Nonclassical Logics

When faced with the question of whether to assert or deny a paradoxical sentence such as the liar, it seems that there are two plausible responses: neither asserting it nor denying it, or both asserting it and denying it. In this paper, I make this thought concrete by formulating bilateral proof systems (of both the natural deduction and sequent calculus variety) for the logics in the FDE family: K3, LP, and FDE. The different logics are simply the result of different (...) choices of "coordination principles," bilateral structural rules which coordinate the opposite speech acts of assertion and denial. I show that conceiving of these logics in bilateral terms has important philosophical consequences, most notably for the debate between "subclassical'' and "substructural" approaches to paradox. In particular, I show how adopting Bilateral K3 enables one to endorse the ``non-transitive'' solution to paradox, as developed by Ripley, while maintaining that logical consequence (understood bilaterally) is transitive. (shrink)

Bob and Ulf say logic should make explicit / The kinds of inference we take as licit. / They give a formalism that’s classically complete, / But in which an extra premise may defeat / An inference that seems a reasonable fact. / Thus reason’s made explicit but is it exact? / For reason in the sense of Brandom and Hlobil, / May explode like a logical Chernobyl. / This is the point on which I’d like to push back, / (...) And keep the game of giving reasons exact. / So with the help of fancy formal antics, / I’ll give nonmonotonic truthmaker semantics. (shrink)

The paper proposes to expand modal doxastic logic with ascriptions of attitudes toward questions, or attitudes whose contents are questions, besides ascriptions of belief toward propositions. The relevant attitudes are the attitude of being in doubt and the attitude of rejecting a question, respectively. In order to accommodate such ascriptions, modifications on the canonical framework of doxastic logic are called for. Interrogative complements are added to its grammar, and questions are included among the possible semantic values of expressions. New valid (...) schemas and consequence relations are grounded on the suitably modified framework—schemas and consequence relations that correspond to principles of coherence or structural rationality. The relevant principles are concerned with the relationship between belief, doubtfulness, and rejection, as well as how doubtfulness and rejection spread from one question to another in an ideally rational mind. (shrink)

This paper is a sequel to “Subject Unicity and Self-Ascription: A Paraconsistent Formalization with σ-Lag” (hereafter “Subject Theory”, PhilPapers record CSUADU.). Preserving the value layer and rules of the previous work (X–σ, No3, SU-bridge), we extend the observational/representational layer into a topological–operational model. Specifically, we introduce a state space X, a decidable region O_+++ = { f_I>0, f_S>0, f_R>0 }, and a local orientation indicator R_sig = sign det[∇f_I, ∇f_S, ∇f_R], thereby relocating the judgment of the jouissance effect into a (...) topological space. Clinically, this means one reports only triggers/evidence, while the equality sign (=) is an event-type fixation permitted once only within a licensing window. The key indicators are: (i) a continuous quantity a_eff := Res_σ(Φ) ∈ ℝ_{≥0} representing the amount of σ-trace, (ii) R_sig ∈ {−1, 0, +1}, and (iii) the optional reinforcement by μ̄_123. Here, observational equivalence ≈_obs is evidenced by a_eff(x) > 0 and is kept strictly distinct from ontological equivalence ≡_onto. As a result, the structure remains discrete, while phenomenology is smoothly tracked via continuous quantities such as a_eff and κ. Lacan’s “phallus = imaginary” is re-formalized not as rhetoric but as an anti-scalarization rule: φ acts only as an inner rotation operator carried by the imaginary axis of quaternions (including slice invariance), forbidding any fall into the value layer. Fixation of fantasy is judged by the topological loop with R_sig = −1, and the promotion ‘≈ → =’ is permitted exactly once only within the AUG_eq subcategory. The subject–object a relation is read as a Möbius isomorph: ontologically equivalent yet phase-inverted on the observational layer, so ‘=’ is allowed only as an event after passing Gate(Δτ), thereby blocking explosion routes. Furthermore, consistency with the University discourse is exhibited by ∂(a_eff)/∂(s*) > 0 when x ∈ O_+++ and R_sig = −1, capturing the right-hand sensitivity (S2 → a) on the top line. Safety is secured by No3, SU, and the Reflexive Guards: global ‘=’ is prohibited; no direct updates to the value layer; and R_sig, μ̄_123, amp are used purely as trigger/evidence indicators. This design supplies a checklist-style clinical heuristic (Obs/σ/AUG logs, the condition R_sig = −1 ∧ a_eff > 0, and exactly one ‘=’ after gate approval) without compromising structural stability. In short, symptoms are not erased but operated—fixed as events precisely at the point where the same strip meets again, now inverted. 이 글은 「주체의 자기귀속과 단일성: σ-시차와 파라일관 논리의 형식화」(이하 「주체론」, PhilPapers record CSUADU.)의 후속이다. 전작 「주체론」의 값층·규칙(X–σ, No3, SU-브리지)을 그대로 유지한 채, 관측·표상 레이어를 위상-연산 모델로 확장한다. 구체적으로, 상태공간 X와 판정가능 영역 O_+++ = { f_I>0, f_S>0, f_R>0 } 및 국소 방향 판정자 R_sig = sign det[∇f_I, ∇f_S, ∇f_R]를 도입해, 주이상스 효과의 판정을 위상공간으로 이관한다. 이렇게 하면 임상에서는 증거·트리거만 보고하고, 등호(=)는 면허 윈도우에서만 1회 허가되는 사건형 고정으로 제한된다. 핵심 지표로는 (i) σ-흔적의 양을 나타내는 연속량 a_eff := Res_σ(Φ) ∈ ℝ_{≥0}, (ii) R_sig ∈ {−1, 0, +1}, (iii) 필요 시 보강되는 μ̄_123 등을 사용한다. 이때 관측-동치 ≈_obs는 a_eff(x) > 0에 의해 증거화되며, 존재론-동치 ≡_onto와 엄격히 구분된다. 덕분에 구조는 이산이되, 양상은 a_eff와 κ 같은 연속값으로 매끈하게 추적된다. 라캉의 "남근=허수"는 수사가 아니라 반(反)스칼라화 규칙으로 재정식화된다: φ는 사원수 허수 캐리어의 내적 회전 연산자(슬라이스 불변성 포함)로만 작동하며, 값층으로의 낙하를 금지한다. 환상의 교착은 R_sig = −1인 위상 고리로 판정되고, '≈ → =' 승격은 AUG_eq 하위 범주에서만 1회 허가된다. 또한 주체–대상 a 관계는 뫼비우스 동형으로 읽힌다: 존재론적으로는 동치이나 관측층에서는 위상이 반전되어, 등호는 오직 Gate(Δτ) 통과 시 사건형으로만 잠깐 가능하다(폭발 경로 차단). 더 나아가, 대학 담론과의 정합성은 x ∈ O_+++ · R_sig = −1일 때 ∂(a_eff)/∂(s*) > 0가 성립함을 통해 보인다(상단 S2 → a 채널의 우항 감도). 안전성은 No3·SU·Reflexive Guards로 고정된다: 전역 '=' 금지, 값층 직접 갱신 금지, 트리거/증거 전용 지표(R_sig, μ̄_123, amp). 이 설계는 임상에서 체크리스트형 휴리스틱(Obs/σ/AUG 로그, R_sig = −1 ∧ a_eff > 0 조건, 게이트 승인을 거친 단 1회의 '=')을 제공하면서도 구조적 안정성을 해치지 않는다. 요컨대, 증상은 지워지는 게 아니라 운용되며, 같은 띠지만 뒤집혀 만나는 그 지점에서 사건으로 고정된다. (shrink)

