The classical debate between determinism and libertarian free will relies on a strict application of the Law of Excluded Middle, forcing human agency into a binary of either being fully determined by antecedent causes or relying on metaphysically uncaused events. This paper proposes a third category of causation via a stochastic, self-modifying model of consciousness. We represent mental states not as discrete symbols, but as continuously updating probability distributions over an effectively infinite-dimensional space. This space is constructed via (...) the tensor product of interacting cognitive domains—such as memory, emotion, intelligence, and environmental input—with thoughts emerging through a stochastic collapse mechanism. Drawing on the epistemological limits demonstrated by the Abel-Ruffini theorem and David Wolpert's formal proofs on the physical limits of embedded inference, we introduce the principle of "structural stochasticity". We demonstrate that while determinism may formally hold at an inaccessible infinite precision, it is operationally vacuous for any finite, self-modifying agent embedded within the system. Consequently, the trajectory of a conscious agent is heavily caused and structurally constrained, yet irreducibly unpredictable and creatively self-generating at every physically accessible level of description. (shrink)

The classical debate between determinism and libertarian free will relies on a strict application of the Law of Excluded Middle, forcing human agency into a binary of either being fully determined by antecedent causes or relying on metaphysically uncaused events. This paper proposes a third category of causation via a stochastic, self-modifying model of consciousness. We represent mental states not as discrete symbols, but as continuously updating probability distributions over an effectively infinite-dimensional space. This space is constructed via (...) the tensor product of interacting cognitive domains—such as memory, emotion, intelligence, and environmental input—with thoughts emerging through a stochastic collapse mechanism. Drawing on the epistemological limits demonstrated by the Abel-Ruffini theorem and David Wolpert's formal proofs on the physical limits of embedded inference, we introduce the principle of structural stochasticity. We demonstrate that while determinism may formally hold at an inaccessible infinite precision, it is operationally vacuous for any finite, self-modifying agent embedded within the system. Consequently, the trajectory of a conscious agent is heavily caused and structurally constrained, yet irreducibly unpredictable and creatively self-generating at every physically accessible level of description. (shrink)

This paper argues that the fundamental laws of logic can be derived through analysis of the structure of existence. Beginning from the first principle that something exists, we show that determinate existence necessarily entails differentiation — distinguishing A from not-A. This primary distinction, as the first and unavoidable consequence of existence, exhibits five structural properties that correspond to the classical laws of logic: identity (stability of what is distinguished), non-contradiction (exclusivity of the distinguished from its background), excluded middle (...) (totality of the distinction), sufficient reason (self-grounding of the distinction), and order (sequential and hierarchical structure). The Law of Order, which governs the inherent sequence and hierarchy in the structure of existence, is proposed as a fifth fundamental law coordinate with the classical three plus the principle of sufficient reason. Crucially, this derivation is epistemic rather than ontological: the laws are conditions for the possibility of existence, not consequences of it; but we discover them by analyzing what determinate existence requires. This approach offers a unified foundation for logic that explains both why these laws hold and why violations produce characteristic errors. The framework has implications for the foundations of mathematics, paradox resolution, and the philosophy of infinity, which are briefly indicated as directions for future work. (shrink)

This article reconstructs the fundamental logical framework originating with Aristotle and its modern developments. It shows the failure of the law of excluded middle (LEM), establishes that proof by contradiction is valid independent of LEM, and introduces a paradox‑free definition of sets that requires no axiomatic restrictions. This framework thus offers a coherent and robust alternative to the currently accepted logical and set‑theoretic foundations.

This article reconstructs the fundamental logical framework originating with Aristotle and its modern developments. It shows the failure of the law of excluded middle (LEM), establishes that proof by contradiction is valid independent of LEM, and introduces a paradox‑free definition of sets that requires no axiomatic restrictions. This framework thus offers a coherent and robust alternative to the currently accepted logical and set‑theoretic foundations.

Various philosophers have long since been attracted to the doctrine that future contingent propositions systematically fail to be true—what is sometimes called the doctrine of the open future. However, open futurists have always struggled to articulate how their view interacts with standard principles of classical logic—most notably, with the Law of Excluded Middle. For consider the following two claims: Trump will be impeached tomorrow; Trump will not be impeached tomorrow. According to the kind of open futurist at issue, (...) both of these claims may well fail to be true. According to many, however, the disjunction of these claims can be represented as p ∨ ~p—that is, as an instance of LEM. In this essay, however, I wish to defend the view that the disjunction these claims cannot be represented as an instance of p ∨ ~p. And this is for the following reason: the latter claim is not, in fact, the strict negation of the former. More particularly, there is an important semantic distinction between the strict negation of the first claim [~] and the latter claim. However, the viability of this approach has been denied by Thomason, and more recently by MacFarlane and Cariani and Santorio, the latter of whom call the denial of the given semantic distinction “scopelessness”. According to these authors, that is, will is “scopeless” with respect to negation; whereas there is perhaps a syntactic distinction between ‘~Will p’ and ‘Will ~p’, there is no corresponding semantic distinction. And if this is so, the approach in question fails. In this paper, then, I criticize the claim that will is “scopeless” with respect to negation. I argue that will is a so-called “neg-raising” predicate—and that, in this light, we can see that the requisite scope distinctions aren’t missing, but are simply being masked. The result: a under-appreciated solution to the problem of future contingents that sees and as contraries, not contradictories. (shrink)

The classical laws of logic—identity, non-contradiction, and excluded middle—are almost universally treated as primitive axioms: presupposed by every argument but grounded by none. This paper proposes a derivation. It argues that any coherent existence-ground must differentiate into two co-primordial, antithetical aspects: bounded structural content (here designated SP, or 1) and its unbounded contextual background (here designated IP, or 0). This binary is not a logical construction but the ontological precondition for any information to exist at all—the structure that (...) logic subsequently describes. From this minimal commitment, the three classical laws follow necessarily: identity from the self-definition of SP against IP; non-contradiction from the antithetical nature of the two poles; excluded middle from the binary’s admission of no third value. The paper engages Graham Priest’s defense of paraconsistent logic and W.V.O. Quine’s contingency thesis as the two most direct challenges, arguing that both presuppose the binary they attempt to revise. No further metaphysical commitments are required beyond the co-primordial structure itself. (shrink)

