Deontic Logic for Halachic Normative Reasoning¶

Abstract¶

Standard Deontic Logic (SDL) proves insufficient for halachic reasoning due to its inability to represent graded obligation levels. This document extends SDL with madrega (normative level) annotations, yielding operators O_m(φ) representing obligation at level m. We define the madrega hierarchy (d'oraita > d'rabanan > minhag > chumra) and establish the formal interaction between deontic operators and stringency levels (lechatchila/bediavad). The resulting logic captures the nuanced normative structure of halachic discourse where the same action may be forbidden at different levels depending on its source, and where post-facto permissibility differs from ab-initio requirements.

1. Introduction¶

1.1 The Problem with Standard Deontic Logic¶

Standard Deontic Logic (SDL), originating with von Wright (1951), provides three primitive operators:

O(φ): φ is obligatory
P(φ): φ is permitted
F(φ): φ is forbidden

These operators are typically interdefinable: - F(φ) ≡ O(¬φ) - P(φ) ≡ ¬O(¬φ)

However, SDL treats all obligations as equivalent in normative force. This fails to capture the halachic distinction between:

D'oraita (Biblical): Violations carry the full weight of Torah law
D'rabanan (Rabbinic): Violations of rabbinic enactments
Minhag (Custom): Violations of established practice
Chumra (Stringency): Going beyond strict requirement

A food might be "forbidden" at one level but "permitted" at another, depending on circumstances.

1.2 Contribution¶

We extend SDL with:

Madrega-annotated deontic operators: O_m(φ), P_m(φ), F_m(φ)
Stringency modalities: lechatchila (ab initio) vs. bediavad (post facto)
Interaction rules between madrega levels and stringency
World-relativized deontic operators for multi-tradition reasoning

2. Preliminaries¶

2.1 Standard Deontic Logic¶

Definition 2.1 (SDL Syntax). The language of SDL extends propositional logic with: $$φ ::= p | ¬φ | φ ∧ ψ | O(φ) | P(φ) | F(φ)$$

Definition 2.2 (SDL Axioms). - (K): O(φ → ψ) → (O(φ) → O(ψ)) - (D): O(φ) → P(φ) [ought implies may] - (N): From ⊢ φ, infer ⊢ O(φ) [necessitation]

Definition 2.3 (Interdefinability). - P(φ) ≡ ¬F(φ) ≡ ¬O(¬φ) - F(φ) ≡ ¬P(φ) ≡ O(¬φ)

2.2 Limitations of SDL for Halacha¶

Problem 2.1 (Graded Obligation). SDL cannot distinguish: - "Eating this is Biblically forbidden" (severe) - "Eating this is Rabbinically forbidden" (less severe) - "Eating this violates custom" (still less severe)

Problem 2.2 (Contextual Permission). In SDL, P(φ) is absolute. But in halacha: - "One should not eat this ab initio" (lechatchila forbidden) - "If already mixed, one may eat it" (bediavad permitted)

Problem 2.3 (Deontic Conflicts). SDL's axiom (D) prevents O(φ) ∧ O(¬φ). But legitimate machloket may have one authority ruling O(φ) and another ruling O(¬φ).

3. Madrega-Annotated Deontic Logic¶

3.1 The Madrega Hierarchy¶

Definition 3.1 (Madrega). A madrega (normative level) is an element of the ordered set: $$\mathcal{M} = {d_oraita, d_rabanan, minhag, chumra}$$

with strict ordering >: $$d_oraita > d_rabanan > minhag > chumra$$

Definition 3.2 (Madrega Strength). For madregot m₁, m₂: $$stronger(m_1, m_2) \text{ iff } m_1 > m_2$$

The transitive closure gives: - stronger(d_oraita, d_rabanan) - stronger(d_oraita, minhag) - stronger(d_oraita, chumra) - stronger(d_rabanan, minhag) - stronger(d_rabanan, chumra) - stronger(minhag, chumra)

3.2 Madrega-Annotated Operators¶

Definition 3.3 (Annotated Obligation). For madrega m ∈ 𝓜: $$O_m(φ) \text{ denotes "φ is obligatory at level } m"$$

Similarly: - F_m(φ): φ is forbidden at level m - P_m(φ): φ is permitted at level m

Definition 3.4 (Annotated Interdefinability). - F_m(φ) ≡ O_m(¬φ) - P_m(φ) ≡ ¬F_m(φ) [at level m]

Axiom 3.1 (Madrega Monotonicity). Stronger madregot imply weaker ones: $$O_{m_1}(φ) ∧ stronger(m_1, m_2) → O_{m_2}(φ)$$

Interpretation: If something is Biblically obligatory, it is certainly Rabbinically obligatory, customarily required, etc.

