If comparativity and reflexivity imply symmetry and transitivity how can the axioms of equivalence be orthogonal?

Question

According to BBFSK a relation with the properties of comparativity and reflexivity satisfies symmetry and transitivity. Comparativity is defined as $x\sim{z}\land{y\sim{z}}\implies{x\sim{y}}$. Apparently the proof of this is as simple as $x\sim{x}\land{y\sim{x}}\implies{x\sim{y}},$ which shows symmetry. From symmetry and comparativity, transitivity clearly follows.

Using the sets $$\mathcal{S}\equiv\left\{1,2,3\right\};\mathcal{P}\equiv\left\{\langle2,1\rangle,\langle3,1\rangle,\langle3,2\rangle\right\};\mathcal{Q}\equiv\left\{\langle1,3\rangle,\langle3,1\rangle\right\};$$

and relations

$$F(x,y)\equiv\text{False};L(a,b)\equiv\langle a,b\rangle\notin\mathcal{P};A(a,b)\equiv\langle a,b\rangle\notin\mathcal{Q};$$

it is possible to show that each relation satisfies exactly two of the axioms of an equivalence relation. Each pair of satisfied axioms is different from that of another of the relations. This demonstrates that the axioms are independent, that is orthogonal.

I know that "orthogonality" is something of a metaphor when applied to axioms. Nonetheless, I find it unsettling that a system of three axioms can be shown to be isomorphic to a system of two axioms. Is this an acceptable state of affairs?

Answer 1

In fact, it's equivalent to a system with one axiom, since we can always take the conjunction of any finite number of sentences.

Of course, you may object that doing so changes the form of the axioms in some way, and there are interesting questions to be asked here. For example, in the context of general algebras (= structures in a purely functional signature), we can ask $(i)$ whether a class of structures can be axiomatized by equations (this has a complete answer, essentially) and $(ii)$ if so, how many equations are needed (e.g. for the class of groups). But this is a side issue; the situation is completely acceptable, some axiomatizations are different from others.

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Separately, it's worth saying that your statement

"orthogonality" is something of a metaphor when applied to axioms

is quite true, and is why the word "orthogonality" really shouldn't be used here: I suspect your worry comes from the sense that somehow we have an object which is both three-dimensional and two-dimensional, but this is only due to a misrepresentation of a logical phenomenon as a geometric one.