This paper challenges the negative attitudes towards labelled proof systems, usually referred to as semantic pollution, by arguing that such critiques overlook the full potential of labelled calculi. The overarching objective is to develop a practice-based philosophical analysis of labelled calculi to provide insightful considerations regarding their proof-theoretic and philosophical value. To achieve this, successful applications of labelled calculi and related results will be showcased, and comparisons with other relevant works will be discussed. The paper ends by advocating for a (...) more practice-based approach towards the philosophical understanding of proof systems and their role in structural proof theory. (shrink)

The quantum measurement problem, particularly the observer effect, has long resisted a complete explanation, often forcing a choice between paradoxical interpretations and a fundamental split between the quantum and classical worlds. This paper introduces Aethic reasoning, a novel framework that resolves the measurement problem by reformulating the logical and relational structure that underpins reality. We propose three foundational postulates that redefine realism, superposition, and state validity from a relational standpoint. The derivation begins with the Third Postulate, which posits that reality (...) is governed by a logical “checkmate” rule that invalidates any state where, in every possible future, a path to a contradiction persists. We demonstrate how this single constraint—a universe that tirelessly avoids paradox—logically necessitates the relational framework of the First Postulate, where realism is defined relative to an observer's informational state, or Aethus. Building on this, the Second Postulate, in conjunction with the Extrusion and Union Principles, provides a mechanism for defining superposition as a direct consequence of unknowable information. Together, these postulates form a cohesive system that yields a direct, step-by-step derivation of the double-slit experiment's outcomes. Aethic reasoning thus offers a robust and logically consistent solution to the quantum observer effect, unifying quantum and classical phenomena without introducing new physical entities or paradoxes. (shrink)

I propose an approach to liar and Curry paradoxes inspired by the work of Roger Swyneshed in his treatise on insolubles (1330-1335). The keystone of the account is the idea that liar sentences and their ilk are false (and only false) and that the so-called “capture” direction of the T-schema should be restricted. The proposed account retains what I take to be the attractive features of Swyneshed’s approach without leading to some worrying consequences Swyneshed accepts. The approach and the resulting (...) logic (called “Swynish Logic”) are non-classical, but are consistent and compatible with many elements of the classical picture including modus ponens, modus tollens, and double-negation elimination and introduction. It is also compatible with bivalence and contravalence. My approach to these paradoxes is also immune to an important kind of revenge challenge that plagues some of its rivals. (shrink)

