Artificial Intelligence 2004 Non-Classical Logics

 Specific Language Constructs added to classic FOPL  Different Types of Logics  Modal Logics most popular ones, e.g.  Deontic Logic (allowed and forbidden; ethics; law)  Epistemic Logic (Knowledge) and Doxastic (Belief) Logic  Possible World Semantics

Non-Classical Logics 1  (many-) sorted logic  individuals pre-arranged in sets = sorts  many-valued logic  more than two truth values (e.g. Lukasiewicz “intermediate truth” = I; "don't know" status)  fuzzy logic  degree of truth between 0 and 1 (predicate corresponds to fuzzy set; membership in set to a certain degree, not just yes or no)  non-monotonic logic  belief modeling; defaults; set of true formulae can change (non-monotonicity); TMS

 higher-order logic quantification over predicates (as variables), like  P:..., or statements about statements, e.g. “This sentence is false.”  modal logics (see later slides) describe “subjunctive” statements in addition to assertional statements, using Modal Operators, i.e. "possible P" and "necessary P" Non-Classical Logics 2

Non-Classical Logics 3  time logics  time as temporal modality  time logic based on time points and relations between them (like “t1 earlier than t2”)  Allen’s model of time intervals  situational logic; situation calculus (McCarthy)  situation as additional parameter to predicate expressions, for describing change due to events  additional axioms describe transformations of ssituations due to actions  used for reasoning with time and planning

Modal Logics 1 Uses additional operators and axioms to describe logic. Includes FOPL assertions, and in addition statements using Modal Operators. Different Modalities express different types of statements, e.g.  alethic modality  “necessary” and “possible” as additional operators  temporal modality  with necessary  “always” and possible  “sometimes”  deontic modality  “permissible” (allowed) and “obligatory” (must)  epistemic modality “knows” and “beliefs” as operators

Alethic Modality Alethic modality Something is necessarily true, or possibly true. Operators: “necessary”  and “possible”  Axioms: A1) necessary(P)  possible(P) A2)  possible(P)   P “If P is necessarily true, then P is also possible.” “If P is not possible, then P cannot be true.”

Temporal Modality Temporal modality Something is always or sometimes true. Operators: “always”  “necessary” “sometimes”  “possible” Axioms: A1) always (P)  sometimes (P) “If P is always true, then P is sometimes true.” A2) always (  P)   sometimes (P) “If not P is always true, then P is not sometimes true.” Also for tenses like “past”, “past perfect”, “future”,...

Deontic Modality Deontic modality (ethics) Something is permitted or obligatory. Operators: “permissible” and “obligatory” Axioms: e.g.obligatory (P)  permissible (P) “If P is obligatory, then P is also permitted.”

Epistemic Modality Epistemic modality Reasoning about knowledge (and beliefs) Operators: “Knows” and “Believes” Axioms: e.g.Knows A (P)  P “If agent A knows P, then P must be true.” Knows A (P)  Believes A (P) “If agent A knows P, then agent A also believes P.” Knows A (P)  Knows A (P  Q)  Knows A (Q) Note: P and Q refer in this context to closed FOPL formulae.

Epistemic Modality - Axioms Most Common Axioms (Nilsson): 1.Modus Ponens Knowledge [Knows A (P)  Knows A (P  Q) ]  Knows A (Q) 2.Distribution Axiom Knows A (P  Q)  [Knows A (P)  Knows A (Q) ] 3.Knowledge Axiom Knows A (P)  P 4.Positive-Introspection Axiom Knows A (P)  (Knows A (Knows A (P)) 5.Negative Introspection Axiom ¬ Knows A (P)  (Knows A (¬ Knows A (P))

Epistemic Modality - Inference Inferential Properties of Agents: Epistemic Necessitation: from |– α infer Knows A (α) Logical Omniscience: from α |– β and Knows A (α) infer Knows A (β) or: from |– (α  β) infer Knows A (α)  Knows A (β)

Epistemic Modality - Problems 1 Problem: "Referential Opaqueness" Different statements refering to the same extension, cannot necessarily be substituted. Agent A knows John's phone number. John's phone number is the same as Jane's phone number. You cannot conclude that A also knows Jane's phone number. Another approach (than ML) is to use Strings instead of plain formulae to model referential opaqueness (cf. Norvig): e.g. Knows A (P)  Knows A ("P=Q")  Knows A (Q)

