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Independence-Friendly Existential Graphs

Ahti-Veikko PietarinenDepartment of Philosophy

University of Helsinki29 April 2004

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Outline

1. Symbolic vs. diagrammatic logic

2. Independence-friendly (IF) logic

3. Existential graphs (EG)

4. IF EGs

5. Conclusions

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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

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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.

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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

( / )x y x Sxy

1 1f x Sxf 1 1( )f x Sxf x

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Henkin quantifiers

• Henkin (1959):

[ , , , ]x y

x y z uz u

true in M iff

1 2 1 2[ , ( ), , ( )]f f x z x f x z f y

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

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Henkin quantifiers

• Krynicki normal forms (1993)

1 1

2 2

1 1 1 11 1

2 2 2 21 1

1 1

... ...

... ...

... ...m m

n n

n n k kij ij

m m m mn n

x x y y

x x y ySx y

x x y y

reduce to

11 1

1

...... ...

...n

n kk

x x ySx x yz z u

z z u

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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 Huyz u

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

Most : ( )( , )

Most : ( )

x Linguist xAdmire x z

z Logician z

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Language of IF logic

• Let

be in the scope of

Given

wffs of

•

• We may even have

, { , }, ( ) , { , }iQx L Q L

1 0 2 1 1 1 1,..., , { ,..., }n n nQ x Q x Q x A x x

, ( / )B A Qx B areIFL

( ( / ) ) ( ( / ) )

Sandu, G. & Pietarinen, A.-V. (2001): “Partiality and Games: Propositional Logic”, Logic Journal of the IGPL 9, 107-127.

( ( / ) )B and

( / ) ,x y z u x Sxyzu 1 2( ( / ) )x S x x S x

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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

0 1 0 0 1... ( ... ... ...)...ix S x S x x

1 2... ( ... )...x y S xy zS z

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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”

5 4 3 2 1 1 2 3... [ ...]v v v v v v v v

1 3 5 2 4 6 1 2 4 6 2 3 4 6 8 4 5 6 8 10.. .. [ ( ..), , ( ..), , ( ..)..]f f f v v v f v v v v f v v v v f v v v

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

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Game-theoretic semantics

• Hintikka (1973): a game between V rifier

F lsifier( ,Mys lf( ,N ture

,...)ros,...)gape

• Non-cooperative, finite, zero-sum games

• Complete but possibly imperfect information and imperfect recall

• FOL, modal logic, dynamic logic,…

( ,M)G

A.-V. Pietarinen (2004): “Some Games Logic Plays”, Logic, Thought and Action, Kluwer

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Game-theoretic semantics

1 2 then chooses {1,2}i

1 2 then chooses {1,2}i

[ ]x x then chooses (M)a dom

[ ]x x then chooses (M)a dom

then and

In IF logic strong game negation, not classical, weak contradictory negation!

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Game-theoretic semantics

• Winning conventions

• Winning strategies

0 1[ ,..., ]nS x x is a win for if M0 1( ,..., )na a S

0 1[ ,..., ]nS x x is a win for if M0 1( ,..., )na a S

is true in iff there exists a winning strategyfor in ( ,M)G

M

is false in iff there exists a winning strategyfor in ( ,M)G

M

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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

1 2 3 4( / ), ( / ), ( / ), ( / )x W x W W W

1 2ih h

( / )x y x Sxy

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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.

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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

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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 619:8, 1909)

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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

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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

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Alpha part

• Sheet of Assertion (SA, universe of discourse)

• Cuts (negations)

• Juxtaposition (conjunction)

: ( )

SA

SA

SA

T

SA

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Alpha part

• Conditional (“the scroll”)

: ( )

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Beta part

• Rhemas (predicate terms)

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

A man eats a man

A phoenix doesn’t exist

Something exists that is not phoenix

If it thunders, it lightens

lightning

thunder

phoenix

phoenix

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Beta part

• Another example, coreference:

man walks in park

whistles

A man walks in the park. He whistles.

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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

1S 2S

1 2( )x S x S x

1 2xS x S x ?

?

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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

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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

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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,…

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Gamma part

[( ) ( )]x y z Px Hy Wz Lxyz Dxyz

You can lead a horse to water, but you can’t make him drink

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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

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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

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Proofs in EGs

• Double cut insertion/deletion

… …

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

…2 1k

… …2 1k

…1 1

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Proofs in EGs

• Graph erasure: any graph may be erased from an even-polarity area

…2k

… …2k

… 1 1

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

… … … … iteration

deiteration

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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.

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Hyperproof

• Given information: a blocks world (toy model, situation) + FOL sentences

• Determine what characteristics hold

of it

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Conceptual Graphs

John Sowa (2000): Knowledge Representation: Logical, Philosophical and Computational Foundations, Brooks/Cole

A cat is on a mat

Every cat is on a mat

Tom believes that Mary wants to marry

a sailor

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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)

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Semantic networks

• Concepts, relationships

• Boxes, arrows, labels

• Database queries, inferences

• Non-monotonicity

• ER graphs, Dataflows, Petri nets, Neural nets,…

• A very heterogeneous field!

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Discourse-Representation Theory

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

( )

( )

( , )

Man x

Park y

WalksIn x y

,x y

z

( )

x z

whistles z

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.

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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

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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 Peirce’s Logic of Existential Graphs”

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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”

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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.

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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)

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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

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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

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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

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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 BergmanErkki KilpinenSami PaavolaAhti-Veikko PietarinenSami Pihlström…

http://www.helsinki.fi/science/commens/