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```74.419 Artificial
Intelligence 2004
Non-Classical Logics
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
Non-Classical Logics 2


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 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)
“If P is necessarily true, then P is also possible.”
A2)  possible(P)   P
“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
Operators: “Knows” and “Believes”
Axioms:
e.g. KnowsA(P)  P
“If agent A knows P, then P must be true.”
KnowsA(P)  BelievesA(P)
“If agent A knows P, then agent A also believes P.”
KnowsA(P)  KnowsA(P  Q)  KnowsA(Q)
Note: P and Q refer in this context to closed FOPL formulae.
Epistemic Modality - Axioms
Most Common Axioms (Nilsson):
1. Modus Ponens Knowledge
[KnowsA(P)  KnowsA(P  Q) ]  KnowsA(Q)
2. Distribution Axiom
KnowsA(P  Q)  [KnowsA(P)  KnowsA(Q) ]
3. Knowledge Axiom
KnowsA(P)  P
4. Positive-Introspection Axiom
KnowsA(P)  (KnowsA(KnowsA(P))
5. Negative Introspection Axiom
¬ KnowsA(P)  (KnowsA(¬ KnowsA(P))
Epistemic Modality - Inference
Inferential Properties of Agents:
Epistemic Necessitation:
from
|– α
infer
KnowsA(α)
Logical Omniscience:
from
α |– β and KnowsA(α)
infer
KnowsA(β)
or:
from
|– (α  β)
infer
KnowsA(α)  KnowsA(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. KnowsA(P)  KnowsA("P=Q")  KnowsA(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 KnowsA(α ).
e.g. From
KnowsA(P) and (P  Q)
not conclude KnowsA(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 Worlds - Example 2
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)]
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, 1986.
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
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)
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