Taxonomy is a subject in traditional and modern logic. Richmond Thomason (1969) approached it using concepts from algebra, and I will follow suit. But my conception is somewhat different from his; I will provide a point-by-point comparison with his below.
1. Porphyry the Phoenician
Systematic taxonomy, like so much else, begins with Aristotle. In Porphyry’s Isagoge, his introduction to Aristotle’s Categories, we come to see that there is a (downward growing) tree structure in the Aristotelian division of nature into genera and species.
Arbor Porphyrii, Purchotius (public domain)
Here we see, to use our terms, Body divided into Animate and Inanimate, with Animate divided into Rational and Non-Rational. While this Tree of Porphyry cast a long shadow in our history, only the tree structure remains prominent as conceptions and language changed.
As Porphyry explains, the distinction between genus and species is relative. For example the relation of Body to Animate is the relation of genus to species, but so is the relation of Animate to Rational. In modern logic, where the notion of genus or species was found too narrow, the new relative distinction introduced was between determinable anddeterminate.[1] This was meant to apply to properties generally. While it is clear in the Isagoge that color is not a species or genus, it is (in the later terminology) a determinable, with red, blue, and yellow some of its determinates. Then red is in turn a determinable, with scarlet and crimson among its determinates. So the tree structure remains, though how we see nature is not through Aristotelian glasses.
2. Families of properties
As Russell explained in his book on Leibniz, treating relations with the device of relative terms is a handicap. In our terms, the tree structure is a specific partial ordering. Let’s introduce our own term and say that scarlet is subsumed in red, or equivalently that red subsumes scarlet, and use “≤ “ as symbol for this relation:
… scarlet ≤ red ≤ color
Then properties divide naturally into families, related as it were genealogically. There is for example also the family of age properties. These are instantiated by colored things, but no age property is a color property. The family of musical properties is also disjoint from these, but are not instantiated by colored things.
To stabilize my own terminology: I will use “category”, despite the other uses it has or has had in philosophy, logic, and mathematics, to refer to such property families. As Porphyry explains, we can plead precedent: Aristotle had felt free to adapt a term that already had a quite different common use (Porphyry 2026: 45). [2] What we have arrived at so far is that properties form a poset (a set with a partial order relation), and that this set has parts, the categories, which are little posets with the structure of a tree.
The groupings that give rise to a taxonomy are the ones that are significant and salient in a given context or for a given purpose. So the term “category” is context-dependent. The context-dependence can be canceled by qualifiers: botanical category, political category, and so on. Such qualifications determine not only what belongs to the category, but its proper divisions. So for example, color may be divided as painters do (the color wheel), or as in physical optics (color spectrum). These are two different categories, classifications of the same subject but in different ways.
Previewing the precise definitions to be made below, we think of a category as a set of properties related to each other by the determinable/determinate relation, so as to form a tree.
3. The question of meets and joins
In contemporary philosophy there is still another distinction inherited from the tradition that focused on Aristotle’s notions of genus and species, namely the distinction between natural kinds or classes and arbitrary or non-natural classes. For example, man and mouse are considered natural while man-or-mouse is not. We can accept this as a correct observation about natural language use, without buying into any metaphysical doctrine. We can make another logical point of this sort with what, after Gilbert Ryle, we call a category mistake. There is for example nothing possible to which both allegro and red could apply, it would be a category mistake to ask for an example.
But there is a difference between these two. To claim of something that it is both allegro and red would be absurd. We can classify it as a self-contradiction ex vi terminorum. That means that in the poset of properties, both allegro and red is the bottom element, the ‘zero’. Now we might be tempted by the logical duality of “or” and “and” to think that we can treat man-or-mouse the same way, mutatis mutandis. But that would mean that this property is at the top, which must be the property that everything necessarily has. The consequences seem too hard to swallow: would it make any sense to say that every deer and wolf is a man-or-mouse? I think not.
