Laws and Phenomena

This post continues the conversation from this other post.

Let’s use our framework to formalize the idea that for any worlds that differ in physical laws, they also differ in perceptual character, that is, the character that they present to the senses. We can think of this as there being a correspondence between propositions about physical laws and propositions about perceptual character, such that, if a proposition about physical laws makes two worlds equivalent (both worlds give the same value to all individuals), then the corresponding propositions about perceptual character must also make those worlds equivalent.

Below is one way to capture, in our matrix-based semantics, the idea that whenever worlds differ in physical laws, they also differ in perceptual character. Informally, we will say there is a correspondence between propositions about physical laws and propositions about perceptual character such that any two worlds that are indistinguishable (i.e., assign the same truth-values to all individuals) with respect to the physical-laws proposition must also be indistinguishable with respect to the corresponding perceptual-character proposition.

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1. Background and Setup

Recall our framework:

• \( W \) is a set of worlds.
• \( D \) is a set of individuals.
• A (unary) proposition \(P\) is represented by a matrix \[ M_P: W \times D \;\to\; \{0,1\} \] where \(M_P[w,d] = 1\) means \(P\) is true of individual \(d\) in world \(w\), and \(M_P[w,d] = 0\) otherwise.

Equivalence of worlds relative to a proposition

Given a proposition \(P\), we say that two worlds \(w\) and \(w'\) are \(P\)-equivalent (written \( w \sim_P w' \)) just in case \(P\) assigns the same truth values to all individuals in those worlds. Formally,

\[ w \sim_P w' \quad \Longleftrightarrow \quad \forall d \in D,\; M_{P}[w,d] \;=\; M_{P}[w',d]. \]

Intuitively, \( w \sim_P w' \) means that there is no individual \(d\) about whom \(P\) differs in \(w\) and \(w'\).

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2. Propositions About Physical Laws vs. Perceptual Character

Let us divide our unary propositions into two families:

1. Physical-Laws Propositions: Denote these by \( \mathcal{L} = \{ L_i \mid i \in I\} \).

   • Each \(L_i\) is intended to capture some aspect of the “laws of nature” holding (or not) for each individual in each world.
   • Matrix representation: \( M_{L_i}[w, d] \in \{0,1\} \).

2. Perceptual-Character Propositions: Denote these by \( \mathcal{C} = \{ C_i \mid i \in I\} \).

   • Each \(C_i\) is intended to capture a particular aspect of “perceptual character” (how things seem perceptually to each individual in each world).
   • Matrix representation: \( M_{C_i}[w,d] \in \{0,1\} \).

We assume there is some mapping (or “correspondence”) \(\phi: \mathcal{L} \to \mathcal{C}\) associating each physical-laws proposition \(L_i\) with a perceptual-character proposition \(C_{\phi(i)}\).

In many philosophical discussions, physical laws (broadly construed) determine or constrain the phenomenal experiences or perceptual characters that individuals can have. Our aim is to say: if two worlds truly differ with respect to a law \(L_i\), then they must also differ in the corresponding perceptual property \(C_{\phi(i)}\). Equivalently, if two worlds appear indistinguishable regarding \(L_i\) (they give the same truth values for all individuals under \(L_i\)), they must also be indistinguishable for \(C_{\phi(i)}\).

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3. Formal Statement of the Dependence

3.1 Basic Statement

For each physical-laws proposition \(L_i\) and its corresponding perceptual proposition \(C_{\phi(i)}\), we require that:

\[ \forall w,w' \in W: \quad  \Bigl( \underbrace{\,\forall d\in D,\; M_{L_i}[w,d] = M_{L_i}[w',d]\,}_{\text{\(w\) and \(w'\) are \(L_i\)-equivalent}} \Bigr) \;\Longrightarrow\; \Bigl( \underbrace{\,\forall d\in D,\; M_{C_{\phi(i)}}[w,d] = M_{C_{\phi(i)}}[w',d]\,}_{\text{\(w\) and \(w'\) are \(C_{\phi(i)}\)-equivalent}} \Bigr). \]

This condition says: whenever \(L_i\) fails to distinguish two worlds \(w\) and \(w'\), the corresponding proposition \(C_{\phi(i)}\) also fails to distinguish them. Equivalently, if \(w\) and \(w'\) are in the same equivalence class under \(L_i\), they must also lie in the same equivalence class under \(C_{\phi(i)}\).

