. It demonstrates cause and effect, and it is immediately observable, predictable, reproducible, and reliable.

Second, there are problems where a known input produces a distribution of possible outcomes in a certain frequency. Such is the case with Mendelian genetics. Seeds of a certain sort, for example, will always produce about 25% red, 25% white, and 50% pink flowers. The exact numbers are not known because there is some apparent “random chance” that has entered into the equation. It is not absolutely deterministic, like gravity, but the variables are all known and controlled for so that a model can be used to predict likely outcomes. Frequentist statistical methods are excellent at understanding these types of problems and understanding experiments related to this type of science. These were the types of problems that Fisher worked with and he could, given the frequency of outcomes, determine the probability of observing the frequency of those outcomes given a particular hypothesis. That is, Fischer calculated the probability of the observed data given the hypothesis.

The third type of problem is much more difficult but also more analogous to the complex problems that we usually deal with in modern science. How do we deal with problems where all of the variables are not understood and the mechanisms controlling the outcomes are not fully elucidated? In other words, what do we do when there are many unknowns and the relationships between those variables is not understood? We must estimate likely outcomes using probability theory.

What’s more, we need to update our estimates as new data emerges or new observations are made. This is where we need to use Bayesian statistical methods and Bayesian inference. Cox’s Theorem is a useful application of this type of statistical inference. What’s more, we need to know what the probabilities of our hypotheses are so that we can determine the utility of our hypotheses for constituting a justifiable belief. It is here where Bayesian inference and Cox’s theorem are irreplaceable and where Frequentist methods prove insufficient. Only Bayesian inference allows for continuous updating based on a new information and only Bayesian inference allows us to answer the real question: What is the probability of our hypothesis given the data (rather than the probability of the data given the hypothesis)?

Such is the situation for most real world problems, whether in medicine, biology, meteorology, psychology, sociology, or even machine learning in computer science or quantum computing. Bayesian inference excels at predictive modeling while Frequentist methods excel at descriptive modeling. In real life, whether in health care, stock markets, or weather prediction, the problems we actually want to solve are usually of a predictive nature. For example, What’s the probability that a patient will develop a heart attack? What’s the probability that one drug versus another will prevent that heart attack? What’s the chance that it will rain tomorrow? What’s the chance that my hypothesis is true? What’s the chance that OJ killed Nicole? These are all problems that need to be answered with Bayesian inference.

Where Aristotle excelled at the first type of problem (a known input producing a known output) and Fisher excelled at the second type (a known input produces a predictable number of different outputs), only Bayes can provide a meaningful logos for the third type of problem.

If we are to generalize a system of logos, it must work for all three types of problems. Here too, only Bayes works for all three. Frequentism has failed miserably for the third type of problem and this incorrect use of Frequentist statistics is responsible for most of the bad science we see today. Problems of the first type are usually over-simplified, and on closer examination what we thought was a complete understanding seemed complete only because we couldn’t detect error on the scale where we normally consider the problem. For example, Newtonian mechanics seems deterministic and fully explains the reality that we normally see; but it is an over-simplification and fails at extremes of scale, where relativity becomes important. It vaguely approximates 99% of reality, but better and more complex theories are now available.

So how does this help us rationalize our understanding of justifiable beliefs? It affects the way in which we rationalize our beliefs that constitute our knowledge. What is our logos? It must be probability theory, or specifically, Bayesian inference. We develop hypotheses that explain our observations, and then we use all available information to decide which hypothesis best describes objective reality. If we are sufficiently confident in the leading hypothesis (we often are not), then that constitutes knowledge; it is a justifiable belief. But our knowledge is not absolute (even though reality is); rather, it is true to a certain degree of probability, and we revise this probability as new data emerges.

Bayes in turn affects our understanding of logic. Logical propositions are no longer true or false, they are true or false to a certain probability. Of course, it is possible for that probability to equal 0 or equal 1 based on all available data and observations, but even those extreme probabilities might need revision given sufficient new information.

