The number of arguments of a relation is called the arity of the relation. A binary (or dyadic) relation has arity 2.
The atoms of a lattice are the elements which cover the minimal element 0. That is, a is an atom if a > 0, and a >= b > 0 implies a = b. So the atoms are the smallest non-zero elements of the lattice.
A state of affairs which is an atom in the lattice of states of affairs, using the partial ordering A <= B if A entails B (B holds in any World state in which A holds.) In set-theoretic terms, the atomic states of affairs consist of a singleton World state.
A one-to-one function from the set of elements of an algebra onto itself which preserves all the algebraic identities. Example: multiplication by -1 is an automorphism of the integers with respect to addition: if x = y + z, then -x = -y + -z, etc.
Any lattice which is isomorphic to a collection of sets closed under complementation and intersection.
One part or component of the World contains another if the state of the former logically determines the state of the latter. That is, there is a function from the larger to the smaller component, mapping states of the former onto states of the latter.
The converse of T, Tc, where T is the temporal relation. Under Tc, time "flows" from T-future to T-past.
The complement of an atomic state of affairs; alternatively, an atom of the lattice of states of affairs ordered by the converse of the partial ordering used to define the atomic states of affairs: A <= B if A is entailed by B. In set-theoretic terms, elementary states of affairs are of the form S - {s} for some World state s. Elementary states of affairs are minimally contingent -- they are only "one state away" from being the universal state of affairs S which always holds.
A reflexive, symmetric, and transitive relation.
The concept of independence for propositions: a set of propositions A in some mathematical system are logically independent if for any p in A, p is not implied by the conjunction of any subset of A which does not include p.
This generalizes to a lattice with conjunction replaced by join: a subset X of a lattice is independent if for any x in X, x is not <= join(Y). Examples are the atomic states of affairs and the elementary states of affairs, each of which form an independent set.
The kernel of a total function f on S is the partition of S = { B | B = (finv o f)(s) for some s in S}. (finv is the inverse of f: finv(s) = {t: f(t) = s}.)
A lattice is a partially ordered set, closed under least upper and greatest lower bounds. The least upper bound of x and y is called the join of x and y, and is sometimes written as x + y; the greatest lower bound is called the meet and is sometimes written as x . y. Examples are:
- the subsets of a set, partially ordered by set inclusion. The join is set union and the meet is set intersection;
- the partitions of a set, partially ordered by refinement. The join of x and y is formed by taking all the non-empty intersections of blocks of x and y. The meet x and y is more complicated -- a brief but opaque definition is that it is the partition corresponding to the transitive closure of the equivalence relations defined by x and y.
The minimal element O of the lattice of partitions of a set X is the partition with just one block = {x: x in X}. It corresponds to the trivial equivalence relation under which all elements are related to each other.
The natural mapping Pnat of World states S onto a part P is given by Pnat(s) = B where B is the unique block of P such that s in B.
A partition of a set X is a set of disjoint subsets of X whose union is X. The subsets are called the blocks of the partition. For every partition, there is a corresponding equivalence relation which relates two elements of X just in case that they are in the same block of X.
By the same token, every equivalence relation on X defines a partition of X whose blocks are the equivalence classes of the relation.
The reduction of a part P on a set of states X is defined to be the kernel of the restriction of the natural mapping Pnat to X and is written P|X.
Alternatively,
P|X = the set of non-empty intersections of B with X, where B is a block in P.
The sense of a proposition is the state of affairs in which it is true. A contradiction has no sense -- there is no state of the World, and hence no state of affairs, in which it is true. A proposition is necessarily true if its sense is the maximal state of affairs, the set of all World states.
Used as a technical term for a collection of World states. States of objects are states of affairs.
States of affairs are the 'truth-makers' for, or senses of, contingent propositions.
In nominalist language, states of affairs are the individual entities of the world having the form of sums (rather than sets) of World states.
X is a sublattice of Y if Y is a lattice, X is a subset of Y and X is a lattice with the same join and meet operations as Y.
A tautology is a proposition which when considered as a Boolean function of its non-logical terms always has the value
true.
Ultra-atoms are atomic objects, that is, dichotomies, one of whose states is a singleton. Thus the form of an ultra-atom is {{s}, S - {s}} or, more simply, {{s}, -{s}}. Ultra-atoms are, in a sense, very small atoms, in that they are only "one state away" from the minimal O element ({S}) in the lattice of objects
If A is a set on which a binary relation R is defined, A is weakly-connected (with respect to R) if for every distinct x and y in A, either x R y or y R x.