3.2 Contraposition

By contraposition, we can read it as:

\[ \forall w,w' \in W: \quad \Bigl(\exists d\in D,\; M_{C_{\phi(i)}}[w,d] \neq M_{C_{\phi(i)}}[w',d]\Bigr) \;\Longrightarrow\; \Bigl(\exists d\in D,\; M_{L_i}[w,d] \neq M_{L_i}[w',d]\Bigr). \]

In words: if two worlds differ in the perceptual-character proposition \(C_{\phi(i)}\), they must differ in the physical-laws proposition \(L_i\).

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4. Intuitive Meaning

Putting it more colloquially:

• Physical difference \(\implies\) Perceptual difference. If you pick a particular way in which physical laws might differ across worlds (encoded by \(L_i\)), and you find that there are indeed two worlds \(w\) and \(w'\) that produce different truth values under \(L_i\) for some individual, then you should also find a corresponding difference in their perceptual character (\(C_{\phi(i)}\)) for at least one individual.
• No difference in perceptual character \(\implies\) no difference in physical laws. If you cannot find any difference in the perceptual character assigned by \(C_{\phi(i)}\) to two worlds \(w\) and \(w'\), then you cannot find any difference in the physical laws assigned by \(L_i\) either.

This formalizes (at least in a simplified, first-order modal setting) the philosophical view that how things appear cannot diverge unless the underlying physical facts (laws) also diverge, and conversely, that any difference in the laws must manifest itself in some difference of appearance.

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5. Beyond Unary Propositions

If one wanted to incorporate more complex statements—e.g., relational claims (binaries or \(n\)-ary predicates) about physical laws or perceptual states—the same matrix approach generalizes naturally to higher-dimensional arrays:

• For a binary predicate \(P\), one works with \(M_P: W \times D \times D \to \{0,1\}\).
• The definitions of world-equivalence relative to \(P\) and the requirement that \(P\)-equivalence imply (or be implied by) \(Q\)-equivalence follow a similar pattern, only now one checks all relevant tuples \((d_1, d_2)\).

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6. Conclusion

In summary, to say that “for any worlds that differ in physical laws, they also differ in perceptual character” is to impose on our two-dimensional (worlds \(\times\) individuals) semantics the constraint:

\[ \forall i \in I,\quad \forall w,w' \in W:\quad (w \sim_{L_i} w') \;\Longrightarrow\; (w \sim_{C_{\phi(i)}} w'). \]

This axiom captures, in the language of matrices and equivalence classes of worlds, a philosophical principle linking differences in the laws of nature to differences in phenomenal or perceptual aspects of experience.

Degree Theory and Continuous Change

This post continues from this other post.

Now expand the semantics to allow for degree-theoretic truth values (any real value from 0 to 1) and write a theory of continuous change over time where a predicate continuously changes from one extreme to the other

Below is a sketch of how to extend the framework of propositions-as-matrices (or functions) to a degree-theoretic or fuzzy semantics, where truth values range continuously in the real interval \([0,1]\). We then apply this to model continuous change over time, illustrating how a predicate’s truth value can vary continuously from one extreme to another.

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1. Degree-Theoretic Semantics

1.1 From \(\{0,1\}\) to \([0,1]\)

In the standard two-valued setting, each proposition \(P\) was a function \[ P : W \times D \,\longrightarrow\, \{0,1\}. \] We now replace \(\{0,1\}\) with the real interval \([0,1]\). Thus, a (unary) proposition becomes \[ P : W \times D \,\longrightarrow\, [0,1]. \] Intuitively, \(P(w,d)\) measures the degree to which \(P\) holds of individual \(d\) at world \(w\). A value of \(0\) means “completely false,” and a value of \(1\) means “completely true,” with intermediate values for partial truths.

1.2 Matrix Representation

We can still represent such a proposition \(P\) by a matrix \(M_P\), whose rows index possible worlds in \(W\) and whose columns index individuals in \(D\). The difference now is that each cell is a real number between 0 and 1: \[ M_P[w,d] \in [0,1]. \]

1.3 Connectives: Fuzzy Logical Operations

Many options exist for defining fuzzy logical connectives (e.g., Łukasiewicz, Gödel, Product t-norms, etc.). A common, simple choice is to use the minimum and maximum operations:

• Negation (\(\neg P\)):
  \[ (\neg P)(w,d) = 1 - P(w,d). \]

• Conjunction (\(P \wedge Q\)):
  \[ (P \wedge Q)(w,d) = \min\bigl(P(w,d),\, Q(w,d)\bigr). \]

• Disjunction (\(P \vee Q\)):
  \[ (P \vee Q)(w,d) = \max\bigl(P(w,d),\, Q(w,d)\bigr). \]

• Implication (\(P \to Q\)):
  In many fuzzy logics,
  \[ (P \to Q)(w,d) = \max\bigl(1 - P(w,d),\, Q(w,d)\bigr). \] (Łukasiewicz fuzzy logic uses an alternative formula, but the above is a common choice.)

