Can an Arcane Trickster Rogue learn all of the wizard spells from the low levels it has slots for?

Question

My take on this is as follows:

1. Arcane Trickster level 17 class ability allows for stealing spells somebody is casting (this isn't restricted to wizard spells). Skill is located on PHB pg. 98, emphasis is mine.

Spell Thief

At 17th level, you gain the ability to magically steal the knowledge of how to cast a spell from another spellcaster.

Immediately after a creature casts a spell that targets you or includes you in its area of effect, you can use your reaction to force the creature to make a saving throw with its spellcasting ability modifier. The DC equals your spell save DC. On a failed save, you negate the spell’s effect against you, and you steal the knowledge of the spell if it is at least 1st level and of a level you can cast (it doesn’t need to be a wizard spell). For the next 8 hours, you know the spell and can cast it using your spell slots. The creature can’t cast that spell until the 8 hours have passed.

Once you use this feature, you can’t use it again until you finish a long rest.

2. This ability lets you know the spell and prepare it for up to 8 hours.
3. Spell progression is up to level 4 spells.

Now, this character requires a level of Wizard to make the spell stealing permanent. From this, it follows that:

Your Spellbook allows you to copy any wizard spell you know so long as you have the spell slots for it and you have the time to decipher and copy it (2 hours and 50 gp per level, conveniently meshing nicely with the 8 hour limitation on the spell you stole.) Emphasis to pertinent sections of the spellbook are mine (PHB pg. 94)

Your Spellbook

The spells that you add to your spellbook as you gain levels reflect the arcane research you conduct on your own, as well as intellectual breakthroughs you have had about the nature of the multiverse. You might find other spells during your adventures. You could discover a spell recorded on a scroll in an evil wizard's chest, for example, or in a dusty tome in an ancient library.

Copying a Spell into the Book. When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a level for which you have spell slots and if you can spare the time to decipher and copy it. Copying a spell into your spellbook involves reproducing the basic form of the spell, then deciphering the unique system of notation used by the wizard who wrote it. You must practice the spell until you understand the sounds or gestures required, then transcribe it into your spellbook using your own notation. For each level of the spell, the process takes 2 hours and costs 50 gp. The cost represents material components you expend as you experiment with the spell to master it, as well as the fine inks you need to record it. Once you have spent this time and money, you can prepare the spell just like your other spells.

Note: I've used the portion on copying a spell into a spellbook because it specifically utilizes the terminology, "When you find a wizard spell....." Whether you steal it from a scroll, a spellbook, a magical device that imparts the knowledge, training from another wizard, or in this case, the mind of another wizard, so long as you as a character can cast the spell, you have "found" a spell.

1. This spell has to be a wizard spell to add it to your spellbook (this is specifically stated in the first sentence of the spellbook description for Copying a Spell into the Book.)
2. Additionally, since a wizard can write down whatever spells they currently have prepared in the event they lose their spellbook, this stands to reason that the entire spell is memorized. Since the Arcane Trickster steals the knowledge of the spell completely from the wizards mind, this also stands to reason that they would be able to reproduce the spell since it says they can directly in the ability.
3. Under the Spellcasting section of the PHB pg. 203, the requirements to cast a spell are neatly summed up and very explicit. Verbal, Somatic and Material components are the only three components that are potentially necessary. In order to pen a spell, you must know in what manner the physical components are used and train to memorize these components for each spell you are going to be utilizing. This is why wizards require extensive preparation. The Arcane Trickster falls under the preparation field in that the caster portion acts like a wizard. This is covered under the Arcane Trickster Spellcasting Ability portion on PHB pg. 98. Unlike a wizard, the AT only has a small number of spells, and thus knows them instead of preparing them. If you dip wizard though, you can expand the list of spells you know, as well as expand the list of spells you can prepare.
4. With that in mind, it stands to further reason that the written portion of a spell are the components and their specific interactions. That means when you steal a spell you have the components necessary to cast it firmly fixed in your mind for the next 8 hours and are able to replicate so long as you have spell slots. With the wizard's ability to record wizard spells (specific) that you have prepared, this closes the circle in that you have the spell firmly fixed in your mind, and can reproduce it. With that in mind, penning the requirements takes at maximum 8 hours (you only have up to level 4 spell slots), and this ability lasts for specifically 8 hours.

Absolute last point:

Functionally, this doesn't change the class or overpower it in any way. As a wizard at level 3, and Arcane Trickster at level 17, you can pen any level 2 or lower wizard spell already so long as you find it. I submit that due to this, stealing a spell from somebody's mind can act as finding a spell, and the character can utilize the multi-class dip to write it down accordingly.

So with that in mind, does the logic follow that so long as the Arcane Trickster takes a single level dip into Wizard they can write down any spell thief'ed wizard spells up to level 2?

If not, can you illustrate why?

