Tooley - the nature of laws.pdf - Ted Sider

Michael Tooley
Thus one could have, for example, a construction function Ksuch that
K( redness, roundness) = the proposition that all red things are round.
Conceived in the most general way, some construction functions will
map ordered n-tuples of universal into propositions that involve as
constituents universals not contained in the original n-tuple. Thus C
could be a function so defined that C(property P) = the proposition
that everything with property Pis green. Other construction functions
will map ordered n-tuples of universals into propositions which do not
contain, as constituents, all the universals belonging to the n-tuple. H
could be a function so defined that /-/(property P, property Q) = the
proposition that everything has property Q. In order to capture the
notion of logical form, one needs a narrower notion of construction
function, namely, one in which something is a construction function if
and only if it is a mapping from ordered n-tuples of universals into
propositions that contain, as constituents, all and only those universals
belonging to the ordered n-tuple. In this narrower sense, K is a
construction function, but C and H are not.
The final idea required is that of a universal being irreducibly of
order k. Properties of, and relations among particulars, are universals
of order one. If nominalism is false, they are irreducibly so. A universal
is of order two if it is a property of universals of order one, or a relation
among things, some of which are universals of order one, and all of
which are either universals of order one or particulars. It is irreducibly
so if it cannot be analyzed in terms of universals of order one. And in
general, a universal is of order (k + 1) if it is a property of universals of
order k, or a relation among things, some of which are universals of
order k, and all of which are either particulars or universals of order k
or less. It is irreducibly of order (Ic + 1) if it cannot be analyzed in terms
of particulars and universals of order k or less.
Given these notions, it is possible to explain the general concept of
a nomological relation - which will include, but not be restricted to,
the relations of nomic necessitation and nomic exclusion. Thus, as a
first approximation:
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R is a nomological relation if and only if
(1)
(2)
R is an n-ary relation among universals;
R is irreducibly of order (k + 1), where k is the order of the
highest order type of element that can enter into relation R;
(3) R is a contingent relation among universals, in the sense that
there are universals Oj, U2, ..., Un such that it is neither
necessary that R(l>|, U2, •••> ^n) nor necessary that not R(U-\,
U2, ..., Un);