Page 117 - Modern Spatiotemporal Geostatistics

98 Modern Spatiotemporal Geostatistics — Chapter 4

which relates the probability function Prob [•] of the prior stage with the prob-
ability function Prob" [•] of the posterior stage. Finally, one may notice that

which relates the probability functions used in the approach of Proposition 4.1
with the probability functions of this example. D

Conditional Probability
of a Spatiotemporal Map and its
Relation to the Probability of Conditionals

The concept of "cause and effect" is of paramount importance in scientific
investigations. Even if the indetermination of modern physics may show that
not every natural process has a cause (certainly not a deterministic one), most
phenomena can, indeed, have causes. Certainly there exist various sorts of
causation, including deterministic causation in which the causes are necessary
and sufficient for their effects (see, e.g., Mellor, 1995) and also probabilistic
or stochastic causation, which includes causes that raise the chances of their
effects.
The sufficiency and necessity of causes and their effects is usually expressed
by conditionals. Within the framework of modern geostatistics we can distin-
guish between truth-functional and non-truth-functional conditionals related
to Spatiotemporal maps. Truth-functional map conditionals include material
and strict conditionals. The most important among the non-truth-functional
conditionals are physical and logical map conditionals. We will now discuss
these concepts in more detail.

Material and strict map conditionals
From the mathematical logic viewpoint, the material conditional of a spa-
tiotemporal map is a structure of the form "if Xdata(S} occurs, then Xk oc "
curs," or "Xdat a(S) implies x fc,"for short, Xdata(S) ~> Xk Material map
conditionals are logical structures based on purely truth-functional concepts
(conjunction A, negation -i, disjunction V, truth tables, etc.; see for example
Burris, 1998), i.e., they express a truth-functional relation. A material map
conditional is equivalent to the statement "it is not the case thatXdataC-^) anc '
not Xk •" in short, ->(Xdata(^} A "'X*)- Indeed, this is how the word "implies"