If I recall this is referenced most often in regards to the Hive's cleavers. It is said that rather than becoming dull from age each kill hones the edge. This is the sword logic at work.

Think of the sword as you would any other entity. Now consider it and it's user as two beings in a state of symbiosis.

The grimoire mentioned the swords as being more personal than say a boomer. It specifically refers to it as being a bridge between wielder and victim.

By the rules of the sword logic to slay another is to grow in power. This holds true for both entities: the user and the blade itself.

This is why Willbreaker and the Sword of Crota specifically are so powerful. The Sword of Crota for instance was bathed in the light of the thousands of guardians it killed. It took life, light, and power.

Similarly Willbreaker was used by Oryx in the destruction of world after world and galaxy after galaxy for billions of years. This is why it's so powerful and why a shard from it could create weapons as powerful as our exotic swords.

Seeing as the sword logic is the basis for all of the Hive's existence, they make this especially true in their ascendant realms. With each minion slain we not only weaken those they are tithed to, but we claim some of that power as our own.