Can Avicenna's Proof of the Truthful (burhān al-ṣiddīqīn) be translated into contemporary modal logic without losing its metaphysical force?

Avicenna's Proof of the Truthful is among the most profound proofs in the history of philosophy of religion, and the question of its translation into contemporary modal logic touches upon precise philosophical and logical issues relating to the nature of necessity, possibility, and the limits of formal expression.

Inadequate Responses to Avoid

From some defenders of the philosophical tradition:

"Avicenna's proof is deeper than any contemporary logical form." A defensive position that ignores the value of logical formalization as a tool for clarification and revelation of argumentative structure. Formalization does not claim to exhaust philosophical meaning, but rather to reveal logical structure.

"Translating the proof into modal logic distorts Islamic philosophy." This confuses translation as an analytical tool with reductionism. One can translate the proof without claiming that the translation exhausts its philosophical content.

"The proof depends on metaphysical intuitions that transcend logic." Partially correct, but this does not negate that the proof has a logical structure amenable to analysis. Metaphysical intuitions appear in the premises, not in the rules of inference.

From some critics:

"Avicenna's proof is mere wordplay; formalization reveals this." A superficial accusation that ignores the depth of existential analysis in Avicenna. Even if a particular formalization attempt fails, this does not mean the proof is "wordplay."

"Contemporary modal logic has proven the invalidity of traditional concepts of necessity and possibility." This confuses the development of logical tools with the invalidity of philosophical concepts. Modal logic clarifies and refines; it does not "invalidate."

"Any proof formalizable in S5 is suspect." An extreme position that rejects the S5 system entirely, while it has strong philosophical justifications in the context of metaphysical necessity.

Why These Responses Are Inadequate

They ignore the precise technical question: What is lost and what is preserved when translating a classical philosophical proof into contemporary logical language? This is a question in philosophy of logic and philosophy of philosophical translation.

The Structure of Avicenna's Proof of the Truthful

The proof in its original formulation (al-Ishārāt wa-l-tanbīhāt, al-Najāt):

1. Tripartite Division of Existence: Every existent is either necessary in itself, or possible in itself and necessary through another, or possible in itself and possible through another.

2. Impossibility of Infinite Regress in Contingents: A series of pure contingents cannot be infinite without a necessary being to support it.

3. Transition from Possibility to Necessity: The existence of any contingent entails the existence of the Necessary Existent.

4. Unity of the Necessary: The Necessary Existent is necessarily one (additional proofs).

Contemporary Formalization Attempts

First Attempt: Simplified Formalization in S5

```
Definitions:
Wx: x is necessary existent
Cx: x is possible existent
Ex: x exists

Premises:
(1) ◇∃x(Ex) → ∃x(Ex) [from possibility to actual existence]
(2) ∀x(Ex → (Wx ∨ Cx)) [binary division]
(3) ∀x(Cx → ∃y(Wy ∧ Causes(y,x))) [every contingent needs a necessary]
(4) ∃x(Ex) [something exists]

Conclusion:
∃x(Wx) [a necessary existent exists]
```

Critique of this attempt: Oversimplified. It loses:
- The precise distinction between intrinsic possibility and receptive possibility
- Analysis of the meaning of "need" and "existential dependence"
- The temporal/eternal dimension of the proof

Second Attempt: More Precise Formalization (Zarepour 2022)

```
Extended Definitions:
NE(x): x is necessarily existent intrinsically
PE(x): x is possibly existent intrinsically
D(x,y): x depends existentially on y
G(X): X is a grounding set

Principles:
(P1) ∀x(PE(x) → ◇Ex ∧ ◇¬Ex)
(P2) ∀x(NE(x) → □Ex)
(P3) ∀x∀y(D(x,y) → (Ex → Ey))
(P4) ¬∃X(∀y∈X(PE(y)) ∧ G(X))

Inference through several intermediate steps to:
∃x(NE(x))
```

Assessment: Better but faces challenges:
- The concept of "grounding" is modern and not entirely Avicennian
- The relationship between logical necessity and metaphysical necessity

Third Attempt: Formalization Considering the Ontological Dimension

By Davlat (2018), attempting to integrate:
- Avicennian modal theories (essence/existence)
- Distinction between mental and extramental existence
- The concept of "priority by essence" not by time

```
Using two-dimensional modal logic:
⟨W₁, W₂, R₁, R₂, V⟩
where W₁ for logical possibilities, W₂ for metaphysical possibilities
```

Deep Philosophical Challenges

First Challenge: The Nature of Avicennian Necessity

Necessity in Avicenna is not merely logical necessity (truth in all possible worlds), but existential ontological necessity. The Necessary Existent "emanates" existence; it is not merely described by a logical predicate.

