DYNAMIC LOGIC WITH POSSIBLE WORLD 
Ruzhan Lu 
I)epartment of Computer Science and Engineering 
Shanghai Jiao Tong University, Shanghai 200030, I'. R. China 
Abstract 
This paper introduces a semantic theo- 
ry I)I,PW, l)ynamic l,ogic with Possible 
World, which extends Groenendijk's I)PI, 
and Cresswell's Indices Semantics. The 
semantics can interpret the temporal and 
modal sense and anaphora. 
Key words: Semantic Model, Dynamic 
l,ogic, Possible Worlds. 
§ 1 Introduction 
At present there are three main aspects 
in semantical field: 
1. Transformation of' sentences or dis- 
courses into formulas in high order logic. 
2. Semantic interpretation of the logical 
formulas. 
3. Semantic ambiguity. 
This paper presents the semantics, 
dynamic logic with possible world, which 
combines DPIJ 3~ with Indices semantics r'~ 
and extends the theories, the theory can be 
used to interpret the temporal and modal 
sense and anaphoric connection. For the 
limitation of space we only give the defi- 
nitions and examples concerned rather than 
present a formalization which should inc- 
lude the axioms and rules concerned like u 
I)pL TM 
Following Montague semantics, the dis- 
cussion on meaning of a sentence started 
with predicate formula, high order logical 
formual with lambda terms, which is trans- 
lated from a sentence S by means of a set of 
rules and reduced to a frst order predicate 
formula A finally. 
Problem= given an expression A, meani- 
ngEA~=? Assume that a model M is an 
ordered pair (D, F) where D is a domain, a 
non-empty set and F an interpretation 
function assigning a semantic value to each 
non-logical constant of the language. A 
value assignment g is a function assigning a 
member of D to each variable of the lan- 
guage. W is a possible world. The inten- 
sion of the expression A is Int\['A~----- 
IIAII M*. The extension of the expression A is 
Ext EA, W\]=Int EA\](W)= IIAII ..... . 
Some semantic evaluations are as fol- 
lows : 
I. Montague Semantics, given M, W, 
evaluates extension IIA IIM'W'~. 
2. Possible World Semantics (M. J. Cre- 
sswell), given M, W, finds the set of 
possible worlds which satisfy the extension 
of h. i.e., {Wl\[lAl\[M'W~-=True}. 
3. I)ynamic Predicate I,ogic (J. Groenendi- 
jk), given M, W, finds dynamic changes 
between the value assignments: 
\]lAll~'W= {(g~.,, go,,tS}. 
4. Dynamic Logic with Possible World, 
given M, finds dynamic changes between 
the ordered pairs of value assignment and 
world: IIAII u=: r t(G~,,, G,,~}lG,~=(gi~, W~,,}, 
(L,,= <g .... w,,o0}. 
The evaluations 1, 2 above are in static 
sense and for use of interpretation of 
sentences without anaphora, while evalu- 
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ations 3, 4 above in dynamic sense and for 
use of interpretation of anaphoric connec- 
tion. 
§ 2 DLPW, Anaphoric Connection 
In this section we hope that DLPW can 
offer a successful application to anaphoric 
connection like the intepretation of the 
famous donkey sentence in Groenendijk's 
DPL TM. The sentence 
(1) There was a key, it is lost. 
can be formalized as 
(2) P-~key(x) & NOW lost(x) (in the 
hope that) having the same semantic inter- 
pretation as the formula 
(3) PB~(key(x) & NOW lost(x)). 
The idea will come true by means of 
dynamic logic DLPW. A state, denoted by n 
(=<a, g)), may be a pair of a sequence of 
time a(a=-{a(O), a(1)}) and a value assign- 
ment g. 
There are revisions of semantic de- 
finitons of operators concerned. 
(4) IINOWall--{(n~, n2)ln~=(~r,, g,), 
n2=({~, gD, <(a,\[l/0\], g,), ~Dell~ll}. 
(5) IIP~II ={<~,, ~)ln,=<~,, g,), 
n2=(a~, g~), ~,(t<a,(0), 
<(a,\[term t/0\], g,), n2)ellall )}. 
(6) II{o&q~ll={<~,, n~)l~...,(n=<~, g), 
n'-----(a', g>, (n~, n>eLl{oll, 
<~', ~Dell~ll)}. 