Other definitions are possible; the exact choice of fuzzy connectives depends on which fuzzy logic one adopts.

1.4 Quantifiers

For quantification, we typically interpret:

• Universal Quantification (\(\forall x\,P\)): the degree of truth at world \(w\) is the infimum (or minimum, if it exists) of \(P(w,d)\) over all \(d \in D\): \[ (\forall x\,P)(w) \;=\; \inf_{d\in D} P(w,d). \]

• Existential Quantification (\(\exists x\,P\)): the degree of truth at world \(w\) is the supremum (or maximum) of \(P(w,d)\) over all \(d \in D\): \[ (\exists x\,P)(w) \;=\; \sup_{d\in D} P(w,d). \]

These definitions capture the intuition that “\(\forall x\,P\)” is “as true as it is true for every \(d\)” (which is the minimum or inf), while “\(\exists x\,P\)” is “as true as it is true for at least one \(d\)” (maximum or sup).

1.5 Modal Operators

If we extend the framework to fuzzy modal operators (necessity \(\Box\), possibility \(\Diamond\)), we similarly shift to an interpretation based on inf and sup over the accessible worlds:

• Necessity:
  \[ (\Box P)(w,d) \;=\; \inf \{\,P(w',d)\,\mid R(w,w') \}, \] where \(R\) is the accessibility relation.

• Possibility:
  \[ (\Diamond P)(w,d) \;=\; \sup \{\,P(w',d)\,\mid R(w,w') \}. \]

(Alternative fuzzy modal semantics exist, but these “inf/sup” clauses are a natural adaptation of Kripke semantics to fuzzy truth values.)

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2. Continuous Change over Time

Having introduced a continuum of truth values, we can now model a continuous change of a predicate’s truth value with respect to time. Assume we focus on a single Lewisian world \(w_L \in W\) that contains a set \(W^{time}(w_L)\) of temporal instants (or intervals), linearly ordered by \(<_t\).

2.1 A Predicate Varying Over Time

Let \(F: W^{time}(w_L) \times D \to [0,1]\) be a proposition (or property) capturing how strongly \(F\) applies to each individual over time. For simplicity, fix a particular individual \(d_0 \in D\), and consider the one-dimensional function: \[ t \;\mapsto\; F(t, d_0), \] where \(t \in W^{time}(w_L)\). Since we treat each \(t\) as a “world” or “instant,” \(F(t,d_0)\) is a real number in \([0,1]\).

2.2 Continuity Conditions

To say that \(F(t,d_0)\) changes continuously over time, we impose a topological or metric structure on \(W^{time}(w_L)\). For instance, interpret each \(t\) as a real number (or an interval in a real line). Then require:

1. Continuity: If \(\varphi: \mathbb{R} \to [0,1]\) is the real-valued function that extends \(t \mapsto F(t, d_0)\) (viewing each temporal instant \(t\) as a real number), then \(\varphi\) is continuous in the usual sense:
   \[ \lim_{t' \to t} \varphi(t') = \varphi(t) \quad\text{for every } t. \]
2. Boundary Conditions (for a shift from “completely false” to “completely true,” e.g., 0 to 1): Suppose we want a scenario in which at early times the property is nearly false, and at later times it is nearly true. We can specify: \[ \lim_{t \to t_{\text{start}}^-} \varphi(t) = 0 \quad\text{and}\quad \lim_{t \to t_{\text{end}}^+} \varphi(t) = 1, \] ensuring a smooth (continuous) transition from 0 to 1.

2.3 Example: Heating Up

Imagine \(F(t,d_0)\) is the predicate “object \(d_0\) is hot”, with values from 0 (totally cold) to 1 (fully hot). Over time, the object is placed in an oven and its temperature gradually rises. We model this as a continuous function \(\varphi(t)\):

• At \(t_0\) (the start), \(\varphi(t_0) \approx 0\) (the object is very cold).
• Over the interval \([t_0, t_1]\), \(\varphi(t)\) increases monotonically from 0 to 1.
• By \(t_1\), \(\varphi(t_1) \approx 1\) (the object is fully hot).