Axiom 3.2 (Madrega Exclusivity at Source). A rule has exactly one source madrega: $$madrega(r, m_1) ∧ madrega(r, m_2) → m_1 = m_2$$

3.3 Semantic Structure¶

Definition 3.5 (Madrega-Annotated Deontic Model). A model is a tuple ⟨𝑊, 𝑅, 𝓜, μ, 𝑉⟩ where: - ⟨𝑊, 𝑅⟩ is a Kripke frame (from Document 01) - 𝓜 is the madrega hierarchy - μ: Rules → 𝓜 assigns madrega to rules - 𝑉: Prop × 𝓜 → 𝒫(𝑊) assigns truth values relative to madrega

Definition 3.6 (Satisfaction with Madrega). For world w, madrega m: $$\mathcal{M}, w ⊩ O_m(φ) \text{ iff } φ \text{ is required by rules of madrega } ≥ m \text{ in } w$$

4. Stringency Levels¶

4.1 Lechatchila and Bediavad¶

Definition 4.1 (Stringency Level). The stringency levels are: $$\mathcal{S} = {lechatchila, bediavad}$$

with ordering: $$lechatchila > bediavad$$

Definition 4.2 (Stringency-Qualified Operators). - O^{lech}(φ): φ is obligatory ab initio - O^{bed}(φ): φ is obligatory even post facto - P^{lech}(φ): φ is permitted ab initio - P^{bed}(φ): φ is permitted post facto

4.2 Interaction Principles¶

Axiom 4.1 (Stringency Hierarchy). $$O^{bed}(φ) → O^{lech}(φ)$$ $$P^{lech}(φ) → P^{bed}(φ)$$

Interpretation: What's required post-facto is certainly required initially. What's permitted initially is certainly permitted post-facto.

Axiom 4.2 (Lechatchila-Bediavad Gap). It is consistent that: $$O^{lech}(¬φ) ∧ P^{bed}(φ)$$

Interpretation: Something may be forbidden lechatchila but permitted bediavad. This is common in basar bechalav (meat-dairy) mixtures.

Definition 4.3 (Full Operator). Combining madrega and stringency: $$O_m^s(φ)$$ denotes "φ is obligatory at level m with stringency s."

4.3 Halachic Status Types¶

Definition 4.4 (Status Taxonomy). We define six basic statuses:

Status	Symbol	Definition
Assur (Forbidden)	F_m(φ)	O_m(¬φ)
Mutar (Permitted)	P(φ)	¬F(φ) at any level
Chiyuv (Obligatory)	O_m(φ)	Must perform
Reshut (Optional)	R(φ)	P(φ) ∧ P(¬φ)
Mitzvah (Meritorious)	M(φ)	Performance is praiseworthy
Sakanah (Dangerous)	S(φ)	Health prohibition (special status)

Theorem 4.1 (Status Exclusivity). For the same action in the same world and context: $$F_m(φ) → ¬P(φ)$$

Proof. By definition, F_m(φ) ≡ O_m(¬φ). If φ were permitted, then ¬O_m(¬φ), contradiction. □

5. World-Relativized Deontic Operators¶

5.1 Multi-World Normative Reasoning¶

Definition 5.1 (World-Relativized Deontic Operator). For world w: $$O_m^w(φ) \text{ iff } O_m(φ) \text{ holds in world } w$$

This allows different traditions to impose different obligations: - O_{d_rabanan}^{mechaber}(¬eat(fish_dairy)) [Mechaber forbids] - P^{rema}(eat(fish_dairy)) [Rema permits]

5.2 Resolving Cross-World Conflicts¶

Definition 5.2 (Deontic Machloket). A deontic machloket on φ between worlds w₁ and w₂: $$machloket(φ, w_1, w_2) \text{ iff } O_m^{w_1}(φ) ∧ O_{m'}^{w_2}(¬φ)$$

Theorem 5.1 (Non-Contradiction within World). Within a single world, SDL's (D) axiom holds: $$∀w ∈ 𝑊: ¬(O_m^w(φ) ∧ O_{m'}^w(¬φ)) \text{ at the same stringency level}$$

Proof. Mistaber enforces the disjointness constraint:

:- forbidden(W, A, F, C), permitted(W, A, F, C).

This integrity constraint prevents stable models with contradictory normative conclusions in the same world. □

5.3 Context-Sensitive Normativity¶

Definition 5.3 (Context). A context c ∈ 𝒞 is a situational modifier: $$\mathcal{C} = {ctx_normal, ctx_hefsed, ctx_shaat_hadchak, ctx_choleh, ctx_pikuach_nefesh}$$

with ordering reflecting permissiveness: $$ctx_normal < ctx_hefsed < ctx_shaat_hadchak < ctx_choleh < ctx_pikuach_nefesh$$

Definition 5.4 (Context-Qualified Operator). $$O_m^{w,c}(φ)$$ denotes "φ is obligatory at level m in world w under context c."

Axiom 5.1 (Context Relaxation). More pressing contexts may relax lower-level obligations: $$O_{d_rabanan}^{w, ctx_normal}(¬φ) ∧ ctx_hefsed → P^{w, ctx_hefsed}(φ)$$

Interpretation: A Rabbinic prohibition in normal circumstances may be permitted when significant loss (hefsed) is involved.

