trees

1/31/20

Reading

Colin Howson, [1997] Logic with trees Chapter 12 Section 2

The Howson [1997] does not expand on modal logic (and modal trees) so a text like

Rod Girle [2000] Modal Logics and Philosophy

would definitely be a help here.

Tutorial

[Modal logic is a vast area, what is being presented here is the briefest of glimpses through the shop window (a book like the Girle would help you go further).]

 

Exercise: Finding a counter-example.

2013

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.]

2013

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 6/20/12

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.

 

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Exercise: Finding a counter-example.

 

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

The Modal Negation (MN) rules

 

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

 

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□ S5 any world, stage 1, your choice

6/2/12

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').

6/3/12

You can try your own exercises here.

Here are a few hints

• You have to use the right (unicode/html) logical symbols. Check Writing symbols
• The symbols in use here are ~ & ∨ →  
  

  ≡

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∃D. The constant, a, must be new to the branch [here the computer will choose for you]

 

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∀D. Any closed term, stage 1, your choice

∀D. Any closed term, stage 2, the constant 'a' chosen

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¬∃D.