A Short Essay on Resemblance and Logic

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:

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:

which seem fine until we start questioning the last case, because why should the result be 'Francesca' instead of the symbol T? Perhaps our explanation is flawed whereas the execution is flawless—we have been tricked again, ironically, by someone who does not know knowledge.

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.

The Infant Language

Let us imagine a language and call it the infant language, because it mimics the ability of infants 3–12 months old to tell whether two simple objects are the same and to do simple reasoning.

Informally, we use the language this way:

b1: "x1 = x2 then conclude x1"
b2: "x1 ≠ x2 then conclude F"

We use the language to define beliefs, mapping events or observations to conclusions—what we expect to follow next. The previous beliefs correctly represent 'AND'. But then the question is: can we express all boolean connectors in this way?

The intuitive answer is yes—in fact, we can represent each connector by means of two beliefs. Let us challenge ourselves with the more complicated 'XNOR' boolean connector.

Instead of asking ourselves why this is called 'XNOR', we should be happy to realize it's just material equality, which we represent as

b1: "x1 = x2 then conclude T"
b2: "x1 ≠ x2 then conclude F"

Similarly, we have XOR:

b1: "x1 = x2 then conclude F"
b2: "x1 ≠ x2 then conclude T"

Finally, instead of covering each possible operator one by one, let us directly show 'NAND', thereby proving the whole point.

b1: "x1 = x2 then conclude outcome ≠ x1"
b2: "x1 ≠ x2 then conclude T"

The last case is interesting. On the one hand, it proves we can express all logical connectors; on the other hand, it relies on the ambiguous syntax "then conclude outcome ≠ x1". To understand why this syntax is ambiguous, suppose that instead of two symbols we have many. Therefore, if I say "x1 ≠ a", I am really saying that x1 could be whatever—it could be 'b', 'c', … and so on—though it can't be 'a'.

This is a consequence of diversity. If we assume that resemblance and diversity are primitives, we are also assuming the existence of some intuitive concept of group or family under which we collect similar cases—whose notion of being similar itself depends on context. Hence, "b, c, d, …" is a family of things that are different from 'a', which is some basic property they share with each other.

Conclusion

In this short essay, I proposed a twist. Why do we always have to start from logic, and how could we possibly justify the use of logic itself?

In fact, without the ability to distinguish things, in my view, there could not be any logic in the first place—because it would be impossible to tell the difference between true and false, undermining the necessity of logic itself. Instead, because either two things are the same or not, logic emerges, implicit in the morphological calculus of resemblances and diversities.

References

The ideas proposed here are mostly related to

• Coliva, Annalisa. "Morphology, Knowledge and Understanding. Prolegomena to a Pluralist Manifesto." Philosophical Studies (2025): 1–21.

The idea of Natural Kind in Quine:

• Quine, Willard V. "Natural Kinds." Essays in Honor of Carl G. Hempel: A Tribute on the Occasion of His Sixty-Fifth Birthday. Dordrecht: Springer Netherlands, 1969. 5–23.

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

• Wittgenstein, Ludwig. Philosophical Investigations.