Logic for Computer Science and Artificial Intelligence

Chapter 2

A Few Thoughts Before the Formalization

2.1. What is logic?

We cannot give a formal answer to this question right away (we will get back to it though). In order to be understood, the answer would require some hindsight on the topic about to be studied.¹

Try to understand what mathematics is about by only relying on its definition: mathematics, the science of quantity and space.

To choose an answer would not be very helpful right now, because a lack of criteria and references to concrete cases make it difficult to judge the pertinence of the answer.

A problem that is closely related to the one under consideration inspired the following thought to a famous philosopher (D. Hume):

The fact that ideas should logically entail one another is nothing but a fact, no more understandable by itself than any fact of the material world.

And this one from another less famous philosopher:

A logical formula is the expression of a natural phenomenon, as is gravitation or the growth of a tree.

2.1.1. Logic and paradoxes

Let us return to the history of logic and try to analyze some of the well-known concepts that are involved.

Logical difficulties arose very early in philosophy, religious “treaties”, and literature. Here are two examples:

– the liar’s paradox, due to Eubulides of Miletus (see digression 2.2): I am lying;

– the sentence: this sentence is false;

– the version of the liar’s paradox due to Epimenides of Knossos (6th Century BC): All Cretans are liars.

Are these really paradoxes? ...