Artificial Intelligence - TU Clausthal June On Consistency Handling Inconsistency Paraconsistent Logic - Many-Valued Logic

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1 1 Artificial Intelligence - TU Clausthal June 2013 On Consistency Handling Inconsistency Paraconsistent Logic - Many-Valued Logic Jørgen Villadsen Associate Professor in Algorithms, Logic and Graphs Section Department of Applied Mathematics and Computer Science Technical University of Denmark algolog.compute.dtu.dk

2 2 Learning Outcomes After the lectures the students can Explain the logical concepts related to consistency and paraconsistency 2. Use a many-valued logic for inconsistency handling in simple situations 3. Provide a brief overview of approaches to inconsistency handling in AI Please feel free to interrupt with questions and/or comments Connections to the course slides will be made Keep an open mind for new ideas The slides will be available

3 3 Table of Contents Introduction - Learning Outcomes / Erasmus Mobility / About Me :-) Part 1 - Classical & Non-Classical Logics - Classical Propositional Logic = Sentential Logic - Explosive Logics - On Consistency & Inconsistency - Gödel s Incompleteness Theorems - Running Example - Paraconsistency - Many-Valued Logics Part 2...

4 4 Jørgen Villadsen A Short Bio Technical University of Denmark (DTU) & Roskilde University (RUC) Born 1965 in Copenhagen MSc & PhD DTU Computer Science DTU s main campus is located about 15 km north of Copenhagen center: World-conquering urban quality of life requires the trickiest of balancing acts between progress and preservation, stimulation and security, global and local. Perfection is unobtainable but Copenhagen is striking one of the best deals right now. monocle.com/film/affairs/most-liveable-city-copenhagen/ Prolog Development Center (1999) Danish Defence Research Establishment ( ) Baltica Insurance Company ( ) Centre for Language Technology ( )

5 5 Jørgen Villadsen Selected Publications Combinators for Paraconsistent Attitudes Springer Lecture Notes in Computer Science 2001 Supra-Logic: Using Transfinite Type Theory with Type Variables for Paraconsistency Journal of Applied Non-Classical Logics 2005 Natural Language Processing Using Lexical and Logical Combinators Springer Lecture Notes in Computer Science 2006 Building Multi-Agent Systems Using Jason Annals of Mathematics and Artificial Intelligence 2010 Reimplementing a Multi-Agent System in Python Springer Lecture Notes in Computer Science 2013 A Comparison of Organization-Centered and Agent-Centered Multi-Agent Systems Artificial Intelligence Research 2013 Research Areas: AI & Multi-Agent Systems / Logic & Type Theory / Natural Language Processing

6 6 Classical & Non-Classical Logics Bivalence means that a declarative sentence is either true or false: Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The intended semantics of classical logic is bivalent. en.wikipedia.org/wiki/classical logic Intuitionistic logic rejects double negation elimination ( A A) Modal logic considers modalities of truth, like possibility ( A) Temporal logic - reasoning about the past and the future Dynamic logic - reasoning about change - Hoare logic φ{p}ψ as φ [P]ψ Fuzzy logic - reasoning about vagueness

7 7 Classical Propositional Logic Recall course slide 353 (chapter on knowledge engineering): We are using logics to describe the world and how the world behaves. Example: If Joe studies then Joe gets good grades Joe studies So Joe gets good grades Basic propositions: S for Joe studies and G for Joe gets good grades (L) Main operators on propositions: (from low to high priority) Corresponding valid formula: (S G) S G Is ( A A) A a valid formula? A is an arbitrary formula here

8 8 Christopher Clavius ( ) - German Mathematician & Astronomer The valid formula ( A A) A is the so-called admirable consequence (consequentia mirabilis) or Clavius Law: It states that if a proposition is a consequence of its negation, then it is true, for consistency. en.wikipedia.org/wiki/clavius Law But the validity must be established... Clavius was instrumental in the development of the now standard Gregorian calendar and wrote highly-acclaimed and well-received textbooks The former Julian calendar had no leap years

9 9 Truth Values & Validity Recall course slide 358: The process of mapping a set of L-formulas into {true, false} is called semantics. Use for true and for false The formula has the fixed truth value The formula has the fixed truth value The symbol is here omitted from classical propositional logic ( A A) A A formula is valid if it has the truth value for all possible valuations