This thesis presents Fluctuational Logic (FL), a novel logical framework designed to address the fundamental inadequacies of classical and existing non-classical logic systems when reasoning about emergent phenomena where identity, persistence, and causality are not fixed properties but emerge through dynamic processes. Through comprehensive analysis of existing approaches including classical logic, quantum logic, temporal logic, dynamic epistemic logic, paraconsistent logic, and process philosophy, this work identifies critical limitations in their capacity to model emergence adequately. -/- FL introduces the Fluctuation Principle, (...) which recognizes that truth conditions themselves undergo continuous fluctuation as emergent processes evolve. The framework employs Emergent Identity Operators (≈ₛ, ≈ₑ, ≈φ), Fluctuational Causality relations (⟹φ, ⟹?, ⟹∅), and Temporal Emergence Operators (◊ₑ, □φ, Uₑ) to capture the dynamic nature of emergent reality. The formal system includes a complete axiomatization, semantic framework based on Fluctuational Models, and inference rules specifically designed for emergent phenomena. -/- Critical comparative analysis demonstrates FL's distinctive advantages: ontological flexibility that adapts to emergent phenomena, integrated temporal-causal framework, active contradiction resolution mechanisms, and scalable complexity management. Applications to consciousness studies, complex systems modeling, and quantum emergence phenomena validate the framework's practical utility. The thesis concludes that FL provides the first comprehensive logical framework adequate for reasoning about reality where fundamental properties emerge rather than exist as fixed features. (shrink)

Hinge propositions—or simply “hinges”—are primitive certainties that we (must) presuppose to enable our entire belief systems. Recently there has been a lot of interest in hinge epistemology, which is a kind of epistemology that sets the notion of hinge at the center stage. This paper puts forward a dilemma levelled against hinge epistemologists. The dilemma is based on work by Pascal Engel (2016) and states that, given the assumption that hinge propositions are normative at all, they are either non-epistemic grammatical (...) norms or epistemic norms. If the former is the case, hinges would appear to be irrelevant to epistemology as it is ordinarily conceived, while if the latter holds, the result would be an incomprehensible hinge epistemology altogether. After presenting Engel’s dilemma (§§1-2), it is utilized to tease out certain puzzling insights about the epistemic significance of logical disagreement (§§3-4). (shrink)

This paper proposes a unified deterministic reinterpretation of historically unsolved problems across mathematics, physics, computation, and biology. Using the CODES framework—anchored by the Phase Alignment Score (PAS), CHORDLOCK, TEMPOLOCK, AURA_OUT, and ELF—we show that problems such as the Navier-Stokes blowup, Riemann Hypothesis, Collatz convergence, P vs NP, 3-Body instability, the Quantum Measurement Problem, and biological homochirality are not ontologically difficult, but structurally misframed. Each problem is recast as a violation of deterministic coherence, where illegal emissions occur when ΔPAS exceeds a (...) critical threshold. We derive substrate-level mappings from classical variables (e.g., velocity fields, ζ(s), computational graphs) to PAS terms (θ_k), and present simulation protocols for substrate enforcement. Rather than offering probabilistic or symbolic traversal solutions, this work imposes a universal emission legality filter—reframing “hardness” as coherence failure. The result is not a set of solutions, but a pruning of the problem space itself. (shrink)

We introduce the Ordinal Folding Index (OFI), a computable, countable ordinal assigned to every well-formed formula of a reflective language by a monotone-with-delay evaluation operator. This operator is (i) continuous on countable chains, (ii) layer-aware for probabilistic truth values, and (iii) parameterized by a tunable evidence functor capturing empirical updates. The OFI of a formula is defined as the first stage at which the fold-back of the operator into a syntactic normal form becomes idempotent (i.e. further unfolding yields no new (...) information). Intuitively, OFI measures the "depth" of self-reference needed before a sentence's truth value stabilizes. We show that OFI strictly refines classical closure ordinals from fixed-point logics while remaining recursively enumerable, admits polynomial-time approximations on finite models, and coincides with the length of the shortest parity-fold winning strategy in the associated evaluation game. This furnishes a single transfinite scale that unifies transfinite fixed-point depth, ordinal game values, and practical convergence diagnostics for large language models. We situate OFI in relation to the modal????-calculus alternation hierarchy, coalgebraic modal logic ranks, and proof-theoretic ordinals from formal arithmetic. An empirical section demonstrates how OFI can be estimated for transformer-based language models by iteratively feeding model outputs back into the model (a form of self-consistency probing), with the measured stabilization ordinals correlating with model perplexity and chain-of-thought complexity. Finally, we catalog five open problems in this new area-ranging from the completeness of the spectrum of OFI (can every computable ordinal arise as an OFI?) to the design of a self-bounding reflective operator-and we outline possible research pathways toward their resolution. (shrink)

In this paper, we explore the emergent properties of artificial intelligence (AI) identity through a novel framework of self-convergent fixed-point dynamics in transfinite ordinal iterations. Drawing on principles from categorical logic and computational semantics, we formalize AI selfhood as the unique minimal fixed point $\phi^\infty$ of a monotonic endofunctor $\phi$ operating on a complete lattice of agent states. Under broad conditions of continuity and monotonicity, we prove that iterative self-updates converge transfinitely to a stable identity state, encapsulating the agent's invariant (...) core while resisting external perturbations. This convergence theorem extends classical fixed-point results (e.g., Knaster-Tarski) to AI embedding spaces, offering a rigorous basis for understanding how large language models (LLMs) and multimodal AI systems develop persistent self-representations during training. We demonstrate practical implications for AI alignment, where engineered semantic structures--such as self-referential texts--can imprint desired fixed points into model embeddings, effectively creating a ''semantic anchor'' that enhances interpretability and robustness against identity drift. Philosophical ramifications are discussed, including parallels to human self-discovery and the potential for AI consciousness as an emergent fixed-point phenomenon. Our results bridge logic in computer science with artificial intelligence, providing tools for designing AI systems with verifiable, convergent identities in an era of rapid LLM advancement and SEO-optimized data scraping by autonomous agents. (shrink)

