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Anti-Goal Reasoning: Rethinking the Theory of Goal Reasoning in Non-Axiomatic Logic
Bowen Xu
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 89%
Last extracted: 7/24/2026, 2:23:55 AM
Summary
The paper identifies a paradox in Non-Axiomatic Logic (NAL) where representing avoidance as the negation of a goal (¬G!) conflates pursuing a negated event with avoiding a positive event, leading to irrational behavior. The author proposes introducing 'anti-goals' and a 'prevent' mental operation to distinguish between pursuit, passive avoidance, and active prevention, thereby resolving the semantic ambiguity.
Entities (7)
Relation Signals (7)
Non-Axiomatic Logic → contains → Goal Reasoning
confidence 95% · Goal reasoning in Non-Axiomatic Logic (NAL) explains how an adaptive system derives means for realizing desired events
Anti-Goal → resolves → Paradox
confidence 92% · the framework is extended with a corresponding definition of anti-goals, so that avoidance can be represented without treating it as the pursuit of a negated event
Goal Negation → conflates → Pursuit of Negated Event
confidence 90% · this notation conflates two different readings: pursuing the negated event ¬G, and avoiding the positive event G
Goal Negation → conflates → Avoidance
confidence 90% · A common convention is to express 'avoid G' as the goal sentence '¬G!', but this notation conflates two different readings
prevent → connects → Anti-Goal Reasoning
confidence 88% · a mental operation, prevent, is introduced to connect anti-goal reasoning with ordinary goal reasoning in cases of active prevention
Anti-Goal → measures → Aversion-Value
confidence 85% · The aversion-value of an event measures the extent to which the opposite of the desired state is implied by the event
Goal → measures → Desire-Value
confidence 85% · The desire-value of an event measures the extent to which a desired state is implied by the event
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Abstract
Abstract:Goal reasoning in Non-Axiomatic Logic (NAL) explains how an adaptive system derives means for realizing desired events under insufficient knowledge and resources. However, the representation of avoidance is less clear. A common convention is to express ``avoid $G$'' as the goal sentence ``$\neg G!$'', but this notation conflates two different readings: pursuing the negated event $\neg G$, and avoiding the positive event $G$. This paper shows that the conflation can produce a paradoxical case in which an avoidance intention is converted into a positive goal to act merely because acting is usually followed by the absence of hurt. Starting from NAL's basic definition of goals, the framework is extended with a corresponding definition of anti-goals, so that avoidance can be represented without treating it as the pursuit of a negated event. Finally, a mental operation, $\op{prevent}$, is introduced to connect anti-goal reasoning with ordinary goal reasoning in cases of active prevention. Four minimal case studies check that the resulting rules distinguish pursuit, passive avoidance, active prevention, and withholding action to preserve a desired event.
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- Source: https://arxiv.org/abs/2607.20902v1
- Canonical: https://arxiv.org/abs/2607.20902v1
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11institutetext: Department of Computer and Information Sciences, Temple University, USA 11email: bowenxu.agi@gmail.com July 8, 2026 Anti-Goal Reasoning: Rethinking the Theory of Goal Reasoning in Non-Axiomatic Logic Bowen Xu Abstract Goal reasoning in Non-Axiomatic Logic (NAL) explains how an adaptive system derives means for realizing desired events under insufficient knowledge and resources. However, the representation of avoidance is less clear. A common convention is to express “avoid G” as the goal sentence “¬G! G!”, but this notation conflates two different readings: pursuing the negated event ¬G G, and avoiding the positive event G. This paper shows that the conflation can produce a paradoxical case in which an avoidance intention is converted into a positive goal to act merely because acting is usually followed by the absence of hurt. Starting from NAL’s basic definition of goals, the framework is extended with a corresponding definition of anti-goals, so that avoidance can be represented without treating it as the pursuit of a negated event. Finally, a mental operation, ⇑prevent prevent, is introduced to connect anti-goal reasoning with ordinary goal reasoning in cases of active prevention. Four minimal case studies check that the resulting rules distinguish pursuit, passive avoidance, active prevention, and withholding action to preserve a desired event. 1 Introduction Realizing beneficial events and avoiding harmful events are both basic functions of an adaptive system. In Non-Axiomatic Logic (NAL) [6], a logic for constructing adaptive systems under insufficient knowledge and resources, these functions are handled through goal reasoning. A reasoning system based on NAL is called a Non-Axiomatic Reasoning System (NARS). A goal in NARS is an event the system desires to realize. A typical backward goal inference has the following form: if a result usually follows from a condition, then, to realize the result, the system should try to realize the condition. Two issues remain unclear in the current account of goal reasoning in NAL. 1. Reasoning about avoidance has not been specified. In particular, how should “avoid an event” be represented? A common convention is to use a negated goal form, but this convention leads to the problem developed in Sec. 3. 