# trees

Topic
Logical System

## Help with Reading a Counter Example [Generic]

Topic

2013

### Reading a counter example from an open branch

[This is a Quicktime Movie, click the Play button to view it. The logical symbols you see in use may be different to the ones you are familiar with (sorry about that, but it is not practical to produce different movies for all the minor variations in symbols). Any differences will not affect the principles being explained here.]

Your browser does not support html5 video.

## The Tree Widget

Topic

2013

### An example of the tree widget in use (and we are showing a different parser here to the earlier ones)

The logical symbols in use are ¬ ∧ ∨ → ↔ ∀ ∃ .

Determine whether these arguments are valid (ie try to produce closed trees for them)

a) ∀x(F(x)→G(x)), ∃x¬G(x) ∴ ∃x¬F(x)
b) ∀x(F(x)→∀yG(y)), F(a) ∴ ∀xG(x)
c) ∀x(A(x)→B(x)), ∀x(¬A(x)→C(x))∴ ∀x(¬B(x)→¬C(x))
d) ∃xF(x),∀x(¬G(x)→¬F(x)),∀xM(x) ∴ ∃xG(x)∧∃xM(x)

## 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.

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

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. Any closed term, stage 2, the constant 'a' chosen

Set Theory (and Russell's Paradox) martin Thu, 05/31/2012 - 14:55
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