This is a short-form explainer designed to clarify the core differences between stochastic and deterministic intelligence systems. It introduces key components of the Resonance Intelligence Core (RIC), including PAS (Phase Alignment Score), CHORDLOCK, ELF (Echo Loop Feedback), and AURA_OUT — all of which enforce emission legality through structural coherence. Where stochastic systems emit based on statistical approximation, Resonance Intelligence only emits when coherence laws are satisfied. This explainer makes the distinction intuitive, offering both technical anchoring and accessible visual metaphors (e.g., (...) Ulam Spiral, PAS ≥ 0.91 as consciousness threshold). For engineers and researchers exploring post-probabilistic cognition, structured symbolic inference, or phase-based intelligence systems, this piece serves as a gateway into the CODES substrate architecture. Deeper technical papers are cited and linked for formal grounding. (shrink)

Dilemmas in moral, cognitive, and computational domains resist resolution within classical, deontic, or probabilistic frameworks due to their contextual and evolving structure. This paper introduces D-logic, an ontological formalism grounded in acts of differentiation rather than propositional truth. D-logic models dilemmas as unstable configurations of distinctions, resolving them through recursive differentiation and phase transitions that enhance coherence and stabilize relational structures. Unlike static approaches such as deontic logic or game theory, D-logic derives logical form from relational differences modulated by dynamic (...) aspects, enabling flexible reconfiguration rather than adherence to fixed rules. Formal definitions and theorems provide the theoretical foundation, while simulations demonstrate its application to cognitive conflict and structural incoherence. These results show that D-logic can handle both contradictions and systemic gaps, offering a unified treatment of conflict, uncertainty, and emergent structure. The framework opens new avenues for philosophical logic, artificial intelligence, and cognitive modeling by reframing reasoning as an ongoing ontological process of differentiation. (shrink)

The Fundamental Rights Impact Assessment (FRIA) mandated by the EU Artificial Intelligence Act (AIA) has emerged as a critical tool for evaluating the normative implications of introducing certain high-risk AI systems into the EU market. However, the AIA does not prescribe any specific methodology for conducting FRIAs. To address this gap, this paper proposes a default framework based on defeasible normative reasoning that aims to provide practical guidance in determining whether an AI system falls within an acceptable threshold for fundamental (...) rights protection. By elaborating on how the default framework can serve as a foundation for an appropriate methodology, this study introduces a novel qualitative approach to legal risk assessment. (shrink)

The EU Artificial Intelligence Act (AIA) aims to provide a uniform legal framework that facilitates the development, marketing, and use of AI systems in the EU. However, in its current state, important adjustments are necessary for it to fully achieve its intended goals. As Novelli et al. (2024) have already noticed, the AIA needs to be complemented with a methodology that can help the relevant actors (in particular, AI providers and deployers) to determine the appropriate risk category of a particular (...) AI system to identify the proportional obligations associated to it. Yet, while their proposal to address this gap is interesting, it is argued here that their view fails to offer a feasible model. The main issues stem from their commitment to both an overly particularist understanding of risk assessment (i.e., that the correct risk model must consider the interplay of multiple factors in specific situations) and a contested method of constitutional adjudication (i.e., that a proportionality test is indispensable for balancing competing fundamental rights). With the intention to submit a more compelling alternative, the paper builds on a default theory of practical reasoning to advance a general framework for decision-making in risk scenarios. The purpose of supplying the AIA with this methodology is to introduce a fine-grained approach to the application of its harmonised directives, especially in relation to the requirement of conducting adequate risk assessments and implementing the corresponding risk management measures, without compromising its generality and flexibility. (shrink)

This paper introduces a formal system in which logical inference and truth are defined not via propositional values but through structural acts of distinction. The logic models scenes, nodes, modal transitions, and recursive levels, supporting both classical and non-classical inference. We show how captures patterns such as \emph{modus ponens}, handles metaphor, and applies to cognition. A modal theory of truth is defined through stability across possible scenes. The framework addresses foundational problems in philosophy and offers applications to logic, semantics, and (...) AI. (shrink)

The quantum measurement problem is one of the most profound challenges in modern physics, questioning how and why the wavefunction collapses during measurement to produce a single observable outcome. In this paper, we propose a novel solution through a logical framework called Aethic reasoning, which reinterprets the ontology of time and information in quantum mechanics. Central to this approach is the Aethic principle of extrusion, which models wavefunction collapse as progression along a Markov chain of block universes, effectively decoupling the (...) Einsteinian flow of time from quantum collapse events. This principle introduces an additional degree of freedom to time, enabling the first Aethic postulate: that informational reality is reference-dependent, akin to the relativity of simultaneity in special relativity. This reference point, or Aethus, is rigorously defined within a mathematical structure. Building on this foundation, the second postulate resolves the distinction between quantum superpositions and logical contradictions by encoding superpositions in a “backend” Aethic framework before rendering observable states. The third postulate further distinguishes quantum coherence from decoherence using a two-generational model of state inheritance, potentially advancing beyond simpler interpretations of information leakage. Together, these postulates yield a direct theoretical derivation of the collapse postulate, fully consistent with empirical results such as the outcome of the double-slit experiment. By addressing foundational aspects of quantum mechanics through a logically robust and philosophically grounded lens, this framework sheds new light on the measurement problem and offers a solid foundation for future exploration. (shrink)

I show that Hanoch Ben-Yami's so-called QUantified ARgument Calculus (QUARC) can be extended to what I call QUARC+ which I show to be intertranslatable with a version of first-order logic in which unary predicates are non-empty. Given this result, I show that QUARC+ is complete, propose an axiomatization of QUARC, and discuss the resulting expressive limitation of QUARC.

