A substance concept is a “complete individual concept”—complete in the sense that it entails everything that would be truly predicated of the corresponding substance were its concept instantiated.
A possible world is a set of complete individual concepts closed under the relation of compossibility, which implies that it is maximal and that its members could exist together in a single, unified world were they instantiated. (Actual substances are the instantiations of those complete individual concepts that belong to the actual world.)
Write out a formal model of these definitions, and then model a way of assigning value to worlds, assuming that “evil is the absence of good”: that is, to every instantiated predicate there corresponds a non-negative real value, and “evil” is simply the instantiation of concepts whose instantiation is incompossible with the instantiation of higher-value concepts. Then, define what it should mean for the actual world to be the best of all possible worlds.
Below is one way to formally model the given notions and definitions. The framework is inspired by a Leibnizian conception of possible worlds, complete individual concepts, compossibility, and the valuation of worlds. This is not the only possible formalization, but it captures the intended ideas.
1. Basic Setup
1.1 Domain of Concepts
Let
1.2 Predication and Instantiation
We assume a set
2. Compossibility
2.1 The Compossibility Relation
Define a binary relation
2.2 Properties of Compossibility
- Symmetry: If
, then . - Reflexivity: For all
, . - The compossibility relation induces a notion of "compossible sets": A set
is compossible if for all , .
3. Possible Worlds
3.1 Definition of a Possible World
A possible world is a maximal compossible set of complete individual concepts. Formally,
is non-empty. is pairwise compossible: for all , .- Maximality: There is no
such that is still compossible. That is, you cannot add another complete individual concept to without violating compossibility.
We denote the set of all possible worlds by
3.2 The Actual World
The actual world
4. Valuation of Concepts and Worlds
4.1 Value Assignment to Predicates
Assign to each predicate
These values are meant to represent the "good" contributed by the instantiation of that predicate. Higher values correspond to greater goodness.
4.2 Value of a Concept
Since each complete individual concept
4.3 Evil as Absence of Good
"Evil" is here understood not as a separate positive quantity but as a deficit of good. Since concepts can block the instantiation of better concepts (due to incompossibility), evil manifests as the "opportunity cost" of including lower-value concepts that exclude higher-value concepts.
In other words, evil arises from the global arrangement of concepts in a world. A concept
5. Valuing Worlds
5.1 Aggregate Value of a World
For a world
5.2 Measuring Evil in a World
Because evil is the absence of higher value (or the impossibility of instantiating better concepts due to compossibility constraints), one way to model evil is as follows:
- Consider the theoretical upper bound of the sum of values of concepts that could possibly be instantiated if there were no compossibility constraints (i.e., the supremum of sums of any subset of
). - The "evil" of a particular world
can then be thought of as: where Here, is the maximal possible value attainable by any compossible set of concepts (i.e., the best possible world). Thus, the evil in a given world is simply the gap between its total realized good and the maximal possible good.
6. Defining the Best of All Possible Worlds
6.1 Best Possible World
The best of all possible worlds is that world
Since
6.2 The Actual World is the Best of All Possible Worlds
To say that the actual world
Equivalently, the actual world is one that realizes the minimal possible evil (the smallest gap from the maximal achievable good), i.e., for all
In Summary:
-
Substance Concepts: Each
is a complete individual concept determining a unique set of predicate truths. -
Possible Worlds: A possible world
is a maximal set of pairwise compossible concepts. -
Value of Concepts and Worlds: A value function
assigns a nonnegative real value to each concept based on the predicates it instantiates. The value of a world is the sum of the values of its constituent concepts. - Evil as Absence of Good: Evil in a world is measured by how far that world falls short of the maximal possible sum of values that could be achieved by some (other) compossible set of concepts.
- Best of All Possible Worlds: The best world is one that attains the maximal total value. Declaring the actual world to be the best means its total value is not exceeded by that of any other possible world.
In the original output, the formula about maximal possible value (that follows the word “where”) was opened with \(
but closed with \]
, which broke MathJax rendering both here and in the original UI, but which I have corrected here.
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