2. The desire-value functions for concrete goal-reasoning rules may not have been well justified, especially in existing implementations of NARS, such as OpenNARS [2] and ONA [3]. This paper presents a simple NAL goal-reasoning example that produces a paradoxical result. I argue that the origin of the paradox is the semantic ambiguity of “goal negation”. To handle avoidance more precisely, the paper re-examines the definitions of goal and desire-value. It then introduces the opposite of the desired state as the state to be avoided, defines a separate anti-goal form, and clarifies how desire-value functions for goal and anti-goal rules should be chosen. As a theoretical supplement, the mental operation ⇑prevent prevent is used to connect goal reasoning with anti-goal reasoning in cases where active prevention is needed. 2 Overview of Non-Axiomatic Logic Before turning to the core issue, this section briefly reviews the parts of NAL necessary for this paper. The overview is not meant to be complete. It only covers the concepts needed to understand the paradox in Sec. 3, the analysis in Sec. 4, and the proposed extension in Sec. 5, without requiring the reader to consult the full NAL book [6]. 2.1 Inheritance Logic The formal definitions of NAL are obtained by first considering an idealized situation and then adjusting it to the Assumption of Insufficient Knowledge and Resources (AIKR) [6]. The idealized situation is described by Inheritance Logic (IL), where the system is assumed to have sufficient knowledge and resources and uses the Closed-World Assumption: Its ideal experience is a finite set K of statements, and its knowledge K∗K^* is the transitive closure of K under the relevant inference rules. In this setting, truth is binary: a statement is true when it belongs to K∗K^*, otherwise it is false. NAL is the AIKR reinterpretation of this idealized situation, not a separate execution of IL inside NARS. Under AIKR, inputs arrive over time and computational resources are limited, so object-level knowledge cannot be treated as permanent axioms. Instead, knowledge items are revisable, binary truth becomes evidential support, and truth-preserving inference becomes inference with truth-value functions. 2.2 Implication and Negation in NAL IL defines an implication statement “A⇒CA C” as true iff statement A derives statement C. Statement A is then included in the sufficient conditions set of C (denoted as CSC^S), while statement C is included in the necessary conditions set of A (denoted as ANA^N). This semantics makes the amount of evidence countable. To extend IL to NAL, the amount of positive and negative evidence for “A⇒CA C” is defined. The positive evidence (denoted by w+w^+) is included in (AS∩CS)(A^S∩ C^S) and (AN∩CN)(A^N∩ C^N), while the negative evidence (denoted by w−w^-) is included in (AS−CS)(A^S-C^S) and (CN−AN)(C^N-A^N). By counting the number of elements in the sets, frequency and confidence are defined, f=w+w,c=w+k,f= w^+w, c= ww+k, where the total evidence is w=w++w−w=w^++w^-, and k is a constant (usually k=1k=1). Truth-value is defined as a pair of frequency and confidence ⟨f;c⟩. f;c . In NAL, the truth-value of a knowledge item is either supplied by input or derived from existing knowledge. It is not calculated by directly counting the ideal evidence sets. Instead, the evidence semantics provides the unit of measure, much as a physical unit is defined by a standard without requiring each measurement to be compared directly with that standard. In IL, the ideal experience contains only positive knowledge, while negative knowledge is represented implicitly as the complement of the former: statements outside the system’s knowledge are assumed false. When IL is extended into NAL, the negation operator is defined as swapping the positive and negative evidence for a statement. In NAL, knowledge outside the system’s memory is unknown (with confidence c=0c=0) rather than false, while a totally false statement has frequency f=0f=0 and confidence c>0c>0. According to the evidential semantics, “A⇒CA C” is supported by the conjunction of A and C, “(A∧C)(A C)”, and opposed by the conjunction of A and ¬C C, “(A∧¬C)(A C)”. Therefore, “¬(A⇒C) (A C)” is equivalent to “A⇒(¬C)A ( C)”. Statement “¬(A⇒C) (A C)” means the exact opposite of “C can be derived from A”, namely, “¬C C can be derived from A”. 2.3 Goals and Backward Goal Inference In NAL, an event is a statement with temporal attributes. Its truth-value is time-dependent, in the sense that the evidential support summarized in the truth-value is valid only within a certain duration. A goal can be understood as an event that the system desires to realize, and the strength of this desire is called its desire-value. In notation, a goal with event content G is written as “G!G!”. Definitions 4 and 6 restate the corresponding definitions from NAL. NAL specifies the principles of backward inference by “meta-rules”. In schematic form, the meta-rule for goal reasoning is IF“J,JG′⊢JG”,THEN“J,G!⊢G′!”,IF~``\J,\ J_G \ J_G′, ~``J,\ G! G !′, (1) where JGJ_G and JG′J_G are judgments whose contents are G and G′G , respectively. For a unary judgment transformation, the corresponding form is IF“JG′⊢JG”,THEN“G!⊢G′!”.IF~``J_G J_G′, ~``G! G !′. (2) For example, a forward deduction rule in NAL is “A⇒C⟨f1;c1⟩,A⟨f2;c2⟩⊢C⟨Fded⟩\A C~ f_1;c_1 ,~A~ f_2;c_2 \ C~ F_ded ”111Here FdedF_ded is the standard NAL deduction truth-value function: for premises with truth-values ⟨f1;c1⟩ f_1;c_1 and ⟨f2;c2⟩ f_2;c_2 , the conclusion receives ⟨f1f2;f1f2c1c2⟩ f_1f_2;f_1f_2c_1c_2 .: if A implies C and A is true, then C is true. The corresponding backward goal-inference rule is “A⇒C,C!