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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.
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:
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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:
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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 \(d \in D\): \[ (\forall x\,P)(w) \;=\; \inf_{d\in D} P(w,d). \]
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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:
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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.)
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:
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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:
\[ \lim_{t' \to t} \varphi(t') = \varphi(t) \quad\text{for every } t. \] - 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 \((0,1)\).
3. Integrating with the Theory of Worlds and Mereology
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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.
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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.
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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.
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.
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