Are a Wizard's “spells known” what they have in their spellbook or what they can prepare?

Question

I'm unclear on what a wizard's “spells known” means. Namely, are they the spells they can prepare, or the spells in their spellbook?

This is probably really obvious, but I'm just not figuring it out. Thanks!

Answer 1

"Spells known" is not a wizard concept

The premise of your question is that wizards do have "spells known": they do not in 5e. Go through the entire PHB: spells known is never used for the wizard.

Wizards only have "Cantrips known". Compare the feature table for the wizard on page 113 PHB, to that of the warlock on page 106, the sorcerer on page 100, the arcane trickster on page 98, the ranger on page 90, the eldritch knight on page 75 and the bard on page 53. All those other classes' tables have a column headed "Spells known". The wizard table does not. It only has a column "Cantrips known".

All these other classes have a subheading in their description titled "Spells Known of 1st Level and Higher", that describes the concept that they know a limited number of different spells. The wizard does not have this. Instead he has a subheding on "Learning Spells of 1st level and Higher", and one on "Preparing and Casting Spells" on page 114.

You prepare the list of wizard spells that are available for you to cast. To do so choose a number of wizard spells from your spellbook equal to your Intelligence modifier + your wizard level

The wizard can add wizard spells to their spellbook without an upper limit on the different number of spells. But he does not "know" them: should he ever lose his spellbook, all the spells he has not currently prepared are lost to him until he regains it. From this spellbook he prepares spells for any given day. Instead of a limit of spells known, he has a limit on how many different spells he can prepare.

The only spells a wizard always knows are their cantrips, as shown under the "Cantrips known" heading in the table and "Cantrips" subsection in his spellcasting feature.

So, the spells known concept just does not apply for the wizard.

Multiclassing

You can also see this in the Multiclassing section. There is a paragraph under the Spellcasting subheading on page 164, that discusses spells:

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

The concepts of spells known and spells prepared are non-overlapping. One class may use the spells known concept, the other the "spells prepared" and you treat them separately for each class.

Answer 2

"Spells Known" refers to the spells the wizard currently has written in their spellbook.

From the PHB Page 114:

Spellbook
At 1st level you have a spellbook containing six 1st-level wizard spells of your choice.

...

Learning Spells of 1st Level and Higher
Each time you gain you gain a wizard level you can add two wizard spells must be of a level for which you have spell slots, as shown on the Wizard table. On your adventures you might find other spells you can add to your spellbook.

Okay so let's assume you're a level 2 Wizard who hasn't learned any external spells. Your Spells Known would total 8.

Now as to spells prepared you can prepare Int Mod + Wizard Level # of spells each day. So let's say you have an intelligence of 16 (mod of +3), at level 2 you'd be able to prepare 5 spells from the list of Spells Known.

As you adventure you can come across other spells and copy them down into your spellbook. Your biggest concern is not to lose your spellbook, or to make sure you have a copy of it. Check the Your Spellbook sidebar on page 114 for doing just that.

TL;DR

Spells Known are spells in your spellbook. From those spells, you prepare a number of spells to cast each day equal to (Intelligence Modifier + Wizard Level). These spells you've selected are your Prepared Spells.

Answer 3

Spells you know refers to a couple of things:

1) The spells which you have written in your spellbooks. Notice the plural. The PHB states:

many wizards keep backup spellbooks in a safe place

The only limitation is the time and money it takes to transcribe the spells.

2) The spells you currently have memorized.

If you lose your spellbook, you can use the same procedure to transcribe the spells that you have prepared into a new spell book

If your spellbooks are destroyed, you can re-transcribe a new spell book from spells that you have in memory. Note that this is almost certainly a small subset of what was in the book, and requires you to not cast anything until you have had a chance to write it down.

Spells you can prepare refers to ones you can memorize and cast. This requires the spell book and the spells known that are written in it. Note that a frugal wizard could have a master spellbook at home under lock and key, and have a cheaper "traveling spellbook" to carry with them, containing only the spells they will likely need while adventuring. It would also keep enemies from gaining their entire spell list should they be captured and looted. Thus "spells you can prepare" does not always equal "spells known". The former is situational.