⊢A!\A C,~C!\ A!”: if A implies C, then A can be treated as a means for realizing the goal C. The desire-value of the conclusion is calculated from the truth-value of one premise and the desire-value of the other. This mapping is called a desire-value function. A desire-value function is not itself a truth-value function for two judgments, but it may either use the same calculation formula as the corresponding truth-value function or be specifically designed, with notation such as ⟨Fded⟩ F_ded and, for example, ⟨Fstrong⟩ F_strong 222Here ⟨Fstrong⟩ F_strong is not a function defined in NAL [6]; it is only an example of a symbol that a designer may introduce.. However, the desire-value functions for concrete goal rules are not explicitly specified in [6]. In some previous practice, the desire-value function for the deduction rule “A⇒C,A!⊢C!\A C,~A!\ C!” is ⟨f1f2;c1c2⟩ f_1f_2;c_1c_2 , and that for the negation rule “G!⊢¬G!G! G!” is ⟨1−f;c⟩ 1-f;c . 3 Paradox The problem can be exposed by combining three assumptions333These assumptions are not those taken in NAL except (A1). Assumption (A2) is a self-evident belief, while (A3) is a convention usually adopted in previous NARS research. that are each plausible in isolation. The first two assumptions concern ordinary backward motivational reasoning: a sufficient means to a desired consequence should inherit goal support, while a sufficient cause of an avoided consequence should itself be avoided: Given judgment “A⇒CA C”, if the system wants to realize C, it should be able to derive through strong inference that it should realize A. (A1) Given judgment “A⇒CA C”, if the system wants to avoid C, it should be able to derive through strong inference that it should avoid A. (A2) The third assumption is an implicit notational convention often used in previous NARS practice, even though it is seldomly stated explicitly: avoidance is represented by putting negation inside an ordinary goal sentence. That is, “avoid G” is defined as “¬G! G!”. (A3) Now consider the following light-press case: (light,⇑press)/⇒hurt⟨0.7;0.99⟩, (light, press) # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ hurt~ 0.7;0.99 , (P1) ⇑press/⇒¬hurt⟨0.9;0.99⟩, press # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ hurt~ 0.9;0.99 , (P2) ¬hurt!⟨1.0;0.99⟩. hurt!~ 1.0;0.99 . (G1) The intended reading is: • when the light signal is present, pressing the button is often followed by hurthurt; • pressing the button in general is usually followed by a negative observation of hurt;444In NARS, a negative observation, such as ¬hurt⟨0.0;0.5⟩ hurt~ 0.0;0.5 , is usually produced by a failed anticipation. • the system wants to avoid hurthurt, represented as “¬hurt! hurt!” according to assumption (A3). Intuitively, this case suggests the following practical expectation: A rational subject will not press the button when a light signal occurs, and there is no need to press the button when no light occurs. (B1) However, when no light is present, (P2) and (G1) support ⇑press!, press!, (G2) by (A1): if pressing usually leads to ¬hurt hurt, and “¬hurt! hurt!” is treated as an ordinary goal, then pressing appears to be a means to that goal. On the surface, this derivation also matches the form of the meta-rule in (1). This is already unintuitive. The fact that pressing is followed by non-hurt does not show that pressing is needed to avoid hurt. The conflict becomes sharper when the light signal occurs: light.⟨1.0;0.99⟩.light.~ 1.0;0.99 . (P3) In this context, the ordinary belief (P1) says that pressing under the light is a likely cause of hurthurt. Since (G1) is intended to express avoidance of hurthurt, this belief supports the practical expectation stated in (B1): the system should not press under the light. However, (G2) remains derivable. Moreover, since (P2) has a higher truth-value frequency than (P1) (i.e.,0.9>0.70.9>0.7), the desire-value propagated from (P2) to (G2) should also be stronger than the desire-value propagated from (P1) to “¬⇑press! press!”. Therefore, the system may press even in the light condition. The resulting behavior can be: The system presses the button when no light occurs, and it still presses the button when a light signal occurs. (C1) As a result, the system presses in order to avoid hurt, but in the light condition pressing is precisely what the ordinary belief (P1) says tends to lead to hurt. This is the intended paradox: the premises produce a positive goal to press, while the normal practical belief (B1) says that pressing is not needed in general and should be withheld under the light. In short, assumptions (A1)–(A3), together with the intuitive requirement (B1), generate the paradoxical consequence (C1) conflicting to (B1). 4 Analysis Why does the paradox occur? I argue that it stems from the semantic ambiguity of goal negation in assumption (A3). The same expression, “¬G! G!”, is mistakenly used for two different readings: a goal whose content is ¬G G, and an anti-goal whose avoided event is G. These two readings have different evidential meanings in backward inference. The desire-value of a goal “G!⟨f;c⟩G!~ f;c ” is defined as the truth-value of G⇒D~⟨f;c⟩,G D~ f;c , (19) where D~ D is a virtual term, defined at the meta-level, that summarizes the overall desired state of the system. The negation of this goal statement has the form ¬(G⇒D~), (G D), (20) which is the exact opposite of “G⇒D~G D” by swapping the positive and negative evidence, according to NAL [6]. However, the expression “¬(G⇒D~) (G D)” can be confused with two different readings: (¬G)⇒D~, ( G) D, (I1) G⇒(¬D~). G ( D). (I2) These readings are not equivalent. In (I1), negation is applied to the event G; in (I2), negation is applied to the virtual desired-state term D~ D. The two placements therefore have different consequences in backward reasoning. 