Kaplanian semantic model

A model of semantics must capture how even what might be called literal meaning might depend on context. For instance:

(1)
He went to the bank yesterday.

This has the context-dependent "he" and "yesterday" as well as the homonymous "bank".

One might distinguish "character" and "content". For instance, these sentences have the same character but different contents:

(2)
Mary: I am hungry.
John: I am hungry.

So sentences have a character, a content/intension, and an extension. Given a character, you resolve references, fix indexicals, and disambiguate to get the content/intension, which is a function from possible worlds to truth values. Then depending on the possible world that is assumed, predicates will have different extensions.

Write out a formal model with these features.

Below is a sketch of a Kaplan‐style formal framework (inspired by David Kaplan’s “Demonstratives” and related work) that distinguishes character, content (or intension), and extension, while also accommodating context‐sensitivity and ambiguity. This kind of framework captures how an expression’s literal meaning can depend on features of the context (e.g., speaker, time, disambiguation parameters), how one gets a full “content” (a proposition, in the case of sentences) once those parameters are fixed, and how that content then evaluates to a truth value in each possible world.

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1. The Ingredients

1.1. Worlds and Domains

1. Let \(W\) be a nonempty set of possible worlds.
2. For each world \(w \in W\), let \(D_w\) be the domain of individuals (or objects) in that world.
   • Depending on one’s semantic framework, one may either assume a single domain \(D\) across all worlds, or allow \(D_w\) to vary with \(w\). We will treat the more general case where \(D_w\) can vary.

1.2. Contexts

A context \(c\) provides all the extra‐linguistic parameters needed to interpret context‐sensitive expressions:

\[ c = \langle \text{Speaker}(c), \text{Time}(c), \text{World}^{\ast}(c), \text{Location}(c), \dots, \text{Disambiguation}(c)\rangle. \]

For example:

• \(\text{Speaker}(c)\) is the individual speaking in context \(c\).
• \(\text{Time}(c)\) is the time of utterance in context \(c\).
• \(\text{World}^{\ast}(c)\) is sometimes called the “world of the context” or the “actual world” as per that context.
• \(\text{Location}(c)\) may specify the location of utterance, if relevant.
• \(\text{Disambiguation}(c)\) can capture which sense of a homonymous word is intended (e.g., “bank” as a financial institution vs. “bank” as a river bank).

Let \(C\) be the set of all possible contexts.

1.3. Character, Content (Intension), and Extension

We will distinguish:

1. Character: A function from contexts to contents.

   • Intuitively, this is the “linguistic meaning” plus rules for how to use context‐sensitive items (indexicals like “I” or “yesterday,” pronominal references like “he,” and so on).

2. Content (or intension, in the case of sentences): A function from worlds (and possibly times, if we treat time as part of the index) to extensions.

   • For a sentence, the content will be a function from worlds to truth values (\(\{\top, \bot\}\))—i.e., a proposition.
   • For a noun phrase, the content might be a function from worlds to individuals (its referent in each world).
   • For a predicate, the content might be a function from worlds to sets of individuals (its extension in each world).

3. Extension: The result of evaluating the content at a particular world (and possibly time). For a sentence, the extension is a truth value; for a singular term, the extension is the referent (an individual) in that world, etc.

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2. A Formal Model

We can wrap these pieces up into a single framework as follows.

2.1. The Model Structure

A Kaplanian Model \(\mathcal{M}\) can be given by a tuple:

\[ \mathcal{M} = \langle W, (D_w)_{w \in W}, C, \mathcal{I}, \mathcal{Char} \rangle, \]

where

1. \(W\) is a nonempty set of possible worlds.

2. For each \(w \in W\), \(D_w\) is the domain of that world.

3. \(C\) is a set of possible contexts.

4. \(\mathcal{I}\) is an interpretation function that assigns basic (non‐context‐sensitive) expressions and constants to their contents (e.g., to individuals, sets of individuals, etc., in each world).

5. \(\mathcal{Char}\) is a function that assigns characters to (possibly complex) expressions. Each character is itself a function:

   \[ \mathcal{Char}(e) \;:\; C \;\to\; \bigl(W \to \text{ExtensionSpace}\bigr), \]

   where “\(\text{ExtensionSpace}\)” depends on syntactic category (for instance, \(\{\top,\bot\}\) for sentences, or \(\bigcup_{w\in W} D_w\) for singular terms, etc.).

