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Tree Tutorial 8 Modal Trees

Logical System
6/1/12

Reading

A text like

Rod Girle [2000] Modal Logics and Philosophy

would definitely be a help here.

Tutorial

[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'.

Reading a Counter Example from the Tree

Topic
Logical System
Last altered 6/20/12
A central use for Trees is to produce a counter example to an invalid argument. To do this, you construct a tree with a complete open branch. You will be able to do this for invalid arguments (but not valid ones). Then you run up that branch assigning all atomic formulas True and all negations of atomic formulas False.

 

This software will let you try a few.

 


 

Exercise: Finding a counter-example.

 

Preliminary [Pre-test]

Logical System
6/20/12

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

Review of K Propositional Rules martin Thu, 06/07/2012 - 12:10
Topic
Logical System

There are the ordinary (non-modal) tree propositional rules plus

The Modal Negation (MN) rules

 

Review of Additional S5 Propositional Rules martin Thu, 06/07/2012 - 11:25
Topic
Logical System



 

◊ S5 world, k, must be new [here the computer will choose for you]

 



 

□ S5 any world, stage 1, your choice

Set Theory (and Russell's Paradox)

Logical System
6/5/12

Tutorial

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