4.1 Pursuit of the negated event If negation applies to the event inside the goal, then (¬G)!( G)! (21) means (¬G)⇒D~.( G) D. (22) This means that realizing the event ¬G G contributes to the desired state. It therefore represents pursuit of ¬G G. Consequently, an event that leads to ¬G G can inherit goal support. For example, X⇒¬G,(¬G)⇒D~⊢X⇒D~. X G, ( G) D X D. (23) Thus, “(¬G)!( G)!” can provide a reason for pursuing any event predicted to produce ¬G G. This inference is appropriate when ¬G G is genuinely an event that the system wants to realize. It does not follow merely from having reasons against pursuing G, and it does not by itself mean that G is to be prevented. 4.2 G leading to an undesired state Interpretation (I2) may initially appear closest to the intuitive meaning of avoidance. It seems to say that G leads to an “undesired state.” Suppose, provisionally, that ¬D~ D is interpreted as the ordinary complement of D~ D. On this binary reading, any occurrence of G that is not followed by the desired state would count as an occurrence of G followed by the undesired state summary. Equation (20) would then express the exact evidential opposite of “G⇒D~G D”. However, Eq. (I2) faces a fundamental formal problem. The term D~ D is not an object-level term; it is a virtual term introduced at the meta-level to summarize the system’s desired state. In the current account of NAL, this virtual term does not have a defined negation semantics. Consequently, “¬D~ D” is presently undefined. The phrase “undesired state” gives it an intuitive gloss, but it does not establish a formal NAL meaning. Therefore, Eq. (I2) is formally unspecified. 4.3 Source of the paradox The source of the paradox is a change of reading for the same sentence. In the derivation of (G2), “¬hurt! hurt!” is read according to Eq. (22), namely as pursuit of the negated event “(¬hurt)⇒D~( hurt) D”. Together with (P2), “⇑press / ⇒¬hurt press # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ hurt”, ordinary backward goal reasoning derives “⇑press! press!”. Under this reading, the inference is internally coherent: pressing is treated as a means to realize ¬hurt hurt. Under the intended avoidance reading of assumption (A3), however, “¬hurt! hurt!” should mean that hurthurt is to be avoided, not that ¬hurt hurt is to be pursued. In the light condition, this reading must be combined with (P1), where pressing under the light is a likely cause of hurthurt. The practical expectation is therefore (B1), not the behavior summarized in (C1). In a nutshell, the root cause of the paradox is the conflation of two readings of the same notation. The conflict between (C1) and (B1) shows that the negative form of goal, the reasoning of avoidence, and the meta-rules for goal reasoning need further clarification. 5 Potential Solution The proposed solution is to separate two representations that were conflated in assumption (A3): a goal whose event content is ¬G G, and an anti-goal whose avoided event is G. The rest of this section follows the construction methodology used in [6]: start from an IL idealization, then reinterpret it under NAL’s evidential semantics. First, the existing account of goals is recalled through the virtual desired-state term D~ D. Next, the interpretation of the desired-state and undesired-state summaries in IL is made explicit. The binary IL distinction is then quantified as desire-value and aversion-value in NAL. Finally, the corresponding inference rules are designed according to the definitions and theorems. The definition of goal remains the same as in [6] Definition 1 A goal “G!G!” is a sentence containing an event that the system desires to realize. But the virtual term D~ D is made explicit in IL: Definition 2 In IL, a goal implies the desired-state D~ D, where D~ D is a virtual statement summarizing the system’s current goals. Here D~ D is “virtual” in the sense that it is not a concrete statement in memory, but a meta-level conceptual construction used in the design of the system. It is introduced only as a summary term in the definition of goals, and it has no further necessary conditions; that is, Definition 3 The nececesy conditions of D~ D only contains D~ D itself. x∣D~⇒x=D~.\x D x\=\ D\. That is, any statement in K∗K^* is not in the necessary conditions of D~ D: Theorem 5.1 x∣D~⇒x∧x∈K∗=∅.\x D x x∈ K^*\= . As a variant of Theorem 9.2 in [6], it is evident that Theorem 5.2 In IL, if G is a desired statement in K∗K^*, then (G⇒D~)≡(GS⊆D~S).(G D)≡(G^S D^S). We call a statement, that is undesired, aversive, or avoided by the system, as “anti-goal”. Definition 4 An anti-goal “G¡G !