Title: Independence-Friendly Existential Graphs
1Independence-Friendly Existential Graphs
- Ahti-Veikko Pietarinen
- Department of Philosophy
- University of Helsinki
- 29 April 2004
2Outline
- Symbolic vs. diagrammatic logic
- Independence-friendly (IF) logic
- Existential graphs (EG)
- IF EGs
- Conclusions
3Symbolic 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
4Independence-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.
5What 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
6Henkin quantifiers
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.
7Henkin quantifiers
- Krynicki normal forms (1993)
reduce to
8Branching 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)
9Language 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.
10Binding 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
11Game-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.
12Game-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
13Game-theoretic semantics
then
chooses
then
chooses
then
chooses
then
chooses
then
and
In IF logic strong game negation, not classical,
weak contradictory negation!
14Game-theoretic semantics
if
is a win for
if
is a win for
is true in
iff
there exists a winning strategy
for
in
is false in
iff
there exists a winning strategy
for
in
15Imperfect 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
16Extensive-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.
17Basic 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
18Existential 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)
19Existential 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
20Alpha, 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
21Alpha part
- Sheet of Assertion (SA, universe of discourse)
- Cuts (negations)
- Juxtaposition (conjunction)
SA
T
SA
SA
SA
22Alpha part
23Beta 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
24Beta part
- Another example, coreference
walks in
man
park
whistles
A man walks in the park. He whistles.
25Beta 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
?
?
26Beta 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
27Beta 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
28Gamma 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,
29Gamma part
You can lead a horse to water, but you cant make
him drink
30Existential 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
31Proofs 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
32Proofs in EGs
- Double cut insertion/deletion
- Graph insertion any graph may be added to an
odd-polarity area
33Proofs in EGs
- Iteration/deiteration any copy of a subgraph may
be added/erased to/from the same or deeper areas
than it
iteration
deiteration
34Heterogeneous 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.
35Hyperproof
- Given information a blocks world (toy model,
situation) FOL sentences - Determine what characteristics hold
- of it
36Conceptual 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
37Conceptual Graphs
- An open-ended enterprise
- Formal concept analysis
- Natural-language processing
- Software specification
- Information extraction
- CGWorld
- PrologCG (integrates Prolog, CGs, OOP and JAVA)
38Semantic networks
- Concepts, relationships
- Boxes, arrows, labels
- Database queries, inferences
- Non-monotonicity
- ER graphs, Dataflows, Petri nets, Neural nets,
- A very heterogeneous field!
39Discourse-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.
40IF 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
41IF 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
42IF 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
43Conclusions
- 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.
44Conclusions
- 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)
45Conclusions
- 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
46The 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
47Projects
- 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
48Commens 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/