Modality and Identity

This conversation was started as a continuation of the previous post.

We should like to define, for our matrix model, what it means for one column of one matrix to designate the same object as a column of another matrix. I’m thinking if individuals are present in the same Lewisian world, then in order to be identical, they must be mereologically coextensive and share the truth-value of all their property propositions (in all worlds that are, mereologically, part of that Lewisian world). But if they don’t share a Lewisian world, then their identity is determined by their both being part of a “rigid designator” set.

Below is a formal outline for how to define cross-matrix and cross-world identity of individuals (as represented by matrix columns) using the concepts of Lewisian worlds, mereological parthood, and rigid designator sets.

Setup:

1. Matrices and Individuals:
   Consider a collection of propositions \(\{P_i\}_{i \in I}\), each represented as a matrix \(M_{P_i}\). For each proposition \(P_i\), we have: \[ M_{P_i}: W \times D \to \{0,1\}, \] where \(W\) is the class of all worlds, and \(D\) is the class of all individuals. The columns of each matrix correspond to individuals, and the rows correspond to worlds.

2. Lewisian Worlds and Mereology:
   Among the worlds \(W\), we single out a subset \(W^{Lewis}\) of Lewisian possible worlds. Each \(w_L \in W^{Lewis}\) is a maximal world, containing within it temporal parts (\(W^{time}(w_L)\)) and spatial parts (\(W^{space}(w_L)\)), and obeying the mereological principles discussed previously.

3. Property Extension of an Individual:
   For any individual \(d\), its “profile” or “extension” within a Lewisian world \(w_L\) is given by the set of all truth values \(M_{P_i}[w', d]\) for every property proposition \(P_i\) and every world \(w' \leq_w w_L\) (where \(\leq_w\) is the mereological parthood relation among worlds). This essentially captures how the individual is characterized throughout the entirety of \(w_L\).

Defining Identity of Individuals Across Columns:

We want to say when two columns \(d_1\) and \(d_2\) (possibly from different matrices, or from the same matrix but considered in different contexts) represent the same individual.

Case 1: The Individuals Share a Lewisian World

• Suppose the two individuals \(d_1\) and \(d_2\) are both instantiated somewhere in the same Lewisian world \(w_L \in W^{Lewis}\). Formally, there exist worlds \(w_{L1}, w_{L2} \leq_w w_L\) (sub-worlds of \(w_L\)) such that: \[ Inst(d_1, w_{L1}) = \text{true} \quad \text{and} \quad Inst(d_2, w_{L2}) = \text{true}. \] Because we are within a single Lewisian world \(w_L\), we have well-defined mereological relations and a full set of property propositions.

  Condition for Identity within a Lewisian World:
  Individuals \(d_1\) and \(d_2\) are identical if and only if:

  1. They are mereologically coextensive in \(w_L\). That is, for all spatial/temporal parts \(w' \leq_w w_L\), \(Inst(d_1, w') = Inst(d_2, w')\). In other words, they occupy exactly the same sub-worlds within \(w_L\).
  2. They have the same property profiles throughout \(w_L\). For every proposition \(P_i\) and every \(w' \leq_w w_L\): \[ M_{P_i}[w', d_1] = M_{P_i}[w', d_2]. \]

  If these two conditions are met, \(d_1\) and \(d_2\) designate the same individual. Intuitively, if two individuals share the same spatiotemporal footprint and agree in all properties throughout a Lewisian world, they are indistinguishable and hence identical.

Case 2: The Individuals Do Not Share a Lewisian World

• If \(d_1\) and \(d_2\) never appear together in any single Lewisian world, we cannot directly compare their extensions within one world. In this scenario, trans-world identity is not determined by overlapping presence in a single world.

  Rigid Designator Sets:
  We introduce the notion of a rigid designator set: a set of columns/individuals grouped together based on a stipulated naming or reference scheme that picks out the same individual across distinct Lewisian worlds (and possibly across worlds that do not share a common Lewisian super-world). This notion mirrors the idea of “rigid designators” from modal logic and metaphysics: certain terms or designators refer to the same individual in every possible world, even if those worlds are disconnected or non-overlapping in the Lewisian sense.

