Visual Logic: Peirce’s Existential Graphs Minds and Machines October 3, 2001
Overview • Existential Graphs • Symbolization • Inference Rules • Questions • Demo • Improvements and Extensions
SymbolizationSheet 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
SymbolizationLocation 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.
SymbolizationJuxtaposition 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 .
SymbolizationCut 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)
SymbolizationExpressive 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).
SymbolizationFrom PL to EG Symbolization in EG Expression in PL P P ~
SymbolizationFrom EG to PL Possible Readings P Q P Q or Q P or P and Q ~(P Q) or ~(Q P) or ~P ~Q or ~Q ~P P Q ~(P ~Q) or ~(~Q P) or ~P Q or Q ~P or P Q P Q ~(~P ~Q) or ~P Q or P Q or ~(~Q ~P) or ~Q P or Q P P Q
Inference RulesInference 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: The graph on the left has the following subgraphs: Q Q Q Q , R , R , , R R R • 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: contain double cuts, P , R Q , and but R Q does not.
Inference Rules Level • The level of any subgraph is the number of cuts around it. Thus, in the following graph: Q Q (the graph itself) is at level 0, R R Q Q , and are at level 1, and R is at level 2 , R R
Inference Rules Nested Level • A subgraph is said to exist at a nestedlevel 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: R exists at a nested level in relation to Q, but not in relation to P. Also: Q P exist at a nested level in relation to each other. Q and R R
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. 2k+1 1 2k+1 1
Inference Rules Erasure • The Erasure rule of inference allows one to erase any graph from any even level. 2k 1 2k 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).
Questions 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 ExtensionsFitch • Have goal in separate window • The user needs to be able to load, save, and edit proofs • User can ‘see’ the proof (steps don’t get lost) • 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 ExtensionsMore Fitch • Helpful Buttons • Buttons for ‘P’, ‘Q’, etc. • Buttons for ‘’, ‘’, ‘’, ‘’, ‘’, and ‘’ • Example: For ‘If P then Q’ graph, click ‘’ button, then ‘P’, and then ‘Q’. • ‘Lift and Replace’ Technique • To work on a subset of premises among many, ‘lift’ them from the main proof area, transform, and have the result replace the original section
Improvements and ExtensionsProofs as Movies • Ability to view and edit proofs • User can play, rewind, stop, fast-forward • The proof can be edited
Improvements and ExtensionsRelation between EG and PL • Translate between PL and EG • Problem: This is a many-to-many-more mapping • Solution: Automatically generate possible generations; User picks preferred one • Can this be used for automatic proof generation?
Improvements and ExtensionsVisual Features • Tree->Graph automatic drawing • Can ‘clean up’ user mess • Automatic Drawing vs User Drawing switch • Exploit 2 dimensional space • Greatly facilitates proof editing as visual operations can now be handled by operations over tree data structures. • Apply common patterns to part of graph • Reduce tedium and repetition • Import from file • All of this can be facilitated through automatic drawing routine and ‘Lift and Replace’ routine
Improvements and ExtensionsWorking Backwards • All rules of EG can easily be reversed: • Double Cut and (De)Iteration are rules of equivalence, so they go both ways already • The inverse of Erasure is Insertion on an even level • The inverse of Insertion is Erasure from an odd level • Therefore, we can work backwards from the conclusion to the premises by applying transformation rules! • Traditional systems do not have this feature!!
Improvements and ExtensionsConsequence Rule • Routine that checks whether a statement is a logical consequence of a set of statements. • Useful for user (Fitch again) • Application for Automatic Theorem Proving!