Independence-Friendly
Existential Graphs
Ahti-Veikko Pietarinen
Department of Philosophy
University of Helsinki
29 April 2004
1
Outline
1.
2.
3.
4.
5.
Symbolic vs. diagrammatic logic
Independence-friendly (IF) logic
Existential graphs (EG)
IF EGs
Conclusions
2
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
3
Independence-friendly (IF) logic
Jaakko Hintikka:
• ”The real source of the
expressive power of firstorder 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.
4
What is IF logic?
• Allow explicit independence between
quantifiers:
x(y / x) Sxy
• For all x there exists y ”independently of” x
• Skolem functions: not f1x Sxf1 ( x)
but f1x Sxf1
• Semantic games of imperfect information
– Arrays of Skolem functions = winning
strategies
5
Henkin quantifiers
• Henkin (1959):
 x y 

 [ x, y, z, u ] true in M iff
 z u 
f1f 2xz [ x, f1 ( x), z, f 2 ( y)]
Leon Henkin (1961): ”Some Remarks on Infinitely Long Formulas”, Infinistic Methods.
Proceedings of the Symposium on Foundations of Mathematics, Pergamon Press, 167-183.
6
Henkin quantifiers
• Krynicki normal forms (1993)
 x11...x1n1 y11...y1n1 
 2

2
2
2
 x1 ...xn2 y1 ...yn2 
k k
Sx


ij yij


 x m ...x m y m ...y m 
nm
1
nm 
 1
reduce to
 x1...xn y 

 Sx1...xn yz1...zk u
 z1...zk u 
7
Branching quantifiers
• “Some relative of every villager and some friend
of every townsman hate each other” (Hintikka
1974)
 x y 

 [(Vx  Tz  Rxy  Fzu )  ( Hyu  Huy )]
 z u 
• “Most linguists and most logicians admire each
other” (Barwise 1979)
 Most x : Linguist ( x) 

 Admire( x, z )
 Most z : Logician( z ) 
8
Language of IF logic
• Let Qxi  L , Q {, }, (  )  L , {, }
be in the scope of Q1 x0Q2 x1 ,..., Qn xn1 , A  {x1 ,..., xn1}
Given B  A, (Qx /B)  and ( ( /B) ) are
IF
wffs of L
• xyz (u/x) Sxyzu, x ( S1 x ( / x) S2 x)
• We may even have
( ( /) )  ( ( /)  )
Sandu, G. & Pietarinen, A.-V. (2001): “Partiality and Games:
Propositional Logic”, Logic Journal of the IGPL 9, 107-127.
9
Binding vs. priority scope
• Binding scope: The reach of any single
instantiation of values
... x 0( S1x 0 
... ...  Si x 0 x...)
1...
• Priority scope: Logical ordering of quantifiers
... x y( S1xy... zS2...
z)
• In FOL these go together, in IF logic not
• Limitation of the Frege-Russell concept of logic
10
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”
...v5v4 v3v2v1[v1v2v3 ...]
f1 f3 f5 ..v2v4v6 ..[ f1 (v2v4v6 ..), v2 , f3 (v4v6v8 ..), v4 , f5 (v6v8v10 ..)..]
Leon Henkin (1961): ”Some Remarks on Infinitely Long Formulas”, Infinistic Methods.
Proceedings of the Symposium on Foundations of Mathematics, Pergamon Press, 167-183.
11
Game-theoretic semantics
• Hintikka (1973): a game G ( , M) between
V rifier ( Myslf , ros,...)
F lsifier ( N ture, gape,...)
• 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
12
Game-theoretic semantics
  1  2 then  chooses i  {1, 2}
  1  2 then  chooses i  {1, 2}
  x [ x] then  chooses a  dom(M)
  x [ x] then  chooses a  dom(M)
 
then 
 and  
In IF logic strong game negation, not classical,
weak contradictory negation!
13
Game-theoretic semantics
• Winning conventions
S[ x0 ,..., xn1 ] is a win for  if (a0 ,..., an 1 )  S M
M
if
S[ x0 ,..., xn1 ] is a win for  (a0 ,..., an 1 )  S
• Winning strategies