This paper establishes the Law of Order—the principle that logic possesses inherent sequential and hierarchical structure—as the fifth fundamental law of logic, coordinate with identity, non-contradiction, excluded middle, and sufficient reason. We derive the law from the structure of primary distinction: the act by which anything determinate is differentiated from what it is not. The derivation shows that distinction necessarily exhibits sequence (from undifferentiated to differentiated) and hierarchy (the distinguished presupposes the act of distinguishing). We introduce the E/R/R (...) Framework (Elements/Roles/Rules) as the structural decomposition of any determinate system: Elements (what exists—grounded in L1), Roles (why signifficant—grounded in L4), and Rules (how structured—grounded in L5). This framework provides both a method for analyzing systems and a diagnostic tool for paradoxes. We derive the L5-Resolution Principle: when multiple elements qualify for the same role, L5 demands deterministic resolution through leftmost selection—not as arbitrary convention but as the unique order-respecting choice. This transforms a philosophical principle into a concrete computational rule. We provide a formal treatment in type theory, demonstrating that paradoxes are blocked structurally at the level of type-checking. The explanatory power of the Law of Order is shown through the E/R/R diagnosis of paradoxes: Russell's paradox, the Liar, and related constructions all involve confusing E/R/R categories—attempting to place an entity at two levels simultaneously. The framework provides a unified philosophical foundation for formal solutions including Russell's type theory, Tarski's hierarchy, and ZFC's axiom of regularity. (shrink)

In this multi-disciplinary investigation we show how an evidence-based perspective of quantification---in terms of algorithmic verifiability and algorithmic computability---admits evidence-based definitions of well-definedness and effective computability, which yield two unarguably constructive interpretations of the first-order Peano Arithmetic PA---over the structure N of the natural numbers---that are complementary, not contradictory. The first yields the weak, standard, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically verifiable Tarskian truth values to the formulas of PA under the interpretation. (...) The second yields a strong, finitary, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically computable Tarskian truth values to the formulas of PA under the interpretation. We situate our investigation within a broad analysis of quantification vis a vis: * Hilbert's epsilon-calculus * Goedel's omega-consistency * The Law of the Excluded Middle * Hilbert's omega-Rule * An Algorithmic omega-Rule * Gentzen's Rule of Infinite Induction * Rosser's Rule C * Markov's Principle * The Church-Turing Thesis * Aristotle's particularisation * Wittgenstein's perspective of constructive mathematics * An evidence-based perspective of quantification. By showing how these are formally inter-related, we highlight the fragility of both the persisting, theistic, classical/Platonic interpretation of quantification grounded in Hilbert's epsilon-calculus; and the persisting, atheistic, constructive/Intuitionistic interpretation of quantification rooted in Brouwer's belief that the Law of the Excluded Middle is non-finitary. We then consider some consequences for mathematics, mathematics education, philosophy, and the natural sciences, of an agnostic, evidence-based, finitary interpretation of quantification that challenges classical paradigms in all these disciplines. (shrink)

This dissertation examines several of the problems that Hilbert discovered in the foundations of mathematics, from a metalogical perspective. The problems manifest themselves in four different aspects of Hilbert’s views: (i) Hilbert’s axiomatic approach to the foundations of mathematics; (ii) His response to criticisms of set theory; (iii) His response to intuitionist criticisms of classical mathematics; (iv) Hilbert’s contribution to the specification of the role of logical inference in mathematical reasoning. This dissertation argues that Hilbert’s axiomatic approach was guided primarily (...) by model theoretical concerns. Accordingly, the ultimate aim of his consistency program was to prove the model-theoretical consistency of mathematical theories. It turns out that for the purpose of carrying out such consistency proofs, a suitable modification of the ordinary first-order logic is needed. To effect this modification, independence-friendly logic is needed as the appropriate conceptual framework. It is then shown how the model theoretical consistency of arithmetic can be proved by using IF logic as its basic logic. Hilbert’s other problems, manifesting themselves as aspects (ii), (iii), and (iv)—most notably the problem of the status of the axiom of choice, the problem of the role of the law of excluded middle, and the problem of giving an elementary account of quantification—can likewise be approached by using the resources of IF logic. It is shown that by means of IF logic one can carry out Hilbertian solutions to all these problems. The two major results concerning aspects (ii), (iii) and (iv) are the following: (a) The axiom of choice is a logical principle; (b) The law of excluded middle divides metamathematical methods into elementary and non-elementary ones. It is argued that these results show that IF logic helps to vindicate Hilbert’s nominalist philosophy of mathematics. On the basis of an elementary approach to logic, which enriches the expressive resources of ordinary first-order logic, this dissertation shows how the different problems that Hilbert discovered in the foundations of mathematics can be solved. (shrink)

Nicomachean Ethics/NE, Book III, Chapters 1-5, provides Aristotle’s account of “Voluntary Movement.” It, thus, draws the Passion-Action distinction, only posited earlier in Categories, while also serving as the linchpin of NE’s discussion of Virtue, in explicitly connecting it to Right Reason. My explication of this text renders its terminology consistent with the Law of Excluded Middle and rebuts two criticisms of the Eudaimonistic Axiology on which it is based. These results are shown to be entailments of Aristotle’s doctrine (...) that Voluntary is (a) Relative. That categorization is then used to help treat a problematic case of Deliberation beyond his purview. (shrink)

Chrysippus often talks as if there is a third option when we might expect that two options in response to a question are exhaustive. Things are true, false or neither; equal, unequal, or neither; the same, different, or neither.. and so on. There seems to be a general pattern here that calls for a general explanation. This paper offers a general explanation of this pattern, preserving Stoic commitments to excluded middle and bivalence, arguing that Chrysippus employs this trichotomy (...) move when he wishes to argue that apparent contradictories are only contraries, and wishes to endorse a third option. This general explanation of the pattern of trichotomies sheds light on a number of interpretive puzzles, including on Chrysippus's response to the paradox of the cone. The purpose of these trichotomies is also discussed, and it is suggested that they originate from the dialectical context in which philosophical problems were posed. (shrink)

