Semantic Worlds

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

Awesome, let’s use a particular view of possible worlds, henceforth simply “worlds”. By a world, we are going to understand “something propositions are true in, independently of their being true in other worlds”. For this, there is no relevant difference between points or locations in space, instants or intervals in time, and Lewisian possible worlds. But due to the restriction that the truth-conditions of a proposition in a world are independent of the proposition’s being true in other worlds, the situations from Barwise & Perry’s situation semantics are excluded from being worlds in this sense.

Given this understanding, write formal definitions of what it means for a world to be a point or location in space, an instant or interval in time, or a Lewisian possible world, in terms of its relationships with other worlds. For instance, worlds which are instants or intervals in time are related by their position along a nondenumerable “time line”, while worlds which are positions or locations in space are related spatially.

Also, account formally for mereological parthood relations among worlds, and among individuals, assuming Classical Extensional Mereology. Worlds can be composed of other worlds: an interval is composed of its instants, and a volume is composed of its points, and each interval, instant, volume, and point is a world in this theory—a world that has other worlds as its parts. A Lewisian world has spaces and times as parts, but is not part of any spaces and times. Also, individuals (objects) can have other individuals as their mereological parts, subject to conditions on the worlds that they are instantiated in. (An individual that has parts is still considered “individual”, undivided, with respect to a proposition that defines truth-conditions for that individual at worlds.)

Below is a set of formal definitions and characterizations of different kinds of “worlds” and their relations, as well as mereological structures on worlds and individuals, consistent with the given conceptual framework. All of these definitions proceed within a broader theory in which we have:

• A set \(W\) of worlds.
• A mereological parthood relation \(\leq_w\) on \(W\).
• Additional relational structures on \(W\) to encode temporal, spatial, and modal characteristics.
• A set \(D\) of individuals (objects) and a mereological parthood relation \(\leq_d\) on \(D\).

1. General Constraints on Worlds

1.1 Independence Condition
For any world \(w \in W\), the truth-conditions of any proposition at \(w\) are independent of the truth-conditions of that proposition at any other world \(w' \in W, w' \neq w\). Formally, if \(P: W \times D \to \{\text{true}, \text{false}\}\) is a proposition, then for each fixed \(d \in D\), \(P(w,d)\) is determined solely by \(w\) (and \(d\)) and not by \(P(w',d)\) for any \(w' \neq w\).

This independence excludes “situations” as defined by Barwise & Perry, since in that framework the truth of a proposition at a situation can depend on truths at other situations.

2. Kinds of Worlds

We distinguish different kinds of worlds by the relational structures they participate in. Each kind of world is associated with a certain underlying ordering or relational system.

2.1 Worlds as Instants or Intervals in Time

• Assume a dense, linear ordering \((W^{time}, <_t)\) on a subset \(W^{time} \subseteq W\) that represents times. Here \(<_t\) is a strict linear order without gaps, and potentially uncountable.
• Temporal Instant: A world \(w \in W^{time}\) is a temporal instant if it is a minimal “atom” in this temporal ordering structure. Formally, \(w\) is a temporal instant if for every nonempty interval \(I \subseteq W^{time}\) such that \(w \in I\), if \(I\) is convex with respect to \(<_t\), then \(I = \{w\}\). Another way: \(w\) is an instant if it has no nontrivial proper temporal parts.
• Temporal Interval: A world \(w \in W^{time}\) is a temporal interval if it is composed (via the mereological relation \(\leq_w\)) of multiple temporal instants forming a convex subset of \((W^{time}, <_t)\). Formally, if \(I \subseteq W^{time}\) is a convex set under \(<_t\), then let \(\bigoplus I\) denote its mereological fusion. A world \(w\) is a temporal interval if \(w = \bigoplus I\) for some convex \(I\) containing more than one element.
• The parthood relation \(\leq_w\) for temporal worlds respects the ordering \(<_t\):
  For all \(w_1, w_2 \in W^{time}\), \(w_1 \leq_w w_2\) if and only if whenever \(w_2\) is an interval or instant, and \(w_1\) is included as a temporal sub-region, \(w_1\) corresponds to a subset of times included in the set of times composing \(w_2\).

Thus, temporal worlds are arranged along a nondenumerable timeline, and instants are minimal elements under mereological decomposition, while intervals are mereological fusions of instants.

