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