  Formally, we assume there is a function: \[ R: D \to \mathcal{P}(D) \] that maps each individual to a “rigid designator set” of which it is a member. Each such set \(R(d)\) is specified by the interpretation of some rigid designator (a proper name, say) in the model.

  Condition for Identity without a Shared Lewisian World:
  If \(d_1\) and \(d_2\) do not co-inhabit any Lewisian world, then: \[ d_1 = d_2 \iff R(d_1) \cap R(d_2) \neq \emptyset. \] More simply, \(d_1\) and \(d_2\) are identical if they belong to the same rigid designator set. This enforces a metaphysical stipulation: the identity of individuals that cannot be compared within a single Lewisian world is determined by an external naming scheme or an intensional device that keeps track of cross-world identity.

Summary:

• Within a shared Lewisian world: Two columns represent the same individual if they occupy the same sub-worlds and have identical property profiles throughout that world.

• Without a shared Lewisian world: Two columns represent the same individual if and only if they are both designated by the same rigid designator, i.e., they belong to the same rigid designator set.

This two-pronged approach respects the idea that identity within a common metaphysical framework (like a single Lewisian world) can be determined by mereological and property-theoretic considerations, while identity across disconnected worlds requires an additional external criterion, provided by rigid designation.

Some people may endorse the idea that individuals are identical to the mereological sum of their spatiotemporal parts. An objection is sometimes drawn to this from Leibniz’s Law, since, in different Lewisian possible worlds, individuals may have had more or fewer temporal parts, by perishing sooner or later than they actually did, or growing/shrinking more or less than they actually did.

Using the formal definitions from our model, explain how rigid designators can allow for different mereological sums of spatiotemporal parts to be trans-world identical: the sums are both assigned the same designator. Formally characterize how, even so, our model does not require individuals to be identical to the mereological sum of their spatiotemporal parts, either: mereological coextensiveness is necessary but not sufficient for intraworld identity, and the model leaves open whether, within a world, one may define properties that are not equally true of all spatiotemporally colocated individuals, e.g., equally true of a wall and of the sum of all the bricks within it.

Formally characterize the axiom of “strong composition-as-identity”, which requires that individuals that are mereologically coextensive in a Lewisian world have the same properties in that Lewisian world and in all worlds that are, mereologically, part of that Lewisian world. A denier of strong composition-as-identity may define that, in the case of a wall the the sum of its bricks, the bricks are “many”, while the wall is “one”, so they have different properties; a defender may argue that counting is illegitimate without units, so the wall is “many bricks” (with the counting unit “brick”) but “one wall” (with the counting unit “wall”), and equally, the sum-of-bricks is also “many bricks” but “one wall”, and there is no property in which they differ.

Below is a formal and conceptual treatment of the issues raised:

1. Rigid Designators and Trans-World Identity

We have defined that individuals are represented as columns in matrices and that their identity across worlds can be secured by rigid designators. Formally:

• Let \(D\) be the set of all individuals across all worlds, and let \(R: D \to \mathcal{P}(D)\) be a function assigning to each individual \(d\) a “rigid designator set” \(R(d)\). Two individuals \(d_1\) and \(d_2\) are identified across distinct Lewisian worlds if and only if \(R(d_1)\) and \(R(d_2)\) intersect (i.e., they share the same name/designation).

This allows for cases where the mereological sum of the spatiotemporal parts that constitute an individual may differ from one Lewisian world to another, but the same rigid designator applies to both sums. In one world, the individual may have “shorter” temporal extent (perishing earlier) or “less extensive” spatial parts (occupying a smaller volume), while in another world, it may have “longer” temporal extent or “larger” spatial volume. Yet, the same rigid designator links these two variant mereological sums as a single trans-world individual.

Formally, suppose we have two individuals \(d_1\) and \(d_2\) in different Lewisian worlds, \(w_{L1}\) and \(w_{L2}\), with different sets of temporal and spatial parts. We can still have: \[ R(d_1) \cap R(d_2) \neq \emptyset, \] ensuring that \(d_1\) and \(d_2\) represent the same trans-world individual, despite differing mereologically from one world to another.