(7) It~{o11={<~1, ~)1~=, 
(n=<a, g), g=g,\[xl, <n, nDell{011)}. 
(8) II{0(x)ll={<~,, ~DI~,=~, 
g~= g2, g,(x)~F(ep)}. 
Where F(~0) is a set of individuals, F is 
the function in a model M. {p is an atomic 
formula. Here obviously, conjunction is 
treated like composition ( i.e., compound 
statement {S~, S~}). Intuitively, the con- 
junction is treated in the sequential sense. 
The meanings of the formulas a,\[l/0\], 
¢l\[term t/0\], g,\[x\] follow the statements in 
the preceding section. 
Assume that initially for all i, j, 
ai(j)~t0 and a~, gi in n~ denoted components 
concerned. Hence 
(9) <n,, nD~il(2)li iff, by (6), for some 
n3, n3', g.~(=gs'), 
(lO) (n~, n.OellP_~key(x)ll and 
(11) <ns', n2)~liNOW lost(x)ii. 
(10) holds iff, by (5), for some 
t(<al(0)), 
(12) ((al\[termt/0J, g,),n~)~ 
\]\] ~xkey(x)iP iff, by (7), for some h(=g,\[x\]), 
(13) ((a,\[term t/0\], gl\[x\]), 7~)e 
tlkey(x) H iff, by (8), 
(14) ns=<a,\[termt/O\], g,\[x\]) (i.e. a.~ 
-=~r~\[term t/0\], gs=gl\[x\]), and 
g,\[x\](x)eF(key), where g,\[x\](x)=h(x). 
(11) holds, iff, by (4), 
(15) <<~3'\[1/0\], g3'), nDeljlost(x)lt, iff 
by (8), n2=as'\[1/0\], g,~') (i.e., a2=(r3'\[l/0\] 
and g2=g.~'=g.~, (by (6))=g,\[x\] (by (14)), 
and gs(x)eF(lost). 
It means that g.~(x)(=gl\[x\](x)=-indivi- 
dual kn, say) is a key at t(<t0) and gs(x) 
(= k0) is lost at t~. 
On the other hand, for the formula (3), 
(16) (n,, n.0ei\](3)ll iff, by (5), for some 
t(<a,(0)), 
(17) (<(rl\[term t/0\], gl), n2) 
ll~(key(x) & NOW lost(x))H, iff, by (6) 
for some h(:=gl\[x\]), 
(18) ((a,\[term t/0J, g,\[x\]), n2)e 
Jlkey(x) & NOW lost(x)H, iff, by (6) for 
some n3, n.~', g3(---g.~'), 
(19) (@,\[term t/0\], g,\[x\]), us)e 
IIkey(x) el and 
(20) <n.~', nDc\]tNOW lost(x)ll, 
(19) holds iff n.~=<a,\[termt/0\], 
g~\[x\]), and g,\[x\](x)cF(key), i.e., 
~---- a, \[term t/0~ and g3 =- g, \[x\], 
(20) holds iff, by (4), 
(2t) <<as'\[l/0\], g.~'), nD~Hlost(x)\]\[ iff, 
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by (8), 7r2=<m~'\[1/0\], g3'>, and g.,'(x)c 
F(lost), i.e., ~2=~s'\[l/0J and g2=ga '= 
g3 = g, \[x\]. 
It shows formulas (2) and (3) have 
identical meaning. This is just what we 
require. 
Acknowledgement 
The research work is supported in part 
by the National Natural Science Founda- 
tion and The National Key Lab. of Com- 
puter Software New Technology in Nanking 
University. 
Re~rence 
\[1\] Cresswcll. M. J. Entities and Indices, Kluwer 
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\[2\] Dowry. D. R. Introduction to Montague Seman- 
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\[3\] Grocnendijk. J. S. Dynamic Predicate Logic, 
Linguistic and Philosophy, Vol.14. No.l, 1991. 
\[4\] Ruzhan Lu. About Semantic Theory and Under- 
standing Natural Language, TU Berlin, Tcch. 
Iceport. 1992. 
\[5\] Ruzhan Lu. Dynamic Semantic with Indices, TU 
Berlin, Tech. Report. 1992. 
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