10 10 Associativity & Equivalence Valid formula expressing transitivity: (A B) (B C) (A C) All operators associative to the right: (A B) (B C) A C Exercise: Show the validity of the equivalence A B C A B C Same formula (not equivalences): (A B) (B C) A C (no extra parentheses) (A B) (B C) (A C) (some extra parentheses) (A B) ((B C) (A C)) (all extra parentheses) Is A A B a valid formula? Of course A and B are arbitrary formulas here too

11 11 Explosive Logics Recall course slide 363 (ex falso quodlibet) or the so-called principle of explosion: From a contradiction, anything follows. en.wikipedia.org/wiki/principle of explosion A A B or equivalently A A B gives an explosive logic A A B Intuitionistic logic - a fragment of classical logic - is explosive too

12 12 More on Explosive Logics Consider again A A B Note that if F then F B And F can be just, a contradiction A A or a very complex formula! Claim: Difficult to avoid inconsistency, in particular in large knowledge bases A preview of the running example (motivation to follow):... S G... S E... G E... G... Just consider the valid equivalence S G G S (contraposition)

13 13 Hilbert-Type Calculus Recall course slides : Calculus - Soundness & Completeness Elegant calculus for classical propositional logic due to Lukasiewicz Axioms: ( A A) A (Clavius Law) A A B (Principle of Explosion) (A B) (B C) A C (Transitivity of Implication) Rule: From A B and A infer B (Modus Ponens) Exercise: Define other operators from and

14 14 Overview Propositional Logic - Syntax - Semantics - Truth Tables A formula is valid if it has the truth value for all possible valuations A formula is unsatisfiable if it has the truth value for all possible valuations Hilbert-Type Calculus (Axiomatics) Gentzen-Type Calculus (Natural Deduction & Sequent Calculus) Resolution (Unsatisfiablity / Clauses) Recall course slides Tableau Methods (Unsatisfiability) Fact: Classical Propositional Logic Is Explosive

15 15 On Consistency Recall course slides : Hoare logic for program verification The theory of Z is undecidable. The theory of Z builds on classical propositional logic It is easy to prove that propositional logic is decidable and consistent: cannot be proved The theory of Z cannot prove that the theory of Z is consistent Unless it is inconsistent in which case it is useless and decidable This follows from Gödel s second incompleteness theorem

16 16 David Hilbert ( ) - German Mathematician In 1921 Hilbert put forward a new proposal for the foundations of mathematics which has come to be known as Hilbert s Program: A consistency proof is a mathematical proof that a particular theory is consistent. The early development of mathematical proof theory was driven by the desire to provide finitary consistency proofs for all of mathematics as part of Hilbert s program. Hilbert s program was strongly impacted by incompleteness theorems, which showed that sufficiently strong proof theories cannot prove their own consistency (provided that they are in fact consistent). en.wikipedia.org/wiki/consistency Such a theory cannot prove the consistency of a stronger theory either

17 17 Gödel s Incompleteness Theorems (1931) Perhaps the most famous result in logic: Gödel s first incompleteness theorem shows that any consistent effective formal system that includes enough of the theory of the natural numbers is incomplete: there are true statements expressible in its language that are unprovable within the system. en.wikipedia.org/wiki/gödel s incompleteness theorems So what? Truth is not absolute Gödel s second incompleteness theorem explained in words of one syllable George Boolos, Mind 103: Warning: Gödel s Completeness Theorem (1930) - Is about a calculus instead

18 18 George Boolos Quote 1/3 First of all, when I say proved, what I will mean is proved with the aid of the whole of math. Now then: two plus two is four, as you well know. And, of course, it can be proved that two plus two is four (proved, that is, with the aid of the whole of math, as I said, though in the case of two plus two, of course we do not need the whole of math to prove that it is four). And, as may not be quite so clear, it can be proved that it can be proved that two plus two is four, as well. And it can be proved that it can be proved that it can be proved that two plus two is four. And so on. In fact, if a claim can be proved, then it can be proved that the claim can be proved. And that too can be proved.

