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Suppose we're trying to set a catapult on fire while defending a siege (and the nearby Kobolds firing it aren't strong enough to be able to move it).

That catapult can't move or speak, and can't take actions or reactions (incapacitated). It seems like it's paralyzed. Does that mean that attack rolls against it have advantage? So, if our wizard wants to set it on fire with a Fire Bolt spell, does the attack roll against it have advantage?

More generally, do attack rolls against incapacitated creatures that can't move have advantage, even if they're not necessarily paralyzed?

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The answer is on page 246 of the DMG.

Armor Class. An object's Armor Class is a measure of how difficult it is to deal damage to the object when striking it (because the object has no chance of dodging out of the way). The Object Armor Class table provides suggested AC values for various substances.

So in the case of attacking object AC means something different than what it means when it comes to attacking a NPC or monster. So the paralyze condition would not logically apply as an object AC has nothing to do whether it is still or not.

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The rules

I think you are looking at this from the wrong perspective. The main reason this doesn't make sense is that conditions only apply to creatures, not objects.

Appendix A: Conditions (PHB p. 290)

Conditions alter a creature's capabilities in a variety of ways and can arise as a result of a spell, a class feature, a monster’s attack, or other effect.

The solution

If you want to deal damage to an object, you should look at the Statistics for objects section of the DMG (p. 246) which deals with assigning AC and HP to destructible objects (and is too long to quote here in it's entirety).

The formal logic arguments

RAW, a condition applies various effects on a creature, but suffering from those effects doesn't mean that you are subject to that condition. So if you are incapacitated you cannot take actions or reactions, but if you can't take actions or reactions it doesn't mean you're necessarily incapacitated. This is a logical fallacy known as affirming the consequent:

If P implies Q, the only inference that can be made is non-Q implies non-P. (Non-P and non-Q designate the opposite propositions to P and Q.) This is known as logical contraposition.

Even so, ignoring everything said so far, picking some of the effects of a condition ("it cannot take actions and reactions"), suggesting that that infers the condition ("it seems like it's paralyzed"), and then deciding to apply the rest of the effects for that condition ad-hoc (giving advantage to attack rolls) is not a very sound train of thought.

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