Stephen Barker presents a novel approach to solving semantic paradoxes, including the Liar and its variants and Curry’s paradox. His approach is based around the concept of alethic undecidability. His approach, if successful, renders futile all attempts to assign semantic properties to the paradoxical sentences, whilst leaving classical logic fully intact. And, according to Barker, even the T-scheme remains valid, for validity is not undermined by undecidable instances. Barker’s approach is innovative and worthy of further consideration, particularly by (...) those of us who aim to find a solution without logical revisionism. As it stands, however, the approach is unsuccessful, as I shall demonstrate below. (shrink)

I use the principle of truth-maker maximalism to provide a new solution to the semantic paradoxes. According to the solution, AUS, its undecidable whether paradoxical sentences are grounded or ungrounded. From this it follows that their alethic status is undecidable. We cannot assert, in principle, whether paradoxical sentences are true, false, either true or false, neither true nor false, both true and false, and so on. AUS involves no ad hoc modification of logic, denial of the (...) T-schema's validity, or obvious revenge. (shrink)

In this paper, we will discuss what is called the “Manifestation Challenge” to semantic realism, which was originally developed by Michael Dummett and has been further refined by Crispin Wright. According to this challenge, semantic realism has to meet the requirement that knowledge of meaning must be publically manifested in linguistic behaviour. In this regard, we will introduce and evaluate John McDowell’s response to this anti-realistic challenge, which was put forward to show that the challenge cannot undermine realism. According to (...) McDowell, knowledge of undecidable sentences’ truth-conditions can be properly manifested in our ordinary practice of asserting such sentences under certain circumstances, and any further requirement will be redundant. Wright’s further objection to McDowell’s response will be also discussed and it will be argued that this objection fails to raise any serious problem for McDowell’s response and that it is an implausible objection in general. (shrink)

If the conclusion of the Tarski Undefinability Theorem was that some artificially constrained limited notions of a formal system necessarily have undecidable sentences, then Tarski made no mistake within his assumptions. When we expand the scope of his investigation to other notions of formal systems we reach an entirely different conclusion showing that Tarski's assumptions were wrong.

The generalized conclusion of the Tarski and Gödel proofs: All formal systems of greater expressive power than arithmetic necessarily have undecidable sentences. Is not the immutable truth that Tarski made it out to be it is only based on his starting assumptions. -/- When we reexamine these starting assumptions from the perspective of the philosophy of logic we find that there are alternative ways that formal systems can be defined that make undecidability inexpressible in all of these formal (...) systems. (shrink)

Pluralists maintain that there is more than one truth property in virtue of which bearers are true. Unfortunately, it is not yet clear how they diagnose the liar paradox or what resources they have available to treat it. This chapter considers one recent attempt by Cotnoir (2013b) to treat the Liar. It argues that pluralists should reject the version of pluralism that Cotnoir assumes, discourse pluralism, in favor of a more naturalized approach to truth predication in real languages, which should (...) be a desideratum on any successful pluralist conception. Appealing to determination pluralism instead, which focuses on truth properties, it then proposes an alternative treatment to the Liar that shows liar sentences to be undecidable. (shrink)

This article is primarily concerned with the articulation of a defensible position on the relevance of phenomenological analysis with the current epistemological edifice as this latter has evolved since the rupture with the classical scientific paradigm pointing to the Newtonian-Leibnizian tradition which took place around the beginning of 20th century. My approach is generally based on the reduction of the objects-contents of natural sciences, abstracted in the form of ideal objectivities in the corresponding logical-mathematical theories, to the content of meaning-acts (...) ultimately referring to a specific being-within-the-world experience. This is a position that finds itself in line with Husserl’s gradual departure from the psychologistic interpretations of his earlier works on the philosophy of logic and mathematics and culminates in a properly meant phenomenological foundation of natural sciences in his last major published work, namely the Crisis of European Sciences and the Transcendental Phenomenology. Further this article tries to set up a context of discourse in which to found both physical and formal objects in parallel terms as essentially temporal-noematic objects to the extent that they may be considered as invariants of the constitutional modes of a temporal consciousness. (shrink)