Thinking probabilistically about truth and knowledge helps us be more comfortable with uncertainty (since everything is uncertain to some degree). Donald Knuth, an influential computer scientist, said, “Science is what we understand well enough to explain to a computer. Art is everything else we do.” In other words, he is comfortable with classical logic. Computers only work well when true and false propositions are unequivocally true or false. We can program a computer to execute billions of mathematical calculations per second, and the answers are certain and deterministic (I’m obviously not talking about quantum computing). But how do we program a computer to deal with ambiguity and uncertainty? For Knuth, this uncertainty creates discomfort. Humans, however, are not computers, and it is fallacy to try to think like a computer in most circumstances (though in some circumstances, it’s ideal). Indeed, we think using a mixture of two distinctly different processes.

Modern psychologists describe these two processes using the ideas of Dual Process Theory (DPT). In DPT, System 1 thinking describes decision making that is quick, associative (a lumper), unconscious, implicit (very susceptible to bias), and non-logical. System 2 thinking is explicit, conscious reasoning that is based on rules and is logical (a splitter). System 1 thinking describes how we make most decisions, through heuristics and gestalt. System 2 thinking requires more work and is slower, requiring more analysis and data.

We all like to think that we are logical, but almost all of our decisions are made using System 1 thinking. Where we eat lunch, who we vote for, who we love, how we invest money, and most other major decisions in life are made as a result of System 1 cognition and these decisions are usually illogical and affected by our biases and emotions. Don Knuth doesn’t know how to program a computer to make the stupid decisions that humans normally make (but that doesn’t make it an art).

This idea of two different ways of thinking is certainly not a modern one. Recent interest in this concept has been sparked from the research and popular psychology work of Daniel Kahneman, but Williams James in 1890 laid the groundwork for DPT. He described “associative” thinking, which drew on past experiences, and “true reasoning” which could be used for new experiences. Obviously, associative thinking is subject to all of the mistakes and cognitive biases which hang upon our past experiences. Forcing “true reasoning” onto familiar situations might mitigate this problem. James recognized the structure that Darwin called “lumpers and splitters.” But 250 years before Williams James, Blaise Pascal recognized these same cognitive processes.

Pascal famously said, “The heart has its reasons, which reason knows not.” Pascal realized that men are often unable to logically defend their beliefs (and are unwilling to give up those beliefs even though they have no defense). Pascal, in the beginning of the Pensées, discusses what he calls the mathematical mind (the logical, rational, System 2 mind) and the intuitive mind (the heuristic-driven, biased, System 1 mind). He starts,

“The difference between the mathematical and the intuitive mind.—In the one, the principles are palpable, but removed from ordinary use; so that for want of habit it is difficult to turn one’s mind in that direction: but if one turns it thither ever so little, one sees the principles fully, and one must have a quite inaccurate mind who reasons wrongly from principles so plain that it is almost impossible they should escape notice.

But in the intuitive mind the principles are found in common use and are before the eyes of everybody. One has only to look, and no effort is necessary; it is only a question of good eyesight, but it must be good, for the principles are so subtle and so numerous that it is almost impossible but that some escape notice. Now the omission of one principle leads to error; thus one must have very clear sight to see all the principles and, in the next place, an accurate mind not to draw false deductions from known principles.

All mathematicians would then be intuitive if they had clear sight, for they do not reason incorrectly from principles known to them; and intuitive minds would be mathematical if they could turn their eyes to the principles of mathematics to which they are unused.”

Pascal delineated Dual Process Theory long before modern psychology took an interest in it. When Pascal speaks of seeing clearly, he is describing in his own words what we now call cognitive bias or logical fallacies. He is right – if one is not subject to any bias or logical error, and if the facts one sees are unquestionably true, then we would all come to the same conclusions about the same knowledge. But, clearly, we do not see facts without bias and/or logic is perverted by our cognitive predispositions to respond. Further, the facts we see are not always true (but are true to some degree or probability).

Scott Adams, the creator of Dilbert, has summed all of this up in the following way,

“Smart, well-informed people disagree on nearly all major issues. So being smart and well-informed doesn’t help you grasp reality as much as you would hope. If it did, all of the smart, well-informed people would agree. They don’t.

