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A fellow DM brought an interesting question to my attention. WE were looking at the wizard class and were wondering if technically the wizard can add spells from other arcane classes to his spellbook copying them from a scroll.

The PHB states:

Wizards can add new spells to their spellbooks through several methods. If a wizard has chosen to specialize in a school of magic, she can learn spells only from schools whose spells she can cast.

It goes on to say:

Spells Copied from Another’s Spellbook or a Scroll A wizard can also add a spell to her book whenever she encounters one on a magic scroll or in another wizard’s spellbook.

Does this mean a Wizard could copy a spell from any arcane class off of a scroll? Even if he CAN is he able to cast the spell as if it is now on his class spell list?

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The Player's Handbook on Spells Copied from Another’s Spellbook or a Scroll says

A wizard can also add a spell to her book whenever she encounters one on a magic scroll or in another wizard’s spellbook.… [T]he wizard must first decipher the magical writing…. Next, she must spend a day studying the spell. At the end of the day, she must make a Spellcraft check (DC 15 + spell’s level).… If the check succeeds, the wizard understands the spell and can copy it into her spellbook…. (179)

(Link mine.) So, by the core rules, a wizard can copy any written spell into her spellbook.

However, the Rule's Compendium says, "Spellcasters who use spellbooks can add a spell to their book whenever they find one on a scroll or in another caster’s spellbook. The spell to be copied must be on the copier’s class spell list" (160), and, here, the Rules Compendium agrees with one of the game's designers (see this answer). Ask your DM. This DM has found the game easier to manage for both the DM and the wizard if everyone agrees to limit wizards to scribing into spellbooks only spells that are on the sorcerer/wizard spells list, but another DM's experience may be different.

A typical single-classed wizard can't cast spells that don't appear on the sorcerer/wizard spell list even if they are in the wizard's spellbook

The presence of a spell in a wizard's spellbook doesn't enable that wizard to cast that spell.

The Player's Handbook in the description of the wizard class feature spells says, "A wizard casts arcane spells (the same type of spells available to sorcerers and bards), which are drawn from the sorcerer/wizard spell list…" (56). For example, the typical single-classed wizard just can't cast a spell that's exclusive to the cleric spell list even if the wizard understands that cleric-spell-list-only spell and has it in her spellbook.

In short, a spell must be on the sorcerer/wizard spell list for a wizard to cast it. Alternatively, an individual wizard must uniquely add the spell to her own wizard spell list.

(For example, this DM has always assumed that a creature adds the new spells to the creature's appropriate spell list when the creature succeeds at Researching Original Spells (DMG 198)—as always, bear in mind that researching original spells is vaguely defined, easily abused, and almost entirely at the DM's discretion.)

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No

Reading the description of the wizard class on page 57 of the PHB (my emphasis added):

Spells: A wizard casts arcane spells (the same type of spells available to sorcerers and bards) which are drawn from the sorcerer/wizard spell list (page 192).

The general principle is that classes can only learn and cast spells from their own class. Certain exceptions may be noted for prestige classes, and Use Magic Device skill allows use of scrolls from other classes, but the general rule is that a spell must be on the class spell list for it to be learnt.

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    \$\begingroup\$ This rule does not (explicitly) prevent the Wizard from copying the spells in its spellbook, however, it only prevents them to cast them afterward. There may be value for a Wizard in having spells they cannot cast in their spellbook regardless: such spells can be studied, sold to others, ... \$\endgroup\$ – Matthieu M. Jun 2 '18 at 17:42

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