The function \(\varphi\) might be something like:

\[ \varphi(t) \;=\; \begin{cases} 0 & t \leq t_0, \\ \text{some smooth increasing function} & t_0 < t < t_1, \\ 1 & t \geq t_1. \end{cases} \]

In each case, \(\varphi\) remains continuous, capturing the intuition that “hotness” does not jump abruptly from 0 to 1, but passes through all intermediate values in \((0,1)\).

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3. Integrating with the Theory of Worlds and Mereology

1. Worlds: We still have a set \(W\) of worlds (possibly including Lewisian worlds, times, spaces, etc.). In each world \(w \in W\), a proposition can have a real-valued truth degree.

2. Individuals: The parthood relation on individuals remains classical mereology. Each individual \(d\) can be extended in time, and at each instant \(t\) we have a (possibly) different real-valued truth for each property.

3. Continuous Trajectories: When we say “a property changes continuously,” we rely on the fact that we treat temporal worlds \(\{w_t\}\) as points in a continuum (e.g. \(\mathbb{R}\)), and the map \(w_t \mapsto P(w_t, d)\) is a continuous function into \([0,1]\). This is a natural extension of classical possible worlds semantics to a fuzzy temporal semantics with continuity constraints.

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4. Conclusion

By allowing truth values in \([0,1]\) and imposing continuity conditions over an ordered set of temporal worlds, we obtain a framework in which properties (or propositions) can gradually transition between degrees of truth, rather than flipping abruptly from 0 to 1. This combines:

• Fuzzy or degree-theoretic semantics for partial truth,
• Classical or fuzzy connectives to combine propositions,
• Quantification by inf/sup over individuals,
• Temporal structure to model continuous variation of truth values over time.

Hence, we have a robust formal theory that accommodates both gradual changes of properties and the standard logical apparatus of (fuzzy) connectives and quantifiers.

What is argmax?

what is argmax? explain, give examples

The term argmax stands for "argument of the maximum." It is a concept commonly used in mathematics, computer science, and machine learning, particularly in optimization and decision-making problems.

Explanation

• Definition: The argmax of a function is the input value (or values) where the function achieves its maximum output.
• Notation: \[ \text{argmax}_{x} f(x) \] This means "the value of \( x \) that maximizes \( f(x) \)."

Unlike \(\max f(x)\), which gives the maximum value of the function, \(\text{argmax}\) gives the input \( x \) that produces this maximum value.

Examples

Example 1: Simple Numeric Array

Suppose you have an array: \[ \text{A} = [3, 1, 4, 1, 5, 9, 2] \] The maximum value is 9, and its index (position) is 5 (0-based indexing).

• \(\max(\text{A}) = 9\)
• \(\text{argmax}(\text{A}) = 5\)

Example 2: Mathematical Function

Consider the function: \[ f(x) = -x^2 + 4x \] To find \(\text{argmax}_x f(x)\), we solve for \( x \) where \( f(x) \) is maximized. First, take the derivative and set it to zero: \[ f'(x) = -2x + 4 = 0 \] \[ x = 2 \] The maximum value of \( f(x) \) occurs at \( x = 2 \), so: \[ \text{argmax}_x f(x) = 2 \]

Example 3: Machine Learning (Softmax Output)

In machine learning, argmax is often used in classification tasks to determine the predicted class.

For example, consider the output of a softmax layer: \[ \text{P} = [0.1, 0.3, 0.6] \] The values represent probabilities for three classes. The maximum probability is \( 0.6 \), which corresponds to class 2 (0-based indexing). Thus: \[ \text{argmax}(\text{P}) = 2 \]

Applications

1. Machine Learning:
   • Used to select the class with the highest predicted probability in classification models.
2. Optimization:
   • Finds the value of a variable that maximizes a given objective function.
3. Game Theory and Economics:
   • Determines strategies that yield the highest payoff.

Key Differences: argmax vs. max

• argmax gives the input \( x \).
• max gives the output value \( f(x) \).

Let me know if you'd like to see more examples or specific applications!