… Facts don’t influence decisions. Humans decide first, then rationalize their irrational choices with cherry-picked data. You see this all the time with the people who disagree with your brilliance. Just remember that they see the same irrationality in you that you see in them.”

It’s not that facts don’t matter, it’s more the case that most humans don’t really consider them. This is the danger of solipsism (why consider facts in the first place if they can’t be known or are subjective?). We must recognize that our tendency is to think and decide irrationally but that we also have access to what Pascal called the mathematical mind. Our intuitive minds work better (System 1 thinking) if our mathematical minds are exercised more (System 2 thinking). Our brains cannot work without implicit bias, but the more information that we learn in a logical way, most closely modeling reality, the more accurate our biases will be, and the better our intuitive minds will work. We must practice good thinking. Pascal later says,

“Those who are accustomed to judge by feeling do not understand the process of reasoning, for they would understand at first sight and are not used to seek for principles. And others, on the contrary, who are accustomed to reason from principles, do not at all understand matters of feeling, seeking principles and being unable to see at a glance.”

If the only exercise of our faculties is in the form of System 1 thinking, then we will reject altogether System 2 reasonings; but if we are too caught up in System 2 thinking, then we will be ineffective advocates for our beliefs, using logos too much and pathos too little. More practically, we would not be able to make all of the decisions necessary to live our daily lives if we only employed System 2 cognition. System 2 thinking is too slow and too burdensome to carry us through our daily life and work. The best decision is not always the most accurate one.

Thus, we must emphasize learning new knowledge in a structured, System 2 environment, to get the most out of our faulty System 1 decisions later on. The mind is like a parachute: it works best when properly packed.

Unfortunately, much of modern philosophy and education has rejected this approach. Typical of post-modern abandonment of intellectual reason is this quote by the Susan Sontag,

“The kind of thinking that makes a distinction between thought and feeling is just one of those forms of demagogy that causes lots of trouble for people.”

Such pseudo-intellectual inanity does not help advance justified belief. Often, our goal in all aspects of life is persuading others. Life is an argument. But convincing others of your opinion is nearly impossible, regardless of how much logic or how many facts you might bring to bear.

Pascal stated, “People are generally better persuaded by the reasons which they have themselves discovered than by those which have come into the mind of others.” Aristotle realized that pathos was a much more effective tool for persuasion than logos and the great rhetoricians from Cicero to Gerry Spence have perfected techniques for persuasion using things other than logos. But this is simply because most people don’t really utilize System 2 thinking, or, if they do, they do so poorly.

System 1 thinking is what we are stuck with unless there is a deliberate attempt to learn to think in a System 2 way. What separates man from animals is not thought, but System 2 thought. But System 2 thought is not essential to survival and often has no evolutionary advantage. System 2 thought takes work, and that work can only be done in a resource rich environment. It requires education; it requires time; it requires expensive effort – but 99%+ of all people who have ever lived haven’t had the resources to dedicate to it.

Today, we still do not emphasize what could be called critical thinking in education. We emphasize regurgitation of facts without context, algorithms without understanding, and coloring in the lines without question. We teach people to quote others out of context and we teach children to blindly follow orders without questioning. We emphasize laws and rules that are binary and don’t accommodate nuance. We emphasize form over function. We are lumpers rather than splitters. We reduce complicated problems down to simple factors. We find false causation wherever we see association. We focus on effect rather than cause. We judge action rather than intent. We focus on the ends rather than the means. In short, we are primitive, because we think primitively.

We think like children rather than like adults. We think selfishly because we only see facts that self-justify our world view. We see the world as we want it to be, not as it is.

All of this to say that we need to have an awareness of how we think in order to understand what constitutes knowledge: Knowledge is justified belief, and that justification must come from logical inference. For complex problems, only Bayesian inference will suffice. Knowledge is the result of System 2 reasoning. System 1 thoughts are often correct, but they are not justified and therefore are not knowledge, even if correct – they are opinion.

Echinech Pathway to opinion Pathway to knowledge
Aristotle Pathos Logos
Pascal Intuitive mind Mathematical Mind
James Associative True reasoning
Darwin Lumpers Splitters
Kahneman System 1 System 2
Knuth Art Science