We now know that human intelligence is generated by System 1 and System 2 thinking: System 1 is our intuitive brain while System 2 is our rational brain. In this post, we are going to take a deeper dive in how we use System 2 to create meaning from the world around us by studying inference.

Inference is the ability to reach a conclusion about something through facts and reasoning. We use the information we receive from the world and apply reasoning and logic to understand what that informations means to us. Without inference, our knowledge would be limited to what we observe in the world or what we directly experience. It would almost hard to generate a cause and effect understanding of the world. So inference is our most important cognitive tool, yet was not taught in schools when I grew up. My primary source is Barbara Minto’s outstanding book The Pyramid Principle on which this is based. The actual source is American philosopher Charles Sanders Peirce (1839-1914).

Peirce is best known for being the father of pragmatism, a philosophical movement that judged the value of beliefs by its practical use. He placed great value on the value of knowledge that was actionable, connecting that knowledge with experience, and using inquiry as the main logical process. Peirce studied extensively about how we construct beliefs about the world and deconstructed the human thought process down into three core elements. The first element are the rules that we carry around inside of us that explains how the world works. We now call these schemas, consisting of frames and scripts.  Rules are guaranteed to be true and are used to explain the world around us. They are extremely powerful things, derived from the hundreds of thousands of years humans have been around. These rules are the first principles of reasoning. 

The next two elements are case and result. A case is a fact that exists in the world or our head. Result is what should happen if we apply the rule to this particular case. You can also use the terms Cause and effect in lieu of case and result. Where you start (rule, case, result) indicates the type of reasoning you are doing. Deduction starts at the rule, induction starts with the case and result, and abduction starts with the result. We will examine each one in turn.

The first method  we will explore  in our cognitive toolkit is inductive reasoning. Induction is used to create a rule out of a series of observations we have made of case result pairs. Take for instance the following logical argument:

• Socrates is a man (case)
• Socrates is mortal (result)
• Therefore all men are mortal (rule)

In inductive reasoning we start with case result pairs and seek to find or create a rule that will explain it. Here we see Socrates being a man and being mortal. We also see this with Plato and Aristotle. In fact, we can see this with all men and women we know. We also see this with all living things. Armed with this information, we can therefore construct a rule that all men (and women) are mortal. This then becomes one of the foundational principles we use in deductive reasoning. 

But there is a catch. Induction is not guaranteed to be true. This is called the induction problem is highlight by Nassim Taleb in his book The Black Swan. In 1697, Dutch explorer Willem de Vlamingh was exploring rivers in western Australia when he came upon a swan, The problem was the swan was black instead of white. Up until that point in time, the only swan ever witnessed in Europe was white so a rule was created that all swans are white based on these countless observations. De Vlamingh’s single observation negated every observation before it and falsified the rule that all swans are white. This is essence of the induction problem. It means that rules we assume are true could be invalidated at any moment in time. When something that is not supposed to happen does,  the induction problem is at work. The only disciplines that do not suffer from the induction problem are those based on logic, such as mathematics and classical logic. Nevertheless, not withstanding this problem, induction is valuable as it guides us to candidate rules that we can then use in deductive reasoning. 

With deductive reasoning, you start with a rule to explain cases and results in the world. It is this explanatory effect that makes deductive reasoning the most powerful tool in our arsenal. Take for instance, the following logical argument which is deductive: 

• All men are mortal (major premise or rule)
• Socrates is a man (minor premise or case)
• Therefore Socrates is mortal (conclusion or result)

This is classic deductive reasoning which is rule driven. It starts with a rule, applies that rule to a case which yields a result. This approach is logically consistent and if done properly always creates logical accuracy, not necessarily factual though. For example, lets take the following:

• All men are rabbits
• Socrates is a man
• Therefore Socrates is a rabbit

From a pure logic point of view, this is valid because it is not possible to accept the premises and deny the conclusion. It is absurd though because men are not rabbits. So in this case the major premise is false. This type of reasoning is also the source of bigotry and bias. The point here though is that deductive reasoning is rule driven. We go from the general to the specific. We blindly follow the rule, no matter where it takes us.

While deductive and inductive reasoning are always discussed in books as our two modes of reasoning, it is actually adductive reasoning that drives most problem solving today. In adductive reasoning, we start with the result and try to figure out what caused it. It is the scientific method at work in our minds. Take for example the following:

• Socrates is mortal (result)
• All men are mortal (rule)
• Maybe Socrates is a man  (case)

Here we start with an observation that Socrates died. He is mortal. We know that all men are mortal so perhaps Socrates is a man. Notice that he could also be a parakeet, a sheep, a dog, or anything else that is mortal. By the process of adding more facts, we can determine what species Socrates is and come to a definite conclusion about what happened. Abductive thinking is hypothesis driven and is our best guess. When we reason abductively about something, we are seeking either  the best explanation for the facts on hand or the best path forward given the current situation. It is in this sense that scientific method strategic thinking are intertwined via abductive reasoning. Both use it to determine the best guess. In a twist of irony, Sherlock Holmes, who very name is associated with deductive reasoning, used adductive thinking to solve his cases. He collected facts and then developed theories to explain them. In one of his most famous quotes to Watson, he said:

“Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth.”

