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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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)
Last altered 6/20/12A 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.
There are the ordinary (non-modal) tree propositional rules plus
The Modal Negation (MN) rules
◊ S5 world, k, must be new [here the computer will choose for you]
□ S5 any world, stage 1, your choice
Review of Tree Propositional Rules, shown as patterns
Here the letters 'P' and 'Q' are being used to stand for entire well formed formulas (so, on a particular occasion, 'P' might stand for the atomic formula 'F' and, on another occasion, it might stand for the compound formula 'F&G').
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