Is a thing just a class with only one member?

Question

Imagine this ontology:

Thing > Language > European Language > Nordic Language > Scandinavian Language

Each of the above are a type of thing, which seems to imply that there can be one or more instances of such a thing.

An instance of a Scandinavian language could be Norwegian.

On the one hand, you can consider an instance of a type to be an orthogonal type of relation to merely being a subtype. So you would say that Norwegian is an instance of a Scandinavian language, not a subtype.

Each of the above classes can have associated properties. An instance of a class instantiates values for its properties. So maybe Norwegian is defined by those choices of values for the properties of a Scandinavian language. If a language has a lexicon and a grammar, then Norwegian must have a lexicon (the set of words in Norwegian) and a grammar (the set of grammar rules of Norwegian).

Because Norwegian is defined by certain properties, what if we created a type, Norwegian, and specified that it must have properties like a Norwegian lexicon, and a Norwegian grammar (both of which are also classes). We could say that a Norwegian lexicon is any lexicon which has the following words: [list the words of Norwegian].

Now it appears that Norwegian is a type of thing defined by certain properties and relationships, and it just happens to have only one instance (the language Norwegian).

It seems like any instance can be converted into a class with only one member.

This might also seem more accurate if we consider contexts in which things commonly taken to be unique no longer are. If there are parallel universes, there can be another person with the exact same properties as me, but we are not the same instance. We are both “a Julius”, an instantiation of type “Julius”, with my exact characteristics and personal history.

Does this imply that we don’t need the concept of an “instance”? Can everything be defined as a collection of properties or relationships?

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It reminds me a bit of:

• a Borges story about a secret society of “idealists” who only believe in the reality of mental representations
• the Ship of Theseus in that to exist in time is to be a sequence of physical states where there is no inherent way to claim they form a single “thing”; every moment in time the physical states in the world update, and you are a re-instantiation of a thing with nearly identical properties to the corresponding thing in the moment before
• dependent type theory, where I think every term of a type is also a type in its own right.

What philosophical theory is this and who has developed this into deep conclusions, one way or the other?

Answer 1

The 1991 book "Parts of classes" by David Lewis is about set theory and mereology. Lewis says his "Main Thesis" is the following proposal: The parts of a class are all and only its subclasses. He goes on to conclude that singletons are atoms, i.e., in Lewis's set theory, he has identified the one-element class {x} with its sole member x. In this system, the answer to your question is yes: a thing is precisely a class with only one member.

For a fast but substantial description of Lewis's project, John Burgess wrote a nice review of "Parts of Classes" available from his website, here: https://www.princeton.edu/~jburgess/Lewis.pdf

About the mathematical consequences: Hamkins and Kikuchi wrote a nice paper about Lewis's system, here: https://arxiv.org/abs/1601.06593 As I recall, Hamkins and Kikuchi show that Lewis's system is decidable, and consequently cannot serve as a foundational system for much of modern mathematics, in which many problems (e.g. the word problem in groups) are known to be undecidable.

Answer 2

You ask:

Does this imply that we don’t need the concept of an “instance”? Can everything be defined as a collection of properties or relationships?

These are questions that are fundamentally ontological in nature and have to do with the classic problem of universals. For a very long time, it has been self-evident that there seem to be instances and classes in which we can lump instances together. Let's say you own two dogs: Casper and Wilfred. It is natural to consider Casper and Wilfred as individuals, but given their differences and similarities, attempt to create a word for what they have in common (dog). And a look around the world means there are all sorts of ways we can create categories (SEP). The two dogs are both male. The two dogs are different breeds. One dog is young and the other is old.

It's difficult to see how instances aren't necessary to thought, because it is from the instance that we derive the notion of identity. From a naturalized epistemology, it seems our minds are evolved to identify instances, in fact. The rest of the abstractions of classifications creates troubles. Should a breed be a concrete thing? What about a dog? What about an animal. These are the classic questions of how to deal with universals and particulars or which one can have a realist, conceptualist, or nominalist bent.

In fact, categories are so dominant in our ways of grouping together instances in life, there's a part of speech dedicated to assisting nouns in dealing with classes and subclasses: subsective modifier. From WP:

In linguistics, a subsective modifier is an expression which modifies another by delivering a subset of its denotation. For instance, the English adjective "skilled" is subsective since being a skilled surgeon entails being a surgeon. By contrast, the English adjective "alleged" is non-subsective since an "alleged spy" need not be an actual spy.

