1 Logic Notations Frege’s Begriffsschrift (concept writing) : assert P not P if P then Q for every x, P(x) P P Q P P(x) x

2 Logic Notations Frege’s Begriffsschrift (concept writing) : Every ball is red Some ball is red red(x) x ball(x) red(x) x ball(x)

3 Logic Notations Algebraic notation - Peirce, 1883: –Universal quantifier:  x P x –Existential quantifier:  x P x

4 Logic Notations Algebraic notation - Peirce, 1883: Every ball is red:  x (ball x —< red x ) Some ball is red:  x (ball x red x )

5 Logic Notations Peano’s and later notation: Every ball is red:  x (ball(x)  red(x)) Some ball is red:  x (ball(x)  red(x))

6 Logic Notations Existential graphs - Peirce, 1897: –Existential quantifier: a link structure of bars, called line of identity, represents  –Conjunction: the juxtaposition of two graphs represents  –Negation: an oval enclosure represents 

7 Existential Graphs If a farmer owns a donkey, then he beats it: farmerownsdonkey beats

8 Existential Graphs EG’s rules of inferences: –Erasure: in a positive context, any graph may be erased. –Insertion: in a negative context, any graph may be inserted. –Iteration: a copy of a graph may be written in the same context or any nested context. –Deiteration: any graph may be erased if a copy of its occurs in the same context or a containing context. –Double negation: two negations with nothing between them may be erased or inserted.

9 Existential Graphs Prove: ((p  r)  (q  s))  ((p q)  (r s)) is valid

10 Existential Graphs Prove: ((p  r)  (q  s))  ((p q)  (r s)) Erasure: in a positive context, any graph may be erased. Insertion: in a negative context, any graph may be inserted. Iteration: a copy of a graph may be written in the same context or any nested context. Deiteration: any graph may be erased if a copy of its occurs in the same context or a containing context. Double negation: two negations with nothing between them may be erased or inserted.

11 Existential Graphs Prove: ((p  r)  (q  s))  ((p q)  (r s)) Erasure: in a positive context, any graph may be erased. Insertion: in a negative context, any graph may be inserted. Iteration: a copy of a graph may be written in the same context or any nested context. Deiteration: any graph may be erased if a copy of its occurs in the same context or a containing context. Double negation: two negations with nothing between them may be erased or inserted. r p s q

12 Existential Graphs Prove: ((p  r)  (q  s))  ((p q)  (r s)) Erasure: in a positive context, any graph may be erased. Insertion: in a negative context, any graph may be inserted. Iteration: a copy of a graph may be written in the same context or any nested context. Deiteration: any graph may be erased if a copy of its occurs in the same context or a containing context. Double negation: two negations with nothing between them may be erased or inserted. r p s q r p s q r p

13 Existential Graphs Prove: ((p  r)  (q  s))  ((p q)  (r s)) Erasure: in a positive context, any graph may be erased. Insertion: in a negative context, any graph may be inserted. Iteration: a copy of a graph may be written in the same context or any nested context. Deiteration: any graph may be erased if a copy of its occurs in the same context or a containing context. Double negation: two negations with nothing between them may be erased or inserted. r p s q r p r p s q r p q

14 Existential Graphs Prove: ((p  r)  (q  s))  ((p q)  (r s)) Erasure: in a positive context, any graph may be erased. Insertion: in a negative context, any graph may be inserted. Iteration: a copy of a graph may be written in the same context or any nested context. Deiteration: any graph may be erased if a copy of its occurs in the same context or a containing context. Double negation: two negations with nothing between them may be erased or inserted. r p s q p q s q r r p s q r

15 Existential Graphs Prove: ((p  r)  (q  s))  ((p q)  (r s)) Erasure: in a positive context, any graph may be erased. Insertion: in a negative context, any graph may be inserted. Iteration: a copy of a graph may be written in the same context or any nested context. Deiteration: any graph may be erased if a copy of its occurs in the same context or a containing context. Double negation: two negations with nothing between them may be erased or inserted. r p s q p q s q r r p s q r s

16 Existential Graphs Prove: ((p  r)  (q  s))  ((p q)  (r s)) r p s q r p s q r p r p s q r p q r p s q s q r r p s q r s

17 Existential Graphs  -graphs: propositional logic  -graphs: first-order logic  -graphs: high-order and modal logic

18 Semantic Networks Since the late 1950s dozens of different versions of semantic networks have been proposed, with various terminologies and notations. The main ideas: –For representing knowledge in structures –The meaning of a concept comes from the ways it is connected to other concepts –Labelled nodes representing concepts are connected by labelled arcs representing relations

19 Semantic Networks Mammal Person Owen Nose RedLiverpool isa instance has-part uniform color team

20 Semantic Networks John H1H2 height Bill height greater-than 1.80 value

21 Semantic Networks Dog db isa Bite assailant Mail-carrier m isa victim The dog bit the mail carrier

22 Semantic Networks Dog db isa Bite assailant Mail-carrier m isa victim Every dog has bitten a mail-carrier g GS isaform 

23 Conceptual Graphs Sowa, J.F Conceptual Structures: Information Processing in Mind and Machine. CG = a combination of Perice’s EGs and semantic networks.

24 Conceptual Graphs 1968: term paper to Marvin Minsky at Harvard. 1970's: seriously working on CGs 1976: first paper on CGs : meeting with Norman Foo 1984: the book coming out CG homepage:

25 Simple Conceptual Graphs CAT: tunaMAT: *On 12 conceptrelationconcept type (class) relation type individual referentgeneric referent

26 Ontology Ontology: the study of "being" or existence An ontology = "A catalog of types of things that are assumed to exist in a domain of interest" (Sowa, 2000) An ontology = "The arrangement of kinds of things into types and categories with a well- defined structure" (Passin 2004)

27 Ontology top-level categories domain-specific

28 Ontology Being Substance Accident Property Relation Inherence Directedness MovementIntermediacyQuantityQuality Aristotle's categories Containment Activity Passivity HavingSituated SpatialTemporal

29 Ontology Geographical-Feature AreaLine On-LandOn-Water Road Border Mountain Terrain Block Geographical categories Railroad Country Wetland Point Power-Line River Heliport Town Dam Bridge Airstrip

30 Ontology Relation

31 Ontology ANIMAL FOOD Eat

32 Ontology ANIMAL FOOD Eat PERSON: johnCAKE: * Eat

33 CG Projection PERSON: johnPERSON: *Has-Relative 12 PERSON: johnWOMAN: maryHas-Wife 12

34 Nested Conceptual Graphs CAT: tunaMAT: * On Neg CAT: tunaMAT: * On  It is not true that cat Tuna is on a mat.

35 Nested Conceptual Graphs CAT: *MAT: * On  CAT: *  coreference link Every cat is on a mat.

36 Nested Conceptual Graphs PERSON: julianPLANET: mars Fly-To  Poss Past Julian could not fly to Mars.

37 Nested Conceptual Graphs SAILOR: * Marry Want Believe PERSON: mary PERSON: tom PERSON:* Tom believes that Mary wants to marry a sailor.

38 Exercises Reading: Sowa, J.F Knowledge Representation: Logical, Philosophical, and Computational Foundations (Section 1.1: history of logic). Way, E.C Conceptual Graphs – Past, Present, and Future. Procs. of ICCS'94.