Independence-Friendly Existential Graphs

1
Independence-Friendly Existential Graphs

• Ahti-Veikko Pietarinen
• Department of Philosophy
• University of Helsinki
• 29 April 2004

2
Outline

1. Symbolic vs. diagrammatic logic
2. Independence-friendly (IF) logic
3. Existential graphs (EG)
4. IF EGs
5. Conclusions

3
Symbolic vs. diagrammatic logical representations

• 20th century Mainly symbolic logic
• 19th century Lots of diagrammatic logics (Venn,
  Kempe, Sylvester, Peirce,)
• Earlier (Euler, Bruno, Vives,)
• 21st century ?
• Diagrams are not conventional like symbols, but
  iconic

4
Independence-friendly (IF) logic

• Jaakko Hintikka
• The real source of the expressive power of
  first-order logic lies not in the notion of
  quantifier per se, but in the idea of a dependent
  quantifier
• Hintikka, Jaakko (1996 47)
• The Principles of
• Mathematics Revisited,
• CUP.

5
What is IF logic?

• Allow explicit independence between quantifiers
• For all x there exists y independently of x
• Skolem functions not
• but
• Semantic games of imperfect information
• Arrays of Skolem functions winning strategies

6
Henkin quantifiers

• Henkin (1959)

true in M iff
Leon Henkin (1961) Some Remarks on Infinitely
Long Formulas, Infinistic Methods. Proceedings
of the Symposium on Foundations of Mathematics,
Pergamon Press, 167-183.
7
Henkin quantifiers

• Krynicki normal forms (1993)

reduce to
8
Branching quantifiers

• Some relative of every villager and some friend
  of every townsman hate each other (Hintikka 1974)

• Most linguists and most logicians admire each
  other (Barwise 1979)

9
Language of IF logic

• Let
• be in the scope of
• Given
• wffs of
• We may even have

are
and
Sandu, G. Pietarinen, A.-V. (2001) Partiality
and Games Propositional Logic, Logic Journal of
the IGPL 9, 107-127.
10
Binding vs. priority scope

• Binding scope The reach of any single
  instantiation of values
• Priority scope Logical ordering of quantifiers
• In FOL these go together, in IF logic not
• Limitation of the Frege-Russell concept of logic

11
Game-theoretic semantics

• Henkin (1961)
• Imagine, for instance, a game in which a
  First Player and a Second Player alternate in
  choosing an element from a set I the infinite
  sequence generated by this alternation of choices
  then determines the winner

Leon Henkin (1961) Some Remarks on Infinitely
Long Formulas, Infinistic Methods. Proceedings
of the Symposium on Foundations of Mathematics,
Pergamon Press, 167-183.
12
Game-theoretic semantics

• Hintikka (1973) a game between

• Non-cooperative, finite, zero-sum games
• Complete but possibly imperfect information and
  imperfect recall
• FOL, modal logic, dynamic logic,

A.-V. Pietarinen (2004) Some Games Logic
Plays, Logic, Thought and Action, Kluwer
13
Game-theoretic semantics
then
chooses
then
chooses
then
chooses
then
chooses
then
and
In IF logic strong game negation, not classical,
weak contradictory negation!
14
Game-theoretic semantics

• Winning conventions

if
is a win for
if
is a win for

• Winning strategies

is true in
iff
there exists a winning strategy
for
in
is false in
iff
there exists a winning strategy
for
in
15
Imperfect information

• In any
• player chooses without knowing previous
  choices in W
• Induces equivalence relations
• between game histories
• Information sets in extensive-form games
• Non-determined formulas

16
Extensive-form games

• Interactive move-by-move setting
• Provides derivational histories
• Explicit representation of information flow
• Imperfect recall (memory)
• Partial semantics

A.-V. Pietarinen (2004) Semantic Games in Logic
and Epistemology, Logic, Epistemology and the
Unity of Science, Kluwer Academics.
17
Basic properties of IF logic

• Agrees with the -fragment of the second-order
  logic
• Compactness
• Downwards Löwenheim-Skolem
• Not recursively axiomatisable
• Expresses NP-complete properties on finite models

18
Existential Graphs

• Charles S. Peirce (1839-1914)
• I do not think I ever reflect in words I employ
  visual diagrams, firstly, because this way of
  thinking is my natural language of
  self-communion, and secondly, because I am
  convinced that it is the best system for the
  purpose
• (MS 6198, 1909)

