Help with Reading a Counter Example [Generic]



Reading a counter example from an open branch

[This is a Quicktime Movie, click the Play button to view it. The logical symbols you see in use may be different to the ones you are familiar with (sorry about that, but it is not practical to produce different movies for all the minor variations in symbols). Any differences will not affect the principles being explained here.]

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The Tree Widget



An example of the tree widget in use (and we are showing a different parser here to the earlier ones)

The logical symbols in use are ¬ ∧ ∨ → ↔ ∀ ∃ .

Determine whether these arguments are valid (ie try to produce closed trees for them)

a) ∀x(F(x)→G(x)), ∃x¬G(x) ∴ ∃x¬F(x)
b) ∀x(F(x)→∀yG(y)), F(a) ∴ ∀xG(x)
c) ∀x(A(x)→B(x)), ∀x(¬A(x)→C(x))∴ ∀x(¬B(x)→¬C(x))
d) ∃xF(x),∀x(¬G(x)→¬F(x)),∀xM(x) ∴ ∃xG(x)∧∃xM(x)

Reading a Counter Example from the Tree

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.


Review of K Propositional Rules martin Thu, 06/07/2012 - 12:10
Logical System

There are the ordinary (non-modal) tree propositional rules plus

The Modal Negation (MN) rules


Review of Additional S5 Propositional Rules martin Thu, 06/07/2012 - 11:25
Logical System


◊ S5 world, k, must be new [here the computer will choose for you]



□ S5 any world, stage 1, your choice

Set Theory (and Russell's Paradox) martin Thu, 05/31/2012 - 14:55
Logical System


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


Axiom Schema of Abstraction (or Specification or Comprehension). The Set Builder Axiom Schema.

And a number of other axioms