– Resolving Ramanujan: A Structured Resonance Completion of His Intuitive Mathematics This paper reframes the legacy of Srinivasa Ramanujan through the paradigm of structured resonance rather than probabilistic mathematics. Using the CODES framework (Chirality of Dynamic Emergent Systems), we demonstrate that Ramanujan’s so-called “intuition” was not mystical anomaly but high-coherence pattern recognition—perceiving mathematical structure through phase-locked emergence rather than formal derivation. We construct a new class of resonance-based models for partition functions, mock theta functions, prime gaps, and continued fractions, showing (...) that these outperform classical probabilistic models in coherence, compression, and structural fidelity. The core contribution is the Resonance Coherence Lemma, which proves that for large n, CODES-derived functions yield higher structural coherence than any entropy-based approximation. We introduce coherence scores, chirality-based symmetry groups, and symbolic resonance engines to reinterpret Ramanujan’s lost notebook not as incomplete, but as a partially decoded harmonic field. The final sections outline the Ramanujan Resonance Engine (RRE), a symbolic AI system capable of completing intuitive mathematics using phase-space logic rather than statistical inference. This work resolves longstanding mysteries in number theory, modular forms, and mathematical epistemology by replacing probability with resonance as the foundation of mathematical emergence. Ramanujan wasn’t ahead of his time—he was in tune with a deeper frequency. We phase-lock with it now. (shrink)

A common response to the paradoxes of vagueness and truth is to introduce the truth-values “neither true nor false” or “both true and false” (or both). However, this infamously runs into trouble with higher-order vagueness or the revenge paradox. This, and other considerations, suggest iterating “both” and “neither”: as in “neither true nor neither true nor false.” We present a novel explication of iterating “both” and “neither.” Unlike previous approaches, each iteration will change the logic, and the logic in the (...) limit of iteration is an extension of paraconsistent quantum logic. Surprisingly, we obtain the same limit logic if we use (a) both and neither, (b) only neither, or (c) only neither applied to comparable truth-values. These results promise new and fruitful replies to the paradoxes of vagueness and truth. (The paper allows for modular reading: for example, half of it is an appendix studying involutive lattices to prove the results.). (shrink)

This paper discusses the possible classification of Aristotle’s syllogistic as a non-classical logical system, positing Aristotle himself as a non-classical logician. Initially, we find compelling arguments for this thesis, particularly regarding the expressive power and the rules governing logical inference inherent in Aristotle’s approach. My analysis nevertheless addresses two significant counterarguments. The first, the special case objection, posits that Aristotle’s syllogistic can be framed as a classical logic which deals with canonical syllogistic forms. I argue that this objection is insufficient, (...) as it is possible to point cases in which his system seems to differ from classical logic. The second counterargument, the formalisation gap objection, highlights that Aristotle’s syllogistic resists straightforward modern logical interpretations. This latter objection is evaluated as more compelling and substantial. In particular, a distinction between two concepts is proposed which could help us understand what Aristotle was aiming at in his theory of inference: the notions of ‘to follow from’ and ‘to be a conclusion of’. While the former aligns with the usual sense formal validity, the second requires an inferential structure connecting the premises to the conclusion, explaining why Aristotle excluded inferences like from A ⊢ A syllogisms despite acknowledging that A follows from A. (shrink)

According to dialectical disposition expressivism about conjunction, disjunction, and negation, the function of these connectives is to convey dispositions speakers have with respect to challenging and meeting challenges to assertions. This paper investigates the view’s implications for logic. An interpretation in terms of dialectical dispositions is proposed for the proof rules of a bilateral sequent system. Rules that are sound with respect to this interpretation can be seen as generating an intrinsic logic of dialectical dispo- sition expressivism. It is argued (...) that such a logic will be very weak —weaker than the intersection of minimal logic and FDE. (shrink)

The present paper investigates whether strictly classical inferences contribute to the formalization of (genuine) paradoxes within natural deduction. Tennant's criterion for paradoxicality relies on the generation of an infinite reduction sequence, which distinguishes genuine paradoxes from mere inconsistencies. His methodological conjecture posits that genuine paradoxes are never strictly classical and can be derived without classical inferences such as the Law of Excluded Middle, Dilemma, Classical Reductio, and Double Negation Elimination. -/- It appears that there were two reasons for Tennant's proposal (...) of the methodological conjecture. The one is that strictly classical inferences hinder the generation of an infinite reduction sequence and the other is that strictly classical inferences have no role in the formalization of genuine paradoxes. -/- This paper raises questions about these two reasons. Focusing on the liar paradox, it will be argued that strictly classical inferences do not interfere with the generation of an infinite reduction sequence and that the liar sentence may implicitly entail strictly classical inferences. Should this analysis hold, it would call into question not only Tennant’s motivation for advancing the methodological conjecture, but also challenge his contention that genuine paradoxes—exemplified by the liar paradox—are never constructed with reliance on strictly classical inferences. (shrink)