The thesis that the two-valued system of classical logic is insufficient to explanation the various intermediate situations in the entity, has led to the development of many-valued and fuzzy logic systems. These systems suggest that this limitation is incorrect. They oppose the law of excluded middle (tertium non datur) which is one of the basic principles of classical logic, and even principle of non-contradiction and argue that is not an obstacle for things both to exist and to not (...) exist at the same time. However, contrary to these claims, there is no inadequacy in the two-valued system of classical logic in explanation the intermediate situations in existence. The law of exclusion and the intermediate situations in the external world are separate things. The law of excluded middle has been inevitably accepted by other logic systems which are considered to reject this principle. The many-valued and the fuzzy logic systems do not transcend the classical logic. Misconceptions from incomplete information and incomplete research are effective in these criticisms. In addition, it is also effective to move the discussion about the intellectual conception (tasawwur) into the field of judgmental assent (tasdiq) and confusion of the mawhum (imaginable) with the ma‘kûl (intellegible). (shrink)

Conditional excluded middle (CEM) is the following principe of counterfactual logic: either, if it were the case that φ, it would be the case that ψ, or, if it were the case that φ, it would be the case that not-ψ. I will first show that CEM entails the identity of indiscernibles, the falsity of physicalism, and the failure of the modal to supervene on the categorical and of the vague to supervene on the precise. I will then (...) argue that we should accept these startling conclusions, since CEM is valid. (shrink)

There is a standard quantificational view of generic sentences according to which they have a tripartite logical form involving a phonologically null generic operator called 'Gen'. Recently, a number of theorists have questioned the standard view and revived a competing proposal according to which generics involve the predication of properties to kinds. This paper offers a novel argument against the kind-predication approach on the basis of the invalidity of Generic Excluded Middle, a principle according to which any sentence (...) of the form ⌜Either Fs are G or Fs are not G⌝ is true. I argue that the kind-predication approach erroneously predicts that GEM is valid, and that it can only avoid this conclusion by either collapsing into a form of the quantificational analysis or otherwise garnering unpalatable metaphysical commitments. I also show that, while the quantificational approach does not validate GEM as a matter of logical form, the principle may be validated on certain semantic analyses of the generic operator, and so, such theories should be rejected. (shrink)

Lewis (1973) gave a short argument against conditional excluded middle, based on his treatment of ‘might’ counterfactuals. Bennett (2003), with much of the recent literature, gives an alternative take on ‘might’ counterfactuals. But Bennett claims the might-argument against CEM still goes through. This turns on a specific claim I call Bennett’s Hypothesis. I argue that independently of issues to do with the proper analysis of might-counterfactuals, Bennett’s Hypothesis is inconsistent with CEM. But Bennett’s Hypothesis is independently objectionable, so (...) we should resolve this tension by dropping the Hypothesis, not by dropping CEM. (shrink)

In this chapter we consider three philosophical perspectives (including those of Stalnaker and Lewis) on the question of whether and how the principle of conditional excluded middle should figure in the logic and semantics of counterfactuals. We articulate and defend a third view that is patterned after belief revision theories offered in other areas of logic and philosophy. Unlike Lewis’ view, the belief revision perspective does not reject conditional excluded middle, and unlike Stalnaker’s, it does not (...) embrace supervaluationism. We adduce both theoretical and empirical considerations to argue that the belief revision perspective should be preferred to its alternatives. The empirical considerations are drawn from the results of four empirical studies (which we report below) of non-experts’ judgments about counterfactuals and conditional excluded middle. (shrink)

This paper presents a range of new triviality proofs pertaining to naïve truth theory formulated in paraconsistent relevant logics. It is shown that excluded middle together with various permutation principles such as A → (B → C)⊩B → (A → C) trivialize naïve truth theory. The paper also provides some new triviality proofs which utilize the axioms ((A → B)∧ (B → C)) → (A → C) and (A → ¬A) → ¬A, the fusion connective and the Ackermann (...) constant. An overview over various ways to formulate Leibniz’s law in non-classical logics and two new triviality proofs for naïve set theory are also provided. (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)

In a recent paper, Thomas Ferguson argues that the source of Plumwood's homogenization, that is, the suppression of individual differences of members of a certain class, should be tied to some formal feature of classical negation and concludes that the culprit is the law of excluded middle (LEM). In this paper, we will show that an alternative argument can be made that leads to the law of double negation elimination (DNE) instead. And as the interderivability of LEM and (...) DNE may fail in a nonclassical context, this suggests that LEM and DNE might independently enforce homogenization. Furthermore, we show that there are two different kinds of homogenization that can be identified in classical and intuitionistic logic. (shrink)

This paper develops a semantic solution to the puzzle of Free Choice permission. The paper begins with a battery of impossibility results showing that Free Choice is in tension with a variety of classical principles, including Disjunction Introduction and the Law of Excluded Middle. Most interestingly, Free Choice appears incompatible with a principle concerning the behavior of Free Choice under negation, Double Prohibition, which says that Mary can’t have soup or salad implies Mary can’t have soup and Mary (...) can’t have salad. Alonso-Ovalle 2006 and others have appealed to Double Prohibition to motivate pragmatic accounts of Free Choice. Aher 2012, Aloni 2018, and others have developed semantic accounts of Free Choice that also explain Double Prohibition. -/- This paper offers a new semantic analysis of Free Choice designed to handle the full range of impossibility results involved in Free Choice. The paper develops the hypothesis that Free Choice is a homogeneity effect. The claim possibly A or B is defined only when A and B are homogenous with respect to their modal status, either both possible or both impossible. Paired with a notion of entailment that is sensitive to definedness conditions, this theory validates Free Choice while retaining a wide variety of classical principles except for the transitivity of entailment. The homogeneity hypothesis is implemented in two different ways, homogeneous alternative semantics and homogeneous dynamic semantics, with interestingly different consequences. (shrink)

Todd (2016) proposes an analysis of future-directed sentences, in particular sentences of the form 'will(φ)', that is based on the classic Russellian analysis of definite descriptions. Todd's analysis is supposed to vindicate the claim that the future is metaphysically open while retaining a simple Ockhamist semantics of future contingents and the principles of classical logic, i.e. bivalence and the law of excluded middle. Consequently, an open futurist can straightforwardly retain classical logic without appeal to supervaluations, determinacy operators, or (...) any further controversial semantical or metaphysical complication. In this paper, we will show that this quasi-Russellian analysis of 'will' both lacks linguistic motivation and faces a variety of significant problems. In particular, we show that the standard arguments for Russell's treatment of definite descriptions fail to apply to statements of the form 'will(φ)'. (shrink)

I show that standard dynamic approaches to the semantics of epistemic modals invalidate the classical laws of excluded middle and non-contradiction, as well as the law of epistemic non-contradiction. I argue that these facts pose a serious challenge.

