Propositional Probes in Neural Models

Updated 10 November 2025

Propositional probes are methodologies that extract explicit, compositional, symbolic propositions from neural activations to assess structured knowledge.
They employ classifier banks, binding subspaces, and autoencoders to decode latent world states and perform thought logging in AI systems.
Key evaluation metrics include syntactic and semantic accuracy, selectivity, and monosemanticity, driving advances in interpretability research.

Propositional probes are methodologies and tools designed to extract, interpret, and evaluate propositional content—explicit, compositional, and often symbolic propositions—within the internal activations of neural architectures. These techniques are developed to assess whether neural models genuinely represent structured knowledge, logical rules, or latent world states, as opposed to merely capturing surface correlations. By leveraging explicit logical structure, compositional binding mechanisms, and information-theoretic principles, propositional probes play a central role in mechanistic interpretability and the emerging challenge of "thought logging" in artificial intelligence systems.

1. Formal Foundations and Definitions

Propositional probes are rooted in the formalism of propositional logic and propositional interpretability. The foundational elements consist of:

Alphabet: A finite set of Boolean variables $\{x_1, x_2, \ldots, x_n\}$ and logical connectives such as negation $(\neg)$ , conjunction $(\wedge)$ , and disjunction $(\vee)$ , with extensions to bi-implication $(\leftrightarrow)$ and exclusive-or $(\oplus)$ (Langedijk et al., 10 Jun 2025).
Formula Syntax: Defined by a context-free grammar,

$\Phi(x_1, \dots, x_n) ::= x_i \mid \neg\Phi \mid (\Phi \wedge \Phi) \mid (\Phi \vee \Phi)$

Interpretation Task: Given the internal state $A_t$ (typically a vector of activations) of a system $M$ at time $t$ , a propositional probe attempts to decode propositional content $p$ , producing judgments or attitudes (such as belief or desire) about $p$ , generally denoted as $(\tau, p, w)$ where $\tau$ is an attitude type, $p$ a proposition, and $w$ a degree (numeric or binary) (Chalmers, 27 Jan 2025).

A propositional probe, in its most explicit form, consists of (a) a bank of domain-specific classifiers (one per attribute or logical variable) and (b) a mechanism—often a low-rank binding subspace—for reconstructing predicate-argument structure by matching token-wise activations (Feng et al., 27 Jun 2024).

2. Methodologies and Model Architectures

Several methodologies for propositional probing are established:

2.1 Domain Probes and Binding Subspaces

Domain Probes: Each probe $P_k$ assigns an activation $Z_s \in \mathbb{R}^d$ at position $s$ to a value in domain $D_k$ (e.g., name, country, occupation, food) or a null symbol. Classification is performed as $P_k(Z) = \operatorname{argmax}_{i \in D_k}(u_k^{(i)} \cdot Z)$ , thresholded by $h_k$ (Feng et al., 27 Jun 2024).
Binding Mechanism: A symmetric, low-rank bilinear form $H \in \mathbb{R}^{d \times d}$ , identified via the Hessian of a binding-score function $F$ , defines a binding subspace. The binding score is $d(Z_1, Z_2) = Z_1^\top U_{(k)} S_{(k)}^2 U_{(k)}^\top Z_2$ for low-rank SVD factors.

2.2 Diagnostic Classifiers and Autoencoders

Diagnostic Probes: Linear or shallow MLPs trained with binary cross-entropy loss detect the presence of a target proposition $p$ in activations $A_t$ . For a finite candidate set $\{p_j\}$ , one can train a bank $\{g_{p_j}\}$ with $g_{p_j}: \mathbb{R}^m \rightarrow \{0,1\}$ or $[0,1]$ (Chalmers, 27 Jan 2025).
Sparse Autoencoders: Models $E_\theta:\mathbb{R}^m \rightarrow \mathbb{R}^k$ and $D_\psi:\mathbb{R}^k \rightarrow \mathbb{R}^m$ jointly minimize reconstruction loss and sparsity. Post-training, each code dimension is analyzed for monosemanticity against candidate features.

2.3 Chain-of-Thought Self-Reports

Chain-of-Thought (CoT) Probing: Prompting models to explicate intermediate reasoning steps, then parsing these into formal propositions, provides a means for partial ground-truthing and thought logging. Alignment is assessed via intervention experiments and stepwise accuracy.

2.4 Control Tasks and Selectivity

Control Task Construction: For each genuine task, a matching control task is created with randomized labels sampled independently for each input type, ensuring no genuine signal is available. Selectivity is defined as

$\text{Selectivity} = \text{Accuracy}_{\text{linguistic}} - \text{Accuracy}_{\text{control}}$

Quantitatively high selectivity indicates that a probe leverages encoded structure rather than superficial memorization (Hewitt et al., 2019).