Answer 1

Yes, but only 1st-level spells

PurpleVermont has already argued why such a multiclass could only write down spells of a level they could cast as a Wizard (so actually up to 2nd level, if they take Wizard 3). I disagree with Vermont's argument that stolen spells could not be transcribed at all, however.

The rules don't clearly state that you have to have seen a spell written down to be able to transcribe it, that's a matter of interpretation--and I disagree with the interpretation:

Copying a spell into your spellbook involves reproducing the basic form of the spell, then deciphering the unique system of notation used by the wizard who wrote it. You must practice the spell until you understand the sounds or gestures required, then transcribe it into your spellbook using your own notation.

The fact that different wizards use different notation implies that the written form of the spell is not an inherent part of the magic, merely an aide-memoire. And the requirement to practice the spell until you understand its V/S requirements -- precisely the thing which the Arcane Trickster can instantly intuit -- strongly suggests that this is the most important part of "learning" a spell, and the rest is merely paperwork that your first-level Wizard training would absolutely prepare you for.

Answer 2

No, Knowing a spell is not enough to transcribe or copy it

All the methods given for putting spells into a Wizard's spellbook require that Wizard has seen the spell in written form. Their options are transcribing a spell that they have prepared (by reading it from their spellbook), or copying spells they have found in other written sources such as scrolls, dusty old tomes, or other wizards' spellbooks.

The Arcane Trickster feature say that:

For the next 8 hours, you know the spell and can cast it using your spell slots

Knowing a spell is not the same as finding or preparing a spell, and the Wizard has to have done one of those two things before writing it to a spellbook.

Transcribing a prepared spell:

The Wizard spellbook transcription feature says that:

you can transcribe the spells you have prepared into a new spellbook

The Arcane Trickster ability does not say you have the spell prepared because Arcane Tricksters do not prepare spells. They know spells in a different way than Wizards prepare them.

Copying a found spell:

The Wizard spell copying feature says:

Copying a Spell into the Book. When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a level for which you have spell slots and if you can spare the time to decipher and copy it. Copying a spell into your spellbook involves reproducing the basic form of the spell, then deciphering the unique system of notation used by the wizard who wrote it.

The description of finding a spell states:

You might find other spells during your adventures. You could discover a spell recorded on a scroll in an evil wizard's chest, for example, or in a dusty tome in an ancient library.

Note that in the examples, found spells are always written.

I don't believe that stealing a spell from a Wizard's mind counts as finding a spell since the examples in the PHB of places a spell might be found are examples of written spells. There is nothing in the PHB to suggest that a Wizard can teach another Wizard a spell without letting him copy it from his spellbook, or that a magic item can impart the knowledge necessary to transcribe a spell into a spellbook, or that merely knowing how to cast a spell allows the wizard to transcribe it into his spellbook.

Moreover, the use of the verb copy indicates transcription from one written source to another, and this is reinforced by reference to the need to decipher the unique writing system of the wizard who originally wrote it.

Since the Arcane Trickster has never seen the spell written down, they cannot copy or reproduce it in written form. There is nothing to suggest that an Arcane Trickster's innate knowledge of a spell would allow them to have any idea what the spell should look like in written form.

In any case, a first level Wizard cannot prepare or cast a 2nd, 3rd or 4th level spell

Even if you allow that the Arcane Trickster somehow steals not only the knowledge of how to cast the spell, but also how to prepare and transcribe it when stealing from a Wizard, this trick will only work for first level spells, because a first level Wizard can only prepare first level spells from his spellbook. This is because the multiclassing rules specify:

You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class.

For more discussion on that issue, see: Can a multiclass Wizard use any Wizard spell? Some answers there (including mine) suggest that the multiclassed Wizard may be able to copy (or transcribe) spells of a higher level, but that does him no good since he can't prepare or cast them.

An Arcane Trickster cannot prepare or cast spells from a Wizard spellbook either. Although they do learn spells as Wizards do initially, they are only allowed to have a fixed number learned, and only allowed to change the spells they have learned when they level up. They cannot prepare additional spells from a spellbook.

Answer 3

It depends on the DM.

Clearly, a level 20 character multiclassed as LV17 Arcane Trickster and a LV3 Wizard should have no problem "stealing" a spell from an unwilling or fully voluntary subject two or more times, and knowing the spell well enough to use it. In developing a parallel understanding of how to bend the flow of magic for the original spell through a different process, Wizards are able to develop new spells or copy written formats off of the original spell. Arcane Trickster's use intelligence for the spells they cast, as do Wizards.

All a Wizard has to do is understand how it is done and the methods employed to develop something off of the original.
With this approach, you can have the Wizard understand and know how to use the spell through his Arcane Trickster skills, then use his training as a Wizard to put it into writing.

Refrences found in D&D 5e PHB on pages 95, 96, 97, 112, 113, 114, 115, 201, 202, 203, 204,(205 This page is critical to understanding The Weave Of Magic).