Two systems of belief change based on paraconsistent logics are introduced in this article by means of AGM-like postulates. The first one, AGMp, is defined over any paraconsistent logic which extends classical logic such that the law of excluded middle holds w.r.t. the paraconsistent negation. The second one, AGMo, is specifically designed for paraconsistent logics known as Logics of Formal Inconsistency (LFIs), which have a formal consistency operator that allows to recover all the classical inferences. Besides the three (...) usual operations over belief sets, namely expansion, contraction and revision (which is obtained from contraction by the Levi identity), the underlying paraconsistent logic allows us to define additional operations involving (non-explosive) contradictions. Thus, it is defined external revision (which is obtained from contraction by the reverse Levi identity), consolidation and semi-revision, all of them over belief sets. It is worth noting that the latter operations, introduced by S. Hansson, involve the temporary acceptance of contradictory beliefs, and so they were originally defined only for belief bases. Unlike to previous proposals in the literature, only defined for specific paraconsistent logics, the present approach can be applied to a general class of paraconsistent logics which are supraclassical, thus preserving the spirit of AGM. Moreover, representation theorems w.r.t. constructions based on selection functions are obtained for all the operations. (shrink)

I investigate why Frege rejected the theory of types, as Russell presented it to him in their correspondence. Frege claims that it commits one to violations of the law of excluded middle, but this complaint seems to rest on a dogmatic refusal to take Russell’s proposal seriously on its own terms. What is at stake is not so much the truth of a law of logic, but the structure of the hierarchy of the logical categories, something Frege seems (...) to neglect. To come to a better understanding of Frege’s response, I proceed to investigate his conception of the nature of the logical categories, and how it differs from Russell’s. I argue that, for Frege, our grasp of the logical categories cannot be severed from our grasp of the Begriffsschrift notation itself. Russell, on the other hand, attaches no such importance to notation. From Frege’s point of view, Russell has not succeeded in presenting an alternative conception of the logical hierarchy, since such a conception must go in tandem with the development of a notation. Moreover, Frege has good reasons to think that Russell’s proposal does not admit of a suitable notation. (shrink)

The determinism-free will debate is perhaps as old as philosophy itself and has been engaged in from a great variety of points of view including those of scientific, theological, and logical character. This chapter focuses on two arguments from logic. First, there is an argument in support of determinism that dates back to Aristotle, if not farther. It rests on acceptance of the Law of Excluded Middle, according to which every proposition is either true or false, no matter (...) whether the proposition is about the past, present or future. In particular, the argument goes, whatever one does or does not do in the future is determined in the present by the truth or falsity of the corresponding proposition. The second argument coming from logic is much more modern and appeals to Gödel's incompleteness theorems to make the case against determinism and in favour of free will, insofar as that applies to the mathematical potentialities of human beings. The claim more precisely is that as a consequence of the incompleteness theorems, those potentialities cannot be exactly circumscribed by the output of any computing machine even allowing unlimited time and space for its work. The chapter concludes with some new considerations that may be in favour of a partial mechanist account of the mathematical mind. (shrink)

Language acts as a cognitive constraint, creating an inertia that pulls fallible, scalar thoughts toward binary assertions. This paper argues that the grammar of natural languages forces speakers to collapse gradient epistemic states into discrete ontological claims. While the human mind operates on an "Analog Epistemology" of probabilities, our linguistic tools are "Digital," constructed around the Law of Excluded Middle. This mismatch creates a "Binary Inertia" that distorts discourse and obscures calibration. Furthermore, we argue that even softening approaches (...) like "provisional adoption"-while pragmatically useful-remain dangerously tethered to the binary structure. True epistemic resolution requires a more radical embrace of scalar uncertainty, where provisional stances are recognized merely as functional compression artifacts, not accurate reflections of cognitive states. 2 Contents. (shrink)

Logical pluralism faces the collapse problem, as formulated by Stei (2023): any position that simultaneously affirms a plurality of correct logics, competition among them at the level of application, and the normativity of logical consequence will inevitably collapse into monism or nihilism. Existing responses—whether appealing to derivative normativity (Tajer, 2024) or metalinguistic stipulation—leave a more fundamental question unanswered: in virtue of what can different logical systems coexist and be compared at all? This paper argues that the root of the dilemma (...) lies in the fact that every logical system has always presupposed distinctions—true and false, premise and conclusion—yet has never asked about the conditions that make distinction possible. Through an argument from operational self-refutation and an analysis of asymmetric existential dependence, this paper establishes a minimal ontological structure: the possibility of distinction (the self-sufficient primal ground), distinction itself, and products of distinction. From this structure, three Primal Laws are derived by ontological reductio—the Primal Law of Non-Contradiction (the condition under which distinction is meaningful), the Primal Law of Excluded Middle (the binary exhaustion of distinction modalities), and the Primal Law of Identity (the non-self-sufficiency of distinction products)—together with a further derived Mutual Exclusion Principle. It is then shown that Leibniz’s principle of the Indiscernibility of Identicals secretly presupposes all three Primal Laws. On this basis, the paper argues that Primal Logic constitutes the shared ground protocol underlying all logical systems, and provides a preliminary demonstration of its positive inferential capacity, thereby grounding logical pluralism without recourse to monism. Keywords: Primal Logic; the possibility of distinction; logical pluralism; collapse problem; Indiscernibility of Identicals. (shrink)