2. Identity vs. Mereological Coextensiveness

Within a single Lewisian world \(w_L\), one might be tempted to identify an individual with the mereological sum of its spatiotemporal parts. The framework presented allows for the following condition within \(w_L\):

• Mereological Coextensiveness: Let \(d_1\) and \(d_2\) be individuals both instantiated within \(w_L\). They are mereologically coextensive in \(w_L\) if and only if, for every sub-world \(w' \leq_w w_L\), \[ Inst(d_1, w') = Inst(d_2, w'). \] In other words, \(d_1\) and \(d_2\) occupy exactly the same spatiotemporal regions within \(w_L\).

Mereological coextensiveness is often viewed as a necessary condition for intraworld identity: if two individuals differ in their spatiotemporal presence, they are distinct. However, the model does not require that mereological coextensiveness alone suffices for identity. Two distinct individuals could be perfectly colocated mereologically—think of the classical puzzle of the statue and the lump of clay, or the wall and the sum of its bricks—yet differ in other properties.

This is possible if there are properties in the model that differentiate entities that are mereologically coextensive. For example, at a single instant:

• Let \(P_{one}\) be a property representing “is one object.”
• Let \(P_{many}\) be a property representing “is many objects.”

One might define that the wall is “one object,” while the sum of bricks is “many objects,” and thus even though they are mereologically coextensive, they differ with respect to \(P_{one}\) and \(P_{many}\).

3. Strong Composition-as-Identity Axiom

The “strong composition-as-identity” view would impose an additional constraint on properties:

• Axiom (Strong Composition-as-Identity): For all individuals \(d_1, d_2\) within a Lewisian world \(w_L\), if \(d_1\) and \(d_2\) are mereologically coextensive in \(w_L\) (i.e., they occupy exactly the same spatiotemporal parts of \(w_L\)), then \(d_1\) and \(d_2\) must agree on all properties in all worlds that are mereological parts of \(w_L\). Formally: \[ (\forall w' \leq_w w_L)(Inst(d_1, w') = Inst(d_2, w')) \implies (\forall i, \forall w'' \leq_w w_L: M_{P_i}[w'', d_1] = M_{P_i}[w'', d_2]). \]

This axiom ensures not only mereological coextensiveness but also property indistinguishability within the entire Lewisian world structure containing them. Under strong composition-as-identity, the wall and the sum of bricks are not merely colocated; they are indiscernible with respect to all properties defined on that world and its sub-worlds.

4. Debating Strong Composition-as-Identity

• The Denier’s Strategy:
  If one denies strong composition-as-identity, one may allow that even though two individuals are mereologically coextensive in \(w_L\), they differ with respect to some properties. For example, a denier might say:

  • The wall is “one,” while the sum of bricks is “many.”

  Here “one” and “many” are taken as genuine properties that can differ even for mereologically coextensive entities. Thus, not all properties are purely extensional with respect to mereological composition. By rejecting the axiom, the model remains open to the idea that different conceptualizations, counting practices, or conventions can yield distinct property assignments to entities that share the same parts.

• The Defender’s Strategy:
  A defender of strong composition-as-identity would argue that such counting properties are either not well-formed intrinsic properties or can be “units-relative.” For example:

  • Counting in bricks: the wall and the sum of bricks are both “many bricks.”
  • Counting in walls: the wall and the sum of bricks are “one wall.”

  From this perspective, once one normalizes the property definitions (ensuring that all such counting is relative to a chosen unit or category), there remains no intrinsic property by which the wall and the sum of bricks could differ. Thus, under normalized property definitions, strong composition-as-identity holds.

5. The Model’s Neutrality

Our matrix model does not mandate strong composition-as-identity. It allows for:

• Rigid designators that ensure trans-world identity of varying mereological sums of spatiotemporal parts.
• The possibility that mereological coextensiveness within a world is necessary but not sufficient for identity, leaving room for intensional properties that differentiate colocated entities.
• The option, through an additional axiom (strong composition-as-identity), to rule out such divergences and require that coextensive entities have identical properties.