In abductive thinking, you are likely to think of many hypotheses to explain the facts at hand. Each hypothesis has its own branch on our thinking tree. What Holmes highlights here is that you remove all hypotheses except one which must be the truth, particularly as we are looking at historical fact. The development of strategy is also hypothesis driven, however in this case, the hypothesis acts more like a bet; we are betting that the strategy is correct. Sometimes we have to revisit our strategy if in fact it is not working as anticipated. In this case, we either have to identify why our hypothesis is not working (perhaps we made an incorrect assumption or we just didn’t execute the strategy correctly) and potentially adjust course.  Here is a topical example.
 

• Prices have gone up (result)
• Tariffs can increase prices (rule or in this case our hypothesis)
• Yes, we have enacted tariffs which caused prices to rise (case and result 

In this case, we hypothesize about what could increase prices. But other things affect prices such as energy prices, resource shortages, and price manipulation. If prices fall when tariffs are in place, then it seems something else is at play.

This is inference in a nutshell. The triumvirate of rule case result gives us a roadmap to organize our thinking. The three modes of thinking can thus be summarized by the following table:

Mode	Process	Driver	Result
Deductive	Rule Case Result	Rule	Certainly true
Inductive	Case Result Rule	Data	Probably true
Abductive	Result Rule Case	Hypothesis	Best guess

There is one more part of logical inference that we should discuss and that is the evaluation of evidence and the forming of beliefs. Fortunately we have a formula for that, developed by Thomas Bayes back in 1763 called Bayes Theorem. It multiples conditional probabilities calculating the probability of a hypothesis to be true given the presence of evidence and is used in all types of filtering applications including medicine and spam filters. The formula is:

P(H | E) = P(E | H) P(H) / P(E)

• P(H | E) is the probability the hypothesis is true given the evidence. This is the posterior. The vertical bar stands for “given the”. This reads the probability the hypothesis is true given the evidence.
• P(E | H) is the likelihood of the evidence given the hypothesis being true. This is the likelihood.
• P(H) is the probability of the hypothesis being true. It is the base rate called the prior.
• P(E) is the probability of the evidence being true regardless of the hypothesis. This is the marginal .

You always start off with a base rate which is the probability of something being true regardless of the evidence. What we want to find out is when a piece of evidence emerges, does make the hypothesis more or less likely given that base. Does it go up or down?

Imagine a disease that affects 1 out of every 100 people. Its base rate is 1 percent or .01. Now imagine you taking a test that indicates you have that disease. Tests can be wrong so should you be worried about having that disease? Our hypothesis is that we have the disease given the test. We find out two things about the test. First, it correctly indicates the disease 80% of the time when the disease is present. In general, the test reports positives 10% of the time which given the base rate of the disease would include 9 false positives. What is the probability you have the disease?

The likelihood you have the disease, which is P(H | E) is 80% since the test detects the disease 80% of the time. This is P(E | H). The base rate of the disease is 1%. This is P(H). The test itself comes back with positives 10% of the time including false positives. This is P(E). This is the marginal rate. The calculation is then .8 x .001 / .10 which is .08 or 8 %. This is the likelihood you have the disease. It has increased from 1% to 8% so redoing the test makes sense.

While the math is simple, Bayes Theorem is incredibly powerful in daily life. And while we do not necessarily do the calculations in our head, we nevertheless follow its logic.

• If the evidence is more likely to occur given the hypothesis than by itself, P(E | H) > P(E), then the chances of the hypothesis being true increase.
• If the evidence is more likely to occur regardless of the hypothesis, the chances of the hypothesis being true decrease.

Its all about the base rate and where we set it. If we set it too high, then no amount of evidence will convince us otherwise. If we are locked into our beliefs, then we will ignore anything to the contrary. If we set the base rate low then, then we can evaluate evidence fairly and adjust our rate depending on what we find. I will close this post with 3 rules of thumb about Bayesian thinking Julia Galef who has many outstanding videos on YouTube regarding this subject. The comments in parenthesis are mine.

1. Remember your priors (keep your base rate in mind)
2. What would the world look like if my theory was wrong? (have I set the base rate too high?)
3. Update incrementally using snowflakes of evidence (Its okay to adjust your opinion gradually over time instead of having wild swings over time).

Combining Bayesian thinking with abductive reasoning is a powerful combination of tools for thinking about complex problems, where the sample size if small. By using these tools, we engage System 2 thinking and relegate System 1 thinking to its proper role of guidance.