In more familiar (less abstract) terms, for each expression \(e\), \(\mathcal{Char}(e)\) is the mapping: \[ \mathcal{Char}(e) : \text{context } c \;\;\mapsto\;\; \text{content } \bigl[\![ e ]\!]_c. \] And then each content \(\bigl[\![ e ]\!]_c\) is a function: \[\bigl[\![ e ]\!]_c : W \;\;\mapsto\;\; \text{extension } \bigl[\![ e ]\!]_{c,w}.\]

Hence, for each context \(c\in C\) and world \(w \in W\), we can speak of the extension \(\bigl[\![ e ]\!]_{c,w}\).

2.2. Illustrating the Process

1. Character \(\chi_e\): Given an expression \(e\), its character \(\chi_e\) says how to use context to determine the content.

   • For example, the character of “I am hungry” is a function that takes a context \(c\) and returns the proposition “\(\text{Speaker}(c)\) is hungry at \(\text{Time}(c)\).”

2. Content \(\bigl[\![ e ]\!]_c\): Once a context \(c\) is fixed, we get an intension (for sentences, a proposition) that can vary across worlds.

   • So \(\bigl[\![ \text{I am hungry} ]\!]_c\) is a function from \(W\) to truth values: in each \(w \in W\), it is true iff the speaker in \(c\) is hungry at the time in \(c\) in world \(w\).

3. Extension \(\bigl[\![ e ]\!]_{c,w}\): Finally, evaluating that content at a world \(w\) gives the truth value (for a sentence) or individual (for a referring expression) in that world.

   • Concretely, \(\bigl[\![ \text{I am hungry} ]\!]_{c,w} = \top\) if and only if the speaker in context \(c\) is hungry at time \(\text{Time}(c)\) in the world \(w\); otherwise it is \(\bot\).

2.3. Handling Indexicals, Pronouns, and Ambiguity

• Indexicals: “I,” “now,” “yesterday” each have a character telling us how to fetch a referent from the context \(c\). For instance: \[ \chi_{\text{I}}(c) = \lambda w.\,\text{Speaker}(c), \quad \chi_{\text{yesterday}}(c) = \lambda w.\,\text{Time}(c) - 1\,\text{day}. \] (Here we are eliding many details about how exactly times are modeled, but the idea is that “yesterday” picks out “\(\text{Time}(c)\) minus one day.”)

• Pronouns like “he” or “she” can be anaphoric or demonstrative. A simplistic approach is that their character is a function from context to a specific referent, determined by either an antecedent in discourse or a demonstration. Real theories refine how context determines referents.

• Ambiguity / Homonymy: For words like “bank,” the character would also look at the \(\text{Disambiguation}(c)\) parameter. Roughly: \[ \chi_{\text{bank}}(c) \;=\; \begin{cases} \lambda w.\,\text{river_bank} & \text{if } \text{Disambiguation}(c) = \text{river}\\ \lambda w.\,\text{financial_bank} & \text{if } \text{Disambiguation}(c) = \text{finance} \end{cases} \] This says: if the context indicates the “river” sense, interpret “bank” as \(\text{river_bank}\), etc.

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3. Examples

3.1. Example (1)

“He went to the bank yesterday.”

1. Character step: From the context \(c\), we determine:

   • The referent of “he” in \(c\) (say \(\text{HeRef}(c)\)).
   • The relevant sense of “bank” (financial institution vs. river bank).
   • The time referred to by “yesterday,” i.e. \(\text{Time}(c) - 1\) day.

2. Content (intension): A function \(\bigl[\![ \text{He went to the bank yesterday} ]\!]_c : W \to \{\top,\bot\}\), which, for each world \(w\), says whether \(\text{HeRef}(c)\) went to the contextually chosen “bank” on the day \(\text{Time}(c) - 1\) in \(w\).

3. Extension: For a particular world \(w\), the sentence is true (\(\top\)) if in \(w\), that person performed the relevant “bank‐going” event at the relevant day/time.

3.2. Example (2)

Mary: “I am hungry.”
John: “I am hungry.”

• Both sentences have the same character: the function that, given a context \(c\), picks out the speaker of \(c\) and asserts that this speaker is hungry at \(\text{Time}(c)\).
• But if Mary utters it, the context \(c_{\text{Mary}}\) has \(\text{Speaker}(c_{\text{Mary}})=\text{Mary}\).
• If John utters it, the context \(c_{\text{John}}\) has \(\text{Speaker}(c_{\text{John}})=\text{John}\).