In sum, the model is flexible and can accommodate both views: it can represent scenarios where two mereologically coextensive entities differ in some property (if we reject strong composition-as-identity), or scenarios in which no such property differences are allowed (if we accept strong composition-as-identity). This flexibility reflects the philosophical debates about the nature of identity, composition, and the individuation of objects across possible worlds.

Consider further implications of the idea that a world is anything that a proposition may be true or false in, depending on conditions that are independent of its being true in different worlds. Or rather, that the conditions may be independent, but that one may postulate axioms for dependence between truth in some worlds and truth in others. Given these essentially semantic concerns, consider what kinds of worlds may be admitted besides spatial worlds, temporal worlds, Lewisian worlds, and (given CEM) mereological sums among these.

Below is a conceptual and formal exploration of how the notion of "world" might extend beyond the familiar categories of spatial, temporal, and Lewisian worlds, as well as their mereological sums. This arises from the recognition that a “world” is semantically defined as an entity relative to which propositions have truth-values—potentially independent of truth-values at other worlds.

1. The Semantic Character of Worlds

We have begun with the idea that a world is something at which a proposition can be true or false, according to conditions intrinsic to that world and independent of other worlds. In the classical Kripkean framework, “possible worlds” are maximally consistent states of affairs. In our extended framework, “worlds” need only satisfy the condition that the truth of propositions at them is determined without direct reference to their truth at other worlds—though we may impose axioms that introduce dependencies.

This interpretation is broad and intentionally so. It treats “worlds” as parameters in a semantic evaluation function: \[ v(P,w) \in \{\text{true}, \text{false}\}, \] with \(w\) ranging over a set \(W\). Initially, we considered spatial points, temporal instants/intervals, and entire Lewisian possible worlds, as well as mereological fusions thereof. But the logical and semantic machinery does not limit us to these.

2. Other Kinds of Worlds

Once we recognize that “world” is merely a parameter for evaluating propositions, we can consider many more kinds of worlds that are not strictly spatial, temporal, or Lewisian. These new worlds can be introduced wherever we want to model variation in truth-conditions that is independent along some dimension or criterion.

2.1 Epistemic Worlds
We might introduce epistemic worlds: distinct states of knowledge or information. For instance, each “world” could be a situation representing what an agent (or group of agents) knows. The truth of propositions about what is known or believed depends only on the epistemic state itself, not on whether that state obtains in another epistemic situation. Epistemic worlds are central to epistemic modal logic, where \(\Box\) and \(\Diamond\) range over what is known or possibly known.

2.2 Doxastic Worlds
Similarly, doxastic worlds correspond to belief states. Instead of “who knows what,” we might consider “who believes what.” Each world is a “belief scenario.” Again, the truth of a proposition relative to a doxastic world is fixed by the beliefs constitutive of that world.

2.3 Contextual Worlds
From the tradition of Kaplanian semantics, we have contexts that fix parameters like speaker, place, time, and standards of precision. We can treat each distinct context as a “world-parameter,” and evaluate context-sensitive expressions relative to these worlds. Such worlds are not merely times or places but context-situations that determine referents and interpret indexicals.

2.4 Normative Worlds
Another possibility is normative worlds: possible normative states of affairs (e.g., “legally possible states,” “morally permissible states”). A proposition’s truth in a normative world depends solely on what is permitted or obligatory in that normative scenario, without directly depending on the truth of that proposition in other normative worlds (unless axioms are introduced to impose such dependencies).

2.5 Counterfactual or Counterlegal Worlds
We might introduce counterfactual worlds explicitly as “worlds” in this sense. Although Lewisian possible worlds already cover a large space, one might consider other graded worlds representing alternative lawlike structures or hypothetical changes in physical constants. These are “similarity-based” or “structured” sets of conditions. As long as the truth at such a world is fixed internally, it can serve as a world parameter.