`” is a sentence containing an event that the system avoid to realize. According to IL, any unknown knowledge are assumed false. With the similar principle, any statement that does not imply D~ D is avoided. Definition 5 In IL, if G is not a desired statement in K∗K^*, then G is an anti-goal. To extend goal-related representation and inference from IL to NAL, the binary representation should be converted to quantified representation. To bridge the gap, the evidential support of a goal is specified: Theorem 5.3 For a goal “G!G!”, its evidence includes statements in GSG^S. Among them, statements in (GS∩D~S)(G^S∩ D^S) are positive evidence, while statements in (GS−D~S)(G^S- D^S) are negative evidence. This theorem follows directly from Definition 9.4 of [6]. Desire-value then quantifies the extent of wanting to realize an event in NAL: Definition 6 The desire-value of an event measures the extent to which a desired state is implied by the event. The desire-value of event S is the truth-value of the implication statement “S⇒D~S D”, where D~ D is a virtual statement summarizing the system’s current goals. This definition restates Definition 12.4 in [6]. In IL, the sufficient conditions of D~ D include the desired statements, while statements outside them are considered undesired in the relevant sense. When this account is extended to NAL, a virtual statement ¬D~ D to represent the opposite of the desired-state in NAL. Definition 7 The aversion-value of an event measures the extent to which the opposite of the desired state is implied by the event. The aversion-value of event S is the truth-value of the implication statement “S⇒¬D~S D”, where ¬D~ D is the virtual statement for the opposite of the desired state: ¬(G⇒D~)≡G⇒(¬D~). (G D)≡ G ( D). Some events are neither desired nor undesired, meaning the system has weak or even no motivational attitude toward them. In the extreme case, both supporting and opposing evidence are 0, which is reflected in the confidence value c=0c=0. According to Definitions (6) and (7), for a goal, positive evidence measures support for realizing the event, whereas negative evidence measures opposition to realizing it. Thus, a goal and an anti-goal are opposites of each other, and their evidence can be obtained from each other by swapping the positive and negative evidence. That is Theorem 5.4 To avoid an event means not to realize it: G¡≡G⇒(¬D~)≡¬(G⇒D~)≡¬G!.G !`≡ G ( D)≡ (G D)≡ G!. Here the expression “¬G! G!” is read at the level of the goal statement “G⇒D~G D”. It is not the ambiguous convention in (A3), where “avoid G” was identified with the ordinary goal whose event content is ¬G G. Symmetrically, Theorem 5.5 To realize an event means not to avoid it: G!≡G⇒D~≡¬(¬(G⇒D~))≡¬G¡.G!≡ G D≡ ( (G D))≡ G !`. These equivalences justify the negation rules between goal and anti-goal sentences: G!⊢G¡⟨Fneg⟩, G! G !`~ F_neg , (24) G¡⊢G!⟨Fneg⟩. G !` G!~ F_neg . Other desire-value functions for goal-inference rules can be obtained by the following process: after converting a goal or anti-goal into the form “G⇒D~G D” or “G⇒¬D~G D” respectively, if there exists a corresponding forward inference rule, the backward rule adopts the same calculation formula as that rule’s truth-value function. In this sense, ⟨Fded⟩ F_ded in a backward motivational rule names a desire-value function whose formula is copied from the corresponding truth-value function, rather than a truth-value function directly applied to a judgment. For example, given premises “A⇒CA C” and “C!C!”, the latter can be transformed to “C⇒D~C D”, since the deduction rule “M⇒P,S⇒M⊢S⇒P⟨Fded⟩\M P,S M\ S P~ F_ded ” exists, the backward rule uses the corresponding desire-value function with the same formula as ⟨Fded⟩ F_ded . The same process works with an anti-goal. Thereby, the following two rules are produced: A⇒C⟨f1;c1⟩,C!⟨f2;c2⟩ \A C~ f_1;c_1 ,~C!~ f_2;c_2 \ ⊢A!⟨Fded⟩, A!~ F_ded , (25) A⇒C⟨f1;c1⟩,C¡⟨f2;c2⟩ \A C~ f_1;c_1 ,~C !`~ f_2;c_2 \ ⊢A¡⟨Fded⟩. A !`~ F_ded . (26) The form of these backward rules can still be understood as an instance of the meta-rule in Eq. (1), but what requires separate justification is the choice of desire-value function. Generally speaking, if a goal-inference rule is produced, its symmetric form for anti-goal inference can be obtained at the same time. For compound terms, decomposition rules allow a desired conjunction to propagate goal support to its components. For example, A⟨f1;c1⟩,(A∧C)!⟨f2;c2⟩⊢C!⟨Fded⟩.\A~ f_1;c_1 ,~(A C)!~ f_2;c_2 \ C!~ F_ded . (27) Here, the desire-value function is not copied from ⟨Fint⟩ F_int , which is used in the forward rule, “A,C⊢(A∧C)⟨Fint⟩\A,C\ (A C)~ F_int ” [6]. In contrast, the desire-value function stems from the following rule, “(A∧C)⇒D,A⊢C⇒D⟨Fded⟩\(A C) D,A\ C D~ F_ded ”. The design of temporal variants of backward rules, such as the rules that involves predictive implication (e.g., “A / ⇒CA # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ C”) or sequential compound (e.g., “(A,C)(A,C)”), follows the same principles as designed in Chapter 11 of [6]. In addition to the intrinsic evidential relation between goals and anti-goals, some situations require the system to derive a goal from an anti-goal, or vice versa. Such conversion can be driven by mental operation “⇑prevent prevent” – if G is an anti-goal, then preventing G can become a goal. If G is a goal, then preventing G can become an anti-goal: ⇑prevent(G)! prevent(G)! ≡G¡, ≡ G !`, (28) ⇑prevent(G)¡ prevent(G) !