One of the most fundamental questions in the philosophy of mathematics concerns the relation between truth and formal proof. The position according to which the two concepts are the same is called deflationism, and the opposing viewpoint substantialism. In an important result of mathematical logic, Kurt Gödel proved in his first incompleteness theorem that all consistent formal systems containing arithmetic include sentences that can neither be proved nor disproved within that system. However, such undecidable Gödel sentences can (...) be established to be true once we expand the formal system with Alfred Tarski s semantical theory of truth, as shown by Stewart Shapiro and Jeffrey Ketland in their semantical arguments for the substantiality of truth. According to them, in Gödel sentences we have an explicit case of true but unprovable sentences, and hence deflationism is refuted. -/- Against that, Neil Tennant has shown that instead of Tarskian truth we can expand the formal system with a soundness principle, according to which all provable sentences are assertable, and the assertability of Gödel sentences follows. This way, the relevant question is not whether we can establish the truth of Gödel sentences, but whether Tarskian truth is a more plausible expansion than a soundness principle. In this work I will argue that this problem is best approached once we think of mathematics as the full human phenomenon, and not just consisting of formal systems. When pre-formal mathematical thinking is included in our account, we see that Tarskian truth is in fact not an expansion at all. I claim that what proof is to formal mathematics, truth is to pre-formal thinking, and the Tarskian account of semantical truth mirrors this relation accurately. -/- However, the introduction of pre-formal mathematics is vulnerable to the deflationist counterargument that while existing in practice, pre-formal thinking could still be philosophically superfluous if it does not refer to anything objective. Against this, I argue that all truly deflationist philosophical theories lead to arbitrariness of mathematics. In all other philosophical accounts of mathematics there is room for a reference of the pre-formal mathematics, and the expansion of Tarkian truth can be made naturally. Hence, if we reject the arbitrariness of mathematics, I argue in this work, we must accept the substantiality of truth. Related subjects such as neo-Fregeanism will also be covered, and shown not to change the need for Tarskian truth. -/- The only remaining route for the deflationist is to change the underlying logic so that our formal languages can include their own truth predicates, which Tarski showed to be impossible for classical first-order languages. With such logics we would have no need to expand the formal systems, and the above argument would fail. From the alternative approaches, in this work I focus mostly on the Independence Friendly (IF) logic of Jaakko Hintikka and Gabriel Sandu. Hintikka has claimed that an IF language can include its own adequate truth predicate. I argue that while this is indeed the case, we cannot recognize the truth predicate as such within the same IF language, and the need for Tarskian truth remains. In addition to IF logic, also second-order logic and Saul Kripke s approach using Kleenean logic will be shown to fail in a similar fashion. (shrink)

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. David Elohim examines the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, (...) class='Hi'>undecidable propositions, and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal and hyperintensional cognitivism and modal and hyperintensional expressivism. Elohim develops a novel, topic-sensitive truthmaker semantics for dynamic epistemic logic, and develops a novel, dynamic two-dimensional semantics grounded in two-dimensional hyperintensional Turing machines. Chapter \textbf{3} provides an abstraction principle for two-dimensional (hyper-)intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal $\mu$-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal $\mu$-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} is availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between Elohim's hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter \textbf{9} examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. Elohim countenances a hyperintensional semantics for novel epistemic abstractionist modalities. Elohim suggests, too, that higher observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for two-dimensional hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter \textbf{10} examines the philosophical significance of hyperintensional $\Omega$-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter \textbf{11} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{12} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between interpretational and objective modalities and the truthmakers thereof. This yields the first hyperintensional category theory in the literature. Elohim invents a new mathematical trick in which first-order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals in the category of Set in category theory. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Elohim provides a counter-example to epistemic closure for logical deduction. Chapter \textbf{14} examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

Propositional logics in general, considered as a set of sentences, can be undecidable even if they have “nice” representations, e.g., are given by a calculus. Even decidable propositional logics can be computationally complex (e.g., already intuitionistic logic is PSPACE-complete). On the other hand, finite-valued logics are computationally relatively simple—at worst NP. Moreover, finite-valued semantics are simple, and general methods for theorem proving exist. This raises the question to what extent and under what circumstances propositional logics represented in various (...) ways can be approximated by finite-valued logics. It is shown that the minimal m-valued logic for which a given calculus is strongly sound can be calculated. It is also investigated under which conditions propositional logics can be characterized as the intersection of (effectively given) sequences of finite-valued logics. (shrink)

