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 pSA

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 thesame 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 notalways 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 EGExpression 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

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

QThe graph on the left has the followingsubgraphs:

Q R R

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

R Q

R Q, , and contain double cuts,

but does 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

RQ

R

Q

is at level 2

(the graph itself) is at level 0,

, , and are 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 and R exist at a nested level inrelation 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

12k+1

Inference Rules

Erasure• The Erasure rule of inference allows one to

erase any graph from any even level.

12k

12k

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’ a b

‘At least 2 things exist’

‘ab’ ba

Quantifier Square of Opposition

‘Something is P’ ‘Something is not P’

‘Nothing is P’‘Everything is P’

P P

PP

Traditional Square of Opposition

‘Some P is Q’ ‘Some P is not Q’

‘No P is Q’‘Every P is Q’

PQ

PQ

PQ

PQ

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!