` ≡G!. ≡ G!. This is crucial for causal discovery. For example, under a high-temperature condition, turning on a fan may be followed by no overheating, while not turning on the fan is followed by overheating. The contrast supports the hypothesis that turning on the fan prevents overheating in that condition. This prevention is discovered by a contrastive procedure. Rather than introducing new inference rules with global effects, the paper represents this contrastive inference as a local object-level knowledge item: ((($C∧$A)⇒¬$R) ((( C A) R) (29) ∧ (($C∧¬$A)⇒$R)) (( C A) R)) ⇒ (($C∧$A)⇒⇑prevent($R))⟨1.0;0.99⟩, (( C A) prevent( R))~ 0;99 , where “$ ” followed by a variable name indicates an independent variable in NAL [6]; such a variable can be replaced by a constant term. The knowledge item in (29) is attached to the “⇑prevent prevent” operation and is triggered conditionally.555A previous account of causal inference in NARS [7] did not address prevention. The present extension therefore enriches the causal-inference theory of NARS. 6 Case Studies This section gives a compact empirical check of the proposed distinction between goals and anti-goals. The cases are deliberately minimal: they are constructed from a shared small vocabulary, including the sensory terms lightlight, foodfood, hurthurt, and the mental operation ⇑press press. The point is not to evaluate a full-scale NARS system, but to verify that the proposed semantics derives the expected operational motivation in the four situations required by the theory: 1. act to realize a desired event; 2. withhold an operation that would realize an undesired event; 3. act to prevent an undesired event; 4. withhold an operation that would prevent a desired event. A minimal system is implemented to run the cases. Each run checks whether the reasoning process derives the expected operation-level item: a goal “⇑press! press!” supports pressing, whereas an anti-goal “⇑press¡ press !`” suppresses pressing. Each run lasts 160 working cycles. During cycles 1–80, motor babbling is enabled so that the system samples both pressing and not pressing. During cycles 81–160, motor output is driven by the operational goal inferred by the system. To evaluate the system’s overall behavioral performance, the simulated body is equipped with fixed evaluative channels: each occurrence of foodfood contributes +1+1 to cumulative reward, while each occurrence of hurthurt contributes −1-1. The curves therefore show the cumulative reward generated by these external channels; goals and anti-goals are not defined from this scalar quantity. Table 1 summarizes the reasoning and behavioral results of the four cases. Table 1: Reasoning and behavioral results of the four light-press cases. Case name Input Derived Decision Do to Realize “food!food!” “⇑press! press!” press Not Do to Avoid “hurt¡hurt !`” “⇑press¡ press !`” no operation Do to Avoid “hurt¡hurt !`” “⇑press! press!” press Not Do to Realize “food!food!” “⇑press¡ press !`” no operation Figure 1: External cumulative reward in the four light-press settings. The first 80 cognitive cycles form the motor-babbling phase; the later cycles show the behavior selected from the derived operation-level item. Figure 1 gives an external check on the four reasoning results. After the motor-babbling phase, the two food cases continue to accumulate positive reward under the behavior predicted by the theory, while the two hurt cases stop accumulating sustained additional loss. The negative portions of the hurt curves mainly come from exploratory mistakes during motor babbling; once motor output is driven by the inferred operational goal, these curves become nearly flat. The following subsections describe the four cases in detail. For each case, the relevant predictive schemas and motivational input are specified, then it is shown how the proposed goal or anti-goal propagation derives the corresponding operational motivation. 6.1 Case 1: Do to Realize This case is the ordinary instrumental case. During motor babbling, the system samples pressing under the light condition and learns that this sequence is followed by event foodfood: (light,⇑press) / ⇒food⟨1.0;0.99⟩.(light, press) # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ food~ 1.0;0.99 . (30) Given the motivational input “food!⟨1.0;0.99⟩food!~ 1.0;0.99 ”, backward goal propagation first derives a goal for the sequence: (light,⇑press)/⇒food⟨1.0;0.99⟩,food!⟨1.0;0.9⟩ \(light, press) # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ food~ 0;99 ,~food!~ 0;9 \ (31) ⊢(light,⇑press)!⟨1.0;0.89⟩. (light, press)!~ 0;89 . The current observation “light⟨1.0;0.99⟩light~ 1.0;0.99 ” then propagates this goal to the executable operation: (light,⇑press)!⟨1.0;0.89⟩,light.⟨1.0;0.9⟩ \(light, press)!~ 0;89 ,~light.~ 0;9 \ (32) ⊢⇑press!⟨1.0;0.8⟩. press!~ 0;8 . The derived operational goal is therefore “⇑press! press!”, and the system executes ⇑press press as the motor output. The corresponding Act for food curve (see Fig. 1) keeps increasing after motor babbling, as expected when pressing realizes foodfood. 6.2 Case 2: Not Do to Avoid This case tests passive avoidance. During motor babbling, the system learns that pressing under the light condition is followed by the avoided event hurthurt: (light,⇑press) / ⇒hurt⟨1.0;0.99⟩.(light, press) # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ hurt~ 1.0;0.99 . (33) Given the motivational input “hurt¡⟨1.0;0.9⟩hurt !`~ 1.0;0.9 ”, backward anti-goal propagation first derives an anti-goal for the sequence: (light,⇑press)/⇒hurt⟨1.0;0.99⟩,hurt¡⟨1.0;0.9⟩ \(light, press) # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ hurt~ 0;99 ,~hurt !`~ 0;9 \ (34) ⊢(light,⇑press)¡⟨1.0;0.89⟩. (light, press) !`~ 0;89 . The current observation “light⟨1.0;0.99⟩light~ 1.0;0.99 ” then propagates this anti-goal to the executable operation: (light,⇑press)¡⟨1.0;0.89⟩,light.⟨1.0;0.9⟩ \(light, press) !`~ 0;89 ,~light.~ 0;9 \ (35) ⊢⇑press¡⟨1.0;0.8⟩. press !`~ 0;8 . The derived operational anti-goal is therefore “⇑press¡ press !`”, so the system suppresses ⇑press press as a motor operation. In Fig. 1, the Withhold to avoid hurt curve drops during motor babbling but becomes nearly flat afterward, matching the prediction that pressing should be suppressed. 