Epistemic Modality - Problem 2 Problem: "Non-Compositional Semantics" You cannot determine the truth status of a complex expression through composition as in standard FOPL. From A and α you cannot always determine the truth status of Knows A (α ). e.g. From Knows A (P) and (P  Q) not conclude Knows A (Q) Modal Logic uses a "Possible Worlds Semantics"

Possible World Semantics For modal and temporal logics, semantics is often based on considerations about which “worlds” (set of formulae) are compatible with or possible to reach from a certain given “world” → possible world semantics Relations between “worlds”: accessible necessary A world is accessible from a certain world, if it is one possible follow state of that world. A world is a necessary follow state of a certain world, if the formulae in that world must be true, is a necessary conclusion.

Possible World Semantics - Example 1 Possible World Semantics for Epistemic Logic If Agent A knows P, then P must be true in all worlds accessible from the current world. That means these worlds are not only accessible but necessary worlds (respective to the agent and its knowledge). If Agent A believes P, then P can be true in some accessible worlds, and false in others.

Possible World Semantics for Alethic Modality One of my colleagues, let's call him BG for 'Big Guy', wants to loose weight. When we went for coffee last time, it was my turn to buy. Since he has had a hard day, I offered to buy him a Danish, too, but added, that in the future, he would not get any Danish, if he doesn't go to the Gym, but if he does go to the Gym, he might get a Danish - or not, depending on his weight situation.  Gym (BG)   [  Danish (BG)] Gym (BG)   [Danish (BG)] Possible Worlds - Example 2

Representing Time  Time as temporal modality in modal logic  Time in FOPL add time points and time relations as predicates e.g. "earlier-than" (et) for two time points Axioms: e.g.  x,y,z: (et (x,y)  et (y,z))  et (x,z)  x,y: et (x,y)   et (y,x)

Time Interval Representation (Allen) Allen’s Time Interval Logic Time represented based on Intervals. Relations between time intervals are central : e.g. meet (i,j) for Intervals i and j Interval i Interval j Time points representable as functions on intervals, e.g. start(i) and end(i) specify time points. Axioms: e.g. meet(i,j)  time(end(i))=time(start(j))

Time Interval Relations (Allen) Allen’s Time Interval Logic: Relations (Nilsson)

References R. A. Frost: Introduction to Knowledge-Based Systems. Collins, London, Graham Priest, An Introduction to Non-Classical Logic, Cambridge University Press, 2001 Nils J. Nilsson: Artificial Intelligence – A New Synthesis. Morgan Kaufmann, San Francisco, 1998.

Textbooks on (Modal) Logic Richard A. Frost, Introduction to Knowledge-Base Systems, Collins, 1986 (out of print) Comments: one of my favourite books; contains (almost) everything you need w.r.t. foundations of classical and non-classical logic; very compact, comprehensive and relatively easy to understand. Allan Ramsay, Formal Methods in Artificial Intelligence, Cambridge University Press, 1988 Comments: easy to read and to understand; deals also with other formal methods in AI than logic; unfortunately out of print; a copy is on course reserve in the Science Library.

Textbooks on (Modal) Logic Graham Priest, An Introduction to Non-Classical Logic, Cambridge University Press, 2001 Comments: the most poplar book (at least among philosophy students) on non-classical, in particular, (propositional) modal logic. Kenneth Konyndyk, Introductory Modal Logic, University of Notre-Dame Press, 1986 (with later re-prints) Comments: relatively easy and nice to read; contains propositional as well as first-order (quantified) modal logic, and nothing else.

Textbooks on (Modal) Logic J.C. Beall & Bas C. van Fraassen, Possibilities and Paradox, University of Notre-Dame Press, 1986 (with later re-prints) Comments: contains a lot of those weird things, you knew existed but you've never encountered in reality (during your university education). G.E. Hughes & M.J. Creswell, A New Introduction to Modal Logic, Routledge, 1996 Comments: Location: Elizabeth Dafoe Library, 2 nd Floor, Call Number / Volume: BC 199 M6 H