I have read many recent discussions of the limits of computation and the universe as computer, hoping to find some comments on the amazing work of polymath physicist and decision theorist David Wolpert but have not found a single citation and so I present this very brief summary. Wolpert proved some stunning impossibility or incompleteness theorems (1992 to 2008-see arxiv dot org) on the limits to inference (computation) that are so general they are independent of the device doing the computation, (...) and even independent of the laws of physics, so they apply across computers, physics, and human behavior. They make use of Cantor's diagonalization, the liar paradox and worldlines to provide what may be the ultimate theorem in Turing Machine Theory, and seemingly provide insights into impossibility, incompleteness, the limits of computation, and the universe as computer, in all possible universes and all beings or mechanisms, generating, among other things, a non- quantum mechanical uncertainty principle and a proof of monotheism. There are obvious connections to the classic work of Chaitin, Solomonoff, Komolgarov and Wittgenstein and to the notion that no program (and thus no device) can generate a sequence (or device) with greater complexity than it possesses. One might say this body of work implies atheism since there cannot be any entity more complex than the physical universe and from the Wittgensteinian viewpoint, ‘more complex’ is meaningless (has no conditions of satisfaction, i.e., truth-maker or test). Even a ‘God’ (i.e., a ‘device’with limitless time/space and energy) cannot determine whether a given ‘number’ is ‘random’, nor find a certain way to show that a given ‘formula’, ‘theorem’ or ‘sentence’ or ‘device’ (all these being complex language games) is part of a particular ‘system’. -/- Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my book ‘The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle’ 2nd ed (2019). Those interested in more of my writings may see ‘Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 2nd ed (2019) and Suicidal Utopian Delusions in the 21st Century 4th ed (2019) . (shrink)

Classical interpretations of Goedels formal reasoning, and of his conclusions, implicitly imply that mathematical languages are essentially incomplete, in the sense that the truth of some arithmetical propositions of any formal mathematical language, under any interpretation, is, both, non-algorithmic, and essentially unverifiable. However, a language of general, scientific, discourse, which intends to mathematically express, and unambiguously communicate, intuitive concepts that correspond to scientific investigations, cannot allow its mathematical propositions to be interpreted ambiguously. Such a language must, therefore, define mathematical truth (...) verifiably. We consider a constructive interpretation of classical, Tarskian, truth, and of Goedel's reasoning, under which any formal system of Peano Arithmetic---classically accepted as the foundation of all our mathematical Languages---is verifiably complete in the above sense. We show how some paradoxical concepts of Quantum mechanics can, then, be expressed, and interpreted, naturally under a constructive definition of mathematical truth. (shrink)

In this paper, I consider the contribution of Geoffrey Bennington's book, _Scatter 1_, to the ongoing discussion of the political dimension of deconstruction. Focusing on the resonances between Bennington's "politics of politics" and the notion of infrapolitics, I suggest that Bennington's major contribution revolves around the introduction of undecidability into political action and thought.

In the spatialized Prisoner's Dilemma, players compete against their immediate neighbors and adopt a neighbor's strategy should it prove locally superior. Fields of strategies evolve in the manner of cellular automata (Nowak and May, 1993; Mar and St. Denis, 1993a,b; Grim 1995, 1996). Often a question arises as to what the eventual outcome of an initial spatial configuration of strategies will be: Will a single strategy prove triumphant in the sense of progressively conquering more and more territory without opposition, or (...) will an equilibrium of some small number of strategies emerge? Here it is shown, for finite configurations of Prisoner's Dilemma strategies embedded in a given infinite background, that such questions are formally undecidable: there is no algorithm or effective procedure which, given a specification of a finite configuration, will in all cases tell us whether that configuration will or will not result in progressive conquest by a single strategy when embedded in the given field. The proof introduces undecidability into decision theory in three steps: by (1) outlining a class of abstract machines with familiar undecidability results, by (2) modelling these machines within a particular family of cellular automata, carrying over undecidability results for these, and finally by (3) showing that spatial configurations of Prisoner's Dilemma strategies will take the form of such cellular automata. (shrink)