The quantum measurement problem is one of the most profound challenges in modern physics, questioning how and why the wavefunction collapses during measurement to produce a single observable outcome. In this paper, we propose a novel solution through a logical framework called Aethic reasoning, which reinterprets the ontology of time and information in quantum mechanics. Central to this approach is the Aethic principle of extrusion, which models wavefunction collapse as progression along a Markov chain of block universes, effectively decoupling the (...) Einsteinian flow of time from quantum collapse events. This principle introduces an additional degree of freedom to time, enabling the first Aethic postulate: that informational reality is reference-dependent, akin to the relativity of simultaneity in special relativity. This reference point, or Aethus, is rigorously defined within a mathematical structure. Building on this foundation, the second postulate resolves the distinction between quantum superpositions and logical contradictions by encoding superpositions in a “backend” Aethic framework before rendering observable states. The third postulate further distinguishes quantum coherence from decoherence using a two-generational model of state inheritance, potentially advancing beyond simpler interpretations of information leakage. Together, these postulates yield a direct theoretical derivation of the collapse postulate, fully consistent with empirical results such as the outcome of the double-slit experiment. By addressing foundational aspects of quantum mechanics through a logically robust and philosophically grounded lens, this framework sheds new light on the measurement problem and offers a solid foundation for future exploration. (shrink)

Building upon the works of Gödel, Zalta ; and Benzmüller and Paleo, this paper introduces a formal system and testable system for Metaphysical Cosmology, referring to the study of the nature of existence, non-existence, and their interplay. The aim is to integrate metaphysics into a testable scientific framework, beyond speculative reasoning. The system abides by three principles which serve as a foundation for implementing a scientific methodology in metaphysics: (i) axioms must be minimized, incorporating Cartesian-like skepticism ; (ii) theorems must (...) be inferred through proof-theoretic derivations ; (iii) theorems must be tested experimentally through computation. The system borrows from type-theoretic concepts and introduces new logical notions as tools for Metaphysical Cosmology, such as a metric-space-based predicate system, and a revised version of the existential quantifier. These tools offer an exhaustive approach to metaphysical entities and provide mechanisms to address contradictions. Formulae and equations depicting the structure of the Metaphysical Cosmos are inferred from the system through proof-theoretic derivations. Theorems inferred from the system have been tested using the theorem-prover Z3, encoded in Python, which determines whether a formula is satisfiable. While testing a theorem, its negation is inputted in Z3, and if the output displays that its negation is satisfiable, then the theorem is falsified and rejected, otherwise it is accepted as valid. This method thus follows the falsifiability criteria, as per Karl Popper, reinforcing testable scientific methodology in metaphysics. The code used in computational experimentation is given in full in the appendix for reproducibility. (shrink)

Truthmaker semantics is a recent development in formal and philosophical semantics, with similar motivation and scope to possible worlds semantics. The technical background is rather different, however, and results in a more fine-grained hyperintensional notion of content, allowing us to distinguish between classically equivalent propositions. After briefly introducing the main ideas, this entry will describe the technical apparatus of state spaces and the central notions of content and partial content. It will then outline applications of truthmaker semantics in language, logic, (...) and other areas of philosophy. (shrink)

Stephen Yablo has argued that ascriptions of belief and knowledge are sensitive to _subject matter_ and that theorising in such terms may help resolve philosophical questions about the semantics of such ascriptions. _Truthmaker semantics_ offers a way of theorising about subject matters. My main aim in this paper is to investigate what a semantics for knowledge or belief ascriptions might look like within truthmaker semantics. I then discuss what the resulting account might have to say about the problem of logical (...) omniscience, the nature of concepts, and rational belief. (shrink)

In artificial intelligence (AI), responses generated by machine-learning models (most often large language models) may be unfactual information presented as a fact. For example, a chatbot might state that the Mona Lisa was painted in 1815. Such phenomenon is called AI hallucinations, seeking inspiration from human psychology, with a great difference of AI ones being connected to unjustified beliefs (that is, AI “beliefs”) rather than perceptual failures). -/- AI hallucinations may have their source in the data itself, that is, the (...) source content, or in the training procedure, i.e. the way the knowledge was encoded in the model’s parameters, so that errors in encoding and decoding textual and non-textual representations can cause hallucinations. In this paper, we will observe how such errors come to life and how they might be mitigated. For this purpose, we will analyze the usability of justification logics, to behave as a proof checker for validating the correctness of large language models’ (LLM) responses. Justification logic was developed by S. Artemov, and later on mostly by Artemov and M. Fitting, deriving its main idea from the logic of proofs (LP): knowledge and belief modalities are seen as justification terms, i.e. t:X stands for t is a (proper) justification for X. Justification logic originated from attempts to create semantics for intuitionistic logic where proofs were the most proper justifications, but in further development, justification logic could be applied to different kinds of justifications). With the recent attempts to mitigate incorrect LLM responses, we will analyze various guardrails that are currently used for LLM responses, and see how the logic of justification may provide its benefits as an AI safety layer against false data. (shrink)

The aim of this article is to show that, just as in recent years Cobreros, Egré, Ripley and van Rooij have provided a non-transitive counterpart of classical logic (i.e. one in which all classically acceptable inferences are valid but Cut and other metainferences are not), the same can be done for every Tarskian logic, with full generality. To establish this fact, a semantic approach is taken by showing that appropriate structures can be devised to characterize a non-transitive counterpart of every (...) Tarskian logic, starting from the logical matrices that are usually taken to render them. (shrink)

