Mankind's impulse to symbolize, by metaphor or storytelling, or simply through the various grams (picto-, ideo-, phono-), has come in many times and in various ways.
summary: a musing on set diagrams in exploration of the idea that a proper study of logic will not neglect the production of visual, non-verbal diagrams to represent separation
I would posit that the point of symbolism is not to understand the separation of existing things, but to explain them - either in a record for the self, or for others. Primarily, I would suspect, for the latter. As the Woman on the Island scenario humbly suggests, language is not necessary for a solitary individual. Even basic symbolism external of the mind is unnecessary. Groups hardly require it, but language streamlines interaction.
In logic, an idea or concept is referred to as a "term." The word that expresses that term is not the term. Hence the immensely popular (to logicians and others) painting by Magritte:
The word "pipe" is not the pipe itself. Nor is the word "moth" necessary to the existence of a moth - or even the experience of a moth. It is, however, immensely useful to a rational mind seeking to explain the concept to another.
The concept of a "term" is the first hurdle a student of logic must leap. After all, our understanding of the world in modern cultures is so fundamentally tied to language, it can be hard to consider objects without language. This is why I enjoy having students think about the words "felis," "cat," "gato," and "neko." All describe the same term. The sounds have different feels. I am not instructing students to view language as worthless or a hindrance to understanding (indeed, it is a gateway to understanding), but to understand how language colors our approach to a term. Contemplating the objects described by words is to acknowledge the difference between terms and words.
Terms, however, overlap. The concept of an animal that has legs contains many individual objects.
Of course, it could be argued that this grouping is artificial. Barring a common ancestor or divine ordinance, what relationship does a spider have with a dog?
Perhaps this points to language having an inherent element of expectation. By using a regular set of symbols (spoken, written, gestures) to express what one supposes to be a consistent understanding of a term, one is positing a consistency to reality. We call it a "dog" by presupposing that there is a value to separating this thing we encounter that seems to reproduce after its kind and acts and appears differently than others. It is not the same as the thing we call a spider. However, it does share things in common.
Set diagrams, known often as Venn diagrams or Euler diagrams (which are not the same), are one way to express categories as well as overlap. They are a valuable tool for testing the critical thinking of students at various stages.
A student could be asked what the relationship is between "food," "fruit," "strawberries," "cheese," and "apples." They would hopefully be able to express that the latter four are types of food, and strawberries and apples are types of fruit, whereas cheese is not - but it is a type of food.
Of course, there is an element of artificiality. Consider the tomato - is it a fruit or vegetable? The division is one of popular terminology as opposed to botanical terminology - literally, a tomato is a fruit based on botanical understanding. Given its taste and consistency and general usage in most cultures, it's a vegetable.
Regardless, to even talk of the difference is to talk of the dividing principle of its usage. And this division is, of course, separation.
In set diagrams, it is customary but not universal to surround a set diagram with a box that represents everything.
But the question is: what is 'everything?'
I'm offering that the understanding that the box is things that can be conceived is a good way to define everything. Perhaps "conceiving" isn't the best term, because the word has evolved from its Latin root of "taking in" to an idea of creating, beginning, establishing. Perhaps things that can be contemplated is better, though this may seem like splitting hairs.
Things that can be contemplated are intangible (ideas, abstractions, emotions, plans, etc.) and tangible. "Tangible," able to be touched, would refer to external objects that can be manipulated or experienced directly through the senses.
We interact with reality through our senses, so thus everything we perceive is truly an idea. My sense data tells me that things are hot, cold, blue, green, bright, dim, salty, sweet, rough, smooth, and so forth. My nerves process it to the best of my biological abilities (colorblind, hearing loss, etc.) and provide my brain with an idea of the world.
However, as should be discussed further quite soon, interaction with the world requires either an acknowledgment that an external, independent world exists, or an acceptance that acting as if an external reality exists is the best way to guarantee the continuation of the self.
Elements of that external reality are not the whole of my experience, however. When I contemplate goals, I am not experiencing reality directly. The Woman on the Island may make a plan to scare off a group of birds so she can obtain eggs. As she sits in her domicile, thinking of the best plan of attack, she is entertaining conceptions that are not sense data.
Of course, we can talk of common ideas that have no physical correspondence: love, hate, desire, pride, etc. Many of these require social settings to be experienced, but not all: the Woman on the Island can experience pain, sickness, etc. and contemplate these. She eats a certain thing and grows ill. When the illness passes, she cannot retrieve it, hold it in her hand, and observe it then. But she can recall it, and begin to seek the source. What did she eat recently that may have been out of the ordinary that preceded her illness?