One well known approach to the soritical paradoxes is epistemicism, the view that propositions involving vague notions have definite truth values, though it is impossible in principle to know what they are. Recently, Paul Horwich has extended this approach to the liar paradox, arguing that the liar proposition has a truth value, though it is impossible to know which one it is. The main virtue of the epistemicist approach is that it need not reject classical logic, and in particular the (...) unrestricted acceptance of the principle of bivalence and law of excluded middle. Regardless of its success in solving the soritical paradoxes, the epistemicist approach faces a number of independent objections when it is applied to the liar paradox. I argue that the approach does not offer a satisfying, stable response to the paradoxes—not in general, and not for a minimalist about truth like Horwich. (shrink)

In De interpretatione 9, Aristotle argues against the fatalist view that if statements about future contingent singular events (e.g. ‘There will be a sea battle tomorrow,’ ‘There will not be a sea battle tomorrow’) are already true or false, then the events to which those statements refer will necessarily occur or necessarily not occur. Scholars have generally held that, to refute this argument, Aristotle allows that future contingent statements are exempt from either the principle of bivalence, or the law of (...) excluded middle. This paper offers a new interpretation of Aristotle’s refutation of fatalism. According to this interpretation, each member of a pair of contradictory future contingent statements, in virtue of expressing modal necessity, is simply false. (shrink)

L.E.J. Brouwer's intuitionistic mathematics introduces an element of indeterminacy into mathematics; an indeterminacy stemming from the role of infinity in mathematics and reflected in the rejection of the law of excluded middle by formal intuitionistic logic. There has been recent interest in applying some of the insights of intuitionistic mathematics and logic in order to inject an element of indeterminacy into physical considerations. In this note, I point out that in fact this emerging physical application involves two distinct (...) notions of indeterminacy that interact delicately. In what follows I will set out this distinction, provide the outlines of a formal framework in which we can represent the separate notions, and show how they can interact consistently. The framework, in brief puts a metric temporal logic (a logic with indexed temporal operators) on the base of a free intuitionistic logic. That is a logic whose propositional fragment is intuitionistic and whose quantificational portion is free. Thus the title: Free Intuitionistic Metric Temporal Logic (Free IMTL). I speak of this technical combination as a "framework" for two reasons. First, I mean the machinery to be sufficiently flexible to accommodate different choices for the temporal axioms. Indeed I'll mention a number of options. Secondly, I leave the work of providing an adequate semantics for another occasion. I will also show how this combination elucidates a pair of distinct philosophical approaches to temporal indeterminacy. I call them the "Aristotelian approach" and the "broadly Kantian approach." ("Broadly" Kantian, because this approach itself has two versions: the strictly Kantian view and the Brouwerian view.). (shrink)

This paper explores what the epistemic account of vagueness means for theories of legal interpretation. The thesis of epistemicism is that vague statements are true or false even though it is impossible to know which. I argue that if epistemicism is accepted within the domain of the law, then the following three conditions must be satisfied: Interpretative reasoning within the law must adhere to the principle of bivalence and the law of excluded middle, interpretative reasoning within the law (...) must construe vague statements as an epistemic phenomenon, and epistemicism must be expanded to include normative considerations in order to account for legal theories that are consistent with the first two conditions. The first two conditions are internal to a particular theory of legal interpretation, while the third condition is external to a particular theory of legal interpretation. My conclusion shows that there are legal theories that are internally consistent with the fundamental features of epistemicism. However, within the domain of law—and specifically in the case of legal theories that are internally consistent with epistemicism—I show that vagueness cannot be explained simply by our ignorance of the meaning and use of vague expressions. Rather, epistemicism must also account for ignorance of the requisite normative considerations in legal theories with which it is otherwise consistent. (shrink)

Can an epistemic conception of truth and an endorsement of the excluded middle (together with other principles of classical logic abandoned by the intuitionists) cohabit in a plausible philosophical view? In PART I I describe the general problem concerning the relation between the epistemic conception of truth and the principle of excluded middle. In PART II I give a historical overview of different attitudes regarding the problem. In PART III I sketch a possible holistic solution.

As part of our small contribution in dialogue toward better peace development and reconciliation studies, and following Toffler & Toffler’s War and Antiwar (1993), the present article delves into a realm of logic beyond the traditional confines of negation and the excluded middle principle, exploring the nuances of "Otherness" that transcend classical and Nagatomo logics. Departing from the foundational premises of classical Aristotelian logic systems, this exploration ventures into alternative realms of reasoning, specifically examining Neutrosophic Logic and Klein (...) bottle logic (cf. Smarandache, 2005). The study challenges conventional boundaries and explores the implications of embracing paradoxes and self-reference in logic systems, aiming to redefine approaches to understanding truth and reasoning. The paper investigates how these alternative logics open avenues for philosophical inquiry, redefining entropy, and potentially influencing innovative perspectives in free energy systems. Through this exploration, it seeks to expand the discourse on logic, welcoming a broader spectrum of thought beyond established frameworks; and we also discuss shortly a number of possible implementations including in risk management and also Klein bottle entropy redefinition (Tang et al, 2018). (shrink)

This paper explores the interaction of well-motivated (if controversial) principles governing the probability conditionals, with accounts of what it is for a sentence to be indefinite. The conclusion can be played in a variety of ways. It could be regarded as a new reason to be suspicious of the intuitive data about the probability of conditionals; or, holding fixed the data, it could be used to give traction on the philosophical analysis of a contentious notion—indefiniteness. The paper outlines the various (...) options, and shows that ‘rejectionist’ theories of indefiniteness are incompatible with the results. Rejectionist theories include popular accounts such as supervaluationism, non-classical truth-value gap theories, and accounts of indeterminacy that centre on rejecting the law of excluded middle. An appendix compares the results obtained here with the ‘impossibility’ results descending from Lewis ( 1976 ). (shrink)

The principle of Conditional Excluded Middle has been a matter of longstanding controversy in both semantics and metaphysics. The principle suggests (among other things) that for any coin that isn't flipped, there is a fact of the matter about how it would have landed if it had been flipped: either it would have landed heads, or it would have landed tails. This view has gained support from linguistic evidence indicating that ‘would’ commutes with negation (e.g., ‘not: if A, (...) would C’ is equivalent to ‘if A, would not C’). There is, however, a long list of operators that similarly appear to commute with negation, even though the corresponding excluded middle principles are indefensible. We suggest that the data supporting Conditional Excluded Middle is best explained as a pragmatic effect. (shrink)