The paper aims to weaken a widespread argument against indexical contextualism regarding matters of personal taste. According to indexical contextualism, an utterance of “T is tasty” (where T is an object of taste) expresses the proposition that T is tasty for J (where J is a judge). This argument suggests that indexical contextualism cannot do justice to our disagreement intuitions regarding typical disputes about personal taste because it has to treat conversations in which one speaker utters “T is tasty” and (...) another responds with “T is not tasty” (referred to as ‘judge-non-specific conversations’ in this paper) as being on a par with conversations in which one speaker utters “T is tasty to me” and another responds with “T is not tasty to me” (referred to as ‘judge-specific conversations’). The argument has it that judge-specific conversations, unlike judge-non-specific conversations, do not contain disagreement between speakers. To defend indexical contextualism, some philosophers have proposed accounts (here referred to as ‘dual-proposition theories’) according to which utterances of “T is tasty” are used to communicate both the above kinds of semantically expressed proposition and some other kinds of proposition (like superiority propositions or metalinguistic propositions) that could be used to explain disagreements about taste. The paper defends two claims: First, it is argued that judge-specific conversations, or at least some of them, do contain disagreement between speakers, contrary to what the anti-indexical-contextualist argument supposes. Second, it is argued that dual-proposition indexical-contextualist theories fail to explain judge-specific conversations that are intuitively interpreted as containing disagreement. (shrink)

It is often supposed that, unlike typical axioms of mathematics, the Continuum Hypothesis (CH) is indeterminate. This position is normally defended on the ground that the CH is undecidable in a way that typical axioms are not. Call this kind of undecidability “absolute undecidability”. In this paper, I seek to understand what absolute undecidability could be such that one might hope to establish that (a) CH is absolutely undecidable, (b) typical axioms are not absolutely undecidable, and (c) (...) if a mathematical hypothesis is absolutely undecidable, then it is indeterminate. I shall argue that on no understanding of absolute undecidability could one hope to establish all of (a)–(c). However, I will identify one understanding of absolute undecidability on which one might hope to establish both (a) and (c) to the exclusion of (b). This suggests that a new style of mathematical antirealism deserves attention—one that does not depend on familiar epistemological or ontological concerns. The key idea behind this view is that typical mathematical hypotheses are indeterminate because they are relevantly similar to CH. (shrink)

Some have suggested that certain classical physical systems have undecidable long-term behavior, without specifying an appropriate notion of decidability over the reals. We introduce such a notion, decidability in (or d- ) for any measure , which is particularly appropriate for physics and in some ways more intuitive than Ko's (1991) recursive approximability (r.a.). For Lebesgue measure , d- implies r.a. Sets with positive -measure that are sufficiently "riddled" with holes are never d- but are often r.a. This explicates (...) Sommerer and Ott's (1996) claim of uncomputable behavior in a system with riddled basins of attraction. Furthermore, it clarifies speculations that the stability of the solar system (and similar systems) may be undecidable, for the invariant tori established by KAM theory form sets that are not d-. (shrink)

معمولا تصور می شود که عدم امکان ، بی کامل بودن ، پارامونشتها ، Undecidability ، اتفاقی ، قابلیت های مختلف ، پارادوکس ، عدم قطعیت و محدودیت های دلیل ، مسائل فیزیکی و ریاضی علمی و یا با داشتن کمی یا هیچ چیز در مشترک. من پیشنهاد می کنم که آنها تا حد زیادی مشکلات فلسفی استاندارد (به عنوان مثال ، بازی های زبان) که عمدتا توسط ویتگنشتاین بیش از 80 سال پیش حل و فصل شد. -/- "آنچه ما (...) وسوسه می گویند" در چنین موردی است ، البته ، نه فلسفه ، اما آن مواد خام است. بنابراین ، برای مثال ، چه ریاضیدان تمایل به در مورد عینیت و واقعیت حقایق ریاضی می گویند ، فلسفه ریاضیات نیست ، اما چیزی برای درمان فلسفی است. ویتگنشتاین PI ۲۳۴ -/- "فلاسفه به طور مداوم به روش علم قبل از چشم خود را ببینید و irresistibly وسوسه به درخواست و پاسخ به سوالات در علم راه می کند. این گرایش ، منبع واقعی متافیزیک است و فیلسوف را به تاریکی کامل هدایت می کند.» ویتگنشتاین -/- من ارائه خلاصه ای از برخی از یافته های عمده ای از دو نفر از برجسته ترین دانش آموزان از رفتار دوران مدرن ، لودویگ ویتگنشتاین و جان سرل ، در ساختار منطقی از قصدمندی (ذهن ، زبان ، رفتار) ، با توجه به عنوان نقطه شروع من کشف بنیادی ویتگنشتاین-که همه مشکلات فلسفی آن یکسان هستند-ابهامات در مورد چگونگی استفاده از زبان در یک چارچوب خاص ، و به همین ترتیب تمام راه حل ها یکسان هستند-به دنبال چگونه زبان را می توان در زمینه مورد استفاده قرار گیرد به طوری که حقیقت آن شرایط (شرایط رضایت یا COS) روشن است. مشکل اساسی این است که می توان هر چیزی می گویند، اما یکی نمی تواند به معنای (دولت از COS برای) هر گفته های خودسرانه و معنی تنها در یک زمینه بسیار خاص ممکن است. -/- من برخی از نوشته های چند تن از مفسران عمده در این مسائل از نظر Wittgensteinian در چارچوب دیدگاه مدرن از دو سیستم اندیشه (به عنوان "فکر سریع ، تفکر آهسته ') ، استفاده از جدول جدید قصدمندی سیستم های جدید دوگانه نامگذاری. من نشان می دهم که این یک اکتشافی قدرتمند برای توصیف ماهیت واقعی این مسائل علمی ، فیزیکی یا ریاضی است که واقعا بهترین هستند به عنوان مشکلات فلسفی استاندارد چگونه زبان مورد استفاده قرار گیرد (بازی های زبان در ویتگنشتاین اصطلاحات). -/- این مشاجره من است که جدول از معنایی (عقلانیت ، ذهن ، فکر ، زبان ، شخصیت و غیره) که ویژگی های برجسته در اینجا توصیف بیشتر یا کمتر با دقت ، و یا حداقل به عنوان یک اکتشافی برای ، چگونه ما فکر می کنیم و رفتار ، و پس از آن را شامل نمی صرفا فلسفه و روانشناسی ، اما هر چیز دیگری (تاریخ ، ادبیات ، ریاضیات ، سیاست و غیره). توجه داشته باشید به خصوص که قصدمندی و عقلانیت به عنوان من (همراه با سرل ، ویتگنشتاین و دیگران) آن را مشاهده کنید ، شامل هر دو آگاهانه سیستم زبانی ، 2 و ناخودآگاه خودکار سیستم پیش زبانی 1 اقدامات یا رفلکس. (shrink)

