Humans somehow construct opinions that may conflict with each other. How do you determine that something is true? Seriously, how do you determine that a statement is irrefutably true? If you can say that a particular statement is irrefutably true then those people who disagree can be fairly called illogical. More importantly, we can confidently build knowledge upon it.

To do that, we need a tool. More specifically, we need a new language. Human languages are inherently ambiguous. One could interpret a given statement in different ways (one of the reasons why people construct conflicting opinions). Also there are many human languages and it’s not always possible to precisely translate sentences between them.

The tool that can save us is called *logic*, not the sloppy logic that humans use often, but mathematical logic. Logic is the foundation of mathematics and therefore all of science. Logic enables us to state questions rigorously (thereby eliminating the ambiguity of human languages) and defines rigorous rules that can be used to assert or prove the truth of statements. Logics typically ought to represent some static or dynamic aspect of the real world so that we can use them to understand it.

Ancient Greek philosophers such as Aristotle and Euclid determined that in order to derive new and necessarily true knowledge, we need something to start with. Otherwise, we either get into an infinite regression (something must be based on something else ad infinitum) or a cycle (a statement is assumed to be true in order to prove that it is true). We need a set of statements that can be considered true without any proof. These are called *axioms** or **postulates*. Axioms can be based on common sense such as “the whole is bigger than the part.” They can also be based on experience or empirical evidences such as “it’s possible to draw a straight line between any two points.” We can also define axioms for the sake of convenience such as”a + b = b + a.” Note that such axiom does not represent anything or related in any way to our physical world. In general, an axiom can be any independent, self-standing statement.

A set of axioms together with a logic are called an *axiomatic system*. Every statement that has been proven to be true or false is called a *theorem* and the argument that has been used to prove it is called a *proof*. An axiomatic system together with all derived theorems are called a *theory.* Note that it’s crucial that the set of axioms of a theory to satisfy three conditions: each axiom must be *well-formed* (grammatically correct according to the logic), each axiom must be *independent* (does not depend directly or indirectly on any axiom), and all the axioms in the set together must be *consistent* so that it cannot be used to derive contradictory theorems. That is, the theory must be *sound*. Otherwise, the theory is useless.

Humans may construct opinions (statements) illogically by either using statements that have not been proven to be true or by using an inference rule that is not allowed by the logic. The first case is particularly interesting. There are three reasons why a human may consider a statement to be true without a proof:

- The human just believes in the truth of the statement (religion, anyone?). Unfortunately, according to Gödel’s astonishing incompleteness theorems, there is no universal theory that is powerful enough to prove or disprove every well-formed statement. Therefore, sometimes for the sake of argument, we may need to assume that a statement is true. This is OK as long as that statement does not contradict established theorems. A statement whose truth is believed in is really just an axiom.
- The human inadvertently thought that the statement is necessarily true.
- The human has a poof that the statement is true but the proof is illogical. The statement may be true, though. But it could be false as well.

Constructing opinions illogically is not the only way for humans to disagree. Two humans may each construct a different opinion logically. These opinions may conflict if the humans are using conflicting axioms.

Because humans have emotions, are typically arrogant and wishful, and confuse between intuition and facts, they often tend to think illogically, resulting in a significant amount of conflict among them. Considering also the infinite complexity of the world we are living in, it’s absolutely amazing how humans, sometimes, can make logical or nearly-logical arguments. This is fortunate for us of course because otherwise, we would probably go extinct.

There is a difference between logic and *rationality*. Logic describes statements. On the other hand, you cannot say that “reading an article” is true or false (but you can say that “I’m reading an article” is true). Rationality describes actions and goals or desired outcomes. An action is rational if it gets the subject (such as a human) closer to achieving a particular goal. A goal is rational if it makes the subject better in some sense. There is a spectrum of rationality. Note that logic is also not necessarily binary as well. Rationality is studied in Artificial Intelligence (AI), which itself is built upon mathematical logic. *Intelligence* and rationality can mean different things depending on the context. In AI, they almost mean the same thing. In other contexts, they can have very different meanings.

A human is rational if he/she performs rational actions to achieve a rational goal. A human is logical if he/she determines the truth of statements according to some logic. Greed, impatience, and arrogance are the typical reasons why humans are closer to being illogical and irrational. The extent to which a court judge is logical and rational is directly proportional to the fairness of the verdict. Of all creatures, it is interesting to note that a human can only be rational if his/her goals and actions, at least, do not conflict with other rational humans’ goals.

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