The upshot of this is that the poset of properties – call it A –is, in algebraic terms, a meet semi-lattice bounded below. That is, there is a definable operation meet (∧) and special zero element (⊥) such that:
- if x is any element of A then ⊥ ≤ x
- if x, y are elements of A then there is an element x ∧ y of A such that z ≤ x and z ≤ y if and only if z ≤ (x ∧ y)
In words, (x ∧ y) is the greatest lower bound of x and y. Given this we can define moreover a special relation of contrariety:
Definition. We say that elements e and e′ of A are contrary iff e ∧ e′ = ⊥.
From now on I will refer to A as a property algebra, or algebra of properties. Now I have insisted that A is not closed under join, the dual of meet: it is not in general the case that every two elements have a least upper bound. This does not rule out that sometimes they do. Thus we have the resources to define the special sub-families of A which are the categories.
Definition (Partition of e) If e is an element of A then set X of elements of A is a partition of e iff the members of X are contrary to each other and e is the join of the members of X. A partition is trivial iff it has just one member.
Definition (Category in A) A set PC of elements of A is a category in A iff:
- PC has a largest member, ⊤PC and
- there is a finite or infinite sequence X1, X2, . . . of partitions of ⊤PC such that for each j = 1, 2, 3 . . ., Xj+1 is a refinement of Xj, and X1 has at least two members, and for each member e of PC other than ⊤PC there is a number k such that e is in Xk.
- The zero element ⊥ is not a member of PC.
4. Categories in an algebra of properties
A look inside A may reveal that some of its subsets are categories. Some of these may overlap. None include ⊥, so categories need not overlap.
Optimally for a taxonomic system, A may be entirely divided into categories. Suppose CA is the union of all categories that are part of A. This set is not in general closed under either meet or join. But since A is a meet semi-lattice, the closure of CA under meet in A is a meet semi-lattice which is part of A: call it CA ∧ .
As far as taxonomy is concerned, we need nothing more in the algebra beyond this choice of categories, so we restrict A accordingly to be what I will call categorial:
Definition (Categorial) Algebra A is categorial exactly if A = CA∧.
For illustration, here are examples of structures that can be categorial property algebras, and also structures that can’t be categorial property algebras. Some are lattices, some are not; some are distributive or modular and some are neither.
The ‘diamond lattice’ M3 consists of top 1, bottom 0, and three atoms, and is categorial:
This has only one category, namely the entire lattice except for 0: {1, {x, y, z}}.
The 8-element Boolean algebra B3, isomorphic to the powerset of its atoms {a, b, c}, namely,
is a distributive lattice which is categorial.
If we remove the top from B3 we get a structure which is no longer a lattice but it is a bounded semi-lattice, and is also categorial:
Another structure, N5, is the smallest non-modular (hence also non-distributive) lattice:
There are two overlapping categories in N5, with top 1, and respectively the partitions {x, y} and {z, y}.
Traditional examples of the tree of Porphyry are all binary, that is, any cell in the partition Xk is the join of at most two cells in partition Xk+1. One might think that non-modularity or non-distributivity might occur in algebras with (possibly overlapping) non-binary categories, and not if all the categories are binary. We note that the example of the non-modular N5 refutes the thought. N5 is a bounded lattice; it has only two categories in it, both binary; and it is categorial. This example may well strike one as rather unusual looking. In the comparison with Thomason below I will discuss that.
Examples with more intuitive appeal may be taken from geometry, where the sub-spaces of a given space form a non-distributive lattice. The much-discussed case is that of the logic of quantum mechanics. For a finite example of this sort, take S to be a Euclidean plane, its subspaces are all the lines through the origin plus the set that contains just the origin O (null-space):
The join of any two distinct lines is S and their meet is O. A partition is a set of two orthogonal lines through the origin. If {x, y} and {z, t} are partitions with z at a 45◦ angle to x then the set{S, x, y, z, t, O} is a non-distributive lattice, generated by the two overlapping categories with top S.
