Logical implication and causality

Judea Pearl, one of the prominent researchers in the logic of causality, argues that current-day mathematics and logic does not have mechanisms to represent causality. And this has caused great confusions in several instances. 

Let us consider how causality is represented in current day mathematics and logic. 

One of the most common situations where we interpret causality, is in a mathematical equation. An equation of the form 

y = f(x)

is often interpreted as the value of f(x) causes y to get a given value. This is the interpretation used in several programming languages for example. In a language like C or python, when we say: 

a = b

it means that the value of b causes the variable a to attain a given value. It is not the value of a that changes b, but the value of b that changes a. 

But mathematically, the sign "=" simply means equality. In mathematics, a = b is the same as saying b = a. 

This overloading of the "=" symbol causes complications in programming, when we have to distinguish between logical equality, and assignment. Programming languages hence, distinguish between the two kinds of operations. In languages like C or python, logical equality is represented as "==" that tells the program to check for equality, while the symbol "=" represents an imperative-- that tells the program to do something. 

Even here, there is no separate operator for checking causality. When we say "a == b" in a programming language, we are only asking the program to check whether the values of a and b are the same. We are not asking it to check whether a caused b or b caused a to take on a specific value. 

For instance, if 'a' were to represent the air pressure in a canister, and 'b' were to represent the reading of a barometer attached to the canister, then, whether we say "a == b" or "b == a", the outcome would still be the same. However, we can see that, if we manually change the air pressure to a different value, versus, if we manually change the barometer reading to a different value, the logical equality holds in the former case, but breaks in the latter case. This is because, this is not just an equality, but a causal relation between the two variables. The air pressure is causing the barometer reading, while the barometer reading is not causing the air pressure. 

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Another common operator that is commonly mistaken for causality is the logical implication operator in formal logic. A statement of the form 

P  --> Q

is read as, "if P is true, then, Q is true." 

A commonly used example is if P represents "it is raining" and Q represents "the lawn is wet". In this case, we can see that the logical implication is also interpreted as causality. The statement only says that "If it is true that it is raining, then it is true that the lawn is wet." But it is often interpreted as, "If it is raining, then it causes the lawn to be wet." 

There is this common example to illustrate correlations versus causality. It is often seen that increase in ice-cream sales is correlated with increase in drowning incidents. But of course, it is not the ice-cream sales that is causing the drowning incidents. They are both increased because of the summer season, where people eat more ice-creams, and also go swimming more. 

But, if P were to represent "increase in ice-cream sales" and Q were to represent "increase in drowning incidents" then the implication P --> Q is very much valid. The implication only says that if P is true, then Q is true-- not that P causes Q to be true. 

Another confusion that is caused because of the lack of a causality operator is the famous saying by Rene Decartes: "I think, therefore, I am". 

It is often interpreted as "I think, and therefore it causes me to exist" and leads to several jokes like the following: 

René Descartes walked into a bar. The bartender said “would you like a beer?” René replied “I think not”
He disappeared. 

Or this.. 

But the problem here is of course, that of interpreting implication as causation. "I think, therefore, I am" can be interpreted as "I think, and that causes me to exist" or as "I think, and therefore I can infer that I exist." 

What Decartes meant was the latter. The fact that "I think" leads me to infer that "I exist". In other words, it is my existence that is having a causal dependency on the fact that I think, and not the other way around! 

An implication of the form P --> Q can also legitimately mean that Q is the cause and P is the effect! 

To understand his even better, consider the equivalence between an implication and its contra-positive. An implication of the form P --> Q and another implication of them form ~Q --> ~P (where ~ is interpreted as "not") are both equivalent. 

Using this, we can see that "I think, therefore, I am" is equivalent to saying that "I am not, therefore I do not think". which is consistent. 

Even if Decartes had meant that my thinking is causing me to exist, the implication that "I don't think, therefore I don't exist" is still a fallacy-- called the inverse fallacy. "If P then Q" is not equivalent to either "If Q then P" or "If not-P then not-Q".