We discuss the philosophical implications of formal results showing the con- sequences of adding the epsilon operator to intuitionistic predicate logic. These results are related to Diaconescu’s theorem, a result originating in topos theory that, translated to constructive set theory, says that the axiom of choice (an “existence principle”) implies the law of excluded middle (which purports to be a logical principle). As a logical choice principle, epsilon allows us to translate that result to a logical setting, where (...) one can get an analogue of Diaconescu’s result, but also can disentangle the roles of certain other assumptions that are hidden in mathematical presentations. It is our view that these results have not received the attention they deserve: logicians are unlikely to read a discussion because the results considered are “already well known,” while the results are simultaneously unknown to philosophers who do not specialize in what most philosophers will regard as esoteric logics. This is a problem, since these results have important implications for and promise signif i cant illumination of contem- porary debates in metaphysics. The point of this paper is to make the nature of the results clear in a way accessible to philosophers who do not specialize in logic, and in a way that makes clear their implications for contemporary philo- sophical discussions. To make the latter point, we will focus on Dummettian discussions of realism and anti-realism. Keywords: epsilon, axiom of choice, metaphysics, intuitionistic logic, Dummett, realism, antirealism. (shrink)

The Mandela Effect—the phenomenon where large groups share identical "false memories"—has been inadequately explained through cognitive psychology or speculative metaphysics. This paper proposes a rigorous ontological framework grounded in formal logic and set theory: physical and intersubjective reality constitute distinct ontological domains governed by fundamentally different logical structures. Physical existence follows classical bivalent logic with the law of excluded middle. Intersubjective existence operates under paraconsistent logic where contradictory statements can both hold truth values without trivializing the system. The (...) Mandela Effect represents the phenomenological signature of navigating between these incommensurable domains. This framework extends to provide rigorous foundations for understanding the ontological status and functional efficacy of religious constructs, prayer, and ritual practice as legitimate technologies operating within intersubjective space. We formalize these claims, derive testable predictions, and address standard objections. This is a work in progress. Criticism welcome. (shrink)

هو أول كتاب باللغة العربية يعرض لمراحل وآليات تطور المنطق الرمزي المعاصر متعدد القيم بأنساقه المختلفة، مركزًا على مشكلة الغموض المعرفي للإنسان بأبعادها اللغوية والإبستمولوجية والأنطولوجية، والتي تتجلى – على سبيل المثال – فيما تحفل به الدراسات الفلسفية والمنطقية والعلمية من مفارقات تمثل تحديًا قويًا لثنائية الصدق والكذب الكلاسيكية، وكذلك في اكتشاف «هيزنبرج» لمبدأ اللايقين، وتأكيده وعلماء الكمّ على ضرورة التفسيرات الإحصائية في المجال دون الذري، الأمر الذي يؤكد عدم فعالية قانون الثالث المرفوع في التعامل مع معطيات الواقع الفعلي، واستحالة (...) مشروع إقامة لغة مثالية أو اصطناعية أو كاملة منطقيًا تتجاوز عيوب ونقائص اللغة العادية التي نفكر ونتعامل بها. وينتهي الكتاب إلى نتيجة مؤداها أن ما واجهه المنطق الكلاسيكي – ثنائي القيم – من مشكلات أدت إلى تطويره، لاسيما مشكلة الغموض، لابد وأن يواجهه بالمثل المنطق متعدد القيم، ذلك أن الغموض ظاهرة إبستمولوجية في المحل الأول، مردودها إلى الذات العارفة وقصور إمكاناتها الإدراكية والقياسية، لا إلى الوجود ذاته. وأننا حتى لو سمحنا لأية قضية منطقية بقيمة صدق ثالثة، أو بأكثر من قيمة تتوسط بين الصدق التام والكذب التام، فسوف تظل القضية – كتمثيل لغوي لإحدى وقائع العالم – صادقة أو كاذبة، سواء أدركنا ذلك أو لم ندركه، وذلك تأكيد لنزعة أفلاطون الواقعية القائلة بوجود أزلي وثابت للحقائق في عالم المثل، تحول دون معرفتنا الظنية بظلالها في عالم الحس المتغير. ويعني ذلك بعبارة أخرى أن الشك في قانون الثالث المرفوع هو إسقاط من الذات على الموضوع، مبعثه عدم اكتمال العملية المعرفية ومحدوديتها، وأن ظهور الأنساق المنطقية ذات القيم المتعددة ما هو إلا حلقة من حلقات العلاقة الجدلية اللامنتهية بين الإنسان والطبيعة، أو بيم ما هو مُدرك وما هو موجود بالفعل. والكتاب بصفة عامة هو أول كتاب باللغة العربية يعرض لأنساق المنطق متعدد القيم بعد مرحلة البرينكيبيا ماثيماتيكا. (shrink)

In debates concerning the consequence argument, it has long been claimed that [McKay, T. J., and D. Johnson. 1996. “A Reconsideration of an Argument Against Compatibilism.” Philosophical Topics 24 (2): 113–122] demonstrated the invalidity of rule (β). Here, I argue that their result is not as robust as we might like to think. First, I argue that McKay and Johnson's counterexample is successful if one adopts a certain interpretation of ‘no choice about’ and if one is willing to deny the (...) conditional excluded middle principle. In order to make this point I demonstrate that (β) is valid on Stalnaker's theory of counterfactuals. This result is important and should not be neglected, I argue, because there is a particular line of objection to the revised formulations of the consequence argument that does not succeed against the original version. (shrink)

A multidimensional pseudo-Boolean algebra is constructed based on generalized sequences of classes, drawing on N.A. Vasiliev's philosophical ideas and V.A. Smirnov's formalizations. Vasiliev's non-Aristotelian logics, which challenge the laws of non-contradiction and the excluded middle, are extended to n-dimensional logics where atomic statements represent positive, negative, or indifferent states. Finite linear sequences are generalized into partially ordered sequences, forming an implicative lattice with defined intersection and union operations. This structure satisfies the properties of a pseudo-Boolean algebra, providing a (...) formal framework for multidimensional logics applicable to imaginary and normative scenarios, such as deontic logic. The approach parallels non-Euclidean geometry, offering a novel perspective on logical systems and their ontological implications. (shrink)

