The Tableau Calculus is a formal method to prove statements in First Order Predicate Logic. It is explained in extenso for instance in the scriptum to the lecture formal methods (German). The applet on this page has been developed by Mattias Ulbrich.
Before using the applet you will be asked to trust the applet. You do not have
to do so. You must reply with "yes", however, if you want to save images as 'png' files on
your local file system.
Printing can be done in the unprivileged mode also.
A new proof can be started using the Button. You will be asked to enter one or more formulas spearated by ';'. See description below on how to enter formulas.
Nodes in the proof are colorcoded according to their "type" of formula:
ALPHA  Nonbranching formulas with subformulas (such as A ∧ B)  
BETA  Branching formulas with subformulas (such as A ∨ B)  
GAMMA  Universally quantified formula (∀x. A(x))  
DELTA  Existentially quantified formula (∃x. A(x))  
NEGNEG  Special case: Double negation (¬¬A), resolves to A 
You can drag a formula holding your left mouse button (formula becomes red) onto a leaf of the tree to apply the according rule to this branch. The leaf must not be closed. If you want to expand a formula on the spot, you can click on it twice. Sometimes, it is easier to explicitly specify the formula to expand. You can do this by double clicking on a leaf while holding the CTRLkey pressed.
You can also drag a formula with your right mouse button pressed (formula becomes blue) onto a complementary formula (i.e. A onto ¬ A or v.v.) to close branches. The formulas must be complementary, not complementary after unification. You have to perform the unification by adequate instantiations.
The application of gamma rules create new free variables called
Xn
with n a natural number.
These free variables can be instantiated using the
button. A message box pops up and you
can enter the instantiation for one free variable in the form
Xn = term
.
You can undo rule applications and instantiations using the button. Please note that closed branches may become reopened hereby.
If the formulas are not clearly visible or if you prefer ASCII rather than mathematical formulas, you can switch between ASCII and Unicode characters using the button.
The grammar for formulas is straight forward and will be presented here by examples.
Identifiers can be any character string (case sensitive!) starting with a letter (optionally)
followed by letters or digits. You must not use X1, X2, ...
and
sk1, sk2, ...
however, for these are reserved for free variables and
skolem symbols respectively.
A x. p(x) 
for all x, p(x) holds. p is a predicate here. 
E x. p(f(x)) 
There is a x so that p(f(x)) holds. p is a predicate and f is a function here 
P & Q 
P and Q hold. 
P > Q 
P implies Q 
~P 
not P 
P  Q 
P or Q 
P  Q & R 
Operators have the usual precedence: & then  then >. I.e.
P  Q & R = P  (Q & R)

A x. p(x)  q(x) 
Caution! This is equal to (A x. p(x))  q(x) .
Therefore x appears outside the context of the quantifier and is treated like a constant
function symbol. The applet does not check for such things, so be alert!

There are some examples for typical tableau proofs available here
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environment supported