Question

Consider a language with identity and function symbols, and interpret a \(n\)-ary symbol \(F\) by a function \(F_{k}: D(k)^{n} \rightarrow D(k)\) for each \(k\) in a given Kripke model \(\mathcal{K}\). We require monotonicity: \(k \leq l \Rightarrow F_{k} \subseteq F_{l}\) and preservation of equality, where \(a \sim_{k} b \Leftrightarrow k \Vdash \bar{a}=\bar{b}_{1}: \vec{a} \sim_{k} \vec{b} \Rightarrow F_{k}(\vec{a}) \sim_{k}\) \(F_{k}(\vec{b})\). (i) Show \(K \Vdash \forall \vec{x} \exists ! y(F(\vec{x})=y)\) (ii) Show \(\mathcal{K} \| I_{4}\). (iii) Let \(\mathcal{K} \|-\forall \vec{x} \exists ! y \varphi(\vec{x}, y)\), show that we can define for each \(k\) and \(F_{k}\) satisfying the above requirements such that \(\mathcal{K} \| \forall(\vec{x} \varphi(\vec{x}, F(\vec{x}))\). (iv) Show that one can conservatively add definable Skolem functions. Note that we have shown how to introduce functions in Kripke models, when they are given by "functional" relations. So, strictly speaking, Kripke models with just relations are good enough.

Short answer

(i) Function \( F \) ensures unique output \( y \). (ii) \( \mathcal{K} \| I_4 \): equations hold by definition. (iii) Define \( F_k \) for any Kripke condition. (iv) Define Skolem functions conservatively in \( \mathcal{K} \).

Step-by-step solution

Analyze Part (i) Requirements

We need to show that in the Kripke model \( \mathcal{K} \), for every point \( k \), if you substitute any \( n \)-tuple \( \vec{x} \), there is a unique \( y \) such that \( F(\vec{x}) = y \). This involves demonstrating that \( F \) indeed provides a single output \( y \) for each input \( \vec{x} \) while satisfying Kripke semantics.

Demonstrate Unique Existential Quantification

For each \( k \), because \( F_k \) is interpreted as a function, given any input \( \vec{x} \), it produces exactly one output \( F_k(\vec{x}) \). This implies \( \forall \vec{x} \exists y \), that for every input, there exists an output \( y \) such that \( F(\vec{x}) = y \). The uniqueness comes from the function's definition: \( \forall \vec{x}, \exists ! y \equiv \forall \vec{x}, (\exists y)(F(\vec{x}) = y \land \forall z, (F(\vec{x}) = z \Rightarrow y = z)) \). Therefore, the uniqueness \( \exists ! \) condition is satisfied in \( \mathcal{K} \).

Interpret Part (ii) and Part (iii)

For Part (ii), \( \mathcal{K} \| I_4 \) implies that the Kripke model satisfies a certain set of logical formulae or theories, specifically involving equality preservation and functional interpretation as required by question (i). For Part (iii), you must constructively demonstrate within \( \mathcal{K} \), that we can define functions by notice that for each \( k \), the function \( F_k \) is satisfying the relation where \( \phi(\vec{x}, F(\vec{x})) \equiv (F(\vec{x}) = F_k(\vec{x})) \). This uses the fact that \( \varphi \) represents some functional relationship regarding outputs rather than a loose relation.

Prove Existence of Skolem Functions for Part (iv)

In Kripke models, definable Skolem functions are introduced to guarantee existence constructs are consistent. You must show that starting from relations, you replace existential quantifiers with function symbols (while adhering to monotonicity and preservation of equality). Since you have monotonic models, Skolem functions will remain consistent due to their dependence solely on below relations in the frame order.

Key concepts

Functional Interpretation

The concept of functional interpretation in Kripke models involves assigning specific functional meanings to symbols within the framework of logic. In this context, we explore a symbolic function, denoted as \( F \), which is interpreted by a sequence of actual functions \( F_k \), each defined at a particular point \( k \) in the Kripke model \( \mathcal{K} \).

• This approach requires that for each \( k \), the function \( F_k \) maps a certain domain \( D(k)^n \) to \( D(k) \). It ensures that symbolic functions have a clear and deterministic outcome for each input.
• The challenge lies in corroborating that this interpretation satisfies the particular logical formulas, such as ensuring uniqueness of outputs under conditions defined by the Kripke semantics.

In essence, functional interpretation checks the applicability of logical statements within the structural constraints of Kripke models.

Monotonicity

Monotonicity in Kripke models refers to the idea that as we move from one point \( k \) to another point \( l \) (with \( k \leq l \)) within the model, the interpretation of the function \( F \) should remain consistent or enlarge.

• This means that the application of \( F_k \) is a subset of \( F_l \), implying functions become more inclusive or maintain the outputs as they progress through possible worlds.
• In logical terms, this ensures that any provable facts or derived outputs relative to \( k \) still hold at \( l \), an assurance of logical harmony across the model.

Monotonicity is vital for ensuring that logical interpretations do not lose validity as they get extended through the Kripke model's structure.

Skolem Functions

Skolem functions in the context of logic are used to eliminate existential quantifiers by replacing them with function symbols, asserting that a function exists which can provide witnesses for these existential statements.

• In a Kripke model, introducing Skolem functions involves demonstrating that functions adhere to constructs defined under existential quantification, transforming complex relational expressions into statements about functions.
• The process involves checking whether the foundational conditions of monotonicity and equality preservation are satisfied, ensuring that these functions remain logically consistent throughout the Kripke structure.

Introducing Skolem functions is fundamentally about maintaining the logic's expressiveness while ensuring that its statements remain practically interpretable and manageable.

Existential Quantification

Existential quantification is a concept from logic that expresses the existence of at least one element satisfying a given property within a domain. In Kripke models, it takes a slightly different form given the model's possible world configuration.

• When a statement involves existential quantification, it asserts that there is some choice of other-worldly parameters (beyond the apparent pointed world) aligning with the property's logical constraints at \( k \).
• During interpretation, it’s paramount to make existential assertions align with the Kripke semantics thus, under the function \( F \), to ensure that every \( \vec{x} \) has a corresponding output \( y \) (unique or otherwise).

Existential quantification in Kripke models extends the idea of finding suitable witnesses, adapted to comply with the structure imposed by potential "world" relationships in the logical universe.