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Several attacks get only half the Strength bonus to damage - off-hand attacks or attacks from light weapons come to mind.
Of course, main hand, non-light attacks get the full Strength bonus and two handed attacks get 1-1/2 times the Strength bonus.

When Strength is lower than 10 I'm sure main hand, non-light attacks get the Strength penalty to damage.
What about the other two cases? Do they get the Strength penalty as well or do they get half that? If it's half that, how is half of, let's say, -1 calculated?

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The Strength Penalty to Damage Remains Unadjusted
A Medium creature with Strength 8 who's fighting with two weapons, a longsword in one hand and a short sword in his off-hand, will inflict 1d8-1 points of damage with the longsword and 1d6-1 points of damage with the short sword, for example. The same creature with Strength 6 will inflict 1d8-2 points of damage with the longsword and 1d6-2 points of damage with the short sword.


From the SRD.

Strength Bonus
When you hit with a melee or thrown weapon, including a sling, add your Strength modifier to the damage result.

Off-Hand Weapon
When you deal damage with a weapon in your off hand, you add only ½ your Strength bonus.

Wielding a Weapon Two-Handed
When you deal damage with a weapon that you are wielding two-handed, you add 1½ times your Strength bonus. However, you don’t get this higher Strength bonus when using a light weapon with two hands.

Also from the SRD:

Modifiers
A modifier is any bonus or penalty applying to a die roll. A positive modifier is a bonus, and a negative modifier is a penalty.

Emphasis--except in the headers--mine. Thus while the bonus may be different, the other kind of modifier--the penalty--is unchanged.

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The funny thing is I could have found the solution myself if I'd been searching that SRD page for "1/2" instead of for "half". This is it, exactly for the reasons I expected, and my friend who asked me thanks you very much. –  Zachiel Mar 30 at 14:27

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