Review of K Propositional Rules
There are the ordinary (non-modal) tree propositional rules plus
The Modal Negation (MN) rules
A text like
Rod Girle [2000] Modal Logics and Philosophy
would definitely be a help here.
[Modal logic is a vast area, what is being presented here is the briefest of glimpses through the shop window (a book like the Girle would help you go further).]
In modal logic, there are the additional symbols □ (necessary) and ◊ (possible) and, in the background, the notion of 'possible worlds'.
This software will let you try a few.
You need to know some propositional logic to be able to understand the tree tutorials to come. In particular, you need to know about the symbols used in propositional logic, truth tables, satisfiability, consistency, and semantic invalidity (by counter example). You do not need to know propositional rules of inference and derivations.
Howson [1997] will give you enough background.
Alternatively you could look at the first five propositional tutorials in Easy Deriver
There are the ordinary (non-modal) tree propositional rules plus
The Modal Negation (MN) rules
Here the letters 'P' and 'Q' are being used to stand for entire well formed formulas (so, on a particular occasion, 'P' might stand for the atomic formula 'F' and, on another occasion, it might stand for the compound formula 'F&G').
You can try your own exercises here.
Here are a few hints
- You have to use the right (unicode/html) logical symbols. Check Writing symbols
- The symbols in use here are ~ & ∨ →
≡
[These might help, you need only scan them.]
K.H.Blasius et al. eds.[1990] Sorts and Types in Artificial Intelligence
Maria Manzano [1996] Extensions of First-Order Logic
John Sowa [2000] Knowledge Representation
Set theory is an extensive topic introduced elsewhere. It can be written as a first order theory.
There is one axiom schema, Abstraction (or Comprehension), which can generate infinitely many axioms
∀y(yε{x:Φ[x]}≡Φ[y])
Axiom Schema of Abstraction (or Specification or Comprehension). The Set Builder Axiom Schema.
And a number of other axioms
(x=y) ≡∀z(zεx≡zεy)) Axiom of Extensionality