19 19 George Boolos Quote 2/3 Now: two plus two is not five. And it can be proved that two plus two is not five. And it can be proved that it can be proved that two plus two is not five, and so on. Thus: it can be proved that two plus two is not five. Can it be proved as well that two plus two is five? It would be a real blow to math, to say the least, if it could. If it could be proved that two plus two is five, then it could be proved that five is not five, and then there would be no claim that could not be proved, and math would be a lot of bunk. Bunk: Nonsense

20 20 George Boolos Quote 3/3 So, we now want to ask, can it be proved that it can t be proved that two plus two is five? Here s the shock: no, it can t. Or to hedge a bit: if it can be proved that it can t be proved that two plus two is five, then it can be proved as well that two plus two is five, and math is a lot of bunk. Again: The theory of Z cannot prove that the theory of Z is consistent Unless it is inconsistent Follows by taking the theory of Z to be the whole of math (it is too weak in general but sufficiently strong for the incompleteness theorems to hold) And a fragment of the theory of Z or a stronger theory of Z will not do either So the assumption that the theory of Z is consistent cannot be avoided

21 21 On Consistency Continued Hoare logic assumes that the theory of Z is consistent Fact: Foundation of Mathematics Need Consistency Assumption Further consistency assumptions can be avoid in case of program verification in practice one or more so-called mathematical axioms are added to either first-order logic (FOL) or higher-order logic (HOL) to obtain a theory of N, Z or R The real numbers are needed in engineering After the addition of the mathematical axioms one only allows definitions (which cannot lead to inconsistency) But knowledge engineering is different from program verification!

22 22 Summary So Far Inconsistency handling is mandatory when classical propositional logic is used due to the explosion But there is no problem in case of program verification / system development at least not when only definitions and other conservative extensions are allowed and the mathematical axioms are assumed consistent The theorem provers Isabelle and ACL2 adopt this methodology by providing a language for conservative extensions by definition. en.wikipedia.org/wiki/conservative extension This explains why (in-)consistency is not much discussed is computer science except in logic and AI Back to knowledge engineering...

23 23 Running Example Classical Propositional Logic 1/5 Remember the aim of knowledge engineering: We are using logics to describe the world and how the world behaves. #123 Ann believes that if Joe studies then he gets good grades #456 Ann believes that if Joe does not study then he enjoys college #789 Ann believes that if Joe does not get good grades then he does not enjoy college... S G... S E... G E... S for Joe studies, G for Joe gets good grades and E for Joe enjoys college

24 24 Running Example Classical Propositional Logic 2/5 It is a tiny instance of a multi-agent system (MAS) en.wikipedia.org/wiki/multi-agent system Agents Ann, Bob, Joe,... and their knowledge and beliefs The numbers indicate that the beliefs are revealed over time But this is not important here It is also not important here how the observer obtains Ann s beliefs Perhaps Ann utters her beliefs and the observer thinks that Ann is honest Or the observer infers the beliefs from Ann s actions Agents can be (groups of) humans, robots or computer systems / programs

25 25 Running Example Classical Propositional Logic 3/5 Now consider the following additional information: #999 Ann believes that Joe does not get good grades At this point, Ann s beliefs are inconsistent, but maybe Ann is not aware Even if the observer always keeps its knowledge consistent (this is a strong assumption) it is unclear how to revise Ann s beliefs and avoid the explosion Ussually agents have no direct access to other agents mental states and beliefs can also be nested: Ann believes that Bob believes that... Eventually the observer might be able to revise Ann s beliefs, but in the meantime the inconsistency is going to stay en.wikipedia.org/wiki/belief revision

26 26 Running Example Classical Propositional Logic 4/5 ((((S G ) ( S E )) ( G E )) G )...???????? EXPLOSION

27 27 Running Example Classical Propositional Logic 5/5 #1000 Ann believes that Bob gets good grades #1001 Ann believes that Bob does not get good grades #1002 Ann believes that... EXPLOSION

28 28 Paraconsistency A non-explosive logic is called a paraconsistent logic Inconsistency-tolerant logics have been discussed since at least 1910 (and arguably much earlier, for example in the writings of Aristotle); however, the term paraconsistent ( beside the consistent ) was not coined until 1976, by the Peruvian philosopher Francisco Miró Quesada. en.wikipedia.org/wiki/paraconsistency 5th World Congress on Paraconsistency, Kolkata, India, February st-4th WCP: Belgium 1997 / Brazil 2000 / France 2003 / Australia Handbook of Paraconsistency College Publications 2007

29 29 Many-Valued Logics A particular many-valued logic will be presented In a many-valued logic there are more than two truth values Not all paraconsistent logics are many-valued and not all many-valued logics are paraconsistent en.wikipedia.org/wiki/many-valued logic A Treatise on Many-Valued Logics Siegfried Gottwald Research Studies Press logik/gottwald/treatise.pdf Note that a many-valued logic is not a goal in itself (classical logic is simpler)

