Transcript Modal Logic Semantics

74.419 Artificial Intelligence
Modal Logic
see reference last slide
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 □ .
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}
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.
 ...
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 ;
0 otherwise.
Note: Often, the operator ◊ is defined in terms of □:

◊  □
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 ◊  □ !
Semantics of Modal Logic (□ and ◊)
Other logical operators are interpreted as usual, e.g.
w (□)=1 iff w (□)=0
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.
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.
 ...
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.)
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.
 |= 
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.
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 reprints)
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, 2nd Floor,
Call Number / Volume: BC 199 M6 H85 1996