# default

## 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.

## Preliminary [Pre-test]

Topic
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  will give you enough background.

Alternatively you could look at the first five propositional tutorials in Easy Deriver

## Review of K Propositional Rules

Topic
Logical System

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

The Modal Negation (MN) rules

## Review of Additional S5 Propositional Rules

Topic
Logical System  #### ◊ S5 world, k, must be new [here the computer will choose for you] Topic
Logical System
6/2/12

## Review of Tree Propositional Rules, shown as patterns

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').

## Tree Predicate Exercises: Roll your own

Topic
Logical System
###### 6/3/12

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 ~ & ∨ →

## Review of Tree Predicate Rules

Topic
Logical System

#### ∃D. The constant, a, must be new to the branch [here the computer will choose for you] #### ∀D. Any closed term, stage 1, your choice ## Tree Tutorial 7: Type Labels, Sorts, and Signatures ['Mixed Domains']

Topic
Logical System
###### 6/6/12

[These might help, you need only scan them.]

K.H.Blasius et al. eds. Sorts and Types in Artificial Intelligence
Maria Manzano  Extensions of First-Order Logic
John Sowa  Knowledge Representation

## Set Theory (and Russell's Paradox)

Topic
Logical System

### 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

## Number Theory and Peano Arithmetic

Topic
Logical System

### Tutorial

#### Notation

It is common in this setting (which is arithmetic) to use  functional terms like s(x), s(1), s(0) to mean the successor of x, 1, and 0, respectively. Equally common is the notation x', 1', and 0' to mean the same thing. The latter is quicker and shorter (though not semi-nmemonic)-- we will use it here.