2.6 Abstract Parameter Worlds
We can consider even more abstract dimensions: e.g., “interpretation worlds” that vary the interpretation of non-logical terms, “mathematical worlds” that vary sets or axioms (as in forcing or constructible universes in set theory), “artistic worlds” representing different aesthetic frameworks, or “policy worlds” representing different sets of assumptions in decision theory. The criteria remain the same: the truth of a proposition at a given world is fixed independently by that world’s own conditions.

3. Merging and Mereology

Classical Extensional Mereology (CEM) gave us the ability to form mereological fusions of worlds (e.g., an interval of times, a volume of space). If we treat these new dimensions similarly, we might ask whether and how to define mereological relations among them. Some kinds of worlds (e.g., epistemic states) may have no natural mereological structure. Others might be partially ordered or topologically structured, allowing for meaningful parthood or fusion relations.

• One could, for instance, consider a lattice of epistemic states with a partial order representing information inclusion. Then one might form the mereological sum of several epistemic states to represent their combined information. This extends the notion of a “world” beyond purely spatiotemporal fusion to conceptual fusion.

4. Dependencies and Axioms

Initially, we stated that a world is defined by the independence of truth-conditions. However, one may impose axioms for dependencies between worlds:

• Example: We might say that if a proposition \(P\) is true at an epistemic world \(w_{e}\) (representing one agent’s knowledge), it must also be true at all worlds \(w'_{e}\) representing states where that agent has strictly more information. This would be an axiom enforcing a dependency of truth across the epistemic dimension.

• Similarly, one might introduce axioms that link normative worlds to particular temporal or spatial worlds, or link doxastic worlds to epistemic worlds.

The framework places no fundamental restriction on these additional axioms. They can be introduced as needed, reflecting philosophical or theoretical commitments.

5. Conclusion

By considering the nature of “worlds” strictly in terms of the semantic role they play (parameters of truth evaluation), we open the door to a wide variety of worlds beyond the familiar spatial, temporal, and Lewisian domains. We may have:

• Epistemic and doxastic worlds for states of knowledge and belief.
• Contextual worlds for evaluation of context-sensitive expressions.
• Normative worlds for moral, legal, or other normative evaluations.
• Counterfactual or other specialized possible worlds where laws, interpretations, or frameworks vary.
• Abstract parameter worlds for modeling truth-values under different logical, mathematical, or interpretational conditions.

This does not disrupt the original formalism. The matrix model and the notion of identity, rigid designation, and mereology can be extended or restricted depending on whether the new worlds admit a natural mereological structure or whether we simply treat them as points in a parameter space. The key is that the original semantic idea—propositions have truth-values at worlds independently determined by those worlds themselves—can incorporate an enormous variety of semantic parameters, thus greatly enriching the conceptual landscape of what counts as a “world.”

Formally, define the idea of an essential property of an individual, where an individual’s column comes out either true or empty (indicating non-instantiation) in every world row (although, for a necessarily existent being, it must come out true in at least all Lewisian worlds, and not necessarily in their parts). Contrast it with a contingent property, which comes out false in some worlds. Being a typical physical object (TPO) might be essential to every TPO; suggest other essential properties, and other constraints on identity.

Below is a formal characterization of essential and contingent properties for individuals in the matrix model, followed by an example (being a Typical Physical Object, TPO) and a discussion of other essential properties and constraints on identity.

1. Background

• We have a set of possible worlds \(W\), which may include spatial worlds, temporal worlds, Lewisian worlds, and possibly other kinds of worlds.
• We have a set \(D\) of individuals. Each individual \(d \in D\) may be instantiated in some subset of \(W\). At any world \(w \in W\), the cell \((w,d)\) of a proposition matrix can be:
  • True (1): \(d\) exists at world \(w\) and the proposition holds of \(d\) in \(w\).
  • False (0): \(d\) exists at world \(w\) but the proposition does not hold of \(d\) in \(w\).
  • Empty: \(d\) is not instantiated at \(w\); effectively, \(d\) does not exist at \(w\).

For a necessarily existent individual, instantiation is not empty in all Lewisian worlds (though it may still be empty in some parts of those worlds, if we treat sub-worlds as partial instantiations).

