There are many definitions of logic but we can affirm, without loss of generality, that logic is the study of what is rational and of the inference methods that can be used to achieve a truth.
Logic represents the foundations of philosophical and mathematical thought, we could even go so far as to say that logic is what the totality of human thought is based on. Nowadays, however, logic seems to be nowhere to be found in everyday life: if there is a disease of thought that inflicts the modern world, it is undoubtedly a terrible lack of the former. By this I do not mean that we find ourselves in the most irrational slice of history (Middle Ages, anyone?) and personally I do not believe that human history unfolds "asymptotically", that is from a lesser perfection to greater economic, political, of-thought perfection, towards the final self-realization as Hegel and Marx believed.
I just want to say that we inhabitants of this space-time region we call the 21st century have been pardoned with the possibility of obtaining any information we need with times and costs approaching 0 and the fact that we are still so illogical in spite of everything makes us embarrassing and unworthy. If the tone of the speech makes me seem extremely embittered it is because I really am. Think for example of all the controversies that have arisen in recent years regarding vaccines: they make children autistic, they give rise to other diseases in adulthood etc.
To disprove these statements, an internet connection, a minimum cognitive capacity and a semblance of critical thinking are enough. You just need to discriminate between fake sources and reliable sources. A quick glance at global statistics is sufficient to verify that:
1) Vaccines have saved millions of lives since they were invented.
2) There is no stochastic correlation between autism (or other diseases) and vaccines.
In fact a rapid self-training on the subject should be sufficient to convince anyone that vaccines have been one of humanity's greatest achievements. And this is just one of the millions of indicators on the lack of inductive and deductive capacities of which we suffer.
Let now concentrate on the basic aspects of the logical doctrine. Logic is the study of reasoning and it takes place in a well-defined and consistent formal system.
The basic rules of the formal system are called axioms, ie fundamental prepositions which cannot be proved but which are so intuitive that their truth is accepted a priori. Based on these "atoms", (almost) all theorems can be derived.
Now we will proceed to create our very simple formal system of which we will explore the logical attributes.
Let's call our system S, we define the Axioms of S.
Let P, Q, Z be Propositions of S, then:
1) P=P (identity) - in English: Every thing is equal to itself.
2) P=Q →Q=P (symmetry) - in English: if proposition1 = proposition2 then proposition2 = proposition1 .
3) P=Q and Q=Z →P=Z (transitivity) - in english: if proposition1 = proposition2 and proposition2 = proposition3 then proposition1 = proposition3.
4) ¬¬A=A - in English: saying "not not something" is equal to saying "something".
5) P∨¬ P = True - in English: The statement "one thing or its opposite" is always true (tautology).
This axiom is the basis of the famouse joke: "I asked a logician if he wanted his coffee with or without sugar and he answered "Yes".
6) P∧¬ P = False - in english: The statement "one thing and its opposite" is always false (contradiction).
These are generally the axioms of first-order logic.
In the above rules we have listed the logical connectives (or, and, not) but not the quantifiers, of which no logic of the first order can do without.
Simply put, quantifiers serve to expand the properties of propositions beyond themselves, to all propositions that share the same characteristics:
Existential quantifier (∃) - in English: "There exist".
Universal quantifier (∀) - in English: "For All".
Now let's try to derive some theorems of our system S starting from the axioms and the quantifiers enunciated:
∀(P,Q,Z) ((P∨Q∨Z)∨¬(P∨Q∨Z))) This means that for every
propositions triad either their conjunction by "or" connective is true or the negation of their conjunction is true, this derive from the fifth Axiom (in fact it is nothing but a restatement of that axiom and applies to any grouping of propositions, not only for triads).
∃A∃B∃C ((A∧B)∧(B∧C)∧(C∧A)) which means that there exists three propositions such that the above formula is always true, that is when A and B and C are true.
¬(¬A∨B)∨A=A This is a theorem whose utility is found in the simplification that it gives us, mapping a furmula with 4 connectives and two variables to a single variable of the formula itself.
A∨(A∧B)=A Another simplification.
It is easy to see that the two identities above hold up, just take a look at the following truth table:
0 0 | 0
0 1 | 0
1 0 | 1
1 1 | 1
Where 0 means false and 1 means true. This Boolean truth table reflects the behavior of the above formulas and it can be seen
that the output is always equal to the value of the 'A' variable.
At this point the basic mechanics of logical reasoning should be clear to everyone.
Note that, in logic, we are interested in proving truths and not falsehoods: falsehoods are trivial, truths are not.
What has always fascinated me about logic is its deductive power and its ability to produce tautologies. If you try, as an experiment, to ask a child "Is A equal to A ?" or if "A and not A" is true or false, a meaningful sample will answer correctly, this is because there is something innate behind these reasonings.
Moreover, it is almost magical that, starting from very simple rules, we can get to prove the Poincarè conjecture or Fagin's theorem or Ads/CFT Correspondence or billions of others mathematical milestones (mathematics emerge from logic).
The greatest value that mathematics/logic has for me (and here I could sound a little fundamentalist or Platonic) is to allow us to study eternal truths of priceless beauty.
Think for example of the Pythagorean theorem, it was true before the birth of the Universe, before Pythagoras himself formally proved it, it is true today and will be true even after the thermal death of the universe.
Religion tells us that God created the world in seven days, that God is love, that God is eternal, that God is truth and truth will set us free but never gives us any proof of anything. Logic and mathematics, on the other hand, show us substantial truths that , once proved, they cannot be refuted. To be honest, mathematics is a process of discovery, when a theorem is proved it takes a truth value for us (human beings) but in fact it has always been and always will be true. Religion in comparison is ash. Excuse me for the small and emphatic philosophical digression, we can go back to examining concrete problems and the ways in which logic could eradicate them. In my country, Italy, you can't breathe good air lately. The revival of old and dangerous ideologies
The revival of old and dangerous ideologies has been raging for some time now and this has led to the appearance on the scene of politicians with dubious ethical orientations and even more dubious management skills. The point is that if logic seems innate in man, his inclination to irrationality is innate too, the latter being much more dangerous and contagious than the first. One person, one thought, one irrational event inevitably attracts others like a magnet and, before realizing it, one finds oneself in a stinking social and cultural climate.
I'm thinking that maybe I should have warned you that this post would have political content, but now it's too late. I stress out, however, that these contents are indispensable to the achievement of my point.
I maintain that the blame for the rise of these dubious individuals, who would probably be treated as psychiatric patients in a healthy society, is not to be attributed to them but to the millions of supporters who, acting in an illogical manner, let themselves be misled by their empty words.
Let me give you an example: The propaganda of these presumed politicians is mainly based on racial and gender hatred and more stringent nationalism.
In one of the many public events, an internet channel has interviewed some of the supporters of these ideologies, let me report an answer in particular:
"I am in favor of the death penalty but against abortion because you cannot decide for the lives of others."
Doesn't this sentence cause you some rational annoyance? Doesn't it cause you a cognitive short-circuit? Remeber our fifth Axioms? The above proposition is the is the English equivalent of P∨¬ P
and if you hare receptive readers you will remember that this axiom represents a contradiction, that is a thing that is always false, no matter what the proposition under consideration is.
Let me conclude by saying that I passionately believe in the power of logic as the sole deterrent against our self-destruction but, in order to emerge victorious, we must teach ourselves how to reason logically again and, above all, we must not be tempted by the simplest and most primitive pleasures of irrationality.