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Ethical systems, despite their apparent complexity and variation, can be modeled analytically. The set of possible moral actions in any given context can be represented as a discrete set of context-action pairs, where each pair corresponds to a conditional maxim governing behavior. Even if ethical decision-making appears continuous, it can be expressed as a function $f(x_1,x_2,…,x_n)$, demonstrating that ethical reasoning is reducible to a parametrizable and structured system.

The claim that all ethics are analytical follows from the observation that ethical choices, whether grounded in deontological rules, virtue-based assessments, or existential commitments, can be mapped onto discrete decision frameworks. At any given time, an individual’s moral reasoning is confined to a finite set of value-driven maxims that determine their choices. The apparent fluidity of moral thought does not negate its discrete nature but rather reflects the complexity of its parameterization.

Some may claim that ethical reasoning is inherently vague and resists analytic formulation. Virtue ethics, for instance, relies on context-sensitive judgment, and existentialist ethics prioritizes subjective experience over rigid principles. However, these ethical models remain reducible to structured sets. Virtue ethics, despite its reliance on practical wisdom $(\varphi)$, can be decomposed into a function $f(x)$ where $x$ represents relevant contextual inputs. Similarly, existentialist ethics, though emphasizing subjective choice, still operates within a bounded set of maxims at each moment, forming a discrete decision space.

Others argue that overlapping moral maxims introduce irreducible vagueness, making formal analysis untenable. However, overlapping maxims can be described using $n^{th}$-dimensional Gaussian surfaces, which model the probability distribution of moral decisions across a given set of parameters. Alternatively, a discrete piecewise function can capture the segmentation of moral principles based on context. Thus, moral uncertainty does not imply non-analyticity; rather, it suggests a need for probabilistic or structured modeling.

A further objection is that moral intuition and creativity transcend computational representation. Human moral reasoning appears to involve novel synthesis rather than simple function evaluation. However, the universal approximation theorem suggests that any continuous model of ethical reasoning can be approximated by a sufficiently complex function $f(x_1, x_2, …, x_n)$. Therefore, moral reasoning, even when it appears to evolve, operates within a computable space. New ethical insights or principles do not emerge ex nihilo but are instead refinements of existing parameterized structures.

Reduction of Virtue Ethics and Care Ethics to Deontology

Both virtue ethics and care ethics can be reduced to deontology using set operations. Define a set of context-action pairs $(C, A)$ where $C$ represents the contextual parameters and $A$ represents a possible action. A function $g: C \times A \to \{0,1\}$ defines the acceptability of an action, where $g(C, A) = 1$ indicates moral acceptability and $g(C, A) = 0$ indicates moral unacceptability.

For virtue ethics, let $V$ be the set of virtues $\{v_1, v_2, …, v_n\}$, and let $h: C \to V$ map context to the virtue required in that situation. Then, we define acceptability as: $g(C, A) = 1 \iff A \in D(h(C))$ where $D(v)$ is the set of deontologically permissible actions corresponding to virtue $v$. This establishes a mapping from virtue-based assessments to deontological constraints.

For care ethics, let $R$ be the set of relationships $\{r_1, r_2, …, r_m\}$, and let $k: C \to R$ map context to the relevant relationship. Define a function $p: R \to \mathcal{P}(A)$ mapping relationships to permissible actions. Then, the acceptability of an action is given by: $g(C, A) = 1 \iff A \in p(k(C))$ This formulation embeds care-based reasoning into a deontological framework where relationships determine the binding duties.

The ethical decision surface can be modeled as an $n^{th}$-dimensional function $F: \mathbb{R}^n \to [0,1]$, where the probability of an action being permissible is given by a Gaussian function: $F(x_1, …, x_n) = \exp\left(-\sum_{i=1}^{n} \frac{(x_i – \mu_i)^2}{2 \sigma_i^2}\right)$ where $\mu_i$ represents the idealized virtue or relationship parameter, and $\sigma_i$ represents the variance in moral judgment. This allows for a smooth probabilistic transition between different ethical judgments while maintaining an analytically structured framework.

If all ethical systems can be expressed analytically, this raises meta-ethical implications. It suggests that moral realism, wherein ethical truths exist as formalizable structures, may be more tenable than moral relativism. Alternatively, it supports constructivism, wherein ethical systems, though human-generated, remain reducible to structured sets of decision rules. In either case, ethical reasoning, while complex, is ultimately governed by definable, parameterized rules, affirming that all ethics are analytical.