A Java Implementation of Peirce’s Existential Graphs

A Java Implementation of Peirce’s Existential Graphs Bram van Heuveln Department of Philosophy State University College at Oneonta March 22, 2001

Overview • Background: Logic Systems • Peirce’s Existential Graphs • The Project • Implementation • Demonstration

Logic Systems A Logic Puzzle • The body of Mr. X was found murdered in his bedroom by the housemaid. Who did it? • Inspector Clouseau collects the following information: • Only the butler and the housemaid have a key to Mr. X’s bedroom • Only the butler knows about the secret alarm that Mr.X activates at night in his bedroom • The alarm did not go off.

Logic Systems Our Reasoning • “Either the butler or the housemaid killed Mr. X. However, if the housemaid killed Mr. X, the alarm would have gone off, and the alarm didn’t go off, so the housemaid is in the clear. Therefore, the butler did it.” • Can we formalize our reasoning? Yes. This is what logic systems do.

Logic Systems Step 1: Logical Symbolization • Use symbols to represent simple propositions: • H: The housemaid did it • B: The butler did it • A: The alarm went off • Use further symbols to represent complex claims: • H  B: The housemaid or the butler did it • HA: If the housemaid did it, the alarm would go off • ~A: The alarm did not go off

Logic Systems Step 2: Logical Inference • Transform symbolic representations using basic rules that reflect valid inferences: 1. H  B Assumption(A.) 2. HA A. 3. ~A A. 4. ~H 2, 3 MT 5. B 1, 4 DS

Logic Systems Completeness and Soundness • Logic Systems need to be complete and sound: • Expressive Completeness: The system needs to be able to represent every possible logical expression. • Deductive Completeness: The system needs to be able to infer anything that logically follows. • Deductive Soundness: The system should not be able to infer anything that does not logically follow. • Logic Systems can be proven to be complete and sound.

Logic Systems The Trade-off • The rules in logic systems reflect simple logical inferences. The simpler the inferences, the fewer rules the system will have to have in order to be complete, as more complex rules will reduce to sequences of more simple rules. However, this also means that proofs get longer. In other words, there is a trade-off between the number of rules in the system and the length of a given proof.

Logic Systems Example of the Trade-off 1. H  B A. 10. H A. 2. HA A. 11. ~B A. 3. ~A A. 12. H 10 R 4. H A. 13. ~H 7 R 2,4 E 5. A 14. B 11-13 ~E 1,8-9,10-14 E 6. ~A 3 R 15. B 7. ~H 4-6 ~I 8. B A. 24 rules: 2 steps 11 rules: 12 steps 9. B 8 R

Existential Graphs Peirce’s Existential Graphs • A graphical logic system developed by Peirce almost 100 years ago. • Peirce studied semiotics: the relationship between symbols, meanings, and users. • Peirce found the linear notation and accompanying rules of traditional logic systems (which he helped develop) involved and unintuitive. • Existential Graphs allow the user to express logical statements in a completely graphical way.

Existential GraphsSyntax of EG Traditional EG ‘P’ P P ‘not P’ ~P P ‘P and Q’ P&Q P Q ‘P or Q’ PQ P Q ‘if P then Q’ PQ P Q

Existential GraphsInference Rules of EG P P Double Cut (De)Iteration P P Q P Q Erasure P Q P Insertion Q P Q

Existential GraphsProof in EG H B H A A DE H B H A DE B H A DC E B H A B

Existential GraphsStrength of EG • Compact • Only Propositions and Cuts; Only 4 rules • Easy to use • Less chance of making mistakes • Fast • Transform rather than rewrite • Intuitive • Many logical relationships come for free • Maximum Logical Power • Expressively complete; deductively complete

Existential GraphsStudent Response Personal experience from teaching Existential Graphs in logic class: • Even though students were forced to draw successive snapshots, students were more happy with Existential Graphs than traditional systems: • easier • faster • less mistakes • more fun • Students were very excited at the idea of having an interactive interface

The ProjectMotivation • EG presents an interesting alternative to traditional systems • Interface for construction and manipulation of Existential Graphs can be used in logic class • Software does not seem to exist • Conceptual advantages of the dynamic character of logic proofs in EG remain unexplored • Nice example of cross-curricular collaboration • Nice example of integrating technology into the classroom

The ProjectRequired Functionality • The user should be able to: • Generate Existential Graphs • Draw, delete, move, resize, and copy propositions and cuts • Manipulate Existential Graphs • Apply rules of inference • The system should: • Keep track of the logical relationships as expressed by the Existential Graphs • Check if the rules of inference are correctly applied by the user

The Project Desired Additional Functionality • File I/O • To load and save existential graphs • To load and save proofs as a series of images • Proof Editor • Video buttons to play and rewind proofs • Edit existing proofs • Help and Tutorial • Instructions for use • Examples

The ProjectThe Project Team • Supervisors: • Bram van Heuveln (Philosophy) • Dennis Higgins (Math and Computer Science) • We obtained a TLTC Fast Tech Grant • We invited three upper division Computer Science students to develop this software: • Elizabeth Hatfield • Debbie Kilpatrick • Lut Wong • We held weekly meetings to discuss progress

The ProjectDivision of Labor

The ProjectProject Phasing We decided to implement in two phases: • Phase one: develop a Work Area • Interface with full editing capabilities for generating and editing Existential Graphs • Main problem: Correspondence between graphical operations and internal logical data structure • Phase two: develop a Proof Area • Interface for the manipulation of Existential Graphs • Main Problem: Perform checking to insure user selections are legal

The ProjectCurrent Status • Both phases are now complete, and we have a minimally working system • Additional helpful features still need to be implemented

A Java Implementation of Peirce’s Existential Graphs