This idea of things that can be contemplated may split hairs, but consider that possibility that even things we cannot experience, cannot even fathom in terms of their properties, can be contemplated in a very basic sense. We can contemplate the fact that we cannot properly conceive them. We can conjure up their genus: "Things that cannot be conceived." However, we are staring off into an absence of light that can never be illuminated. This is not to rule out invention and ingenuity. An individual Roman may not have been able to contemplate the idea of a blog, though various information networks before the Internet may have actually served to aid them. "What if we could one day send messages without dispatching a messenger along our roads, or lighting signal-fires or sounding trumpets? And what if this system of messaging could also produce physical versions of our messages that could be stored...?"
However, there are things we cannot conceive. "Like what?" one may ask. Well, that's the point. They are so far out of our experience, that they will never be conceived.
We may have a paradox here. I speak of "things that cannot be contemplated," but I posit that we can contemplate their existence, if not their particulars. Perhaps the idea of things that cannot contemplated is the limit. This may be where I need you, dear readers, to tell me if I have a paradox, or merely a contradiction in my explanation. It may be the latter.
The speculation of what those things may be is both contrary to their definition and unnecessary. We are best concerned with what is real, or effectual on our persons (harmful ideas, such as various destructive political theories, can affect me even if they defy the laws of reasonable thought).
So the box can be understood as those things within grasp of our minds, and anything outside of that box is understood to be those things eternally out of our minds. A literally-minded person might desire to make the box represent everything, even those things we cannot contemplate.
Of course, that raises the question: if the box represents everything that can be conceived, even those things that can only be entertained in regard of our inability to ever form a mental image of them or experience them as part of our external reality, then what belongs outside of that box?
If the box represents everything ever in our grasp, and those things ever outside of our grasp, what lies outside of that box?
It is good that logic has bequeathed so much: linguistic theory, computer science, Socratic dialogues, game theory, etc. Otherwise, it could rightfully be dismissed as needless abstraction. However, it is not. Logic, as the pursuit of consistency in thought, is fundamentally a practical exercise. Because its study must be limited to things that can be conceived, we make a fundamental category excluding endless speculation on the unknowable, and focus on the knowable.
Perhaps the box of set diagrams ought to represents things that can be known. Not things that will be known in our lifetimes but things that can be known.
Deductive logic operates with the things we know already, or rather the prevailing theories and categories in our current web of belief. Inductive and abductive logic are the processes that allow us to clear the fog in the corners of the box.
However, we do not need diagrams to know things. Set diagrams do provide an outlet for explaining what we know, and for categorizing our knowledge.
Perhaps my greatest frustration with the few introductory logic texts aimed at junior highers is that they ignore the process of visually representing separation through means other than mere words. By teaching syllogistic or term logic without including the diagrams that are customary in high school or college courses, they limit the ways a student is equipped to express the data they have.
The reasoning offer to me for some of these texts is that the goal is to teach debate, proper writing, etc. But shouldn't the ability to properly expression categories proceed complicated matters of fallacies?
One logic text I was given to review for a school had a cartoon character laughing off formal logic, saying it was boring. "Let's dive into fallacies!" the little homunculus chortled, "Because they're more fun!" However, the informal fallacies make more sense in light of understanding deductive vs. inductive (and abductive & analogical) reasoning, and the basic procedures of categorizing and expression. Because grammar school forays into logic involve placing set diagrams briefly into math and sentence diagrams briefly into grammar, this idea of understanding the process of understanding is often lost. Students associate various aspects of formal logic with individual subjects, instead of understanding how the processes of logic help guide all areas of study.
Beyond that, visual representations are an essential teaching. For a staggering number of languages, glyphs first come as visual depictions (pictograms). As the world of that culture expands and the number of terms they need to express grows, ideograms and phonograms arrive. Ideograms allow expression of concepts not rooted in direct physical experience, and phonograms allow the expression of a wide body of both ideas and external objects, and combinations of both.
Aristotle noted, as is common knowledge, that mankind is very much a visual creature. Thus, a proper course on logic will include a use of visual symbols, outside of language.
Education should seek to not only fill out the Box of That Which Can Be Known, but to allow students to present the relationships between things in that box.
The study of logic, I would humbly assert, is the way to formally educate students in understanding how to critically and consistently approach any topic.