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)
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').