This paper rejects the intuitionistic redefinition of ‘truth’ and proposes a formal system termed the Axioms of Knowledge (AK), designed to accommodate intuitionistic insights within the framework of standard logic. While intuitionism identifies truth with provability, the AK system maintains a strict distinction between truth (t) and knowability (k), reconciling the two by introducing new sentential operators: kX (it is knowable that X is true) and ¬X (it is knowable that X is false). The development of this logic leads to (...) three principal conclusions. Firstly, truth constitutes an independent domain not subordinate to the cognizing subject or the existence of a proof. Secondly, while the Law of Excluded Middle legitimately holds at the levels of truth and falsity, its validity is not guaranteed at the level of knowability. Finally, by accepting positive introspection while rejecting negative introspection, the system incorporates the inherent limits of human cognition — specifically, our potential inability to recognize our own unknowing. (shrink)

Buddhist Illogic. The 2nd Century CE Indian philosopher Nagarjuna founded the Madhyamika (Middle Way) school of Mahayana Buddhism, which strongly influenced Chinese, Korean and Japanese (Ch’an or Zen) Buddhism, as well as Tibetan Buddhism. Nagarjuna is regarded by many Buddhist writers to this day as a very important philosopher, who they claim definitively proved the futility of ordinary human cognitive means. His writings include a series of arguments purporting to show the illogic of logic, the absurdity of reason. He (...) considers this the way to verbalize and justify the Buddhist doctrine of “emptiness” (Shunyata). These arguments attack some of the basic tenets and techniques of reasoning, such as the laws of thought (identity, non-contradiction and the excluded middle), conceptualization and predication, our common assumptions of self, entities and essences, as well as our beliefs in motion and causation. The present essay demonstrates the many sophistries involved in Nagarjuna’s arguments. He uses double standards, applying or ignoring the laws of thought and other norms as convenient to his goals; he manipulates his readers, by giving seemingly logical form (like the dilemma) to his discourse, while in fact engaged in non-sequiturs or appealing to doubtful premises; he plays with words, relying on unclear terminology, misleading equivocations and unfair fixations of meaning; and he ‘steals concepts’, using them to deny the very percepts on which they are based. Although a critique of the Madhyamika philosophical interpretation and defense of “emptiness”, Buddhist Illogic is not intended to dissuade readers from Buddhism. On the contrary, its aim to enhance personal awareness of actual cognitive processes, and so improve meditation. It is also an excellent primer on phenomenological epistemology. (shrink)

In this paper I outline a comprehensive theory of legal interpretation based on an assumption that legal text, understood as the aggregate of texts of all legal acts in force at a particular time and place, describes one rational and coherent possible world. The picture of this possible world is decoded from the text by interpreters and serves as a holistic model to which the real world is adjusted when the law is applied. From the above premise I will limit (...) myself to drawing two conclusions for how legal interpretation should be carried out. First, I argue that the possible world described by the legal text has to be ‘accessible’ from the real world, i.e. it has to be feasible to transform the actual world into the described one. Were it otherwise, the possible world could not serve as a model for adjustment. The accessibility requirement imposes obligations on the interpreters to secure the rationality of the possible world decoded from the text, amongst other to secure that the description of this world is not contradictory and – as a consequence – the law of excluded middle is obeyed in the possible world described by the legal text. Secondly, I argue for the inevitability of interpretative discretion arising from the requirement to decode a sufficiently ‘saturated’ picture of the possible world., i.e. possessing enough properties to resemble the actual world. As texts have a limited number of sentences and worlds have an unlimited number of properties, interpreters have to supplement the picture of a possible world to achieve its coherence. This involves the inclusion of some additional, non-predetermined features that integrate with the properties of the world predefined by the legal text. This process of saturation consists of filling in so-called ‘places of indeterminacy’ (Roman Ingarden) with content implicated by other features of the possible world. I also argue that the discretion resulting from the necessity of filling in the places of indeterminacy is justified by the requirement of fulfilling the intention of the lawmaker to make the possible world described by the legal text real. The theory presented here is based on contemporary theories of discourse representation and so-called ‘text-world theory’ by J. Gavins. Phenomenalism and causal (historical) theories of reference provide its philosophical background. (shrink)

The principle of bivalence (PB) states that every declarative sentence is either true or false, and the principle of excluded middle (PEM) states that one member of any contradictory pair must be true. According to the standard interpretation of Int. 9, PB fails for future contingents. Moreover, some standardists believe that PEM fails for pairs of contradictory future contingents, whereas other standardists attempt to rescue PEM by applying the method of supervaluations. I argue that PB and PEM are (...) not suspended and that the method of supervaluations fails to explain how PEM holds. I propose an interpretation of Int. 9 according to which neither PB nor PEM fails. (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 paper presents Alfred Tarski’s debate with the semantic antinomies: the basic Liar Paradox, and its more sophisticated versions, which are currently discussed in philosophy: Strengthen Liar Paradox, Cyclical Liar Paradox, Contingent Liar Paradox, Correct Liar Paradox, Card Paradox, Yablo’s Paradox and a few others. Since Tarski, himself did not addressed these paradoxes—neither in his famous work published in 1933, nor in later papers in which he developed the Semantic Theory of Truth—therefore, We try to defend his concept of truth (...) against these antinomies. It is demonstrated that Tarskian theory of truth is resistant to the paradoxes and it is still the best solution to avoid the antinomies and remain within a classical logic, that is, accepting the laws of noncontradiction, excluded middle, and the principle of bivalence. Thus, the goal of the paper is double—firstly, to show that none of the versions of the Liar Paradox’s is a serious threat to Tarski’s concept of truth, and secondly, that Semantic Theory of Truth allows to remain within classical logic, and at the same time, avoid antinomies—which makes it the most attractive among classical theories of truth. (shrink)

