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In an ontology, concrete concepts such as "Dog" or "Person" stand for sets (classes) of invididual instances. Abstract concepts, such as "love", "hate" or "wisdom" don't have any instances, but if they are empty sets, they are all identical. If they are individuals on the other hand, they are not "concepts" in the first place (also, individuals of which class?).
Where is my error in reasoning and how would one model such abstract concepts in an ontology?
You can do it either way. In 'traditional' first-order logic, the only individuals are objects, not properties. In first-order logic, one can say "there is an object that has the property F" or "all objects have the property F" but one cannot "quantify over" properties themselves.
However, in second-order logic this is possible. In fact, Leibniz's Law, which summarizes something important about identity is a second-order formula: It says, "for two objects x and y, [if x = y, then for every property F (Fx if and only if Fy)]."
Long comment
We have in place two dichotomies :
and
If we use the term "concept" not to denote a psychological entities (a mental representation) but as a component of an (objective) thought, it is an abstract and the basic relation is that of "falling under" :
Fido is a dog
is true because my dog Fido (an indivudual) falls under the concept "dog" (an universal).
But a concept can be predicated also of another concept :
prudence is a virtue.
Here we have an universal that is a "subset" of a more wide universal : both are abstracts.
Roughly speaking, particulars are objects and universals "curve out" classes i.e. collections of objects falling under the corresponding concepts.
One "big" issue is :
are there abstract objects ?
If we agree that e.g. numbers are abstract objects, we have that the number 2 is an individual falling under the concept "even".
The notion of a word as a reference to a set of objects fails for obvious reasons best described by Quine's discussion of Natural Kinds.
Late Wittgenstein proposes that instead of a set of objects, a definition is a token in a game that transfers power in the form of knowledge. The power that a definition gives is the ability to predict the behavior of the world around you, including other people.
The notion of love is no different from the notion of dog in this respect. Dogs behave a given way, they cause us to experience various things, and the people around them behave a given way because the dog is present. So does love, it 'behaves' in certain ways, increasing or decreasing and evolving in character, it causes us to experience various things and people behave in given ways when it is present.
So the right way to model both in an ontology is in terms of potential and expected effects.
Abstract ideas are universals - it is also referred to as concepts. A universal is a quality; a particular is a bundle of qualities.The best way to single out a quality is by pointing out what different particulars have in common - this process is called abstraction. All concepts are not equally abstract: the following examples show that there are hierarchies of concepts and that some concepts are extracted from other concepts.
Dog, red and person are not concrete; love, hate or wisdom are not as abstract as one thought.
Red is a class of sensations. No one has ever seen red alone; no one can imagine red independent of other visual properties such as area, shape, depth, brightness, etc, although it is not difficult to imagine a red thing, e.g., Clifford the Big Red Dog. Red is actually abstract - hard to believe, I know - it is the name of a universal. Names standing for particulars are called proper names, e.g., Clifford the Big Red Dog; a name that stands for nothing is a nonsense - from here we depart from the Platonist view. Whether the Platonist view is true or not I personally do not think it's worth considering at all.
Dog stands for a class of animals each of which we call a dog; no one can see or even imagine what dog looks like, although one can easily imagine an instance of dog - Clifford the Big Red Dog, for example. Dog is actually a universal or an "abstract idea" or a concept. On this account, the English language, which puts an indefinite article before a noun, cultivates a very good habit of mind: an indefinite article before a noun indicates that the noun stands for a universal - people have been thinking like this all the time, but it takes a Bertrand Russell to point it out. (English language is a manifestation of a superior mind: there is a famous Chinese paradox "white horse not horse," which is no paradox at all in English, owing to the English indefinite article; this example illustrates how language affects thinking.)
In Mauro Allegranza's answer, Fido, a proper name in the strictest sense, is used to stand for a particular. This is the kind of precision we should at least make an effort to practice in here.
In the case of a particular dog, everything a person knows about it he knows through his sensations in his own head. The dog in itself as an individual is not the origin of our knowledge about it; it is the sensations in our own head that are the origins of our knowledge about the dog. We believe the dog was the beginning of a chain of physical events that lead to our sensations, but everything we can say about the physical world is inferred from sensations; as a matter of fact, the causal chain between the dog and the canine patch of colour in the mind is open to doubt: there is no evidence to suppose we are not dreaming. It follows that in order for a proper name to have meanings, it is only necessary that it invokes mental images; this explains why Hamlet is considered a fiction while Louis XIX is considered meaningless: there is a whole play describing Hamlet in picturesque terms, whereas Louis XIX invokes no mental image.*
Like dog and red, love is also a universal or an abstract idea, but it is not detached from particulars: it is a class of particular mental states or feelings. An instance of love is not intrinsically different from an instance of seeing a dog: they are all mental occurrences, although we believe they both have physical causes.**
All universals derive their meanings ultimately from particulars: some are classes of particulars, some are class of classes of particulars, some others are class of classes of classes of particulars, etc; there are hierarchies of concepts - that is why Bertrand Russell said he discovered the theory of types because that is how people actually think. For example:
Red is the class of all red sensations; red is a colour, but "Clifford the Big Red Dog is a colour" is not true, therefore colour is a class of classes.
2 is the class of classes each of which has only two members; number is a class of numbers and is therefore a class of classes of classes.
A cleat is a member of a carefully selected pair; a pair of cleats is an instance of 2; 2 is an instance of number.
Regarding everything that exists only in thoughts as abstract ideas - many dictionaries are still defining abstract this way - has licensed many nonsenses disguised as "abstract concepts" and has given aura to a mysterious manner of speech that sounds like announcing aphorisms containing profound truths but is actually emitting nonsenses. If a word stands for neither a universal nor a particular, it is a nonsense, or an incomplete symbol at best.
Notes:
** Germans don't say "Ich liebe dich" very often because when they say it they mean it. Americans say "I love you" all the time as a compliment - whether or not this expresses a mental state does not matter - but this kind of lies are fully approved of by Shakespeare.
Source: Russell, Bertrand. "Knowledge by acquaintance And Knowledge by Description." The Problem of Philosophy.
