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 \(\mathcal{C}\) be a set of complete individual concepts, where each \( c \in \mathcal{C} \) corresponds to a unique, maximally specific description of a possible substance. By "complete," we mean that for any predicate \(P\) that could conceivably apply to a substance, it is determined (either affirmed or denied) by \( c \).
1.2 Predication and Instantiation
We assume a set \(\mathcal{P}\) of predicates, each of which can be true or false of any individual substance concept. Each complete individual concept \( c \) entails a truth value assignment to every predicate \( P \in \mathcal{P} \). Formally, we may represent this as a function:
\[
\mathrm{Pred}(c) : \mathcal{P} \to \{\text{true}, \text{false}\}.
\]
Because \( c \) is complete, \(\mathrm{Pred}(c)\) is total—it assigns a truth value to each predicate.
2. Compossibility
2.1 The Compossibility Relation
Define a binary relation \(\mathrel{R}\) on \(\mathcal{C}\) called "compossibility." We write \( c_1 \mathrel{R} c_2 \) if and only if the complete individual concepts \(c_1\) and \(c_2\) could coexist in a single possible world. Intuitively, no contradictions arise from jointly instantiating them.
2.2 Properties of Compossibility
- Symmetry: If \( c_1 \mathrel{R} c_2 \), then \( c_2 \mathrel{R} c_1 \).
- Reflexivity: For all \( c \in \mathcal{C} \), \( c \mathrel{R} c \).
- The compossibility relation induces a notion of "compossible sets": A set \( W \subseteq \mathcal{C} \) is compossible if for all \( c, c' \in W \), \( c \mathrel{R} c' \).
3. Possible Worlds
3.1 Definition of a Possible World
A possible world is a maximal compossible set of complete individual concepts. Formally, \( W \subseteq \mathcal{C} \) is a possible world if and only if:
- \( W \) is non-empty.
- \( W \) is pairwise compossible: for all \( c, c' \in W \), \( c \mathrel{R} c' \).
- Maximality: There is no \( c \in \mathcal{C} \setminus W \) such that \( W \cup \{c\} \) is still compossible. That is, you cannot add another complete individual concept to \( W \) without violating compossibility.
We denote the set of all possible worlds by \(\mathcal{W}\).
3.2 The Actual World
The actual world \( W_{\text{act}} \) is one distinguished element of \(\mathcal{W}\). Actual substances are the instantiations of those complete individual concepts in \( W_{\text{act}} \).
4. Valuation of Concepts and Worlds
4.1 Value Assignment to Predicates
Assign to each predicate \( P \in \mathcal{P} \) a non-negative real number representing its "value":
\[
v_P : \mathcal{P} \to \mathbb{R}_{\geq 0}.
\]
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 \( c \) entails a unique truth value assignment to every predicate, the value of a concept \( c \) is the sum of the values of the predicates it instantiates:
\[
v(c) = \sum_{\{P \in \mathcal{P} \mid \mathrm{Pred}(c)(P) = \text{true}\}} v_P(P).
\]
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 \( c \) might be "evil" not because it has a negative value, but because its presence in a world excludes the presence of some other concept \( c' \) with higher value. Thus, what we call "evil" can be understood as the difference between the maximal attainable goodness and the goodness actually realized.
5. Valuing Worlds
5.1 Aggregate Value of a World
For a world \( W \in \mathcal{W} \), define its total value as the sum of the values of the concepts it contains:
\[
V(W) = \sum_{c \in W} v(c).
\]
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 \(\mathcal{C}\)).
- The "evil" of a particular world \( W \) can then be thought of as: \[ E(W) = M - V(W), \] where \[ M = \sup\{\sum_{c' \in S} v(c') : S \subseteq \mathcal{C} \text{ is compossible}\}. \] Here, \(M\) 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 \( W^* \in \mathcal{W} \) which attains the maximal total value:
\[
W^* = \arg\max_{W \in \mathcal{W}} V(W).
\]
Since \(\mathcal{W}\) is defined as the set of all maximal compossible sets and assuming that the supremum of values is actually attained by some world (which we posit for the sake of this framework), there exists at least one such \( W^* \).
6.2 The Actual World is the Best of All Possible Worlds
To say that the actual world \( W_{\text{act}} \) is the best of all possible worlds means:
\[
V(W_{\text{act}}) = \max_{W \in \mathcal{W}} V(W).
\]
Equivalently, the actual world is one that realizes the minimal possible evil (the smallest gap from the maximal achievable good), i.e., for all \( W \in \mathcal{W} \), \[ E(W_{\text{act}}) \leq E(W), \] with equality only when \( W = W_{\text{act}} \) is also a best world.
In Summary:
- Substance Concepts: Each \( c \in \mathcal{C} \) is a complete individual concept determining a unique set of predicate truths.
- Possible Worlds: A possible world \( W \) is a maximal set of pairwise compossible concepts.
- Value of Concepts and Worlds: A value function \( v(c) \) assigns a nonnegative real value to each concept based on the predicates it instantiates. The value of a world \( V(W) \) 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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