19
Existential Graphs

• Entitative Graphs (1886) ? Existential Graphs
  (EG, 1895)
• The goal is not to have heterogeneous logic but
  iconic, diagrammatic, graphical
• Origins in algebra of relatives and valental
  chemistry
• EGs put before us moving-pictures of thought
  (1906)

A.-V. Pietarinen (2004) Peirces Magic Lantern
I Moving Pictures of Thought, Transactions of
the C.S. Peirce Society
20
Alpha, Beta, Gamma

• Alpha graphs propositional logic
• Beta graphs predicate logic /w identity w/o
  constants, function symbols
• Gamma graphs
• Modalities (possibility, necessity, knowledge,
  time, tinctures, 1908)
• Higher-order assertions
• Graphs of graphs, abstractions
• Interrogatives, imperatives, absurdities
• Delta graphs (1911) to deal with modals

Don D. Roberts (1973) The Existential Graphs of
Charles S. Peirce, Mouton
21
Alpha part

• Sheet of Assertion (SA, universe of discourse)
• Cuts (negations)
• Juxtaposition (conjunction)

SA
T
SA
SA
SA
22
Alpha part

• Conditional (the scroll)

23
Beta part
A man eats a man

• Rhemas (predicate terms)
• Lines of identities (LI, existence, identity,
  predication, subsumption)

phoenix
A phoenix doesnt exist
Something exists that is not phoenix
thunder
lightning
If it thunders, it lightens
24
Beta part

• Another example, coreference

walks in
man
park
whistles
A man walks in the park. He whistles.
25
Beta part

• Binding scope is given by the system of LIs
  (ligatures)
• Priority scope is given by the system of cuts
• In FOL these go together, in Beta they do not
• Beta not isomorphic to FOL
• Rather like dynamic semantics

?
?
26
Beta part

• Different readings of is not logically
  different
• Existence Socrates exists
• Identity L. Carroll is C. Dodgson
• Predication Socrates is mortal
• Subsumption Man is an animal
• Coreference A man walks in the park. He
  whistles.

A.-V. Pietarinen (2004) Signs of Logic Peircean
Themes on the Philosophy of Language, Games, and
Communication, Kluwer
27
Beta part

• Rhemas, graphs, inferences are continuous with
  one another (1908)
• Connectivity between different parts of SAs by
  LIs and juxtaposition gives rise to propositions
• Meaning-preserving transformations as continuous
  deformations give rise to inferential arguments
• Topological system

28
Gamma part

• Modalities (It is possible that it rains)
• Higher-order assertions (Aristotle has all the
  virtues of a philosopher)
• Meta-assertions (You are a good girl is much to
  be wished)
• Non-declaratives Questions, commands,
  absurdities, emotions, music,

29
Gamma part
You can lead a horse to water, but you cant make
him drink
30
Existential Graphs

• Explicit, non-inductive definitions
• Holistic, non-compositional system of meaning
• Semantics in terms of the Endoporeutic Method
  (1905)
• Similar to Game-Theoretic Semantics
• Utterer vs. Interpreter play the game
• Perfect information, winning strategies as
  habits of action

A.-V. Pietarinen (2003) Peirces Game-theoretic
Ideas in Logic, Semiotica 144, 33-47
31
Proofs in EGs

• Four rules of transformation double negation,
  insertion, erasure, iteration/deiteration
• Sound and complete for Alpha Beta
• Natural deduction system, 30 years before Gentzen
  and others

32
Proofs in EGs

• Double cut insertion/deletion

• Graph insertion any graph may be added to an
  odd-polarity area

33
Proofs in EGs

• Iteration/deiteration any copy of a subgraph may
  be added/erased to/from the same or deeper areas
  than it

iteration




deiteration
34
Heterogeneous reasoning systems

• A hundred years later
• Barwise Etchemendys Hyperproof
• John Sowas Conceptual Graphs
• Semantic networks
• Hans Kamps Discourse-Representation Theory
• Spider diagrams (extending Euler-Venn)
• and much more

A.-V. Pietarinen (2004) Diagrammatic Logic and
Game Playing, Multidisciplinary Studies on
Visual Representations and Interpretations,
Elsevier.
35
Hyperproof

• Given information a blocks world (toy model,
  situation) FOL sentences
• Determine what characteristics hold
• of it

36
Conceptual Graphs
A cat is on a mat
Every cat is on a mat
Tom believes that Mary wants to marry a sailor
John Sowa (2000) Knowledge Representation
Logical, Philosophical and Computational
Foundations, Brooks/Cole
37
Conceptual Graphs

• An open-ended enterprise
• Formal concept analysis
• Natural-language processing
• Software specification
• Information extraction
• CGWorld
• PrologCG (integrates Prolog, CGs, OOP and JAVA)

38
Semantic networks

• Concepts, relationships
• Boxes, arrows, labels
• Database queries, inferences
• Non-monotonicity
• ER graphs, Dataflows, Petri nets, Neural nets,
• A very heterogeneous field!