is true in M iff there exists a winning strategy
for  in G ( , M)
is false in M iff there exists a winning strategy
for  in G ( , M)
14
Imperfect information
• In any (x / W1 ), (x / W2 ), ( / W3 ), ( / W4 )
player chooses “without knowing” previous
choices in W
• Induces equivalence relations h1 i h2
between game histories
• Information sets in extensive-form games
• Non-determined formulas x(y /x) Sxy
15
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.
16
Basic properties of IF logic
1
1
• Agrees with the  -fragment of the
second-order logic
• Compactness
• Downwards Löwenheim-Skolem
• Not recursively axiomatisable
• Expresses NP-complete properties on
finite models
17
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 selfcommunion, and secondly,
because I am convinced
that it is the best system for
the purpose”
(MS 619:8, 1909)
18
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): ”Peirce’s Magic Lantern I: Moving Pictures of Thought”,
Transactions of the C.S. Peirce Society
19
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
20
Alpha part
• Sheet of Assertion (SA, universe of
discourse)

• Cuts (negations)
SA
• Juxtaposition (conjunction)

SA
T

SA
 


  : (   )
SA
21
Alpha part
• Conditional (“the scroll”)


   : (   )    
22
Beta part
• Rhemas (predicate
terms)
• Lines of identities (LI,
existence, identity,
predication,
subsumption)
thunder
A man eats a man
phoenix
phoenix
A phoenix doesn’t
exist
Something exists
that is not phoenix
lightning
If it thunders, it lightens
23
Beta part
• Another example, coreference:
man
walks in
park
whistles
A man walks in the park. He whistles.
24
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
S1
S2
x( S1x  S2 x)
?
x S1x  S2 x
?
25
Beta part
• Different readings of “is” not logically
different:
– Existence
– Identity
– Predication
– Subsumption
– Coreference
Socrates exists
L. Carroll is C. Dodgson
Socrates is mortal
Man is an animal
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
26
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
27
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,…
28
Gamma part
You can lead a horse to water, but you can’t make him drink
xyz [( Px  Hy  Wz )  (Lxyz  Dxyz )]
29
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): “Peirce’s Game-theoretic Ideas in Logic”, Semiotica 144, 33-47
30
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
31
Proofs in EGs
• Double cut insertion/deletion
… 


…
 
• Graph insertion: any graph may be added
to an odd-polarity area
…
2k  1 1
…

…

…
2k  1 1
32
Proofs in EGs
• Graph erasure: any graph may be erased
from an even-polarity area

…
2k
…
1

…
2k
…
1
• Iteration/deiteration: any copy of a
subgraph may be added/erased to/from
the same or deeper areas than it
…

…


iteration
deiteration
… 

…
33
Heterogeneous reasoning systems
• A hundred years later…
– Barwise & Etchemendy’s Hyperproof
– John Sowa’s Conceptual Graphs
– Semantic networks
– Hans Kamp’s 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.
34
Hyperproof
• Given information: a
blocks world (toy
model, situation) +
FOL sentences
• Determine what
characteristics hold
of it
35
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
36
Conceptual Graphs
• An open-ended enterprise:
– Formal concept analysis
– Natural-language processing
– Software specification
– Information extraction
– CGWorld
– Prolog+CG (integrates Prolog, CGs, OOP and
JAVA)
37
Semantic networks
•
•
•
•
•
Concepts, relationships
Boxes, arrows, labels
Database queries, inferences
Non-monotonicity
ER graphs, Dataflows, Petri nets, Neural
nets,…
• A very heterogeneous field!
38
Discourse-Representation Theory
• Hans Kamp (1981), Lauri Karttunen (1976)
x,y
Man( x)
Park( y)
WalksIn( ,x y)
A man walks in the park. He
whistles.
z
x z
whistles( z)
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.
39
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): ”Peirce’s Diagrammatic Logic in IF Perspective”, LNAI 2980, 97-111
40
IF EGs
• IF extension of EGs expressive enough so
as to capture much of our mathematics
• IF EGs model good deal of naturallanguage 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 Peirce’s Logic of Existential Graphs”
41
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): “Peirce’s Magic Lantern II: Topology, Graphs and Games”
42
Conclusions
Peirce envisaged some 3D extension:
• “Three dimensions are necessary and sufficient
for the expression of all assertions; so that, if
man’s 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. 43
Conclusions

IF EGs fulfill Peirce’s 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)
44
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
45
The way ahead…
• A semantic web using diagrammatic
representational systems?
• Pragmatics through games
– Abstract vs. strategic meaning
– Speaker’s 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
46
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
47
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/
48