The Reading a Counter Example from a Tree Widget
Topic
Logical System
howson
Notation
martin
Thu, 02/11/2010 - 11:02
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, T₁2 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)
Topic
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
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
Tree Tutorial 6: Functional Terms and First Order Theories
Topic
Logical System
6/5/12
Reading
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
Reading
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
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
Preliminary [Pre-test]
martin
Mon, 03/16/2009 - 22:37
Topic
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
Reading a Counter Example from the Tree
martin
Fri, 01/23/2009 - 16:47
Topic
Logical System
2013
Reading
Colin Howson, [1997] Logic with trees Chapter 2
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.