6. Formal Rules of Inference¶

6.1 Madrega-Based Inference¶

Rule 6.1 (Madrega Subsumption). $$\frac{O_{d_oraita}(φ)}{O_{d_rabanan}(φ)}$$

Rule 6.2 (Madrega-Safek Interaction). See Document 05 for full treatment. $$\frac{safek(O_{d_oraita}(φ))}{O(φ) \text{ (l'chumra)}}$$ $$\frac{safek(O_{d_rabanan}(φ))}{P(φ) \text{ (l'kula)}}$$

6.2 Stringency-Based Inference¶

Rule 6.3 (Bediavad Relaxation). $$\frac{O^{lech}(¬φ) \quad already_occurred(φ)}{P^{bed}(benefit(result(φ)))}$$

Interpretation: If an action was forbidden lechatchila but occurred, the result may be permissible bediavad.

6.3 Context-Based Inference¶

Rule 6.4 (Hefsed Relaxation). $$\frac{O_{d_rabanan}^{ctx_normal}(¬φ) \quad hefsed_merubeh}{may_rely_on_lenient_opinion(φ)}$$

7. Implementation Notes¶

7.1 Madrega Encoding¶

File: mistaber/ontology/base/madrega.lp

% Madrega hierarchy
stronger(d_oraita, d_rabanan).
stronger(d_rabanan, minhag).
stronger(minhag, chumra).

% Transitive closure
stronger_transitive(M1, M2) :- stronger(M1, M2).
stronger_transitive(M1, M3) :- stronger(M1, M2), stronger_transitive(M2, M3).

% Rule priority based on madrega
rule_madrega_stronger(R1, R2) :-
    rule(R1), rule(R2), R1 != R2,
    madrega(R1, M1), madrega(R2, M2),
    stronger_transitive(M1, M2).

7.2 Status Encoding¶

File: mistaber/ontology/schema/sorts.lp

% Normative levels
madrega_type(d_oraita).
madrega_type(d_rabanan).
madrega_type(minhag).
madrega_type(chumra).

% Status modalities
status(assur).
status(mutar).
status(chiyuv).
status(reshut).
status(mitzvah).
status(sakanah).

% Stringency levels
stringency_level(lechatchila).
stringency_level(bediavad).

7.3 Disjointness Constraints¶

File: mistaber/ontology/schema/disjointness.lp

% Normative levels are mutually exclusive per rule
:- madrega(R, M1), madrega(R, M2), M1 != M2.

% Cannot be both forbidden and permitted in same world/context
:- forbidden(W, A, F, C), permitted(W, A, F, C).

7.4 Rule Attribution¶

Rules carry madrega annotations explicitly:

rule(r_bb_beheima_achiila).
makor(r_bb_beheima_achiila, sa("yd:87:1")).
madrega(r_bb_beheima_achiila, d_oraita).
scope(r_bb_beheima_achiila, mechaber).

asserts(mechaber, issur(achiila, M, d_oraita)) :-
    is_beheima_chalav_mixture(M).

8. Worked Example¶

8.1 Chicken and Dairy¶

Consider the prohibition of eating poultry (of) with dairy (chalav).

Biblical Law (D'oraita): The Torah prohibits only beheima (domesticated animal) with milk: $$O_{d_oraita}(¬eat(beheima \land chalav))$$

Rabbinic Extension (D'rabanan): The Rabbis extended this to poultry: $$O_{d_rabanan}(¬eat(of \land chalav))$$

Analysis: - Eating chicken with milk violates a Rabbinic prohibition - In cases of safek (doubt), we may be lenient (safek d'rabanan l'kula) - The result of accidental mixing may be permitted bediavad with certain shiurim

Mistaber Encoding:

rule(r_bb_of_achiila).
makor(r_bb_of_achiila, sa("yd:87:3")).
madrega(r_bb_of_achiila, d_rabanan).  % Note: d_rabanan, not d_oraita

asserts(mechaber, issur(achiila, M, d_rabanan)) :-
    is_of_chalav_mixture(M).

Our madrega-annotated deontic logic builds on:

Graded modalities in deontic logic (McNamara, 2019)
Multi-agent deontic systems (Horty, 2001)
Legal formalization work (Prakken & Sartor, 2015)

The novel contribution is the specific adaptation to halachic normative structure with its unique lechatchila/bediavad distinction and four-level madrega hierarchy.

References¶

von Wright, G.H. (1951). Deontic Logic. Mind, 60(237), 1-15.
McNamara, P. (2019). Deontic Logic. The Stanford Encyclopedia of Philosophy.
Horty, J. (2001). Agency and Deontic Logic. Oxford University Press.
Prakken, H., & Sartor, G. (2015). Law and Logic: A Review from an Argumentation Perspective. Artificial Intelligence, 227, 214-245.
Carmo, J., & Jones, A.J.I. (2002). Deontic Logic and Contrary-to-Duties. In Handbook of Philosophical Logic (Vol. 8, pp. 265-343). Springer.
Makinson, D. (1986). On the Formal Representation of Rights Relations. Journal of Philosophical Logic, 15(4), 403-425.