The unreduced solution to the arbitrary interaction problem, absent in the standard theory framework, reveals many equally real and mutually incompatible system configurations, or "realizations". This is the essence of universal dynamic undecidability, or multivaluedness, and the ensuing causal randomness (unpredictability), non-computability, irreversible time flow (evolution, emergence), and dynamic complexity of every real system, object, or process. This creative undecidability of real-world dynamics provides causal explanations for "quantum mysteries", relativity postulates, cosmological problems, and the huge efficiency of high-complexity phenomena, such (...) as life, intelligence, and consciousness, giving rise to extended applications, far beyond the critical limits of usual science paradigm. (shrink)

Despite an increasing role of machine learning in science, there is a lack of results on limits of empirical exploration aided by machine learning. In this paper, we construct one such limit by proving undecidability of learnability of state spaces of physical systems. We characterize state spaces as binary hypothesis classes of the computable Probably Approximately Correct learning framework. This leads to identifying the first limit for learnability of state spaces in the agnostic setting. Further, using the fact that finiteness (...) of the combinatorial dimension of hypothesis classes is undecidable, we derive undecidability for learnability of state spaces as well. Throughout the paper, we try to connect our formal results with modern neural networks. This allows us to bring the limits close to the current practice and make a first step in connecting scientific exploration aided by machine learning with results from learning theory. (shrink)

A natural problem from elementary arithmetic which is so strongly undecidable that it is not even Trial and Error decidable (in other words, not decidable in the limit) is presented. As a corollary, a natural, elementary arithmetical property which makes a difference between intuitionistic and classical theories is isolated.

The title of the present book suggests that scientific results obtained in mathematics and quantum physics can be in some way related to the question of the existence of God. This seems possible to us, because it is our conviction that reality in all its dimensions is intelligible. The really impressive progress in science and technology demonstrates that we can trust our intellect, and that nature is not offering us a collection of meaningless absurdities. We first of all intend to (...) show with results taken from mathematics and quantum physics: Mathematical Undecidability: man will never have a universal method to solve any mathematical problem. In arithmetic there always will be unsolved, solvable problems. Quantum Nonlocality: certain phenomena in nature seem to imply the existence of correlations based on faster-than-light influences. These influences, however, are not accessible to manipulation by man for use in, for example, faster than light communication. In the various contributions pieces of a puzzle are offered, which suggest that there exists more than the world of phenomena around us. The results discussed point to intelligent and unobservable causes governing the world. One is led to perceive the shade of a reality which many people would call God. (shrink)

