trees

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

6/5/12

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.

6/3/12

Tutorial

Groups can be characterized by three proper symbols {=,+,0} (ie identity, one infix operator, we will use '+', and an identify element '0') and the three proper axioms

∀x∀y∀z((x+y)+z=x+(y+z)),  (*associativity*)
∀x(x+0=x&0+x=x),                (*identity element, right and left*) 
∀x∃y(x+y=0&y+x=0)             (*inverse*)

6/5/12

Tutorial:

The word 'terms' in logic means 'names' and thus far we have met two kinds of terms: constants (or proper names), and variables.

5/30/12

Skills to be acquired in this tutorial:

To become familiar with the new rules for predicate logic trees with identity.

Tutorial:

It is possible to use trees with formulas containing identity. Really there are two different ways to implement it: either there are two new rules, or there is one new rule and a new way of closing a branch from a single formula. Both will be described.

6/3/2012

Skills to be acquired in this tutorial:

To become familiar with the new rules for predicate logic trees.

Tutorial:

Introduction

There are further rules for predicate logic trees (which we will come to shortly).

5/27/12

You can try material of your own below.

Here are a few hints

5/28/12

Skills to be acquired in this tutorial:

To become familiar with the notions of closed and complete trees. To be able to use trees to test for satisfiability and invalidity.

Tutorial:

Closure and completeness

In Tutorial 1, we met the notions of closed and open branches (a closed branch was one containing a formula and also the negation of that formula, an open branch was a branch that was not closed).

5/28/12

A Simple Tree

[This is a Video, click the Play button to view it..]

12/5/20

Skills to be acquired in this tutorial:

To become familiar with the rules for propositional truth trees.

Tutorial:

Truth Tree Rules

Trees are going to be used to 'picture' the truth conditions or requirements for a formula (then, as the technique is developed, for several formulas at once).

Any easy way to remember the rules is to regard them as having been built from the truth tables.

The truth table for 'and' is

Conjunction