1. 74.419 Artificial Intelligence Modal Logic
see reference last slide

2. Syntax of Modal Logic (□ and ◊)
Formulae in (propositional) Modal Logic ML:
The Language of ML contains the Language of Propositional Calculus, i.e. if P is a formula in Propositional Calculus, then P is a formula in ML.
If  and  are formulae in ML, then
, , , , □, ◊ *
are also formulae in ML.
* Note: The operator ◊ is often later introduced and defined through □ .

3. Semantics of Modal Logic (□ and ◊)
The semantics of a modal logic ML is defined through:
a set of worlds W = {w1, w2, ..., wn},
an accessibility relation RWW, and
an interpretation function : {0,1}

4. Semantics of Modal Logic ( and )
The interpretation in ML of a formula P, Q, ... of the propositional language of ML corresponds to its truth value in the "current world":
w (P)=1 iff I(P) is true in w.
w (PQ)=1 iff I(PQ) is true in w.
...

5. Semantics of Modal Logic (□ and ◊)
We extend the semantics with an interpretation of the operators □ and ◊, specified relative to a "current world" w.
For all wW:
w (□)=1 iff w': (w,w')R  w' ()=1 ;
0 otherwise.
w (◊)=1 iff w': (w,w')R  w' ()=1 ;
Note: Often, the operator ◊ is defined in terms of □:
◊  □

6. Semantics of Modal Logic (□ and ◊)
We can also prove the equivalence of □ and ◊ for our definitions above:
w (□)=1 iff (w (□)=1) (or w (□)=0)
iff w': (w,w')R  w' ()=1
iff w': (w,w')R  w' ()=0
iff w': (w,w')R  w' ()=1
iff w (◊)=1
This means: □  ◊
Exercise: Proof ◊  □ !

7. Semantics of Modal Logic (□ and ◊)
Other logical operators are interpreted as usual, e.g.
w (□)=1 iff w (□)=0

8. Semantics of ML - Complex Formulas
The interpretation of a complex formula of ML is based on the interpretation of the atomic propositional symbols, and then composed using the interpretation function  defined above, e.g.
w (□)=1 iff (w': (w,w')R  w' ()=1)
iff w': (w,w')R  w' ()=0
Let's say   (PQ).
w': (w,w')R  w' (PQ)=0
w': (w,w')R  (w' (P)=0  w' (Q)=0)
"P or Q" is not necessarily true in world w, if there is a world w', accessible from w, in which P is false or Q is false.

9. Semantics of Modal Logic - Grounding
The interpretation in ML of a formula P, Q, ... of the propositional language of ML corresponds to its truth value in the "current world":
w (P)=1 iff I(P) is true in w.
w (PQ)=1 iff I(PQ) is true in w.
...

10. Semantics of Modal Logic
A formula  is satisfied in a world w of a Model M=<W,R,>, if it is true in this world wW under the given interpretation , i.e. w ()=1.
M, w |= 
A formula  is true in a Model M=<W,R,>, if it is satisfied in all worlds wW of M.
M |= 
A formula  is valid, if it is true in all Models.
|= 
A formula  is satisfiable, if it is satisfied in at least one world wW of one Model M=<W,R,>. (or: If its negation is not valid.)

11. Semantics of Modal Logic
A formula  is satisfied in a world w of a Model M=<W,R,>, if it is true in this world under the given interpretation , i.e. w ()=1.
M, w |= 
A formula  is true in a Model M=<W,R,>, if it is satisfied in all worlds wW of M.
M |= 
A formula  is valid, if it is true in all Models.
 |= 
A formula  is satisfiable, if it is satisfied in at least one world wW of one Model M=<W,R,>. (or: If its negation is not valid.)
A formula  is a consequence of a set of formulas  in M=<W,r,>, if in all worlds wW, in which  is satisfied,  is also satisfied.
 |= 

12. Semantics of Modal Logic: Terminology
Sometimes the term "frame" is used to refer to worlds and their connection through the accessibility relation:
A frame <W, R> is a pair consisting of a non-empty set W (of worlds) and a binary relation R on W.
A model <F, > consists of a frame F, and a valuation  that assigns truth values to each atomic sentence at each world in W.

13. 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.

14. 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.

15. 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, 2nd Floor, Call Number / Volume: BC 199 M6 H