2. Defining Essential vs. Contingent Properties

Let \(P: W \times D \to \{0,1,\emptyset\}\) represent a proposition that may or may not hold of an individual \(d\) at world \(w\). Here we use \(\emptyset\) to indicate non-instantiation of \(d\) at \(w\).

• Essential Property:
  A property \(P\) is essential to an individual \(d\) if and only if \(P\) holds of \(d\) in every world \(w\) in which \(d\) is instantiated. Formally: \[ (\forall w \in W)(P(w,d) \neq \emptyset \implies P(w,d) = 1). \] In other words, whenever \(d\) exists at world \(w\), \(d\) has property \(P\) at \(w\). This is often taken to mean that \(d\) cannot lose or fail to have \(P\) once \(d\) exists.

  For a necessarily existent being \(d\) (one that is instantiated in all Lewisian worlds \(w_L \in W^{Lewis}\)), essentiality would mean: \[ (\forall w_L \in W^{Lewis}, \forall w \leq_w w_L)(P(w,d) \neq \emptyset \implies P(w,d) = 1). \] Since \(d\) never fails to exist in any Lewisian world, \(d\) necessarily exemplifies \(P\) in all of them.

• Contingent Property:
  A property \(P\) is contingent for an individual \(d\) if \(d\) has \(P\) in some worlds where it is instantiated and not in others. Formally: \[ (\exists w_1, w_2 \in W)(P(w_1,d) = 1 \text{ and } P(w_2,d) = 0), \] with the understanding that \(P(w_1,d) \neq \emptyset\) and \(P(w_2,d) \neq \emptyset\). This means that \(d\) sometimes has \(P\), sometimes lacks it, depending on the world.

3. Essential Properties and TPO Status

• Being a Typical Physical Object (TPO):
  Suppose we have defined a property proposition \(TPO\) that holds of an individual if and only if it is a TPO at the worlds where it exists. If we treat being a TPO as an essential property, then for any TPO individual \(d\): \[ (\forall w \in W)(P_{TPO}(w,d) \neq \emptyset \implies P_{TPO}(w,d) = 1). \] This means that if \(d\) is a TPO in one world, it must be a TPO in every world in which it exists. TPO-hood cannot be lost or gained; it’s essential to the individual’s nature as soon as that individual exists.

• Other Essential Properties:
  Besides TPO-hood, other properties might be essential to certain kinds of individuals. For example:

  • Being a “person” might be considered essential to certain individuals that are persons in every world they inhabit.
  • Being a “set” or a “number” in a mathematical structuralist interpretation could be essential properties for entities in mathematical worlds.

  Whether these properties are truly essential depends on the philosophical or theoretical commitments of the modeler.

4. Other Constraints on Identity

The model we’ve described is neutral about certain philosophical questions. It does not require that individuals be identical to their mereological sums of spatiotemporal parts. One could:

• Accept “strong composition-as-identity” (as previously defined), under which mereological coextensiveness guarantees that no property differs. Under such an axiom, if two individuals are coextensive in a Lewisian world, they share all properties in that world and its parts, making their identity conditions quite strict.

• Deny “strong composition-as-identity” and allow that even coextensive entities might differ in some properties (e.g., “the sum of these bricks” vs. “this wall”). In that case, mereological coextensiveness is necessary but not sufficient for identity. Essential properties can help distinguish individuals that are spatiotemporally coincident but differ in kind.

For instance, if “being essentially composed of these particular bricks” is an essential property of the brick-sum but not of the wall, then the wall and the brick-sum differ in at least one essential property. Likewise, one could impose constraints so that identity hinges not merely on mereological configuration, but on the possession of certain essential properties.

5. Conclusion

Essential properties, defined in the model, are those that an individual never lacks whenever it is instantiated. Contingent properties vary from world to world. Being a TPO may be an example of an essential property for TPOs. This framework allows for flexibility in defining what counts as essential and how identity is constrained, and it can accommodate a range of philosophical positions on the nature of individuals, their persistence, and the role of composition in determining their identity.