Best of All Possible Worlds

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:

1. \( W \) is non-empty.
2. \( W \) is pairwise compossible: for all \( c, c' \in W \), \( c \mathrel{R} c' \).
3. 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.

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In Summary:

1. Substance Concepts: Each \( c \in \mathcal{C} \) is a complete individual concept determining a unique set of predicate truths.
2. Possible Worlds: A possible world \( W \) is a maximal set of pairwise compossible concepts.
3. 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.
4. 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.
5. 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.

Chronology of HTML Versions

chronology of html versions please

Here's a chronology of HTML versions along with key milestones:

1. HTML 1.0 (1993)

   • The first version of HTML, created by Tim Berners-Lee.
   • Basic structure: headings, paragraphs, and links.

2. HTML 2.0 (1995)

   • Standardized the features introduced in HTML 1.0.
   • Added basic features like forms, tables, and text controls.
   • First official W3C Recommendation.

3. HTML 3.2 (1997)

   • Introduced support for scripting languages like JavaScript.
   • Added stylesheets and support for tables.
   • Deprecated some older attributes like <font>.

4. HTML 4.0 (1997)

   • Introduced three "flavors" of HTML: Strict, Transitional, and Frameset.
   • Supported better separation of content and style using CSS.
   • Added support for complex tables and forms.

5. HTML 4.01 (1999)

   • Minor update to HTML 4.0 to fix bugs and inconsistencies.
   • Remained a dominant standard for years.

6. XHTML 1.0 (2000)

   • A reformulation of HTML 4.01 using XML syntax.
   • Enforced stricter rules (tags must be closed, lowercase, etc.).
   • XHTML 1.1 followed in 2001 with further refinements.

7. HTML5 (2014)

   • Introduced major enhancements like native support for audio, video, and canvas elements.
   • Improved semantics with tags like <article>, <section>, <nav>, and <footer>.
   • Enhanced support for APIs (e.g., geolocation, local storage).
   • Marked the shift towards a more flexible and interactive web.

8. HTML5.1 (2016)

   • Added refinements to HTML5, including better accessibility and new attributes.
   • Improved features like native form validation and srcset for responsive images.

9. HTML5.2 (2017)

   • Further refinements to improve security, such as support for Content Security Policy (CSP).
   • Added new attributes and better integration for modern web development.

10. HTML Living Standard (Ongoing since 2014)

    • HTML is now maintained as a "living standard" by WHATWG.
    • Continuous updates, no specific versions anymore.
    • Ensures compatibility with evolving web technologies.

Key Notes:

• After HTML5, version numbers have become less relevant as HTML is now continuously updated.
• The Living Standard approach ensures HTML evolves incrementally, without major version releases.

Let me know if you need a deeper dive into any specific version!

Regência do verbo “indagar”

Explique a regência do verbo indagar

A regência verbal trata da relação entre um verbo e seus complementos, determinando a necessidade ou não de preposição. No caso do verbo indagar, ele pode apresentar diferentes regências, dependendo do contexto e da estrutura da frase.

1. Indagar algo (sem preposição)

O verbo indagar pode ser transitivo direto, ou seja, não exige preposição, quando o complemento é o objeto direto.

• Exemplo:
  • O jornalista indagou o motivo do atraso.
  • O professor indagou a resposta do aluno.

Nesses casos, "o motivo" e "a resposta" são objetos diretos.

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2. Indagar algo a alguém (com preposição "a")

Quando há uma pessoa ou entidade a quem a pergunta é dirigida, o verbo indagar é transitivo direto e indireto ao mesmo tempo:

• O complemento direto (o que se pergunta) + complemento indireto (a quem se pergunta, introduzido pela preposição a).

• Exemplo:

  • A aluna indagou a verdade ao professor.
  • O delegado indagou os detalhes ao suspeito.

Aqui, "a verdade" e "os detalhes" são objetos diretos, enquanto "ao professor" e "ao suspeito" são objetos indiretos.

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3. Indagar sobre algo (com preposição "sobre")

O verbo indagar pode ser usado como transitivo indireto quando está acompanhado da preposição sobre. Neste caso, a ênfase está em buscar informações a respeito de algo.

• Exemplo:
  • O jornalista indagou sobre o acidente.
  • O cliente indagou sobre o prazo de entrega.

Aqui, "sobre o acidente" e "sobre o prazo" são complementos introduzidos pela preposição sobre.

---

Resumo da Regência do Verbo Indagar

1. Indagar algo → Verbo transitivo direto (sem preposição).

   • Exemplo: Ele indagou o motivo da reunião.