30 30 Summary So Far Table of Contents Continued Consistency is mandatory in case of classical logic but it is not easy to ensure Paraconsistency is worth exploring and a many-valued logic is to be presented Part 2 - A Paraconsistent Propositional Logic Online References - Running Example Paraconsistency: The Basic Idea - Definitions - Key Valid Formulas - Truth Tables - Running Example - Summary / Discussion

31 31 A Paraconsistent Propositional Logic Online References Paraconsistent Computational Logic Andreas Schmidt Jensen & Jørgen Villadsen Extended Abstract in Proceedings of 8th Scandinavian Logic Symposium August 2012, Roskilde University, Denmark scandinavianlogic.weebly.com A Paraconsistent Higher Order Logic Jørgen Villadsen 2002 arxiv.org/abs/cs.lo/ www2.compute.dtu.dk/ jovi/poster/

32 32 Running Example Paraconsistency: The Basic Idea We use new truth values in order to handle inconsistent basic propositions Then there are many counter-examples showing that G does not follow from the beliefs expressed in #123, #456 and #789 A few counter-examples: (((S G ) ( S E )) ( G E )) G This is fine; after all we are proposing a weaker logic

33 33 Definitions 1/3 = {, }, the determinate truth values for truth and falsity, respectively = {,,,...}, a countably infinite set of indeterminate truth values The only designated truth value yields the valid formulas None of the indeterminate truth values imply the others and there is no specific ordering of the indeterminate truth values = {} would turn the paraconsistent logic into classical logic

34 34 Definitions 2/3 Key valid formulas motivate the semantic clauses (in addition to [[ ]] = ): [[ ϕ]] = [[ϕ ψ]] = if [[ϕ]] = if [[ϕ]] = [[ϕ]] otherwise [[ϕ]] if [[ϕ]] = [[ψ]] ϕ ϕ ϕ [[ψ]] if [[ϕ]] = ψ ψ [[ϕ]] if [[ψ]] = ϕ ϕ otherwise Abbreviations: ϕ ψ ( ϕ ψ)

35 35 Definitions 3/3 [[ϕ ψ]] = { if [[ϕ]] = [[ψ]] otherwise [[ϕ ψ]] = if [[ϕ]] = [[ψ]] ϕ ϕ [[ψ]] if [[ϕ]] = ψ ψ [[ϕ]] if [[ψ]] = ϕ ϕ [[ ψ]] if [[ϕ]] = ψ ψ [[ ϕ]] if [[ψ]] = ϕ ϕ otherwise Abbreviations: ϕ ψ ϕ ϕ ψ ϕ ψ ϕ ϕ ψ ϕ ϕ ϕ ϕ

36 36 Truth Tables 1/2 Although we have a countably infinite set of truth value we can investigate the logic by truth tables since the indeterminate truth values are not ordered with respect to truth content We consider here 4 truth values as this will allow indeterminate truth values to interact

37 37 Truth Tables 2/2

38 38 Running Example Paraconsistent Propositional Logic 1/4 We now take #999 into account in addition to #123, #456 and #789 We find a counter-example in order to show that a formula is not valid Any of first three previous counter-examples shows that G does not follow: ((((S G ) ( S E )) ( G E )) G ) G

39 39 Running Example Paraconsistent Propositional Logic 2/4 But we have that G follows since the following formula is valid: ((((S G ) ( S E )) ( G E )) G ) G We omit the truth table here and the result is not surprising due to the usual reflexivity and monotonicity of implication or entailment in general By the way, any of the three counter-examples just given also shows that neither B nor B follows B means that Bob gets good grades This is indeed what we want

40 40 Running Example Paraconsistent Propositional Logic 3/4 We obtain some further results The very first counter-example also shows that S does not follow: ((((S G ) ( S E )) ( G E )) G ) S Another counter-example shows that S does not follow either: ((((S G ) ( S E )) ( G E )) G ) S

41 41 Running Example Paraconsistent Propositional Logic 4/4 The third counter-example also shows that E does not follow: ((((S G ) ( S E )) ( G E )) G ) E Again another counter-example shows that E does not follow either: ((((S G ) ( S E )) ( G E )) G ) E Actually this counter-example can be used as well to show that E does not follow (instead of the previous one)

42 42 Summary / Discussion An introduction only: Consistency Inconsistency Paraconsistency The discussed proposal may not solve all issues since it is a quite weak logic Further topics - Some alternative formalizations of the running example - A case study in the domain of medicine - Other paraconsistent logics