A Short Essay on Resemblance and Logic


It is perhaps by intuition or cultural heritage that we think of Logic as the essential means to investigate the validity of our reasoning.

Why?

To some extent, it seems an existential necessity—one that can hardly be explained without the help of other logical concepts, thereby resulting in a self-referential and therefore questionable justification. So, suppose that Francesca is a person new to logic and mathematics, and that we have to explain to her the meaning of the logical operator 'AND' while respecting her initial knowledge. Can we explain the meaning of 'AND' to her without using logic? And, if so, how do we do that?

In this short essay, I argue that one solution to this unnecessary problem is to use the idea of resemblance and diversity. In fact, if Francesca understands that two things in principle either resemble each other or not, then she already has an implicit understanding of the use of the 'AND' operator and of all possible logical connectors. Therefore, I suggest logic as the morphological study of the unintended consequences of resemblance and diversity.

Suppose we find ourselves in the thorny situation of having to explain the logical AND operator to a person new to logic or math and, for the sake of the narrative, let's call her Francesca.

So the problem is: how do we explain to her the meaning of 'AND'? And, by the way, what is the meaning of the word 'meaning' itself—and therefore what does it even mean to understand or know something?

Instead of jeopardizing ourselves by tricking our minds with linguistic games, we might simply say something like:

"The logical AND operator connects two statements and results in a true statement if both statements are true, and in a false statement otherwise."

However, this explanation may not satisfy Francesca at all. In fact, our explanation seems to depend on knowledge of logic itself.

Indeed, Francesca rejects our explanation and requests one that does not rely on logic or algebra in the first place. She explains that she does not understand the meaning of 'true'—and she gaslights us when she asks about the meaning of 'both'.

Instead of entering the rabbit hole of explaining 'both', we decide to buy some time. Maybe we can teach her the meaning of 'AND' by example? So, imagine a convenient blackboard we can use to draw the truth table of the 'AND' operator, which looks like this:

CASES x1 x2 x1 AND x2 — the result
case 1 T T T
case 2 T F F
case 3 F T F
case 4 F F F

If you do not know what a truth table is, you are in good company because we need to explain it to Francesca too. A truth table is really a table in which each row represents a unique event. Either two things are the same or not and, therefore, what is the consequence. Hence, we are just enumerating the possible cases.

Yet, as we begin writing, Francesca warns us that a mere enumeration of facts does not constitute an explanation—and insists that something in common must be found between the examples to account for knowledge.

There is some precious thinking in what Francesca just told us. As a matter of fact, this very idea of commonality itself assumes the ability to distinguish whether two things can be said to be 'common' or not and, at its heart, this implies the ability to tell whether two things are materially equal or not, like the symbols T and F—are they the same symbol or are they not?

Hence, by paying attention to her words carefully, we can try the following trick:

—Look at the table!—

Cases 1 and 4 are similar because, in them, x1, x2, and the result all resemble one another. This means:

i) Whenever x1 and x2 resemble each other, the result is whatever the value of either x is.

Cases 2 and 3 are also similar:

ii) Whenever x1 and x2 do not resemble each other, the result is the constant symbol false.

After a few questions about how to interpret the picture in her mind, she seems satisfied—and how could it possibly not be the case? Everything we explained stems from the fact that either two symbols are the same or not.

As a matter of fact, Francesca now seems pleased—and she demonstrates to us her understanding by showing the following cases:

CASES x1 x2 x1 AND x2 — the result
case 5 Luca Francesca F
case 6 Francesca Francesca Francesca

which seems fine, but what about the last case though? Why should the result be 'Francesca' instead of the symbol T?. That's not a new riddle, it's the riddle itself.

Instead, Francesca breaks our inner conversation, revealing to us that the logical operator AND does not seem to be so different from some uses of the connector and in her own language. She explains that the statement

"Luca and Francesca ARE friends"

makes intuitive sense because both 'Luca' and 'Francesca' are persons and thus resemble each other. Furthermore, because we understand that persons can be friends, the connector and places 'Luca' and 'Francesca' within the same relationship of friendship. By contrast, the statement

"Luca and a stone ARE friends"

does not seem intuitively right. A stone is an object that cannot reciprocate friendship, so in the absence of additional information, the natural conclusion is that the statement lacks reasonable meaning.

Finally, perhaps with some irony, she tells us that we should not bother explaining to her the meaning of the symbol F. She thinks of F as the impossibility of composing something with something else—like Legos, that can be composed only in certain ways—suggesting that either two things can be composed or not and whenever so they resemble each other.

END OF PART I

References

The ideas proposed here are mostly related to

The idea of Natural Kind in Quine:

And the idea of Linguistic Games and Family Resemblance in Wittgenstein: