The Existential Graphs Project Rensselaer Reasoning Group September 12, 2001

Overview Alpha –Symbolization –Inference Rules –Current Java Implementation Beta –Symbolization –The Bird Problem The Future –Questions –Extensions

Symbolization Sheet of Assertion To assert some statement in EG, you put the symbolization  of that statement on a sheet of paper, called the ‘Sheet of Assertion’ (SA). Thus, to assert the truth of some statement p, draw:  where  is the symbolization of p SA

Symbolization Location is irrelevant The location of the symbolization on the SA does not matter: as long as it is somewhere on the SA, it is being asserted. Thus:   states the same as: In fact, the above two graphs are regarded as completely identical.

Symbolization Juxtaposition and Conjunction By drawing the symbolization of two statements on the SA, you are asserting the truth of both statements at once. Hence, the mere juxtaposition of two symbolizations on the SA can be interpreted as the assertion of a single conjunction. Thus:   can be seen as the assertion of both  and , but also as the assertion of   , and also as the assertion   .

Symbolization Cut and Negation You assert the negation of some statement by drawing a cut (circle, oval, rectangle, or any other enclosing figure) around the symbolization of that statement. Thus:  asserts that  is false. (from now on, the SA will not always be drawn)

Symbolization Expressive Completeness Atomic propositions, cuts and juxtaposition are the only symbolization tools of Alpha. However, since conjunction and negation form an expressively complete set of operators for propositional logic, Alpha is expressively complete as well (and Alpha does not need parentheses).

Symbolization From PL to EG P ~~                P      Symbolization in EG Expression in PL

Symbolization From EG to PL P  Q or Q  P or P and Q ~(~P  ~Q) or ~P  Q or P  Q or ~(~Q  ~P) or ~Q  P or Q  P ~(P  ~Q) or ~(~Q  P) or ~P  Q or Q  ~P or P  Q Q QP P QP QP ~(P  Q) or ~(Q  P) or ~P  ~Q or ~Q  ~P Possible Readings

Inference Rules Alpha has four inference rules: –2 rules of inference: Insertion Erasure –2 rules of equivalence: Double Cut Iteration/Deiteration To understand these inference rules, one first has to grasp the concepts of subgraph, double cut, level, and nested level.

Inference Rules Subgraph The notion of subgraph is best illustrated with an example: R Q The graph on the left has the following subgraphs: QRR R Q R Q,,,, In other words, a subgraph is any part of the graph, as long as cuts keep their contents.

Inference Rules Double Cut A Double Cut is any pair of cuts where one is inside the other and with nothing in between. Thus: P RQ RQ,, and contain double cuts, butdoes not.

Inference Rules Level The level of any subgraph is the number of cuts around it. Thus, in the following graph: R Q Q R R R Q R Q is at level 2 (the graph itself) is at level 0,,, andare at level 1, and

Inference Rules Nested Level A subgraph  is said to exist at a nested level in relation to some other subgraph  if and only if one can go from  to  by going inside zero or more cuts, and without going outside of any cuts. E.g. in the graph below: P R Q R exists at a nested level in relation to Q, but not in relation to P. Also: Q andR exist at a nested level in relation to each other.

Inference Rules Double Cut The Double Cut rule of equivalence allows one to draw or erase a double cut around any subgraph. Obviously, this rule corresponds exactly with Double Negation from PL.    

Inference Rules Insertion The Insertion rule allows one to insert any graph at any odd level.   12k+1  1

Inference Rules Erasure The Erasure rule of inference allows one to erase any graph from any even level.   12k  1

Inference Rules Iteration/Deiteration The Iteration/Deiteration rule of equivalence allows one to place or erase a copy of any subgraph at any nested level in relation to that subgraph. 

Inference Rules Completeness and Soundness Although no proof will be given here, it turns out that the 4 inference rules of Alpha form a complete system for truth-functional logic (for a proof, see “The Existential Graphs of Charles S. Peirce” by Don Roberts, 1973). The rules are also sound, although (except for Double Cut), this is not immediately obvious either (for a proof, see the ‘Alpha’ presentation online).

Java Implementation

Existentials, Individuals and Identity ‘Something exists’ ‘Nothing exists’ ‘John exists’ j ‘a=b’ ab ‘At least 2 things exist’ ‘a  b’ ba

Quantifier Square of Opposition ‘Something is P’‘Something is not P’ ‘Nothing is P’‘Everything is P’ PP P P

Traditional Square of Opposition ‘Some P is Q’‘Some P is not Q’ ‘No P is Q’‘Every P is Q’ P Q P Q P Q P Q

The Bird Problem  x (Bx   x Bx)

Questions Does one obtain a deep understanding of logic using EG? How do EG and PL compare and relate? Does EG reveal features that can be exploited in automated theorem proving?

Improvements and Extensions: Fitch Model after Fitch: –Have goal(s) (subgoals?) separate –The user needs to be able to load, save, and edit proofs User can ‘see’ the proof Justifications can be indicated as well Students can hand in proofs as HW –User should be able to make mistakes Program provides feedback User learns from mistakes

Improvements and Extensions: Other Relate to PL: –Translate between PL and EG Automatic? Partly automatic? Helpful Buttons: –To draw a ‘If P then Q’ graph, click on ‘if … then’ button, followed by ‘P’ button and ‘Q’ button (again, like Fitch) Beta!

More Improvements Tree->Graph automatic drawing –Facilitates proof editing as it facilitates pasting of parts of proof –Can ‘clean up’ user mess –Automatic/User Drawing mode Apply patterns to part of graph –Again, pasting can be facilitated through automatic drawing routine –Import patterns Con Rules! –Useful for user –Automatic Theorem Proving!