1. 74.419 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

2. 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

3. 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"

4. 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

5. 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

6. Alethic Modality Alethic modality
Something is necessarily true, or possibly true.
Operators:
“necessary”  and “possible” 
Axioms:
e.g. 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.”

7. 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”, ...

8. 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.”

9. Epistemic Modality Epistemic modality
Reasoning about knowledge (and beliefs)
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.

10. Epistemic Modality - Axioms
Most Common Axioms (Nilsson):
Modus Ponens Knowledge
[KnowsA(P)  KnowsA(P  Q) ]  KnowsA(Q)
Distribution Axiom
KnowsA(P  Q)  [KnowsA(P)  KnowsA(Q) ]
Knowledge Axiom
KnowsA(P)  P
Positive-Introspection Axiom
KnowsA(P)  (KnowsA(KnowsA(P))
Negative Introspection Axiom
¬ KnowsA(P)  (KnowsA(¬ KnowsA(P))

11. 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)

12. 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)

13. 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"

14. 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.

15. 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.

16. 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)]

17. 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)

18. 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))

19. Time Interval Relations (Allen)
Allen’s Time Interval Logic: Relations (after Nilsson)

20. Additional References
R. A. Frost: Introduction to Knowledge-Based Systems. Collins, London, 1986.
Nils J. Nilsson: Artificial Intelligence – A New Synthesis. Morgan Kaufmann, San Francisco, 1998.