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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.

 

Preliminary [Pre-test]

Logical System
6/20/12

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

Tree Tutorial 7: Type Labels, Sorts, and Signatures ['Mixed Domains'] martin Fri, 06/01/2012 - 14:43
Topic
Logical System
6/6/12

Reading

[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)

Logical System
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

 (x=y) ≡∀z(zεx≡zεy))      Axiom of Extensionality

Number Theory and Peano Arithmetic martin Thu, 05/31/2012 - 14:52
Topic
Logical System
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.