Is the notion of a poset (partially ordered set) a first-order abstract concept on a par with other first-order abstract concepts such as group and Boolean algebra?
There is an obvious formal similarity between them, namely, that they are all defined (or definable) top-down, on a presupposed "underlying set" by way in effect of a set of axioms formulated in a first-order language. The concept of a poset, or, that of a partial order ≤ on a set S, is usually defied as follows.
Definition (P)
A binary relation ≤ on S is a partial order on S if it satisfies the following three conditions:
- For all x in S, x≤x. (Reflexivity)
- For all x, y in S, if x≤y and y≤x, then x=y. (Antisymmetry)
- For all x, y, z in S, if x≤y and y≤z, then x≤z. (Transitivity)
But, I suspect that there may be an epistemologically critical difference. Axiomatic definitions Γ of other abstract mathematical concepts (than that of partial order) usually presuppose (among other things) what the symbol = stands for. So, for instance, such Γ usually omits to declare explicitly that the symbol = is to stand for a certain equivalence relation on S (i.e., a relation which is reflexive, symmetric, and transitive on S). It is left to readers to understand this concept this way and to use the symbol = accordingly, as a matter of convention. Similarly, it omits to declare that the equivalence relation = is in addition to satisfy what may be called Γ-substitutivity condition:
So, the relational concept represented by the symbol = qua being used in a given context of Γ, can be said to have two metalinguistically consequential properties, namely, the equivalence-ness and the Γ-substitutivity. Indeed, these properties constitute not only necessary but also sufficient conditions for the axiomatic definition Γ to be a definition of the target abstract concept. The concept represented by = (in a context of Γ) does not have to be anything more specific than an equivalence relation satisfying the condition of Γ-substitutivity (whose specific content is determined relative to the specific content of Γ). This is so at least for the inference-proscribing/prescribing purposes, that is, purposes of enabling us to draw all and only "mathematically correct" inferences from Γ.(2) Thus, the concept represented by = is as abstract as the concept defined by Γ, ant its conceptual content is only to be inferentially articulated (to borrow Brandom's phrase) relative to Γ.
Actually, the common omission of explicit statements of these defining properties of = is rather understandable, if epistemologically misleading.(3) It is understandable because we (native mathematese speakers) all "know" how to use the symbol = appropriately in particular context of Γ whenever we encounter a specific axiom-set Γ. Of course, we "know" this not necessarily in the sense of explicit (verbalized or readily verbalizable) cognizance but primarily in the sense of implicit competence.(4) It is in this sense that usual axiomatic definitions Γ can be said to presuppose what the symbol = stands for (in each context of Γ).
By contrast, the axiomatic definition (P) of partial order can be seen as containing, not presupposing, the equivalence-ness of = on S, if we assume that (P) presupposes the (P)-substitutivity of =. That is to say, all of reflexivity, symmetry, and transitivity of = follow from (P), if we assume (P)-substitutivity of =.
In short, the concept of partial order ≤ can render --- or almost render --- the usually "undefined" concept of = as a "defined" concept, "definable" in terms of ≤. No other axiomatically definable abstract mathematical concept seems to be this way, to the best of my knowledge. (But, see the PS below.) Old news as it may be, I'm inclined to think that this makes the concept of partial order very special, in comparison to other axiomatically definable abstract concepts, because I see the matter from an epistemological point of view, namely, the point of view of epistemological dualism.
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PS
The concept of set-membership may have to be mentioned as another example of such an axiomatically definable abstract concept that can render the concept of = a "defined" concept. But, I think that this concept is far more special, from the point of view of epistemological dualism, than even that of partial order. If I'm right, from this point of view, it is fundamentally inappropriate for us to try to define this concept axiomatically. In other words, the concept of set-membership is not an axiomatically definable abstract concept at all, to begin with. (This is not to preclude a possibility that the concept of set proves to be axiomatically definable even in accordance with epistemological dualism, when we try to define the concept in some other terms than set-membership.)
By contrast, the concept of partial order seems indeed an axiomatically definable abstract concept, although there is something special with this concept, too, from the point of view of epistemological dualism. But, the ways in which these two concepts (partial order and set-membership) are special seem to be different.
Behind all of these claims or ideas of mine, there is this "epistemological dualism" thing. I must articulate what this is as soon as possible, revising the old paper of mine…