Contemporary modal logic deals with necessity as a property of propositions, while Avicenna speaks of necessity in existence itself. This difference is fundamental.

Second Challenge: The Concept of Receptive Possibility

The Avicennian contingent has two aspects:
- Intrinsic possibility (no contradiction in its essence)
- Existential need (permanent dependence on an agent)

Ordinary modal logic struggles to capture this dual dimension. Attempts to use "dependence logic" are promising but complex.

Third Challenge: Metaphysical vs. Logical Causation

Avicenna's proof depends on a metaphysical causal concept: the contingent "needs" the necessary with permanent existential need, not mere temporal precedence.

Contemporary modal logic does not primitively contain the concept of metaphysical causation. Adding it requires technical extensions (modal causal logic).

Fourth Challenge: Divine Simplicity and Logical Composition

The Necessary Existent in Avicenna is absolutely simple, while any logical formalization attributes multiple properties to it (existence, necessity, causation, etc.). How do we reconcile ontological simplicity with logical composition?

Contemporary Positions

The Optimistic Position (McGinnis, Zarepour)
Formalization is possible and useful. It reveals the logical structure of the proof, allows comparison with other proofs (Anselm, Leibniz, Gödel). Deficiency in expressing some metaphysical aspects does not negate value.

The Pessimistic Position (Wisnovsky, Adamson)
Avicenna's proof is rooted in metaphysics that cannot be reduced to formal logic. Formalization loses essential aspects: emanation, priority by essence, the mystical dimension.

The Middle Position (Lizzini, Belo)
Formalization is useful as a preliminary analytical tool, but must be completed with philosophical analysis. Modal logic reveals structure; philosophy fills content.

Evaluative Attempt

From the perspective of rational probability (rajḥān ʿaqlī), one can say:

What is Preserved in Formalization:
- The basic inferential structure
- The relationship between possibility and necessity
- The impossibility of infinite regress of contingents
- The necessity of at least one necessary existent

What is Lost or Weakened:
- The ontological richness of the concept "Necessary Existent"
- The dynamic dimension of the relationship between necessary and contingents
- Precise distinctions in types of possibility
- The connection between the proof and mystical/gnostic experience

Evaluative Conclusion:

The proof can be translated into contemporary modal logic in the sense that one can formulate a formal version that preserves the basic inferential structure and is logically valid in an appropriate system (S5 with additions).

But this translation does not preserve the complete metaphysical force of the original proof. The complete force lies in the deep existential analysis that transcends logical formalization.

This does not mean the formalization fails, but confirms what we know from philosophy of logic: formalization is a tool of clarification and analysis, not a tool for exhausting philosophical meaning. Avicenna's proof retains its philosophical value even if its logical translation is incomplete.

Application to the Site's Methodology

Within the methodology of manifestation and concealment, the proof falls within the philosophical/metaphysical path (maslik). Its amenability to formalization reflects one aspect of divine manifestation through rational structure, while its irreducibility to pure logic reflects an aspect of concealment that preserves the mystery of existence.

Where We Stand in This Discussion Today

The period 2020-2026 witnessed a notable acceleration in attempts to formalize classical Islamic proofs. Zarepour's works (2022-2024) expanded the scope of formalization to include Avicenna's proofs in both theology and natural philosophy, using non-classical modal systems that transcend traditional S5. Meanwhile, researchers like Zamboni (2021) and El-Rouayheb (2023) developed analytical tools that respect the specificity of classical Arabic logic, attempting to build "Avicennian semantics" that do not reduce existential necessity to contemporary logical necessity. A third trend has emerged that exploits developments in grounding logic by Fine and Rosen to offer formalization closer to Avicenna's concept of existential dependence. The most prominent challenge occupying researchers today is not merely the possibility of formalization, but the criteria for its success: when do we say that a logical translation is "faithful" to a classical philosophical proof? This metamethodological question has become central in applied philosophy of logic and remains unresolved. The discussion is open and dynamic, and any claim to a final answer is premature.