The Lore System

Lore skills are an alternate system of knowledge skills designed to reward Intelligence and make knowledge skills more interesting. This system replaces Arcana, Religion, Nature, and History, which are entirely replaced by the Lore skill. The Lore skill has many different subskills. You can have up to 4 “ranks” in the Lore skill, and you can put a rank in Lore in place of regular skill training. If a feat or feature would give you a specific skill training in Arcana, Religion, Nature, or History, you instead gain a rank of Lore.
Whenever you gain a rank in the Lore skill, you choose a number of subskills equal to your proficiency modifier. When your proficiency modifier goes up, you will get additional subskills for each rank of Lore you have.
All characters gain additional Lore subskills equal to your Intelligence Score - 10 (minimum 0).
You do not need any ranks of Lore trained to get these bonus subskills. Any character with an intelligence score above 10 will receive some Lore subskills regardless of their skill trainings/ Lore ranks.
Thus, your total number of Lore subskills at any point are as follows:
Number of Lore subskills = (Lore Ranking x Proficiency Modifier) + (Int. Score - 10) + (any additional subskills from race/class/background or other features)

Lore Subskills

Lore has many different subskills, which you choose for yourself with DM approval. These represent areas of knowledge that you have gained from life experience, informal teachings, or academic study. The more specialized a lore you have, the lower the DC of a given roll will be.
If we take "The Elemental Planes" as an example, there are several different levels of specificity we can get with it:
Very Broad: The Outer Planes
Broad: The Elemental Planes
Narrow: The Elemental Plane of Air
Very Narrow: Nobility of the Elemental Plane of Air

Using Lore Subskills

Your bonus in each of the lore subskills is equal to your Intelligence Modifier + Proficiency Modifier, like a normal skill.

Lore Expertise

You can gain skill expertise in Lore subskills from features and feats. When a feature would give you skill expertise (double proficiency modifier) in a skill of your choice or a knowledge skill (Arcana, Religion, Nature, or History), you can use that to gain a rank of Lore expertise. You can have a maximum Lore expertise ranking equal to your Lore skill ranking, but never higher. When you gain a rank of Lore expertise, choose a number of lore subskills equal to your proficiency modifier. You gain expertise in those subskills. When your proficiency increases, you can gain expertise in additional Lore subskills.
Thus, the amount of Lore subskills you can have expertise is calculated similarly to your number of Lore subskills, as shown below:
Total number of Lore subskill expertise = (Ranks of Lore expertise) x (Proficiency modifier)
If a feat or feature (such as the Arcanist feat) gives you either proficiency with a knowledge skill you are untrained in or double proficiency with a skill you are already proficient in, you instead choose between gaining a rank of the Lore skill or a rank of Lore expertise.

Example

Let's say I'm playing a third level sorcerer. I want to be decently educated magically, so I have a 13 intelligence and have put one of my skill proficiencies into a rank of the Lore skill, and took the Skill Expert feat to give myself a rank of Lore expertise.
I will have 2 Lore subskills from the skill training (from a proficiency modifier of +2 at level 3), and 3 from my int score (13 - 10 = 3).
Based on that, I could have the following lore skills:

• Lore - Arcane Theory
• Lore - Dragons
• Lore - Demonology
• Lore - Magical History
• Lore - The Elemental Planes

If this character also put another skill training in Lore, I may choose to add

• Lore - Angelology
• Lore - Religion, Local

Because I have one rank of Lore expertise, I can now choose two of my Lore subskills (again based on proficiency modifier) to have expertise in. I will choose Arcane Theory and Magical History for my Lore expertises.
My base Lore skill bonus for any Lore subskills will be +3 (+1 Int. mod and +2 proficiency mod). This will be increased to a +5 for Lore subskills I have expertise in. My total Lore subskills will look like this:
Lore skill ranks: 2
Lore expertise rank: 1
Lore Subskills:

• Lore - Arcane Theory (+5)
• Lore - Dragons (+3)
• Lore - Demonology (+3)
• Lore - Magical History (+5)
• Lore - The Elemental Planes (+3)
• Lore - Angelology (+3)
• Lore - Religion, Local (+3)