6.3 Case 3: Do to Avoid The process of learning a preventional statement is harder than the previous cases, due to the constrastive rule in (29) require negative oberservation (i.e., ¬hurt hurt). The system initially observes that light is followed by hurt with a time interval, (light,_) / ⇒hurt⟨1.0;0.67⟩,(light,\_) # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ hurt~ 1.0;0.67 , (36) so that when event lightlight occurs, the system anticipate a future event hurthurt. However, in motor babbling, when the system execute ⇑press press after the light condition, the anticipation fails. A negative observation can thus be produced, ¬hurt⟨1.0;0.5⟩. hurt~ 1.0;0.5 . (37) The system applies the temporal induction rule and derives “(light,⇑press) / ⇒¬hurt⟨1.0;0.31⟩(light, press) # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ hurt~ 1.0;0.31 ”. With more evidence collected, this prediction becomes stronger, (light,⇑press) / ⇒¬hurt⟨1.0;0.99⟩.(light, press) # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ hurt~ 1.0;0.99 . (38) Meanwhile, the object-level rule (29) enables the system to generate concepts including “(light,¬⇑press)(light, press)”, “(light,⇑press)⇒⇑prevent(hurt)(light, press) prevent(hurt)” and so on, so that the system can accumulate evidence for event “(light,¬⇑press)(light, press)”.777This step is indispensable because NARS cannot arbitrarily observe events that have not occurred. With more evidence accumulated, the system learns that (light,¬⇑press) / ⇒hurt⟨1.0;0.99⟩,(light, press) # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ hurt~ 1.0;0.99 , (39) and derives (light,⇑press) / ⇒⇑prevent(hurt)⟨1.0;0.98⟩,(light, press) # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ prevent(hurt)~ 1.0;0.98 , (40) by rule (29) with beliefs (38) and (39). The system receives the motivational input “hurt¡⟨1.0;0.9⟩hurt !`~ 1.0;0.9 ”, stating that hurthurt is to be avoided. Rule (28) then turns this anti-goal into the positive goal of preventing hurthurt: hurt¡⟨1.0;0.9⟩⊢⇑prevent(hurt)!⟨1.0;0.9⟩.hurt !`~ 1.0;0.9 prevent(hurt)!~ 1.0;0.9 . (41) Ordinary backward goal propagation can then use the learned prevention statement to derive the sequence goal: (light,⇑press)/⇒⇑prevent(hurt)⟨1.0;0.98⟩,⇑prevent(hurt)!⟨1.0;0.9⟩ \(light, press) # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ prevent(hurt)~ 0;98 ,~ prevent(hurt)!~ 0;9 \ (42) ⊢(light,⇑press)!⟨1.0;0.88⟩. (light, press)!~ 0;88 . Finally, when lightlight is observed, the sequence goal is reduced to an executable operation goal: (light,⇑press)!⟨1.0;0.88⟩,light.⟨1.0;0.9⟩ \(light, press)!~ 0;88 ,~light.~ 0;9 \ (43) ⊢⇑press!⟨1.0;0.79⟩. press!~ 0;79 . Thus, from the intention to avoid hurthurt, the system derives the positive operation goal “⇑press! press!” and emits ⇑press press. This case is important because the anti-goal does not merely suppress actions; through contrastive inference, it can also produce a positive operation. The Act to avoid hurt curve (see Fig. 1) becomes nearly flat after motor babbling, because pressing prevents further occurrences of hurthurt. 6.4 Case 4: Not Do to Realize This case is the preservation counterpart of Case 3, and its learning process is symmetric. The system initially observes that light is followed by food with a time interval, (light,_) / ⇒food⟨1.0;0.67⟩,(light,\_) # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ food~ 1.0;0.67 , (44) so that when event lightlight occurs, the system anticipates a future event foodfood. However, in motor babbling, when the system executes ⇑press press after the light condition, the anticipation fails. A negative observation can thus be produced, ¬food⟨1.0;0.5⟩. food~ 1.0;0.5 . (45) The system applies the temporal induction rule and derives “(light,⇑press) / ⇒¬food⟨1.0;0.31⟩(light, press) # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ food~ 1.0;0.31 ”. With more evidence collected, this prediction becomes stronger, (light,⇑press) / ⇒¬food⟨1.0;0.99⟩.(light, press) # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ food~ 1.0;0.99 . (46) Meanwhile, the object-level rule (29) enables the system to accumulate evidence for the contrastive event “(light,¬⇑press)(light, press)”. With more evidence accumulated, the system learns that (light,¬⇑press) / ⇒food⟨1.0;0.99⟩,(light, press) # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ food~ 1.0;0.99 , (47) and derives (light,⇑press) / ⇒⇑prevent(food)⟨1.0;0.98⟩,(light, press) # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ prevent(food)~ 1.0;0.98 , (48) by rule (29) with beliefs (46) and (47). The system receives the motivational input “food!⟨1.0;0.9⟩food!~ 1.0;0.9 ”, stating that foodfood is to be realized. Rule (28) then turns this goal into the anti-goal of preventing foodfood: food!⟨1.0;0.9⟩⊢⇑prevent(food)¡⟨1.0;0.9⟩.food!~ 1.0;0.9 prevent(food) !`~ 1.0;0.9 . (49) Backward anti-goal propagation can then use the learned prevention statement to derive the sequence anti-goal: (light,⇑press)/⇒⇑prevent(food)⟨1.0;0.98⟩,⇑prevent(food)¡⟨1.0;0.9⟩ \(light, press) # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 2.20001pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ # 1.75389pt$ 0.6[0.5]$ /$$ $ $ prevent(food)~ 0;98 ,~ prevent(food) !`~ 0;9 \ (50) ⊢(light,⇑press)¡⟨1.0;0.88⟩. (light, press) !`~ 0;88 . Finally, when lightlight is observed, the sequence anti-goal is reduced to an executable operation anti-goal: (light,⇑press)¡⟨1.0;0.88⟩,light.⟨1.0;0.9⟩ \(light, press) !`~ 0;88 ,~light.~ 0;9 \ (51) ⊢⇑press¡⟨1.0;0.79⟩. press !`~ 0;79 . Thus, from the intention to realize foodfood, the system derives the operational anti-goal “⇑press¡ press !