I disagree. It absolutely is. It can't be anything but.
I take
arbitrary to mean something like
unconstrained by reason.
You seem to be using it to mean something like
not admitting of a unique answer.
Prince Valiant doesn't have rules for underwater combat. It does have an Agility skill (p 15):
Agility - how agile and well-trained as an athlete the character is: how well he runs, swims, jumps, dodges, climbs, swings on ropes, and so on. . . . Agility plus Brawn is used when a character tries to move quickly or make a sudden action. . . . It is not used in normal combat, but can be attempted as a throw versus a Difficulty Factor to try a special tactic . . .
A PC was in the water, trying to fight a "dragon" (actually a giant crocodile). The player had a Storyteller Certificate permitting one use of Kill a Foe in Combat as Special Effect. The rules for that include the following (pp 44-46):
The player should not know what Special Effects [NPCs] have, but they should be logical ones for the characters. For example, a beautiful girl is more likely to have the Effects of Incite Lust or Inspire Individual to Greatness than she would be to have Hide or Kill a Foe in Combat. . . . The user states that he is putting into action a Special Effect and reads it into the plot. . . . The Storyteller must create a reasonable explanation for the way in which the Effect takes place, in terms of the current situation. . . . The character must be in combat with the chosen foe at the moment [that Kill a Foe in Combat is used], and not in a disadvantageous situation (surrounded by enemies, injured, his back turned to the enemy). The selected character makes an attack, and the attack is miraculously successful, killing the foe instantly.
What is required for it to be accepted, as part of the fiction, that the PC in question is
in combat with the dragon and not in a disadvantageous situation, such that
the Storyteller can create a reasonable explanation for the way in which the Effect takes place?
There is no unique answer to this question. But it's not immune to reason. Here's how it played out at our table:
The attacking creature was a "dragon" (a giant crocodile, found in the episode "A Dragon" in the core rulebook). It had Brawn 15, and a successful Presence check by Sir Gerren revealed that it had the equivalent of heavy armour (+3 combat dice, for 18 overall) which would make bowfire from the Hunnish bowmen they had recruited largely ineffectual.
<snip>
Sir Morgath, meanwhile, used his Agility skill to stay afloat in the water while he drew the greatsword he had taken (in the previous session) from the Bone Laird in Dacia. The player wanted to use his Storyteler Certificate to Kill a Foe in Combat, but wasn't sure whether he satisfied the requirement that "the benefitted character . . . is not in a disadvantageous situation (surrounded by enemies, injured, back turned to the enemy, etc)". As he tried to avoid the thrashing tail of the dragon (losing some dice but not enough to reduce his Brawn) he saw that it had a soft underbelly, and I ruled that if he was able to swim beneath the dragon and stab it there, he could use his certificate as he wanted to.
The Agility check was made, and hence the dragon was slain by Sir Morgath.
And here is what happened to the other PC knights during the fight against the dragon:
Sir Justin, in the water, was trying to use his silvered dagger to reflect sunlight into the eyes of the dragon to dazzle it (he was in no condition to fight it) but was not able to succeed on a check, except to avoid drowning, which he did. Sir Gerren, using his boar spear from the heaving deck of the ship, tried to strike the dragon but failed - it tried to grab the spear away from him but he kept hold despite the physical strain required (mechanically, when the dragon beat Sir Gerren in their opposed combat checks I gave the player the option of losing his spear rather than taking damage, but he declined).
<snip>
An Oratory check by Sir Gerran enabled him to maintain control over the soldiers still on the boat and that had fallen into the water, so only two Huns of the PCs' entourage were lost. The bones of one was recovered so that they could be placed in the reliquary for martyrs of the Order; and Sir Gerran (with successful Hunting + Brawn) was able to harpoon the dragon so that its body could be carried to Constantinople.
The difference between Sir Morgath swimming beneath the dragon to try and stab it in its soft underbelly, and Sir Gerren standing on the deck of the ship using his spear, was more than just flavour. It mattered to the framing and the resolution of the players' declared actions for their PCs. But it didn't depend on complex combat mechanics - it depended on using the very simple mechanics while having regard to the fictional positioning and fictional consequences.
I agree with
@Umbran that different people might reasonably extrapolate in different ways, but it's not unconstrained, and the constraints are not invisible to the players. The player of Sir Morgath, for instance, had deliberately invested in his PC's Agility skill over multiple sessions, in order to compensate for his PC having lower Brawn than a typical knight. He was aware that this gave his PC a capacity to tackle the dragon that the other PCs lacked, and he was able to trade on that to establish the necessary fictional positioning to use his Storyteller Certificate. I think that's sufficient to show that the role of the fiction in action resolution wasn't arbitrary.