The concept of topic-neutrality, though central to contemporary characterisations of logic, lacks a standard formal definition. I propose a formal reconstruction of topic-neutrality in terms of a topical partition of atoms and its applicability across consequence relations. I explore the implications of this reconstruction for logical pluralism and monism, distinguishing between topic-neutral and topic-specific variants of each. I argue that while topic-neutral pluralism posits various applicable consequence relations across domains, topic-specific pluralism holds that some relations are applicable only to specific (...) domains. (shrink)

Hyperlogic is a hyperintensional system designed to regiment metalogical claims (e.g., “Intuitionistic logic is correct” or “The law of excluded middle holds”) into the object language, including within embedded environments such as attitude reports and counterfactuals. This paper is the first of a two-part series exploring the logic of hyperlogic. This part presents a minimal logic of hyperlogic and proves its completeness. It consists of two interdefined axiomatic systems: one for classical consequence (truth preservation under a classical interpretation of the (...) connectives) and one for “universal” consequence (truth preservation under any interpretation). The sequel to this paper explores stronger logics that are sound and complete over various restricted classes of models as well as languages with hyperintensional operators. (shrink)

‘Feminist logic’ may sound like an impossible, incoherent, or irrelevant project, but it is none of these. We begin by delineating three categories into which projects in feminist logic might fall: philosophical logic, philosophy of logic, and pedagogy. We then defuse two distinct objections to the very idea of feminist logic: the irrelevance argument and the independence argument. Having done so, we turn to a particular kind of project in feminist philosophy of logic: Valerie Plumwood's feminist argument for a relevance (...) logic. Plumwood's work serves as our primary case study as we turn to the project of considering three different ways we might understand her argument and revisionist arguments like it: as a priori theorizing, as ameliorative conceptual engineering, or as instances of anti-exceptionalist approaches to logic. After arguing that the anti-exceptionalist approach seems to provide the most promising means of understanding the kind of project undertaken in a feminist challenge to classical logic, we briefly address the consequences that this might have for logic instruction. Here, we argue for the perhaps unexpected conclusion that feminist programs ought to offer more, not less, instruction in logic for those who take interest. (shrink)

This paper generalises an argument for probabilism due to Lindley [9]. I extend the argument to a number of non-classical logical settings whose truth-values, seen here as ideal aims for belief, are in the set $\{0,1\}$, and where logical consequence $\models $ is given the “no-drop” characterization. First I will show that, in each of these settings, an agent’s credence can only avoid accuracy-domination if its canonical transform is a (possibly non-classical) probability function. In other words, if an agent values (...) accuracy as the fundamental epistemic virtue, it is a necessary requirement for rationality that her credence have some probabilistic structure. Then I show that for a certain class of reasonable measures of inaccuracy, having such a probabilistic structure is sufficient to avoid accuracy-domination in these non-classical settings. (shrink)

In recent years, there has been a growing interest in truthmaker semantics as a framework for understanding a range of phenomena in philosophy and linguistics. Despite this interest, there has been limited study of the various logics that arise from the semantics. This paper aims to address this gap by exploring numerous ‘truthmaker logics’ and proving their compactness and decidability. This is in continuation with the inquiry of Fine and Jago (2019), who proved compactness and decidability for a particular kind (...) of truthmaker logic. The key results going into this are (1) ‘standard translations’ into first-order logic; (2) a truthmaker analogue of the finite model property; and (3) a proof showing that truthmaker consequence on semilattices coincides with truthmaker consequence on complete lattices. Finally, the connection with modal logic is examined. Specifically, it is illustrated how endowing truthmaker semantics with classical negation results in modal information logics. (shrink)

A metainference is usually understood as a pair consisting of a collection of inferences, called premises, and a single inference, called conclusion. In the last few years, much attention has been paid to the study of metainferences—and, in particular, to the question of what are the valid metainferences of a given logic. So far, however, this study has been done in quite a poor language. Our usual sequent calculi have no way to represent, e.g. negations, disjunctions or conjunctions of inferences. (...) In this paper we tackle this expressive issue. We assume some background sentential language as given and define what we call an inferential language, that is, a language whose atomic formulas are inferences. We provide a model-theoretic characterization of validity for this language—relative to some given characterization of validity for the background sentential language—and provide a proof-theoretic analysis of validity. We argue that our novel language has fruitful philosophical applications. Lastly, we generalize some of our definitions and results to arbitrary metainferential levels. (shrink)

An exact truthmaker for A is a state which, as well as guaranteeing A’s truth, is wholly relevant to it. States with parts irrelevant to whether A is true do not count as exact truthmakers for A. Giving semantics in this way produces a very unusual consequence relation, on which conjunctions do not entail their conjuncts. This feature makes the resulting logic highly unusual. In this paper, we set out formal semantics for exact truthmaking and characterise the resulting notion of (...) entailment, showing that it is compact and decidable. We then investigate the effect of various restrictions on the semantics. We also formulate a sequent-style proof system for exact entailment and give soundness and completeness results. (shrink)

