howson

Tree Tutorial 8 Modal Trees

Topic
Logical System
1/31/20

Reading

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).]

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.

Set Theory (and Russell's Paradox) martin Sat, 12/19/2009 - 08:49
Logical System

2013

Reading

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.

And a number of other axioms

Number Theory and Peano Arithmetic

Topic
Logical System

2013

Reading

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

Reading

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

Preliminary [Pre-test]

Logical System

2013

You need to know some propositional logic to be able to understand the 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