What is logic? Is there something in common between logic and curiosity? Well, logic tastes like something boring. It might take us back to school where we had to ponder such things as all men are mortals, Socrates is a man; therefore, Socrates is mortal. But what great news is there here? There is nothing new, and nothing interesting, in knowing something we already know! And yet, logic is concerned with questions like this situation. Maybe your first impression is that logic is the science of obvious.

We don’t fare much better with conceptual inferences either, such as “Xanthippe (Socrates’ wife) is more intelligent than Socrates, therefore Socrates is less intelligent than Xanthippe.” Yet, it was disciple of Socrates’ greatest disciple, Aristotle, who claimed that philosophy arises from wonder, from the curiosity we naturally feel towards reality. And Aristotle himself is the founder of that discipline, logic, which bored us with its schemas. Something doesn’t add up.

So, listen to this little story. Mrs. Parry announces to her students that there will be a surprise exam next week. Nora, who loves puzzles, starts reasoning: Mrs. Parry is reliable, she would never lie to us. But then the exam cannot be on Saturday (the last day of the week). Because in that case, on Friday, we would know that the exam would be the next day, and therefore, it wouldn’t be unexpected at all. So, Nora concludes that it won’t be on Saturday. But, just a moment! Nora repeats the reasoning for Friday as well. It can’t be on Friday either, otherwise, on Thursday, we would know that it would be on Friday, and again, it wouldn’t be unexpected. And so, one after another, all the days fall under Nora’s ironclad logic. She solemnly declares to her classmates, “Guys, don’t worry, I have proof that there will be no exam next week.” The class is happy and relaxed. And the following week, on Wednesday, there it is, the exam. Completely unexpected, just as Madame Parry had predicted. Now, is there something wrong with Nora’s is it? reasoning? And if so, what?

While you’re thinking about it, Ken, a friend of Nora’s, is thinking whether or not to go to the party tonight. Sure, he could have a good time. But he could also watch five episodes of his favorite series. However, Ken is free to make his choice. Now, regardless of his decision, it can already be said that it is true that Ken will or will not go to the party. One of the two alternatives will occur – even if nobody may be aware of it. But let’s think about it for a moment: if it is already true (let’s say) that Ken will go to the party, how can Ken be free not to go? Let’s imagine that things are exactly like that: it is true that Ken will go to the party. However, shortly before 9 p.m., Ken doesn’t want to go, he doesn’t feel well. Now, if Ken is free (and we have no reason to think otherwise), Ken will decide not to go to the party. But it was true that Ken would have gone to the party! Ken’s decision cannot change things in the past; if it was true that Ken would ultimately have gone to the party, it means that Ken will go to the party, by necessity. But then it seems that Ken is not free to decide how to spend the evening.

There are many stories like these; some are very ancient (Ken’s story is an updated version of one that is 2,500 years old and is about a sea battle), while others are more recent. They are, and it may seem strange, among the deepest questions of all human knowledge. If they have piqued your curiosity, if you feel a kind of mental tension, welcome to the realm of paradoxes. My thesis is simple: logic and curiosity are the engines of human knowledge, and paradoxes are its most brilliant examples. 

In a paradox, usually, the premises from which we start are entirely plausible, and equally plausible are the logical steps, the reasoning steps, that allow us to derive new information from the premises (remember Nora’s reasoning?). However, and here lies the beauty, the conclusion is entirely implausible (Ken is not free to go to the party!) or it turns out to be false (There will be no exam!). How is this possible? There must be a flaw somewhere. Perhaps our premises are not as solid as we thought; we check them, and everything seems okay. So, the malfunction must be in the logical machinery itself, but even here, every single gear is functioning. Yet, things don’t add up. Faced with a paradox, we experience a sensation similar to what we feel when encountering a magic trick, an illusion: come on, that girl can’t have disappeared! But we are unable to explain our disbelief. Paradoxes trigger our curiosity because they produce some cognitive dissonance (there must be something wrong with Nora’s reasoning, and yet, everything seems to be in order).

And if we have understood the paradox, then we backtrack, we think again. However, we do it in a different way: to satisfy our curiosity, to unveil the mystery, we think more slowly, we think in slow motion. Our racing intuition is restrained by the constraints of logic. Everything that was taken for granted and, somehow, trivial, becomes problematic: we navigate through a minefield.

Paradoxes teach us to think because they combine logic and curiosity. Faced with Nora and Ken’s stories, we will investigate with precision the semantic sphere of the concept of “unexpected,” or we will question the nature of rational decision-making, free will, and the possibility of doing otherwise. Pure intellectual curiosity, without logic, would be exhausted in a conceptual oddity, a kind of small prodigy for its own sake, like those strange artifacts of the seventeenth-century Wunderkammer. Logic informs, in the etymological sense of the term, i.e. it gives shape (form) to our curiosity. The history of humanity’s intellectual achievements is nothing more than the tale of human curiosity and its rational garb; it is a marvelous story that may have begun tens of thousands of years ago in the slightly astonished gaze of our ancestor, attracted by a strange phenomenon, intrigued by a glitch within the landscape. It is a story that continues with the strenuous and titanic effort to understand the deep nature of what surrounds us, to enjoy its beauty and enchantment. 

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Ciro De Florio is associate professor of Logic at the Faculty of Economics, at Università Cattolica of Milan. His research mainly focuses on philosophy of logic and applied logic.  He is a member of the scientific committee of the Humane Technology Lab of Università Cattolica and he is investigating the relationship between AI and Human Interface. Among his recent publications, Divine Omniscience and Human Free Will. A Logical and Metaphysical Analysis; A Conceptual Characterization of Autonomy in Philosophy of Robotics; Reflections on Logics for Assertion and Denial; Future, Truth, and Probability. He likes hiking and climbing, especially with his family.