# howson

## Tree Tutorial 8 Modal Trees

Topic
Logical System
###### 1/31/20

Colin Howson, [1997] Logic with trees Chapter 12 Section 2

The Howson [1997] does not expand on modal logic (and modal trees) so 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).]

Topic
Logical System

## Notation

Logical System

2013

The Colin Howson book uses a notation like R(a,b,c) for the application of a predicate R to the arguments or terms a, b, c.

It employs the upper case letters A-Z, perhaps followed by subscripts, to be predicates, so, for example, R, S₁, T₁ are all predicates.

The software supports this.

But the software makes an extension.

Often, when working informally, authors will write Red(x) to mean that the predicate Red is applied to the variable x.

## Tree Tutorial 7: Type Labels, Sorts, Order Sorted Logic ['Mixed Domains']

Topic
Logical System

### [Howson[1997] mentions this in Chapter 5, under 'mixed domains'.]

[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 (and Russell's Paradox)

Topic
Logical System

2013

Colin Howson, [1997] Logic with trees Chapter 11

### 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.

## Number Theory and Peano Arithmetic

Topic
Logical System

2013

Colin Howson, [1997] Logic with trees Chapter 9 & 11

### 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.

## Groups

Topic
Logical System

2013

Colin Howson, [1997] Logic with trees Chapter 9

### 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*)

## Tree Tutorial 6: Functional Terms and First Order Theories

Topic
Logical System
###### 6/5/12

Colin Howson, [1997] Logic with trees

### Tutorial:

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

## Tree Predicate Exercises: Roll your own

Topic
Logical System

2013

Colin Howson, [1997] Logic with trees Chapter 2

### Exercises

Howson [1997] has a number of exercises. Many of them you will be able to do in the Widget below.

Here are a few hints

Topic
Logical System