Is initiative ordinal or numerical?

Question

Is the term “initiative” in D&D 5e combat the number that you roll, or your order in the combat turn?

Answer 1

Initiative is both, neither and either ordinal or numeric

Is the term “initiative” in D&D 5e combat the number that you roll, or your order in the combat turn?

Neither; initiative is the overarching term that encompasses the whole concept. Use in context will determine which of the sub-concepts you are using. This is similar to other terms used like "combat" and "spellcasting".

It is a mistake to expect English to behave like Boolean logic; words have different meanings in different contexts.

Player's Handbook Definition

Initiative

Initiative determines the order of turns during combat. When combat starts, every participant makes a Dexterity check to determine their place in the initiative order. The DM makes one roll for an entire group of identical creatures, so each member of the group acts at the same time.

The DM ranks the combatants in order from the one with the highest Dexterity check total to the one with the lowest. This is the order (called the initiative order) in which they act during each round. The initiative order remains the same from round to round.

If a tie occurs, the DM decides the order among tied DM-controlled creatures, and the players decide the order among their tied characters.

It is clear that "initiative" is all of the above: it determines the order, it encompasses the Dexterity check, it includes the initiative order as well as the tie breaking mechanism.

The Monster Manual refers to an "initiative count" which the Player's Handbook doesn't:

If a legendary creature has lair actions, it can use them to harness the ambient magic in its lair. On initiative count 20 (losing all initiative ties), the creature can use one of its lair action options, or forgo using any of them that round.

Does introducing this concept (which is a hangover from earlier editions) create confusion? Yes. Does it create ambiguity? No; it simply means that creatures with lair actions get two "turns" their normal one and a lair one which fits in after all the creatures who rolled 20 and above and before all the creatures who rolled 19 and below.

Answer 2

Initiative is numerical

Initiative determines the order of turns during combat.

a Dexterity check to determine their place in the initiative order.

When determining the order that characters act in combat, we look to the “initiative order” as described in these rules.

Thus, “the order of turns during combat” is equal to “the initiative order.” We can thus replace “the order of turns during combat” with “the initiative order” in the first sentence:

Intiative determines the initiative order.

If initiative determines the initiative order, as it must, then it must be defined as the thing that defines the initiative order:

a Dexterity check determines the initiative order.

Here we can clearly see that “Initiative” must be “a Dexterity check,” specifically the one used to determine initiative order.

In conclusion, initiative is a Dexterity check, and thus numerical. The intiative order that is derived from initiative is ordinal.

Answer 3

Initiative is wholly ordinal

Your “initiative” is your order in the turn.

I didn't believe this at first, until I tried to conclusively demonstrate the opposite in an answer to Does a controlled mount share its rider's turn? and discovered that I couldn't. Here's how it went:

I thought to myself, “…initiative is clearly the number, while initiative order is what you get after resolving your initiative into a ranking. So, when it says the mount shares your initiative, it means the number, not the order, and you have to resolve the tie to get the order. I will prove this with citations!”

I started writing the answer that would thunderously prove that, and started pulling citations. Except in trying to use these citations to prove the argument, I realised something: “Initiative” did not mean the roll result.

I tried to show that initiative order is the ranking order, and initiative is the number you roll. Except I found nothing to support the latter equation. Here's (most of) the definition of Initiative (Player's Basic Rules v3.4, p. 69), with my emphasis on the three things and superscripts to indicate their separate identities (before trying to determine which are the same thing, that is):

Initiative

Initiative¹ determines the order of turns during combat². When combat starts, every participant makes a Dexterity check³ to determine their place in the initiative order². The DM makes one roll for an entire group of identical creatures, so each member of the group acts at the same time.

The DM ranks the combatants in order² from the one with the highest Dexterity check total³ to the one with the lowest. This is the order² (called the initiative order²) in which they act during each round. The initiative order² remains the same from round to round.

If a tie³ occurs, the DM decides the order² among tied DM-controlled creatures, and the players decide the order among their tied characters.

The slam-dunk citation I was expecting to find was initiative¹ equalling the Dexterity check result³. As you can see though, if I tried to argue that with any part of the definition under “Initiative”, I would be transparently failing to support my point. The rules simply never say that the term “initiative” means the result of the die roll. To the contrary, the rules studiously avoid equating the check with a technical term of initiative.

In actual fact, the definition uses the technical term initiative¹ exactly twice: in the title, and when saying that “Initiative” is everything that follows. That can only mean that “initiative” is not any one part, such as the die roll — initiative is the whole procedure and result of the procedure!

So, it turns out that when the rules refer to initiative¹, it doesn't mean a number, it means your order in the round. Or in other words, when initiative appears elsewhere in the rules it should always be read as shorthand for initiative order²; not the numerical Dexterity Check result³, which the procedure doesn't dignify with an initiative-specific name and actually discards once used to determine the ranking.

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Of course, this has significant implications everywhere else that the rules refer to “initiative” without specifying. When the rules use this shorthand, they are referring to the ordinal initiative order, not the (long since discarded) numerical result of a roll.