3. Evaluation Protocols and Datasets

Standardized evaluation is central to propositional probes:

Datasets: PropRandom35 and Prop35Balanced consist of millions of random or balanced propositional formulas in prefix notation, paired with satisfying assignments via SAT solvers. Balanced extensions eliminate branching biases by subtree rotation (Langedijk et al., 10 Jun 2025). Synthetic “world state” datasets—e.g., facts about names, countries, occupations—are templated and paraphrased for robust out-of-domain testing (Feng et al., 27 Jun 2024).
Metrics:
- Syntactic (exact-match) accuracy: strict matching of the output assignment or extracted proposition set.
- Semantic accuracy: any valid satisfying assignment suffices (for logical tasks).
- Jaccard index: intersection-over-union for predicted vs. true proposition sets.
- Probe selectivity: difference between true-task and control-task accuracy.
- Monosemanticity scores (autoencoder probes): regression $R^2$ or classification accuracy for latent dimension–to–feature mapping.
Experiment Regimes: In-distribution, held-out operator combinations (systematic generalization), length productivity, and stress tests under adversarial prompt manipulation or contamination (prompt injections, backdoors, sociolinguistic bias).

4. Architectural Inductive Biases and Limitations

The probe’s ability to reconstruct or extract propositional structure is heavily influenced by model and probe architecture:

Neural Architectures: Transformer encoder–decoders equipped with absolute or tree-structured positional encodings, Graph Convolutional Networks (GCNs) exploiting explicit tree structure, and LSTM encoders with sequence recurrence, all present distinct patterns of compositional generalization (Langedijk et al., 10 Jun 2025).
Inductive Bias Findings:
- Tree-structured positional encodings improve systematic and productivity generalization.
- GCNs and LSTMs demonstrate superior performance on negation-generalization splits, outperforming standard Transformers, attributed to better structural bias for recursive operators.
Observed Limitations:
- All studied architectures fail on novel operator–operator combinations involving negation, with semantic accuracy falling to zero for specific patterns (e.g., “ $\neg\wedge$ ”) unless equipped with structural enhancements.
- Behavioral examination reveals that models may ignore critical logical structure (e.g., negation) in held-out configurations, yielding the same assignment regardless of operator presence (Langedijk et al., 10 Jun 2025).
- Current propositional probes are limited to closed-world or small-vocabulary settings and rely on linear decompositions; scaling to more complex, realistic domains remains challenging (Feng et al., 27 Jun 2024, Chalmers, 27 Jan 2025).

5. Applications and Interpretability Case Studies

Propositional probes serve multiple roles:

Learning Compositionality: Controlled propositional tasks, with fine-grained operator and pattern splits, allow for diagnostic investigation into models’ ability to learn and apply symbolic rules compositionally (Langedijk et al., 10 Jun 2025).
Monitoring World State Representations: In transformer LMs, propositional probes extract latent “world state” as discrete propositions that remain robust even under prompt injection, backdoor attacks, or bias manipulations—revealing a dissociation between internal latent state and final decoded output (Feng et al., 27 Jun 2024).
Thought Logging and Attitude Attribution: As in Chalmers’s “thought logging” architecture, probes continuously extract propositional attitudes (beliefs, credences, etc.) at each model step, creating a log for downstream analysis or auditing. Designs emphasize recording not just binary attitudes but also degrees and provenance (layer, probe, confidence) (Chalmers, 27 Jan 2025).
Feature Discovery: Autoencoder-based probes uncover code directions (features) that are monosemantic for certain concepts, enabling richer, scalable interpretability (e.g., 34 million binary features with ~50% monosemanticity found in large LMs).

6. Design Principles and Guidelines

Effective deployment of propositional probes requires several best practices:

Data and Task Design: Employ balanced datasets to eliminate trivial solution shortcuts, and create systematic train/test splits that withhold key operator–operator or attribute–attribute patterns to diagnose true compositional failures (Langedijk et al., 10 Jun 2025).
Structural Encodings: Where possible, use graph or tree-structural input encodings to enhance inductive bias toward recursive reasoning (Langedijk et al., 10 Jun 2025).
Probing Methods: Combine information-based (mutual information) and use-based (intervention/counterfactual) criteria to justify attribution of propositional content to internal activations—purely observational probes risk capturing spurious correlations (Chalmers, 27 Jan 2025).
Selectivity Enforcement: Always calibrate probe complexity and reporting by probe selectivity: high test accuracy without control-task discrimination is not meaningful. Regularization by hidden size, weight decay, and explicit control tasks increases selectivity (Hewitt et al., 2019).
Compositional Binding: Prefer compositional probe designs—decode primitive attributes and then bind them via low-dimensional subspaces—over proposition-specific classifier banks, which do not scale and do not generalize to novel predicate-argument combinations (Feng et al., 27 Jun 2024, Chalmers, 27 Jan 2025).
Interpretability Audits: Log provenance data (layer, probe, method, and confidence) for every probe output, facilitating robust mechanism logging and auditability (Chalmers, 27 Jan 2025).

7. Significance and Ongoing Directions

Propositional probing occupies a central position at the intersection of AI interpretability, cognitive modeling, and philosophy of mind. These methods provide concrete means to read out not just surface labels, but structured, compositional properties and attitudes embedded in deep models. Key open challenges include:

Scaling propositional probes to open-world and open-vocabulary settings, including free-form natural language and unbounded knowledge domains (Feng et al., 27 Jun 2024, Chalmers, 27 Jan 2025).
Developing richer binding mechanisms for roles, temporal chains, and non-symbolic representations.
Ensuring robustness across prompts, paraphrases, languages, and adversarial attacks, thereby reliably disentangling latent knowledge from unfaithful outputs.

A plausible implication is the convergence toward robust, causally justified interpretability of AI systems, enabling systematic, real-time logging of model "thoughts" and beliefs—a step toward transparent and accountable deployment of future AI.