The pie example appears to treat "fair" as a 1-place word

'Beautiful' needs 2 places because our concept of beauty admits of perceptual variation. 'Fairness' does not grammatically need an 'according to whom?' argument place, because our concept of fairness is not observer-relative. You could introduce a function that takes in a person X who associates a definition with 'fairness,' takes in a situation Y, and asks whether X would call Y 'fair.' But this would be a function for 'What does the spoken syllable FAIR denote in a linguistic community?', not a function for 'What is fair?' If we applied this demand generally, 'beautiful' would became 3-place ('what objects X would some agent Y say some agent Z finds 'beautiful'?'), as would logical terms like 'plus' ('how would some agent X perform the operation X calls "addition" on values Y and Z?'), and indeed all linguistic acts.

intuitions reflect their ancestral environment, and [...] those intuitions can be variable.

Yes, but a given intuition cannot vary limitlessly, because there are limits to what we would consider to fall under the same idea of 'fairness.' Different people may use the spoken syllables FAIR, PLUS, or BEAUTIFUL differently, but past a certain point we rightly intuit that the intension of the words, and not just their extension, has radically changed. Thus even if 'fairness' is disjunctive across several equally good concepts of fairness, there are semantic rules for what gets to be in the club. Plausibly, 'fairness is whatever makes RobbBB happiest' is not a semantic candidate for what English-speakers are logically pinpointing as 'fairness.'

This article reads like you're trying to move your stone tablet from your head into the world of logic, where it can be as universal as the concept of primes.

You hear 'Oh no, he's making morality just as objective as number theory!' whereas I hear 'Oh good, he's making morality just as subjective as number theory.' If we can logically pinpoint 'fairness,' then fairness can be rigorously and objectively discussed even if some species find the concept loathsome; just as if we can logically pinpoint 'prime number,' we can rigorously and objectively discuss the primes even with a species S who finds it unnatural to group 2 with the other primes, and a second species S* who finds it unnatural to exclude 1 from their version of the primes. Our choice of whether to consider 2 prime, like our choice of which semantic value to assign to 'fair,' is both arbitrary and unimpeachably objective.

Or do you think that number theory is literally writ into the fabric of reality somewhere, that Plato's Heaven is actually out there and that we therefore have to be very careful about which logical constructs we allow into the club? This reluctance to let fairness into an elite Abstraction Club, even if some moral codes are just as definable in logical terms as is number theory, reminds me of Plato's neurotic reluctance (in the Parmenides) to allow for the possibility that there might be Forms "of hair, mud, dirt, or anything else which is vile and paltry." Constructible is constructible; there is not a privileged set of Real Constructs distinct from the Mere Fictions, and the truths about Sherlock Holmes, if defined carefully enough, get the same epistemic and metaphysical status as the truths about Graham's Number.

If your sense of elegance is admittedly subjective, why are we supposing a Platonic form of elegance out in the world of logic?

You're confusing epistemic subjectivity with ontological subjectivity. Terms that are defined via or refer to mind- or brain-states may nevertheless be defined with so much rigor that they admit no indeterminacy, i.e., an algorithm could take in the rules for certain sentences about subjectivity and output exactly which cases render those sentences true, and which render them false.

Isn't this basically the error where one takes a cognitive algorithm that recognizes whether or not something is a horse and turns it into a Platonic form of horseness floating in the world of logic?

What makes you think that the 'world of logic' is Platonic in the first place? If logic is a matter of mental construction, not a matter of us looking into our metaphysical crystal balls and glimpsing an otherworldly domain of Magical Nonspatiotemporal Thingies, then we cease to be tempted by Forms of Horsehood for the same reason we cease to be tempted by Forms of Integerhood.