2/40/20 20 Software

### An example of the tree applet in use

The logical symbols in here 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)

*[In the applet below:- Click once to select. Click once, then command-click (or clover leaf click) to select two. Select one item to extend the Tree, select two to close a branch. [Select two for extending with identity.]] *

Answers: Yes, Yes

You will be able to do your own examples in the Applet below.

Here are a few hints

- You have to use the right (unicode/html) logical symbols. Check Writing symbols
- The symbols in use here are ¬ ∧ ∨ → ↔ ∀ ∃ (so copy and paste or drag and drop these).
- When entering from a selection, the software will take a single formula (say, A) or a comma separated list of formulas (say, A,B,C) or a possibly empty comma separated list of formulas followed by ∴ and another formula (say, A,B,C ∴ D). In the last case it will load the negation of the conclusion.
- Just type, cut and paste, or drag and drop, whatever you wish, into the lower text box. Then make a selection and hit the Start button.