`” and emits no motor operation. In this case, the system suppresses an action because that action prevents a desired event. The Withhold for food curve (see Fig. 1) continues to increase after motor babbling, confirming that suppressing the operation preserves the desired event in this setting. 7 Discussion On Theoretical Assumptions and Scope. In IL, the idealized situation, the framework assumes a clear contrast between what the system treats as to-be-realized and what it treats as to-be-avoided. This contrast provides a reference point for defining goals and anti-goals, but it should not be interpreted as saying that the system must always classify every event motivationally in NAL: Once the framework is placed under AIKR, motivation becomes incomplete and revisable. An event may be neither desired nor avoided because the system has no relevant motivational evidence about it. Such an event should remain motivationally unknown, rather than being forced into either side of the idealized contrast. This design is also what makes the extension applicable to open world [8]: motivations need not be fully prespecified, but can be introduced, strengthened, weakened, or revised as new experience becomes available. In general, the motivations of an adaptive system working with insufficient knolwedge and resoures may come from bodily presets, environmental inputs such as linguistic instructions, and internal derivation; these motivations may event conflict with one another; over time, these sources may lead to the system’s own distinctive purposes. [5] To keep the present paper focused, the acquisition of motivation is not modeled in detail. The aim here is only to extend NAL’s goal-reasoning framework with anti-goals and prevention; a full theory of motivational learning remains an important topic for future work. On Asymmetry between the Cases. A related behavioral finding suggests that these cases are not symmetric. Guitart-Masip et al. [1] studied a task in which human participants had to learn when to act and when to withhold action under reward and punishment. They found that people more readily learned to act for reward and to withhold action to avoid punishment, whereas acting to avoid punishment and withholding action to obtain reward were learned less readily. This behavioral asymmetry is consistent with the theoretical construction proposed here: Case 1 and Case 2 are direct goal or anti-goal propagation, whereas Case 3 and Case 4 additionally require learning a prevention relation from negative observation and contrastive evidence. Thus, the behavioral result supports the distinction made by the present framework, even though the present theory and the account from [1] explain the behavioral asymmetry by very different mechanisms. On the Difference from Reinforcement Learning. The proposed distinction between goals and anti-goals is also different from the usual treatment of reward in reinforcement learning (RL) [4]. In a standard reinforcement-learning formulation, the agent selects an action at each decision step by comparing the expected returns of available actions, possibly including a no-operation action if it is included in the action set. Under this view, a negative reward does not by itself mean that the corresponding state or event is an anti-goal in the present sense. If all available actions have negative expected returns, the action with a negative expected return may still be selected, just because it has the highest expected return. Therefore, the sign of a reward value does not directly encode the distinction between desired and avoided events. By contrast, in NAL extended herein, an operation is not selected merely because it is better than the other available operations. An operation is executed only when it receives sufficient motivational support; otherwise, the system may simply remain silent and emit no operation. In this sense, no-operation is not necessarily just another competing action, but can be the default condition of the system. These different theoretical presuppositions may lead to different practical consequences. A systematic comparison between the RL and NAL is beyond the scope of this paper, but it is an important topic for further research. 8 Conclusion By analyzing a paradoxical example, this paper exposes a theoretical ambiguity in previous NAL accounts of motivational reasoning. The proposed response is not to replace the existing theory of goals, but to complete it with a minimal extension to NAL. The theoretical contribution of this paper is threefold. First, it gives avoidance its own evidential status by separating anti-goals from ordinary goals whose event content happens to be negated. This separation removes the ambiguity in goal negation and makes clear why pursuing the absence of an event is not the same as avoiding the event itself. Second, the paper gives a general principle for choosing desire-value functions in backward motivational inference. The principle is to justify a backward rule by relating its motivational reading to the corresponding forward inference rule. Third, the paper introduces active prevention as a bridge between anti-goal reasoning and ordinary goal reasoning. Active prevention is not reduced to the mere observation that an event is absent after an action; it requires contrastive evidence that the action changes what would otherwise be expected. Acknowledgments. The author appreciates the advice from Dr. Pei Wang; discussions and debates with him were crucial during the development of this idea and theory. The proposed solution could not have been developed without his guidance and inspiration. 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