Thus:

1. Same character: \(\chi_{\text{I am hungry}}\).

2. Different contents: \[\bigl[\![ \text{I am hungry} ]\!]_{c_{\text{Mary}}} = \lambda w.\,\text{Mary is hungry at Time}(c_{\text{Mary}})\text{ in }w.\]\[\bigl[\![ \text{I am hungry} ]\!]_{c_{\text{John}}} = \lambda w.\,\text{John is hungry at Time}(c_{\text{John}})\text{ in }w.\]

3. Extension in a specific world \(w\): Each of these contents could be true or false depending on whether Mary or John is hungry in \(w\) at that time.

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4. Summary of the Formal Steps

1. Define a set of contexts \(C\). Each context \(c\) carries all relevant parameters for interpreting indexical and ambiguous expressions (speaker, time, location, “disambiguation,” etc.).
2. Define a set of worlds \(W\) and for each \(w \in W\), a domain \(D_w\).
3. Assign a character \(\chi_e\) to each expression \(e\). The character is a function: \[ \chi_e : C \;\to\; (W \;\to\; \text{ExtensionSpace}). \]
4. Given a context \(c\), the content of \(e\) in \(c\) is \(\bigl[\![ e ]\!]_c := \chi_e(c)\), which is an intensional object (a function from worlds to extensions).
5. Given \(c\) and \(w\), the extension of \(e\) is \(\bigl[\![ e ]\!]_{c,w} := \chi_e(c)(w)\).

This completes a high‐level, formal outline of a semantic model that captures how literal meaning (character) can depend on context, how that yields a context‐fixed content (intension), and how evaluation in a given world yields an extension (truth value or referent).

Russian Nobility and Enlightenment

strange question possibly but was the russian nobility "enlightened"

The Russian nobility during the 18th and 19th centuries was, in some ways, influenced by the ideas of the European Enlightenment, but whether they were truly "enlightened" depends on how one defines the term and which aspects of the nobility one examines.

Enlightenment Influences

1. Education and Ideas:

   • Many Russian nobles were well-versed in the writings of European Enlightenment thinkers like Voltaire, Montesquieu, and Rousseau. Some even corresponded with these figures (e.g., Catherine the Great famously exchanged letters with Voltaire and considered herself an "enlightened despot").
   • Western European education and philosophy were fashionable among the Russian elite, leading to reforms and intellectual movements.

2. Reforms:

   • During Catherine the Great’s reign, some Enlightenment principles, like the codification of laws and limited attempts to modernize institutions, were introduced. Catherine's "Nakaz" (Instruction) of 1767 reflected her engagement with Enlightenment ideas, emphasizing rational governance and individual rights, albeit in a limited way.
   • Alexander I also flirted with reforming Russian society and government along Enlightenment lines, such as plans to abolish serfdom, though these efforts were largely unrealized.

3. Cultural Flourishing:

   • Russian nobles played a key role in fostering literature, science, and art, aligning with Enlightenment values of promoting human achievement and reason. Figures like Pushkin and Tolstoy were products of this cultivated environment.

Limitations and Contradictions

1. Serfdom:

   • The Russian nobility was deeply reliant on serfdom, a system fundamentally at odds with Enlightenment ideals of liberty and equality. Most nobles were not willing to sacrifice their privileges to liberate the peasants, even as they debated abstract notions of freedom and justice.
   • Serfdom persisted until 1861, despite calls for reform by certain enlightened individuals within the nobility.

2. Autocratic Context:

   • The Russian nobility operated within an autocratic system that did not prioritize the broad dissemination of Enlightenment principles. While some nobles supported reforms, they generally did so in ways that preserved their own power and privilege.

3. Selective Adoption:

   • The nobility often adopted Enlightenment ideas superficially, embracing the cultural and intellectual prestige of the movement while neglecting its revolutionary implications.

Examples of Enlightened Nobles

• Prince Dmitry Golitsyn and Count Nikolay Novosiltsev were known for their reformist tendencies.
• Alexander Herzen and Pyotr Chaadaev were critical of Russia’s social order and sought change inspired by Western ideas.

Conclusion

While many Russian nobles were exposed to and influenced by the Enlightenment, their actual "enlightenment" was uneven and constrained by their social and economic interests. Enlightenment ideals were often subordinated to maintaining the status quo of autocracy and serfdom, making their "enlightened" status both complex and debatable.

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!