本文把“人的行为之因”这一追问，化简为一个穷尽的二分。平凡的答案无人争议：一个行为，由在先的物质状态（脑、身、环境）依物理律所致，是一个有因的、物质的果，与任何别的发生无异。但当有人坚持人的行为另有一 个特别的因——一个使行为“真正源出于意愿／自我”、从而逃出单纯物质相继的因——他所要求的，本质上只可能是两样东西之一。本文据排中律导出这个二分：一个行为，要么没有充分的在先之因，要么有。若没有，它就（ 至少部分地）是一个无因的果——一个不被在先状态所决定的开端。若有，则那充分的因，要么全是物质的（于是行为只是物质相继的平凡产物，意愿不过是这相继的一个高层别名、并非特别的源头），要么含有非物质的成分（ 于是存在一个非物质的因）。故“特别的能动性”所需者，舍此二支别无他途。而两支皆落空。第一支：无因的果，纵真有（如把“无因”兑现为量子的非决定），也不是作者身份、而是偶然——一个无因发生的事件，不被任何 东西所“作”（包括意愿），意愿“作”一个无因之果，并不比“作”一次掷骰更名副其实；“无因”恰恰意味着不归任何作者，故它给不出能动性、只给出偶然。第二支：非物质的因，是机器里的鬼——一个在物理秩序之外、 却向物质注入因果的灵魂式实体；它撞上互动难题（非物质如何推动物质），更根本地，意愿是物质活动的一个高层花样、不是一个作用于物质之上的别样实体，没有证据、也无位置容纳一个非物质的因。有人会以“涌现的心理 因果”回避：但涌现的高层因果若是真因果，要么由物质相继所实现、即物质相继的重述（仍是平凡那一支，非特别之因），要么在物质之外另做因果之功（即第二支，附带因果的过度决定）——二分依旧。两支皆空，便只剩平 凡：人的行为是有因的、物质的果，是粒子在物理律下的相继、是它的一个高层花样，与齿轮之转同类，只复杂亿万倍；意愿不是特别的因，是这相继的一个名。本文守一处分寸：说行为没有特别之因，不是说它无因、随机—— 恰相反，它是充分地、物质地有因的，只是寻常地有因、非自由地有因；这也不否认审议、理由、意愿作为花样之真实，只否认它们是一个非物质的因或一个无因的开端；伦理留作开放问题。看清这个二分，不是把能动性证伪得 更玄，而是把它祛魅：所谓“人的行为之因”的全部神秘，浓缩成两个一旦摆明便难以认领的断设——一个无因的果，一个非物质的因。 Abstract This paper reduces the question of “the cause of human action” to an exhaustive dichotomy. The mundane answer is undisputed: an action, brought about by prior material states (brain, body, environment) under physical law, is a caused, material effect, no different from any other occurrence. But when someone insists that human action has a special cause—one that makes the action “genuinely issue from the will/self” and so escape mere material succession—what they demand is, in essence, only one of (...) two things. The paper derives the dichotomy by excluded middle: an action either lacks a sufficient prior cause or has one. If it lacks one, then it is (at least partly) an uncaused effect—a beginning not fixed by prior states. If it has one, then that sufficient cause is either wholly material (whereupon the action is just the mundane upshot of material succession, the will only a high-level alias of that succession, not a special source) or contains a non-material element (whereupon there exists a non-material cause). So special agency requires one of these two branches and no other. And both fail. The first: an uncaused effect, even if real (e.g., cashing out “uncaused” as quantum indeterminacy), is not authorship but chance—an event that happens uncaused is “done” by nothing (the will included); the will “authoring” an uncaused effect is no more apt than its “authoring” a dice-roll; “uncaused” means precisely belonging to no author, so it yields not agency but chance. The second: a non-material cause is a ghost in the machine—a soul-like entity outside the physical order yet injecting causation into matter; it meets the interaction problem (how does non-matter push matter), and, more fundamentally, the will is a high-level figure of material activity, not a separate substance acting upon it, with no evidence and no room for a non-material cause. One might dodge with “emergent mental causation”: but emergent high-level causation, if it is real causation, is either realized by the material succession—a redescription of it (still the mundane branch, not a special cause)—or does causal work over and above the material (the second branch, with causal overdetermination in tow)—the dichotomy stands. Both branches empty, only the mundane remains: human action is a caused, material effect, a succession of particles under physical law, a high-level figure of it, of a kind with a gear’s turning, only a billionfold more complex; the will is not a special cause but a name for that succession. The paper keeps a measure: to say the action has no special cause is not to say it is uncaused or random—on the contrary it is amply, materially caused, only ordinarily caused, not freely; nor does this deny the reality of deliberation, reasons, or the will-as-figure—only that any of these is a non-material cause or an uncaused beginning; ethics is left an open question. Seeing this dichotomy does not make agency more mysteriously disproved; it disenchants it: the whole mystery of “the cause of human action” condenses into two posits that, once laid bare, are hard to own—an uncaused effect and a non-material cause. (shrink)

The theme of this study is about establishing a purely logical theory about the Universe. Logic is the premier candidate for the reality behind phenomena. If there is a final theory, the Universe must be logic itself, called pure logic, elements of which include not only logic and illogic but also logical and illogical manipulations between them. The kernel is the revised law of the excluded middle: between two basic concepts are four possible manipulations, three logical and one (...) illogical, whereas only one is realized. The key parameter for an element is aim, including reason, outcome, and neutrality. When an element is a contradiction, it can be located at three positions: reason, outcome, and manipulation, corresponding to three colors of quark. Thus, there is interaction with the SU(3) symmetry keeping the number of the three elements balanced locally. A proposition without contradiction has two independent aims. When one aim changes, there is interaction with the U(1) symmetry. When two aims change together, there is interaction with the SU(2) symmetry. (shrink)

N. Vasiliev’s early 20th-century concept of multidimensional logic, developed alongside L. Brouwer and J. Lukasiewicz, proposes a non-Aristotelian framework. It distinguishes logical laws into external (gussiological) and internal levels. The external level follows multidimensional principles and the law of the excluded middle, while the internal level depends on endalogous assumptions about the cognizable world. In a one-dimensional world, only positive statements are atomic, with negative statements arising from inference, underscoring the constraints of directly expressing negation.