Finally, here is a non-trivial, infinite example. In the set N of natural numbers, the co-finite subsets (subsets whose complement in N is finite) form a join semilattice with top N but no bottom. The elements are sets and the partial order is set-inclusion; meet and join are intersection and union.
From this we form bounded meet semilattice L by taking as elements the co-finite subsets of N plus the empty set and the unit sets {1}, {2}. This has a bottom and a top N, but is not a lattice because it does not contain the union of {1} and {2}.
Since intersections of co-finite sets are co-finite, a partition of N in L, or of any subset of N, can only have one member that is a co-finite set. Besides that, the partition can only have {1} and/or {2}as members. So for the tops of categories we can choose three sorts of elements: co-finite sets that include either 1 or 2 or both. These, and these alone, have non-trivial partitions in L. For example, {1, 2, 10, 11, …} has the partition {{1}, {2}, {10, 11, . . .}}.
Let us accept all these categories. Their union generates (by closure under intersection) the entire lattice L, for if co-finite set X contains neither 1 nor 2, it is the intersection of X ∪{1} and X ∪{2}, which are co-finite. So L is a categorial meet semilattice.
Finally, there are certainly meet semilattices that are not categorial. Here are two. The first contains only one category, with top c and partition {a, b}, but the closure of this category under meet does not include the lattice top 1:
The second contains no categories at all, since there is no non-trivial partition, due to the location of the bottom.
5. Comparison with Thomason
Thomason bases his approach to the definition of a taxonomic system on the philosophical concept, of a natural kind. According to this concept, which traces its genesis to Aristotle, and which finds ample support in features of natural language, nature is divided into kinds, corresponding to a select class of properties, “carving nature at the joints”.
When it comes to modeling the form of a taxonomic system, Thomason mentions the genus-to-species relation, and submits that
“the natural kinds of a system of classification may be conceived of as obtained by a process of division. The universe is first divided into disjoint sorts (e.g., animal, vegetable, and mineral), then these are further divided into disjoint sorts, and so forth.” (Thomason 1969: 96)
Thomason defines a taxonomic system to be a couple S = <S, ≤ > with ≤ a reflexive, transitive, antisymmetric relation on set S, whose members represent natural kinds. (I will keep using “subsume” for this, as above, and stick with my own symbolism, rather than Thomason’s.)
Thomason stipulates that S is closed under join (lub), with top V. This stipulation is introduced “to obtain a mathematically tidier structure”. But he notes a problem about joins: it is ruled out that, if man and mouse are natural kinds, then man-or-mouse must be as well. How exactly is this point accommodated? They do have an upper bound, namely V, and perhaps one lower down, like animal. Perhaps we can only say that, surprisingly, the join man–or–mouseis identical with the join giraffe-or- mouse. But this is ‘from outside’, it is not a structural feature of the taxonomy.
Above I took the opposite tack of not stipulating that a taxonomic system has a top, which seems more natural to me, foregoing mathematical ‘tidiness’. My motive was to give no room for such a consequence as “Everything is a man-or-mouse”.
In addition to closure under join, Thomason stipulates closure under meet (glb, greatest lower bound) and that there is a zero element (0), which is subsumed by all the kinds. Therefore man-and-mouse also exists, for if nothing else it can be 0. So the structure taken as a whole is a lattice that is bounded both above and below.
His example to prove that the lattice is not in general distributive relies on the identifications mouse-and-beetle = 0, while man-or-beetle = V.
An example of a taxonomic system, using my terms, would be a categorial algebra consisting of a category of which V is the top element, and with 0 added as bottom. There are more such examples, but they are constrained by Thomason’s principle of Disjointness:
(D Disjointness) No natural kinds a and b of a taxonomic system overlap unless a ≤ b or b ≤ a.
