The seeds of the modern era could arguably be traced to the Enlightenment and the invention of rationality. I say invention because although we may be universal computers and we are certainly capable of applying the rules of logic, it is not what we naturally do. What we actually use, as coined by E.T. Jaynes in his iconic book Probability Theory: The Logic of Science, is plausible reasoning. Jaynes is famous for being a major proponent of Bayesian inference during most of the second half of the last century. However, to call Jaynes’s book a book about Bayesian statistics is to wholly miss Jayne’s point, which is that probability theory is not about measures on sample spaces but a generalization of logical inference. In the Jaynes view, probabilities measure a degree of plausibility.
I think a perfect example of how unnatural the rules of formal logic are is to consider the simple implication
which means – If A is true then B is true. By the rules of formal logic, if A is false then B can be true or false (i.e. a false premise can prove anything). Conversely, if B is true, then A can be true or false. The only valid conclusion you can deduce from is that if B is false then A is false. Implication is equivalent to the logical statement , where means negation and means logical OR.
However, people don’t always (seldom?) reason this way. Jaynes points out that the way we naturally reason also includes what he calls weak syllogisms: 1) If A is false then B is less plausible and 2) If B is true then A is more plausible. In fact, more likely we mostly use weak syllogisms and that interferes with formal logic. Jaynes showed that weak syllogisms as well as formal logic arise naturally from Bayesian inference.
We can see this more clearly in a concrete example. Let’s say that A is the statement it is raining and B is the statement it is cloudy. So A B is equivalent to if it is raining then it is cloudy. The only logical inference that can be drawn is that if it is not cloudy then it is not raining. In particular, it is not raining does not imply that it is not cloudy but it does imply that being cloudy is less plausible. Conversely, it is cloudy doesn’t mean it is raining but it does imply that rain is more plausible. This is why if it is cloudy we may bring an umbrella and if is not raining we may reach for some sunblock. If we operated purely using formal logic, we would never draw such conclusions.
Jaynes showed that we can quantify weak syllogisms using Bayesian inference if we let probability measure the degree of plausibility. Let the probability that A is true be given by P(A) and the probability that A is not true be given by P(A). The probability that A is true if B is true is represented by P(A|B). There are similar relations for B by reversing A and B. The relationship between joint and conditional probability
leads immediately to Bayes theorem
where P(B)= P(B|A)P(A)+P(B|A)P(A). In Bayesian lingo, P(A|B) is called the posterior probability, P(A) is called the prior probability and P(B|A) is called the likelihood function. We can similarly rearrange variables to obtain
We can now use these formulas to parse logic and plausible reasoning. The statement that A is true implies B is true implies that P(B|A)=1. We can see this using equation (2) since if A is true then P(A|B)P(B)=P(A) (since the probability that A is true is independent of B). Now suppose that B is false. Then we can use equation (3) to find out about A. Here the probability that B is false and A is true is zero so P(B|A)P(A)=0. Hence, we conclude that P(A|B)=0, or A is false.
Now what about the weak syllogisms? Well let’s consider what happens to our probability for A if B is true. We can use equation (1). We know that P(B|A)=1 from before. Since P(B) then P(A|B) P(A). Hence knowing that B is true increased the probability that A is true. Finally what happens if A is false. Here we can use equation (4) to find out about B. From P(A|B) P(A) we can obtain P(A|B) P(A) (since P(x) = 1 – P(x)), which leads to P(B|A) P(B) i.e. B is less plausible if A is false.
I think this strongly implies that the brain is doing Bayesian inference. The problem is that depending on your priors you can deduce different things. This explains why two perfectly intelligent people can easily come to different conclusions. This also implies that reasoning logically is something that must be learned and practiced. I think it is important to know when you draw a conclusion, whether you are using deductive logic or if you are depending on some prior. Even if it is hard to distinguish between the two for yourself, at least you should recognize that it could be an issue.