And categories that seem to exist independently and universally of people are called natural kinds (SEP). Forces, particles, organisms, and species are the sorts of things that seem to have structure independent of human thought. As far as their properties, when those properties have, well, the same property, then they are considered natural properties (SEP).

Are objects just bundles of properties? Hume thought so. He argued there was no substance that unified properties, and that there really wasn't anything to an object above and beyond its properties, and certainly no objects without properties. In fact, Hume's argument created a crisis of sorts because he argued there was nothing that unified the mind other than this bundle of perceptions of properties. From Britannica's article Bundle theory:

[The theory] advanced by David Hume to the effect that the mind is merely a bundle of perceptions without deeper unity or cohesion, related only by resemblance, succession, and causation. Hume’s well-argued denial of a substantial or unified self precipitated a philosophical crisis from which Immanuel Kant sought to rescue Western philosophy.

So, there is no canonical answer to your question, because you are asking after fundamental questions that present themselves to an ontologist. Where do we draw the line between classes and instances? How do we determine if something is real or a category of the mind? Do properties exist independently of objects? Each of these questions can absorb a thinker for years.

Answer 3

I think that if you look closely, anything you could call an "instance" is actually also a class having multiple instances. "Norwegian" is a good example because, would you believe it, there is North Norwegian which is distinct from South, West or East Norwegian, and each of these again can be divided into finer instances of Norwegian. Which reminds me of the old joke about "Northern Conservative Baptist Great Lakes Region Council of 1879, or Northern Conservative Baptist Great Lakes Region Council of 1912?".

Even individuals (or any other discernible objects) differ from place to place (family vs. work persona) and over time, and not only in the actual matter that makes them up: "You are not the man I married anymore"... So there are actually multiple instances of "me" scattered through time, probably an infinite number of them.

So yes: There are no instances. Everything we consider an "instance" is actually another collective, and we just decide to stop dissecting at some point because it is a good enough approximation, and to not get lost in infinities.

Answer 4

This answer is going to take the form of a critique, of your initial request: that we imagine a certain seemingly linear ontology. Obviously, that is too naive a way to read your intent, so I should qualify that we would actually be speaking of parallel lines of typology (to use a recondite word for it), of trees of types. But...

... now I want you to consider the type-token-occurrence distinction, as per the SEP entry on the topic. Note that, perhaps even more originally, typology was a matter of types and things called anti-types, where this was understood in the sense of "on opposite ends of a given line" rather than on lines themselves opposed to each other. Anyway, so consider the most general type of the letter "t" in English. Now consider how this can be put into boldface, italicized, and switched around as to its typeface/font indefinitely, yet will remain a type itself (of all concrete tokens of "t" per this variation), or rather an ensemble of types, with further ensembles of tokens, etc. So would this look like even a set of parallel typological lines, or a fuzzier, more freely branching thing?

The upshot is not that you are wrong to think that every concrete token is itself a general type in its own way as well. One might say something about there being abstract degrees of generality and particularity, so that we could take some term and compare its degree of generality to its degree of particularity. A deeper (or higher?) type would at least tend to fall on the side of greater generality, both in relative and absolute terms:

• Let F be the degree of generality, and f the degree of particularity. Then firstly we can talk about:

1. F > f
2. F < f
3. F = f

• But also, suppose that there is a specific (range of) value(s) for F or f such that, if that is the dominating value, this makes a difference to our classification scheme. What I mean is, let's say if something's degree of generality is omega₀ (the initial countably infinite ordinal), and its value of f is less than that, then that is a type, whereas if F were some n and so was f but still F > f, then instead of calling this a type, we'd call it a property, maybe. Or we could flip the terms of the definition around (or sideways) and construct some other relevant pattern. The gist of these results would be that differentiating ur-types (types that are just types, and not tokens, not even of themselves) from ur-tokens (tokens that are just tokens, and not types, not even of themselves) becomes a much hazier matter indeed.

Answer 5

The following is an answer to your question from the viewpoint of Object-Oriented Programming, more general Object-Oriented Design and even more fundamental Object-Oriented Thinking, a type of ontology.

The main principle of object-orientation is to combine

• properties,
• methods,
• and lifecycle

of objects to a class, if the objects are similar in these respect.

Hence the strategy to avoid redundancy and its benefit is to form classes with more than one object. Singleton classes are not excluded, they have their use. Only the context and the intended goal can decide.

Ontologies are not discovered but constructed (= designed).