Anti-Goal Reasoning: Rethinking the Theory of Goal Reasoning in Non-Axiomatic Logic

Technical Summary: Anti-Goal Reasoning in Non-Axiomatic Logic

Problem Statement
The paper identifies a critical theoretical ambiguity in the current implementation of Goal Reasoning within Non-Axiomatic Logic (NAL), the logical foundation of the Non-Axiomatic Reasoning System (NARS). While NAL effectively handles the realization of desired events under insufficient knowledge and resources (AIKR), the representation of avoidance is ill-defined. The prevailing convention treats "avoiding event $G$" as the goal sentence "$\neg G!$" (pursuing the negated event). The author demonstrates that this conflation leads to a paradox: a system may derive a positive goal to perform an action merely because that action is usually followed by the absence of a negative outcome, even when the action is also a known cause of that negative outcome under specific conditions.

This paradox arises from the semantic ambiguity of "goal negation." The notation "$\neg G!$" is used to represent two distinct logical readings:

1. Pursuit of a negated event: The system desires the state where $G$ does not occur (i.e., $(\neg G) \Rightarrow \tilde{D}$).
2. Avoidance of a positive event: The system wishes to prevent $G$ from occurring (i.e., $G \Rightarrow \neg \tilde{D}$).

In the "light-press" case study presented, the system incorrectly infers that pressing a button is a means to achieve "no hurt" ($\neg \text{hurt}!$) because pressing generally leads to no hurt. However, this inference ignores the specific context where pressing (under a light signal) causes hurt. The system paradoxically decides to press the button to "avoid hurt," contradicting the practical expectation that it should withhold action.

Methodology
The paper proposes a minimal extension to NAL that separates the representation of goals and anti-goals, moving from an idealized Inheritance Logic (IL) framework to the evidential semantics of NAL.

1. Formal Separation:

   • Goal ($G!$): Defined as an event the system desires to realize, formally represented as $G \Rightarrow \tilde{D}$, where $\tilde{D}$ is a virtual term summarizing the system's desired state.
   • Anti-Goal ($G¡$): Defined as an event the system desires to avoid. This is formally represented as $G \Rightarrow \neg \tilde{D}$, where $\neg \tilde{D}$ represents the opposite of the desired state.
   • The paper clarifies that "avoiding $G$" is not equivalent to "pursuing $\neg G$." Instead, avoiding $G$ is the logical negation of the goal statement itself: $\neg(G \Rightarrow \tilde{D})$.

2. Desire-Value and Aversion-Value Functions:

   • The paper redefines desire-value functions for backward inference. For a goal, the desire-value is the truth-value of $G \Rightarrow \tilde{D}$. For an anti-goal, the aversion-value is the truth-value of $G \Rightarrow \neg \tilde{D}$.
   • Inference rules for anti-goals are constructed symmetrically to goal rules. If a forward inference rule exists (e.g., deduction), the corresponding backward rule for anti-goals adopts the same calculation formula as the forward truth-value function.

3. Active Prevention ($\uparrow\text{prevent}$):

   • To bridge the gap between passive avoidance and active intervention, the paper introduces a mental operation $\uparrow\text{prevent}$.
   • If $G$ is an anti-goal, preventing $G$ becomes a positive goal ($\uparrow\text{prevent}(G)! \equiv G¡$).
   • This operation relies on contrastive evidence: the system must learn that an action leads to the absence of an event that would otherwise occur (e.g., pressing prevents hurt, whereas not pressing leads to hurt). This is formalized via a specific object-level rule (Eq. 13) requiring negative observations and contrastive induction.

Key Contributions
The paper makes three primary theoretical contributions:

1. Resolution of Semantic Ambiguity: It distinguishes between the pursuit of a negated event and the avoidance of a positive event. By introducing the explicit anti-goal form ($G¡$), the framework prevents the paradox where avoidance intentions are incorrectly converted into positive action goals based on general statistical correlations.
2. Principle for Desire-Value Functions: It establishes a general principle for deriving desire-value functions in backward motivational inference. Rather than arbitrarily assigning functions, the paper argues that backward rules should inherit their calculation formulas from their corresponding forward inference rules, ensuring logical consistency between truth-value and desire-value propagation.
3. Integration of Active Prevention: It formalizes the mechanism by which an anti-goal can generate a positive action plan. Through the $\uparrow\text{prevent}$ operation and contrastive inference, the system can derive a goal to act specifically to prevent an undesired outcome, distinguishing this from merely withholding action.

Results
The proposed framework was validated through four minimal case studies involving a simulated system with motor babbling and cumulative reward evaluation:

1. Do to Realize: The system correctly derives a goal to press a button to obtain food.
2. Not Do to Avoid: The system correctly derives an anti-goal to not press a button to avoid hurt (passive avoidance).
3. Do to Avoid: The system learns a prevention relation (pressing prevents hurt) and correctly derives a positive goal to press the button to actively avoid hurt.
4. Not Do to Realize: The system learns that pressing prevents food and correctly derives an anti-goal to not press the button, thereby preserving the desired event.

In all cases, the system's behavior aligned with theoretical expectations. Specifically, in the "Do to Avoid" and "Not Do to Realize" cases, the system successfully utilized contrastive evidence to switch between passive avoidance and active intervention or suppression, a capability that was paradoxical under the previous "negated goal" convention.

Significance
The paper claims that this extension completes the NAL goal-reasoning framework without replacing existing theories. By separating goals and anti-goals, the system can now handle avoidance with the same evidential precision as pursuit. The distinction allows the system to:

• Differentiate between "doing nothing" (withholding action) and "doing something to prevent."
• Avoid the logical error where a general correlation (action $\to$ no hurt) overrides a specific causal risk (action under condition $X$ $\to$ hurt).
• Provide a formal basis for active prevention, which is essential for causal discovery and adaptive behavior in open-world environments where motivations are incomplete and revisable.

The author notes that while the framework explains behavioral asymmetries observed in human learning (e.g., acting for reward vs. withholding for punishment), the current work focuses strictly on the logical extension of NAL, leaving the detailed modeling of motivational acquisition for future research.