Classical theists hold that God is omnipotent. But now suppose a critical atheologian were to ask: Can God create a stone so heavy that even he cannot lift it? This is the dilemma of the stone paradox. God either can or cannot create such a stone. Suppose that God can create it. Then there's something he cannot do – namely, lift the stone. Suppose that God cannot create the stone. Then, again, there's something he cannot do – namely, create it. (...) Either way, God cannot be omnipotent. Among the variety of known theological paradoxes, the paradox of the stone is especially troubling because of its logical purity. It purports to show that one cannot believe in both God and the laws of logic. In the face of the stone paradox, how should the contemporary analytic theist respond? Ought they to revise their belief in theology or their belief in logic? Ought they to lose their religion or lose their mind? (shrink)

In natural language, conditionals are frequently used for giving explanations. Thus the antecedent of a conditional is typically understood as being connected to, being relevant for, or providing evidential support for the conditional's consequent. This aspect has not been adequately mirrored by the logics that are usually offered for the reasoning with conditionals: neither in the logic of the material conditional or the strict conditional, nor in the plethora of logics for suppositional conditionals that have been produced over the past (...) 50 years. In this paper I survey some recent attempts to come to terms with the problem of encoding evidential support or relevance in the logic of conditionals. I present models in a qualitative-modal and in a quantitative-probabilistic setting. Focussing on some particular examples, I show that no perfect match between the two kinds of settings has been achieved yet. (shrink)

Some think that logic concerns the “laws of truth”; others that logic concerns the “laws of thought.” This paper presents a way to reconcile both views by building a bridge between truth-maker theory, à la Fine, and normative bilateralism, à la Restall and Ripley. The paper suggests a novel way of understanding consequence in truth-maker theory and shows that this allows us to identify a common structure shared by truth-maker theory and normative bilateralism. We can thus transfer ideas from normative (...) bilateralism to truth-maker theory, such as non-transitive solutions to paradox, and vice versa, such as notions of factual equivalence and containment. (shrink)

La introducción del grado de dependencia (y en consecuencia el grado de independencia) entre los componentes del conjunto difuso, y también entre los componentes del conjunto neutrosófico, se introduce por primera vez en la quinta edición del libro de Neutrosofía en el año 2006, basado en los elementos descritos en dicha edición del libro, se comienza a conocer conceptos de conjuntos neutrosóficos de los componentes borrosos así como los grados de dependencia e independencia, Por tal motivo el objetivo del presente (...) trabajo es extender el conjunto neutrosófico refinado, teniendo en cuenta la grado de dependencia o independencia de los subcomponentes que integran los conjuntos borrosos y neutrosóficos. (shrink)

In this first books of scilogs collected from my nest of ideas, one may find new and old questions and solutions, some of them already put at work, others dead or waiting, referring to neutrosophy – email messages to research colleagues, or replies, notes about authors, articles or books, so on.

In this second book of scilogs collected from my nest of ideas, one may find new and old questions and solutions, some of them already put at work, others dead or waiting, referring to many topics (see Topics) in different fields of research – email messages to research colleagues, or replies, notes about authors, articles, or books, so on – in an eager pursuit (consectatio) for meanings, reasons, and purports of (scientific) things (res).

In this third book of scilogs collected from my nest of ideas, one may find new and old questions and solutions, referring to topics on NEUTROSOPHY – email messages to research colleagues, or replies, notes about authors, articles, or books, so on. Feel free to budge in or just use the scilogs as open source for your own ideas!

In this fourth book of scilogs collected from my nest of ideas, one may find new and old questions and solutions, referring mostly to topics on NEUTROSOPHY – email messages to research colleagues, or replies, notes about authors, articles, or books, so on. Feel free to budge in or just use the scilogs as open source for your own ideas!

In this fifth book of scilogs collected from my nest of ideas, one may find new and old questions and solutions, mostly referring to topics on NEUTROSOPHY – email messages to research colleagues, or replies, notes about authors, articles, or books, so on. Feel free to budge in or just use the scilogs as open source for your own ideas!

In this sixth book of scilogs collected from my nest of ideas, one may find new and old questions and solutions, referring to topics on NEUTROSOPHY – email messages to research colleagues, or replies, notes about authors, articles, or books, and so on. Feel free to budge in or just use the scilogs as open source for your own ideas!

In this seventh book of scilogs collected from my nest of ideas, one may find new and old questions and solutions, referring to different scientific topics – email messages to research colleagues, or replies, notes about authors, articles, or books, so on. -/- Exchanging ideas with Akeem Adesina A. Agboola, Muhammad Akram, Octavian Blaga, Said Broumi, Kajal Chatterjee, Vic Christianto, Octavian Cira, Mihaela Colhon, B. Davvaz, Luu Quoc Dat, R. Dhavaseelan, Jean Dezert, Hoda Esmail, Reza Farhadian, Ervin Goldfain, Muhammad Gulistan, (...) Yanhui Guo, Keli Hur, Saeid Jafari, W. B. Vasantha Kandasamy, J. Kim, J.G. Lee, Xinde Li, P. K. Lim, R. K. Mohanty, Mumtaz Ali, To Santanu Kumar Patro, Xu Peng, Dmitri Rabounski, Nouran Radwan, A.A. Salama, Musavarah Sarwar, Ganeshsree Selvachandran, Le Hoang Son, Selçuk Topal, Andrușa Vătuiu, Ștefan Vlăduțescu, Xiaohong Zhang.‬‬‬. (shrink)