WinKEA Pedagogic Tool for Teaching Logic and Reasoning

WinKE displays a simple KE proof for the equivalence of the two formulas P Q and Q P.

The WinKE system features an interactive interface via menus, dialogues, and graphic tools, on- and off-line proof-checking, and is completed by the option to automatically perform proofs (or parts of a proof), and to build up countermodels. User support is provided by bookkeeping facilities, hints, various "undo" utilities, and a detailed on-line help system. The system provides several files with example problems. New ones can be edited directly within WinKE.

The interface consists of four windows: the main window providing menus and buttons for quick access to the most basic functions, the graphic window to display and manipulate KE proof trees, a viewer which displays a scaled-down view of the virtual drawing board, allowing to focus on a particular portion of it, and a tool box containing the graphic tools. The design of the interface is close to that of Windows standard software, which makes it very easy to learn how to use the system.

WinKE is supportive of an introductory textbook on classical logic (M. D'Agostino, M. Mondadori, Logica, Edizioni Bruno Mondadori, 1997), but may also be used independently.

WinKE is a new interactive theorem proving assistant based on the KE calculus, a refutation system which combines features from Smullyan's analytic tableaux and Gentzen's natural deduction. It has been developed to support teaching logic and reasoning to undergraduate students.

The WinKE process of constructing a proof tree is as faithful as possible to the pen-and-paper procedure. Running under Windows 95, it is easy to use and visually satisfying.

University of Ferrara Imperial College London

• Every country is either red, green, or blue.• Countries in the neighborhood of green countries are blue.• Countries in the neighborhood of blue countries are red. Show that no country is green.

TreesProblems

Rules

Proofs & Countermodels

• Show that Peirce's law ((P Q) P) P is a tautology.• Show that the following set of formulas is inconsistent: P Q, P Q, P Q, P Q. • Is Socrates mortal?

If there are two complementary formulas on a branch, it is said to be closed. A theorem is proven to hold, if the tree generated by its negation is closed. Branches that cannot be closed represent countermodels.

P QP———— Q

(P Q) P————Q

P QP———Q

P QQ————P

P Q———PQ

(P Q)————PQ

(P Q)————— PQ

P——— P

P QP———Q

P QP————Q

(P Q) P—————Q

(P Q)P————— Q

————

P P

x : P———P[x/t]

x : P————P[x/t]

x : P———P[x/c]

x : P————P[x/c]

for any closed term t

for a new constant c

Alpha & double negation

Beta

Eta

Gamma

Delta

Principle of Bivalence (PB)

For more information visit our web-page:

http://dns.unife.it/dgm/WinKE/

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Understanding Logic

WinKE HelpLectures WinKETextbook

Marco Mondadori & Marcello D’Agostino, Logica, Edizioni Bruno Mondadori, 1997.

x: y: R(x,y) R(y,x)

Imagine you're in Transylvania and you're meeting a suspicious-looking passenger. You ask him what he knows about Dracula. He answers: "Everybody is afraid of Dracula and I am the only person Dracula is afraid of."

What can you conclude from this statement?

Transform the problem into First Order Logic

Create a new problem in WinKE

Apply KE rules to expand the proof tree

manual deduction

automateddeduction

The branch is saturated, i.e. there are no further possible proof steps (that could lead to a closure). Therefore, a countermodel exists.

Building up countermodels is very easy in KE. All you have to do is to collect the literals on a saturated branch.

IsAfraidOf(Dracula,Dracula)

SamePerson(Dracula,Passenger)

Countermodel

An Example

x : IsAfraidOf(x,Dracula)

x : IsAfraidOf(Dracula,x) SamePerson(x,Passenger)

applied to no. 1

applied to no. 2

applied to no. 4 & 3This means: Draculaand the passenger are

the same person!