This is made precise as
(D’) Either (a ∧ b) = 0 or (a ∧ b) = a or (a ∧ b) = b
This allows a taxonomic system of several categories, with V as (perhaps artificial) top element of each. For example, the elements just below V are animate and inanimate, each further divided so as to form a category. But they cannot overlap without violating (D). For example, if animate and inanimate both had virus as one of their subdivisions, then they would have a non-zero overlap although neither is subsumed by the other.
As Thomason points out, (D) implies that the taxonomic system is a modular lattice, it satisfies:
Definition. A lattice is modular iff for any elements a, b, c, if a ≤ b then
a v (b ∧ c) = (a v c) ∧ b.
One my examples of a categorial algebra of properties was N5, the smallest non-modular lattice. The difference between us here comes from the fact that I admit categories with non-trivial overlap. In N5 there are two overlapping categories {1, {x, y}} and {1, {z, y}}, and while x v (z ∧ y) = x, (x v y) ∧ z = 1 ∧ z = z. There is no single category of which both are parts.
Is this egregious? Thomason may well complain that in this scheme, kinds x and z are distinct, although there is in the taxonomy no differentiating property to explain how x could have members that z could not have. Arguably, this makes no sense if all the elements are to be natural kinds.
But in my approach, the algebra of properties is to be represented in such a way that it can harbor radically different taxonomies, not restricted to the tradition of nature ‘carved at the joints’ into natural kinds. A single subject may be categorized in radically different ways, although the background is provided by a single family of properties that can enter the classification.
The design of a specific taxonomy may involve different goals and factual assumptions, even entire scientific theories. Think of the difference between the Linnaean classification and a classification based on genetics or evolutionary relationships.
A guiding principle involved might for instance be that classification should respect what is physically possible. Then two different designs could be based on different beliefs about what is physically possible. And both would differ from one that respects only and all conceptual distinctions, regardless of what is physically possible.
A toy example will fit N5. Imagine two designers who intend to respect physical possibility. There is just one difference: one admits the physical possibility of horses that are unicorns, the other does not. For the latter, “horse or unicorn” is necessarily equivalent to “horse”. When it comes to what is physically possible for things that are not horses or unicorns, there is no difference at all in what they admit. So N5 can model this: x = (horse or unicorn), z = horse, y = (neither horse not unicorn). In this way the categorial algebra of properties represented by N5 accommodates these two distinct but overlapping categories.
Thomason’s paper concludes with an important exploration of how taxonomies can be made to play a role in the semantics of modal logics. I will not try to follow suit here, but see Beall and van Fraassen (forthcoming).
References
Beall, Jc and Bas C. van Fraassen (forthcoming) “A neglected approach to exemplification”. Journal of Philosophy.
Eaton, Ralph M. (1931) General Logic: An Introductory Survey. New York: Charles Scribner’s Sons.
Porphry the Phoenician (1975) Isagoge. Tr. Edward W. Warren. Toronto: Pontifical Institute of Medieval Studies.
Porphyry. Isagoge: An Introduction to Aristotle’s Categories (2026). Tr. Lloyd O’Hara. Kindle Edition. Note: unlike Porphyry 1975, this includes Porphyry’s commentary in question and answer form.
Thomason, Richmond H. (1969) “Species, Determinates and Natural Kinds”. Nous 3: 95-101.
Notes
[1] Credited to W. E. Johnson; see Eaton (1931: 271 n2).
[2] “Question. Since in common usage ‘category’… is the name for an accusation in a legal defense, to which the ‘apology’ … is opposed, why did Aristotle choose to entitle the book Categories, although he did not set out to teach how we accuse opponents in courts of law, but introduced something else, which is not called by this name among the Greeks? Answer. Because common usage, being indicative of everyday matters, has adopted words that signify these matters in its widespread practice, whereas philosophers, being interpreters of matters unknown to the many, needed newer names for the presentation of the matters discovered by them. They either themselves made new and uncommon words, or they used established ones in a special sense[8] to signify the matters discovered by them.” (Porphyry 2026: 45)
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