Introduction to Propositional Logic

Warning

This is an experimental project and a work in progress. It is intended for educational and demonstration purposes.

This is a platform for learning propositional logic. By using a knowledge graph to track what you know, the system schedules reviews using the FSRS v6 algorithm.

Why this exists

1. I was fascinated by the Math Academy system and wanted to create something similar.
2. I wanted a cool project to help land an internship (hiring? moaaz.ae@proton.me).
3. For this to be a resource for others who want to start using the "Math Academy way" to build the most effective platforms for learning.

Credits

• Math Academy: The main inspiration for this system.
• Justin Skycak: For the book The Math Academy Way, which provides the pedagogical foundation for the platform.

Core Architecture

The system uses two main data structures.

Knowledge Graph (DAG)

The curriculum is a directed acyclic graph containing 29 topics. Each topic links to its prerequisites via edges with weights from 0 to 1; a weight of 1 means the prerequisite is practiced completely when the lesson is practiced.

graph LR
    sentences["Sentences"] -- 0.3 --> atomic["Atomic Sentences"]
    sentences -- 0.3 --> arguments["Arguments"]
    atomic -- 0.3 --> tfunc["Truth Functions"]
    atomic -- 0.5 --> conj["Conjunction"]
    tfunc -- 0.5 --> conj
    conj -- 0.3 --> wff["Well-Formed Formulas"]
    wff -- 0.7 --> scope["Scope"]
    scope -- 0.2 --> cond["Conditional"]

Loading

Student Model

The database stores the state for every user and topic, tracking your progress in the topic_state table.

• Stability: The number of days until the system predicts a 90% chance of recall.
• Difficulty: A value from 1 to 10. It represents how hard the topic is for you.
• Due Date: The next time you should review the topic.

Algorithms

The platform uses five core algorithms.

1. Adaptive Diagnostic

The placement test finds your level quickly by using a binary search on milestones.

1. The system identifies milestones in the topological order of the graph.
2. It picks the middle milestone and gives you a problem.
3. If you pass, the search continues in the later half of the graph.
4. If you fail, it searches the earlier half.
5. When the pass/fail boundary is found, the system marks all topics below it as "Review" state.

2. FSRS v6 Scheduling

The Free Spaced Repetition Scheduler (FSRS) predicts when you will forget a topic. It uses the DSR model (Difficulty, Stability, and Retrievability) to calculate the probability of recall.

• Retrievability ($R$): The probability you remember a topic. It follows a power law.
• Formula:

$$R = \left(1 + \frac{\text{factor} \times \text{days}}{\text{stability}}\right)^{-0.1542}$$

• Weights: The algorithm uses 21 trainable parameters ($w_{0 \dots 20}$) to update stability and difficulty after every answer.

3. FIRe (Fractional Implicit Repetition)

FIRe updates the stability of parent topics when you solve a child topic, which prevents redundant reviews of simple concepts.

1. You answer a problem correctly.
2. The system calculates a "chain weight" by multiplying the edge weights of the prerequisites.
3. It applies a stability boost to all ancestor topics in the graph.
4. Boosts decay as the algorithm moves further away from the original topic.

4. Scheduler and Task Compression

The scheduler decides which topics to show on your dashboard.

• Unlocking: A topic is "New" only if all its prerequisites are in the "Review" state.
• Compression: If a topic is due and its child is also due, the system only shows the child.
• Relearning: If you fail a prerequisite, the system hides all dependent topics until you fix the prerequisite.

5. Logic Parser and Evaluator (src/lib/logic.ts)

The src/lib/logic.ts module parses LaTeX logic formulas and evaluates them. It checks formula equivalence by generating and comparing truth tables.

• Parser: A recursive descent parser converts LaTeX strings into an Abstract Syntax Tree (AST).
• Operators: It supports $\neg$, $\land$, $\lor$, $\to$, and $\leftrightarrow$.

Tech Stack

• ts-fsrs
• SvelteKit (Svelte 5)
• PostgreSQL with Kysely
• Better Auth
• TailwindCSS v4
• Remark/Rehype ecosystem with KaTeX
• Kamal & Docker
• Vitest (Unit) and Playwright (E2E)

Project Structure

• src/lib/curriculum: Lesson files and problem definitions.
• src/lib/server/services: Implementation of the diagnostic, scheduler, and FIRe.
• src/lib/server/fsrs: Thin wrapper that configures and re-exports the ts-fsrs scheduler.
• src/lib/logic.ts: The parser and truth table evaluator.

Development

Install dependencies:

pnpm install

Start the database and sync the curriculum:

pnpm db:start:dev
pnpm db:migrate:dev
pnpm db:seed:dev

Run the server:

pnpm dev

License

MIT