2.2 Worlds as Positions or Locations in Space

• Assume a topological or metric structure on another subset \(W^{space} \subseteq W\), representing spatial locations. For simplicity, consider a three-dimensional Euclidean space \(\mathbb{R}^3\). Each world in \(W^{space}\) is identified with a region of space.
• Spatial Point: A world \(w \in W^{space}\) is a spatial point if it corresponds to a single point in \(\mathbb{R}^3\), i.e., a singleton set \(\{p\}\) where \(p \in \mathbb{R}^3\). Points are minimal spatial worlds with no proper spatial parts (other than themselves).
• Spatial Volume (Region): A world \(w \in W^{space}\) is a spatial region (or volume) if it is the mereological fusion of possibly infinitely many points forming a (possibly extended) region \(R \subseteq \mathbb{R}^3\). For a set of points \(S \subseteq \mathbb{R}^3\), let \(\bigoplus S\) be the fusion of all point-worlds in \(S\). If \(w = \bigoplus S\) for some set \(S \subseteq \mathbb{R}^3\), \(w\) is a spatial volume.
• The mereological parthood \(\leq_w\) for spatial worlds corresponds to set inclusion of their underlying point-sets: \[ w_1 \leq_w w_2 \iff \text{If } w_1 = \bigoplus S_1 \text{ and } w_2 = \bigoplus S_2, \text{ then } S_1 \subseteq S_2. \]

Spatial worlds are thus structured according to spatial containment relations.

2.3 Lewisian Possible Worlds

• A Lewisian possible world, in David Lewis’s sense, can be understood as a maximal isolated “whole” world—often a maximal spatiotemporally connected region that does not mereologically overlap with any other distinct maximal region that could serve as a separate “world”.

• Formally, let \(W^{Lewis} \subseteq W\) be the set of Lewisian possible worlds. For each \(w \in W^{Lewis}\), there are subsystems isomorphic to spatial and temporal structures within \(w\): it contains a (possibly vast) mereological fusion of spatial regions and times. However, crucially, each Lewisian world is maximal and not part of any larger structure that would count as another “world” in the same sense.

• Thus, for a Lewisian world \(w\):

  1. Maximality: There is no \(w' \in W\) such that \(w \leq_w w'\) and \(w' \neq w\). That is, \(w\) is not a proper part of another world.
  2. Internal Structure: Within \(w\), there exist sub-worlds that correspond to spatial regions and temporal intervals. Thus, inside a single Lewisian possible world \(w\), we have:
     • A temporal ordering \(<_t\) restricted to parts of \(w\) that represent times.
     • A spatial structure corresponding to points and volumes that are parts of \(w\).

  However, \(w\) itself is not part of any larger temporal or spatial structure encompassing other Lewisian worlds. Each Lewisian world stands alone as a self-contained totality.

3. Mereological Structure Among Worlds

We assume Classical Extensional Mereology (CEM) for the parthood relation \(\leq_w\) on worlds. CEM includes the following axioms:

• Reflexivity: For all \(w \in W\), \(w \leq_w w\).
• Antisymmetry: If \(w_1 \leq_w w_2\) and \(w_2 \leq_w w_1\), then \(w_1 = w_2\).
• Transitivity: If \(w_1 \leq_w w_2\) and \(w_2 \leq_w w_3\), then \(w_1 \leq_w w_3\).
• Unrestricted Fusion: For any set of worlds \(\mathcal{W} \subseteq W\), there exists a world \(w = \bigoplus \mathcal{W}\) that is a fusion of all and only the parts of worlds in \(\mathcal{W}\).

In the temporal case, intervals are fusions of instants; in the spatial case, regions are fusions of points; in a Lewisian world, the entire world is a maximal fusion of its spatial and temporal parts and all individuals contained therein.

4. Mereological Structure Among Individuals

We also have a parthood relation \(\leq_d\) on individuals \(D\), again assumed to satisfy CEM. Individuals can have other individuals as parts. For example, a car may have an engine as a part. Conditions on the instantiation of individuals across worlds:

• Each individual \(d \in D\) is present (instantiated) in some subset of worlds \(W_d \subseteq W\).

• If \(d_1 \leq_d d_2\), then in any world \(w\) where \(d_1\) is instantiated, \(d_2\) is also instantiated. Formally: \[ \forall w \in W: (P(w,d_1) = \text{true} \implies P(w,d_2) = \text{true}) \] for any proposition \(P\) that tracks the mere existence or presence of these individuals at worlds.

• The notion of an “individual” is extensional in the sense of CEM, but the classification as an “individual” does not preclude having parts. Being “undivided with respect to a proposition” means that the proposition’s truth conditions treat the entire individual as a single unit, even if that individual is mereologically complex.

5. Integrating the Framework

• Temporal worlds: These form a linear ordering. Instants are minimal elements; intervals are fusions of these instants.
• Spatial worlds: These form a spatial structure (e.g., subsets of \(\mathbb{R}^3\)). Points are minimal elements; volumes are fusions of these points.
• Lewisian worlds: These are maximal, self-contained aggregates of spatiotemporal structure. They are not proper parts of any larger spatial or temporal system. Within each Lewisian world, sub-worlds corresponding to times and volumes exist as parts.