2. Indagar algo a alguém → Verbo transitivo direto e indireto (com a preposição "a").

   • Exemplo: Ele indagou a situação ao chefe.

3. Indagar sobre algo → Verbo transitivo indireto (com a preposição "sobre").

   • Exemplo: Ele indagou sobre o projeto.

Portanto, o verbo indagar pode ser transitivo direto, transitivo direto e indireto, ou apenas transitivo indireto, dependendo do complemento e do sentido desejado.

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.

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

Propositions as Matrices

Write out a formal theory of propositions as matrices, where the rows are possible worlds and the columns are individuals that the proposition is true or false of at that world

Below is an outline of a formal theory in which propositions are represented as two-dimensional matrices whose rows correspond to possible worlds and whose columns correspond to individuals. Each cell in the matrix encodes the truth value of the proposition for a given individual at a given world. While this framework can be aligned with standard possible worlds semantics, it extends the notion of a proposition from a simple set-of-worlds view to a richer structure accommodating both worlds and individuals simultaneously.

1. Preliminaries

1.1 Logical Vocabulary

• We consider a standard first-order modal language with:
  • A countable set of individual variables \( x, y, z, \ldots \)
  • A countable set of individual constants \( c_0, c_1, c_2, \ldots \)
  • Predicate symbols \( P, Q, R, \ldots \) of various arities (for simplicity, consider only unary predicates here).
  • Logical connectives: \(\neg\), \(\wedge\), \(\vee\), \(\rightarrow\), and modal operators \(\Box, \Diamond\).
  • Quantifiers \(\forall, \exists\).

1.2 Possible Worlds and Domains

• Let \(W\) be a nonempty set of possible worlds. Elements of \(W\) are denoted by \(w, w', w_1, \ldots\).
• Let \(D\) be a nonempty set of individuals. Elements of \(D\) are denoted by \(d, d', d_1, \ldots\).

1.3 Intended Semantics

• In traditional possible worlds semantics, a proposition \(p\) is a function \( p: W \to \{\text{true}, \text{false}\}\).
• Here, we refine this idea. Rather than just tracking the truth of a proposition at each world, we consider propositions as potentially intensional properties that may apply or fail to apply to individuals at each world. Thus, a proposition (or more accurately, a unary property proposition) will be modeled as a function: \[ P: W \times D \to \{\text{true}, \text{false}\}. \]

2. Propositions as Matrices

2.1 Matrix Representation

• We represent each proposition \(P\) as a matrix \(M_P\) with rows indexed by worlds \(W\) and columns indexed by individuals \(D\).
• For each world \(w \in W\) and each individual \(d \in D\): \[ M_P[w, d] = \begin{cases} 1 & \text{if } P(w,d) = \text{true}\\ 0 & \text{if } P(w,d) = \text{false} \end{cases} \]

2.2 Constant Domain vs. Varying Domain

• For simplicity, assume a constant domain \(D\) across all worlds. The framework can be adapted to varying domains by partiality in rows or by designating certain columns as "nonexistent" at given worlds.

2.3 Examples

• Suppose \(W = \{w_1, w_2, w_3\}\) and \(D = \{d_1, d_2\}\). Consider a proposition \(P\) such that:

  • At \(w_1\), \(P(d_1) = \text{true}, P(d_2) = \text{true}\).
  • At \(w_2\), \(P(d_1) = \text{true}, P(d_2) = \text{false}\).
  • At \(w_3\), \(P(d_1) = \text{false}, P(d_2) = \text{false}\).

  The matrix is: \[ M_P = \begin{pmatrix} 1 & 1 \\ % row for w1 1 & 0 \\ % row for w2 0 & 0 % row for w3 \end{pmatrix} \] with rows corresponding to \(w_1, w_2, w_3\) and columns to \(d_1, d_2\).

