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 $$P:W\times D⟶\{0,1\}.$$ We now replace $\{0,1\}$ with the real interval $[0,1]$. Thus, a (unary) proposition becomes $$P:W\times D⟶[0,1].$$ 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 ($\mathrm{\neg }P$):
  $$(\mathrm{\neg }P)(w,d)=1-P(w,d).$$

• Conjunction ($P\wedge Q$):
  $$(P\wedge Q)(w,d)=min(P(w,d),Q(w,d)).$$

• Disjunction ($P\vee Q$):
  $$(P\vee Q)(w,d)=max(P(w,d),Q(w,d)).$$

• Implication ($P\to Q$):
  In many fuzzy logics,
  $$(P\to Q)(w,d)=max(1-P(w,d),Q(w,d)).$$ (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 ($\mathrm{\forall }xP$): the degree of truth at world $w$ is the infimum (or minimum, if it exists) of $P(w,d)$ over all $d\in D$: $$(\mathrm{\forall }xP)(w)=\underset{d\in D}{inf}P(w,d).$$

• Existential Quantification ($\mathrm{\exists }xP$): the degree of truth at world $w$ is the supremum (or maximum) of $P(w,d)$ over all $d\in D$: $$(\mathrm{\exists }xP)(w)=\underset{d\in D}{sup}P(w,d).$$

These definitions capture the intuition that “$\mathrm{\forall }xP$” is “as true as it is true for every $d$” (which is the minimum or inf), while “$\mathrm{\exists }xP$” 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 $◻$, possibility $◊$), we similarly shift to an interpretation based on inf and sup over the accessible worlds:

• Necessity:
  $$(◻P)(w,d)=inf\{P(w^{\prime },d)\mid R(w,w^{\prime })\},$$ where $R$ is the accessibility relation.

• Possibility:
  $$(◊P)(w,d)=sup\{P(w^{\prime },d)\mid R(w,w^{\prime })\}.$$

(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 $\phi :\mathbb{R}\to [0,1]$ is the real-valued function that extends $t\mapsto F(t,d_0)$ (viewing each temporal instant $t$ as a real number), then $\phi$ is continuous in the usual sense:
   $$\underset{t^{\prime }\to t}{lim}\phi (t^{\prime })=\phi (t)\text{for every }t.$$
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: $$\underset{t\to t_{\text{start}}^{-}}{lim}\phi (t)=0\text{and}\underset{t\to t_{\text{end}}^{+}}{lim}\phi (t)=1,$$ 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 $\phi (t)$:

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

The function $\phi$ might be something like:

$$\phi (t)=\{\begin{array}{ll}0& t\le t_0,\\ \text{some smooth increasing function}& t_0<t<t_1,\\ 1& t\ge t_1.\end{array}$$

In each case, $\phi$ 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.