This system allows for a unified treatment of different conceptions of “worlds” under a single mereological and relational framework. Worlds are always entities in which propositions hold independently; how they stand in relation to one another (linearly ordered in time, topologically arranged in space, or isolated complete totalities) determines their classification.

Use these parthood conditions across space and time to formally define a typical physical object (TPO): An individual is a TPO if, and only if, the worlds in which its parts are instantiated comprise, within each Lewisian world, at most a single contiguous interval of time and, in every instant of time in which it the object is instantiated, it occupies a single contiguous spatial volume.

Below is a formal definition of a typical physical object (TPO), integrating the previously discussed framework of worlds, mereology, and the classification of worlds as temporal instants and intervals, spatial points and volumes, and Lewisian possible worlds. We assume the notations and structures established:

• \(W\): the set of all worlds.
• \(W^{Lewis} \subseteq W\): the set of Lewisian possible worlds.
• For each \(w_L \in W^{Lewis}\), there exist subsets \(W^{time}(w_L) \subseteq w_L\) representing temporal worlds (instants or intervals within \(w_L\)) and \(W^{space}(w_L) \subseteq w_L\) representing spatial worlds (points or volumes within \(w_L\)).
• Each individual \(d \in D\) is instantiated in some subset of worlds. Write \(Inst(d,w)\) for “\(d\) is instantiated at world \(w\)”, with truth values in \(\{\text{true},\text{false}\}\).
• \(\leq_w\) is the mereological parthood relation on worlds, and it obeys Classical Extensional Mereology (CEM).
• The temporal worlds in \(W^{time}(w_L)\) form a linear order \((W^{time}(w_L), <_t)\) and the spatial worlds in \(W^{space}(w_L)\) inherit a structure from \(\mathbb{R}^3\) or a similar topological/metric space, making it meaningful to talk about “contiguous” or “connected” spatial regions.

Definition (Typical Physical Object):
An individual \(d \in D\) is a typical physical object (TPO) if and only if it satisfies the following conditions in every Lewisian possible world \(w_L \in W^{Lewis}\):

1. Temporal Contiguity Condition:
   Consider the set of temporal worlds within \(w_L\) at which \(d\) is instantiated: \[ T_d(w_L) := \{ w_t \in W^{time}(w_L) : Inst(d, w_t) = \text{true} \}. \] The set \(T_d(w_L)\) must be either empty or form a single contiguous (convex) interval under the temporal ordering \(<_t\). Formally:

   • If \(w_t, w_t' \in T_d(w_L)\) and \(w_t <_t w''_t <_t w_t'\) for some \(w''_t \in W^{time}(w_L)\), then \(w''_t \in T_d(w_L)\).

   In other words, there are no temporal gaps: once the object appears in time, it continues without interruption until it ceases to exist in that Lewisian world.

2. Spatial Contiguity Condition at Each Instant:
   For each temporal instant \(w_t \in T_d(w_L)\), consider the spatial worlds that \(d\) occupies at that instant. Define: \[ S_d(w_t) := \{ w_s \in W^{space}(w_L) : Inst(d, w_s) = \text{true} \text{ and } w_s \leq_w w_t \}. \] Here, \(S_d(w_t)\) represents the spatial region (or set of spatial points and volumes) occupied by \(d\) at the temporal instant \(w_t\).

   The requirement is that at each such instant \(w_t\), the occupied spatial region is a single mereologically connected volume. Formally:

   • Let \(w_{space}(w_t) = \bigoplus S_d(w_t)\) be the mereological fusion of all spatial worlds in \(S_d(w_t)\).
   • The region \(w_{space}(w_t)\) must be connected in the mereological sense: there is no way to partition \(w_{space}(w_t)\) into two nonempty disjoint subregions \(w_{s1}\) and \(w_{s2}\) such that: \[ w_{space}(w_t) = w_{s1} \oplus w_{s2}, \quad w_{s1} \cap w_{s2} = \emptyset, \quad w_{s1}, w_{s2} \neq \emptyset. \] In spatial terms, the object does not appear as multiple disjoint “blobs” scattered in space at the same instant. It always occupies a single continuous “chunk” of space.

In Summary:
A TPO is an individual whose mereological parts appear, within any given Lewisian world, as a single continuous presence that stretches through time as one contiguous interval, and at each instant within that interval, appears as a single continuous spatial volume. This definition models the intuitive notion of a physical object as something that “hangs together” in space at any given time and persists continuously through some portion of time without spontaneous temporal gaps.

I have made two posts branching from this point in the conversation:

• Modality and Identity
• Degree Theory and Continuous Change