3. Logical Operations on Matrices

3.1 Boolean Connectives

• Given two propositions \(P, Q: W \times D \to \{\text{true}, \text{false}\}\) with matrices \(M_P\) and \(M_Q\), we define:
  • Negation: \((\neg P)(w,d) = \text{true}\) iff \(P(w,d) = \text{false}\). In matrix form: \[ M_{\neg P}[w,d] = 1 - M_P[w,d]. \]
  • Conjunction: \((P \wedge Q)(w,d) = \text{true}\) iff \(P(w,d) = \text{true}\) and \(Q(w,d) = \text{true}\). In matrix form: \[ M_{P \wedge Q}[w,d] = M_P[w,d] \cdot M_Q[w,d]. \]
  • Disjunction: \((P \vee Q)(w,d) = \text{true}\) iff \(P(w,d) = \text{true}\) or \(Q(w,d) = \text{true}\). In matrix form (using logical OR): \[ M_{P \vee Q}[w,d] = \max(M_P[w,d], M_Q[w,d]). \]
  • Implication: \((P \rightarrow Q)(w,d) = \text{false}\) only if \(P(w,d) = \text{true}\) and \(Q(w,d) = \text{false}\). Otherwise, it’s \(\text{true}\). In matrix terms: \[ M_{P \rightarrow Q}[w,d] = 1 \text{ unless } (M_P[w,d] = 1 \text{ and } M_Q[w,d] = 0), \text{ in which case } 0. \]

3.2 Quantifiers

• Universal Quantification: \((\forall x\, P(x))(w) = \text{true}\) iff for all \(d \in D\), \(P(w,d) = \text{true}\). In matrix form, for the row \(w\): \[ M_{\forall x\,P(x)}[w] = \begin{cases} 1 & \text{if for all } d, M_P[w,d] = 1\\ 0 & \text{otherwise} \end{cases} \] This collapses a row into a single truth value.

• Existential Quantification: \((\exists x\, P(x))(w) = \text{true}\) iff there is some \(d \in D\) for which \(P(w,d) = \text{true}\). In matrix form: \[ M_{\exists x\,P(x)}[w] = \max_{d \in D}(M_P[w,d]). \]

3.3 Modal Operators

• To interpret modal operators, we consider an accessibility relation \(R \subseteq W \times W\). For any proposition \(P\):
  • Necessity: \((\Box P)(w,d) = \text{true}\) iff for all \(w' \in W\) with \(R(w,w')\), \(P(w',d) = \text{true}\). In matrix terms, for each fixed \(d\): \[ M_{\Box P}[w,d] = \begin{cases} 1 & \text{if for all } w' (R(w,w') \implies M_P[w',d]=1)\\ 0 & \text{otherwise} \end{cases} \]
  • Possibility: \((\Diamond P)(w,d) = \text{true}\) iff there exists \(w' \in W\) with \(R(w,w')\) such that \(P(w',d) = \text{true}\). \[ M_{\Diamond P}[w,d] = \max_{\{w' \mid R(w,w')\}}(M_P[w',d]). \]

4. Identity of Propositions

4.1 Intensional Identity

• Two propositions \(P\) and \(Q\) are intensionally identical if, and only if, for every \(w \in W\) and \(d \in D\), \(P(w,d) = Q(w,d)\). Equivalently, their matrices are identical: \[ P = Q \iff M_P = M_Q. \]

4.2 Partial Extensions and Hyperintensions

• Because each proposition is a complete mapping \(W \times D \to \{0,1\}\), even finer distinctions are preserved than when we treat propositions simply as sets of worlds. Differences in how a proposition applies to individuals at each world can yield a richer notion of hyperintensional distinctions.

5. Applications and Further Extensions

5.1 Multi-place Predicates

• For \(n\)-ary predicates, we would generalize to \(P: W \times D^n \to \{\text{true},\text{false}\}\). Matrices become \(n\)-dimensional arrays indexed by worlds and \(n\)-tuples of individuals.

5.2 Varying Domains

• If the domain of individuals varies with worlds (as in some modal frameworks), we can represent this by allowing “undefined” cells or by imposing a partial function. Logical rules must then be adjusted accordingly (e.g., quantification restricted to those individuals available in each world).

5.3 Nominalism vs. Realism about Possible Worlds

• This matrix framework is neutral concerning metaphysical questions about what worlds or individuals are. It merely provides a formalism for representing and manipulating propositions as structured sets of truth values.

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Conclusion

By modeling propositions as matrices indexed by possible worlds and individuals, we create a framework that captures both the intensional dimension of modality and the extensional variation across individuals. This theory allows for a rich algebra of logical connectives and quantifiers, mirroring standard possible worlds semantics but at a finer grain. The matrix representation clarifies the internal structure of intensional entities and lends itself to rigorous treatment of logical operations, modal distinctions, and hyperintensional differences.

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

• Semantic Worlds
• Laws and Phenomena