39
Discourse-Representation Theory

• Hans Kamp (1981), Lauri Karttunen (1976)

A man walks in the park. He whistles.
T. Janasik, A.-V. Pietarinen and G. Sandu (2003)
Anaphora and Extensive Games, Papers from the
38th Meeting of the Chicago Linguistic Society,
Chicago Linguistic Society.
40
IF EGs

• Can we increase the expressive power of the Beta
  part of EGs without introducing any new signs?
• Yes ? make EGs Independence-friendly

A.-V. Pietarinen (2004) Peirces Diagrammatic
Logic in IF Perspective, LNAI 2980, 97-111
41
IF EGs

• IF extension of EGs expressive enough so as to
  capture much of our mathematics
• IF EGs model good deal of natural-language
  utterances, including discourse and branching
  quantifiers
• It illustrates the different logical priorities
  between LIs, forbidden in graphs on 2D SAs

A.-V. Pietarinen (2004) Compositionality,
Relevance, and Peirces Logic of Existential
Graphs
42
IF EGs

• Non-compositional system local vs. global
  contexts
• Topological distinction between open/closed sets
  area of the cut / area the cut
• Distinction between strong, game-theoretic
  negation () as a role switch and classical,
  contradictory negation ( ) as complementation
• The latter requires a meta-level definition,
  whereas the former is processual

A.-V. Pietarinen (2004) Peirces Magic Lantern
II Topology, Graphs and Games
43
Conclusions

• Peirce envisaged some 3D extension
• Three dimensions are necessary and sufficient
  for the expression of all assertions so that, if
  mans reason was originally limited to the line
  of speech (which I do not affirm), it has now
  outgrown the limitation
• (MS 654 6, 1910)

Peirce Manuscripts at Harvard University
Helsinki, microfilmed 1967, catalogued by R.
Robin.
44
Conclusions

• IF EGs fulfill Peirces dream
• At great pains, I learned to think in diagrams,
  which is a much superior method to algebraic
  symbols. I am convinced that there is a far
  better one, capable of wonders but the great
  cost of the appatatus forbids my learning it. It
  consists in thinking in stereoscopic moving
  pictures.
• (MS L 231, 1911)

45
Conclusions

• Insufficiency of FOL/Beta Graphs
• Symbolic vs. disgrammatic representations
• Reasoning with non-linguistic forms
• Multi-modal reasoning (perception, tactile etc.
  stimuli, tinctured EGs)
• Free rides
• Corollarial vs. theorematic reasoning

46
The way ahead

• A semantic web using diagrammatic
  representational systems?
• Pragmatics through games
• Abstract vs. strategic meaning
• Speakers vs. literal meaning (interpretants)
• The Web iconic, symbolic, indexical signs
• Putting questions to the Web interrogative games

A.-V. Pietarinen (2003) The Semantic
Pragmatic Web the Semiotic Web, Proc.
International IADIS/WWW Conference, IADIS Press,
981-984 A.-V. Pietarinen (2004) Peircean and
Historical Pragmatics, Journal of Historical
Pragmatics
47
Projects

• 2002-2003 the Academy of Finland Project
  Game-theoretical Semantics and its Applications,
  Director Jaakko Hintikka
• 2003-2005 the Academy of Finland Project Logic
  and Game Theory, A.-V. Pietarinen (Post-Doc
  Fellow)
• 2003-2004 the Academy of Finland Project
  Communications in the 21st Century The Relevance
  of C.S. Peirce

48
Commens is a Finnish Peirce studies website,
which promotes and supports investigation of
Peircean philosophy and sign theory. The Commens
pages include introductions to Peirce and his
philosophy, original papers, various
bibliographies, and other study aids. ...that
mind into which the minds of utterer and
interpreter have to be fused in order that any
communication should take place ... may be called
the commens. It consists of all that is, and must
be, well understood between utterer and
interpreter, at the outset, in order that the
sign in question should fulfill its function."
(Charles S. Peirce, 1906.)
Mats Bergman Erkki Kilpinen Sami
Paavola Ahti-Veikko Pietarinen Sami Pihlström
http//www.helsinki.fi/science/commens/