In the last forty years Dummett, Hanks, Künne, and Bobzien have claimed that when Frege wrote ‘Über Sinn und Bedeutung’, he thought that the sense of a sentence-question [Satzfrage] was not a thought [Gedanke] ((Dummett 1981: 307–308); (Hanks 2007: 142–143); (Künne 2010: 427–429); (Bobzien 2021: 163–164)). Recently, Textor has claimed that when Frege wrote ‘Über Sinn und Bedeutung’, he thought that the sense of a sentence-question was not a question [Frage] (Textor 2021: 227 fn.2). I think it is possible that (...) when Frege wrote ‘Über Sinn und Bedeutung’, he thought that the sense of a sentence-question was a question and a thought. To show that this possibility should not be overlooked, I have written a dialogue in which Frege responds to Dummett, Hanks, Künne, Bobzien, and Textor. -/- Bobzien seems to think that when Frege wrote ‘Die Verneinung’, he thought that the sense of a sentence-question was not a question (Bobzien 2021: 163–165). I think it is possible that when Frege wrote ‘Die Verneinung’, he thought that the sense of a sentence-question was a question and a thought. Hence, I have also written the dialogue to show that this possibility should not be overlooked either. (shrink)

To eliminate incompleteness, undecidability and inconsistency from formal systems we only need to convert the formal proofs to theorem consequences of symbolic logic to conform to the sound deductive inference model. -/- Within the sound deductive inference model there is a (connected sequence of valid deductions from true premises to a true conclusion) thus unlike the formal proofs of symbolic logic provability cannot diverge from truth.

The five English words—sentence, proposition, judgment, statement, and fact—are central to coherent discussion in logic. However, each is ambiguous in that logicians use each with multiple normal meanings. Several of their meanings are vague in the sense of admitting borderline cases. In the course of displaying and describing the phenomena discussed using these words, this paper juxtaposes, distinguishes, and analyzes several senses of these and related words, focusing on a constellation of recommended senses. One of the purposes of this paper (...) is to demonstrate that ordinary English properly used has the resources for intricate and philosophically sound investigation of rather deep issues in logic and philosophy of language. No mathematical, logical, or linguistic symbols are used. Meanings need to be identified and clarified before being expressed in symbols. We hope to establish that clarity is served by deferring the extensive use of formalized or logically perfect languages until a solid “informal” foundation has been established. Questions of “ontological status”—e.g., whether propositions or sentences, or for that matter characters, numbers, truth-values, or instants, are “real entities”, are “idealizations”, or are “theoretical constructs”—plays no role in this paper. As is suggested by the title, this paper is written to be read aloud. -/- I hope that reading this aloud in groups will unite people in the enjoyment of the humanistic spirit of analytic philosophy. (shrink)

In this paper Timothy Williamson’s argument that the knowledge norm of assertion is the best explanation of the unassertability of Morrean sentences is challenged and an alternative account of the norm of assertion is defended.

Chaitin’s incompleteness result related to random reals and the halting probability has been advertised as the ultimate and the strongest possible version of the incompleteness and undecidability theorems. It is argued that such claims are exaggerations.

In ‘Godel’s Way’ three eminent scientists discuss issues such as undecidability, incompleteness, randomness, computability and paraconsistency. I approach these issues from the Wittgensteinian viewpoint that there are two basic issues which have completely different solutions. There are the scientific or empirical issues, which are facts about the world that need to be investigated observationally and philosophical issues as to how language can be used intelligibly (which include certain questions in mathematics and logic), which need to be decided by looking at (...) how we actually use words in particular contexts. When we get clear about which language game we are playing, these topics are seen to be ordinary scientific and mathematical questions like any others. Wittgenstein’s insights have seldom been equaled and never surpassed and are as pertinent today as they were 80 years ago when he dictated the Blue and Brown Books. -/- Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my book ‘The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle’ 2nd ed (2019). Those interested in more of my writings may see ‘Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019), The Logical Structure of Human Behavior (2019), and Suicidal Utopian Delusions in the 21st Century 4th ed (2019). (shrink)

One of the central criteria for free will is “Could I have done otherwise?” But because of a temporal asymmetry in human choice, the question makes no sense. The question is backward-looking, while human choices are forward-looking. At the time when any choice is actually made, there is as of yet no action to do otherwise. Expectation is the only thing to contradict (do other than). So the ability to do something not expected by the ultimate expecter, Laplace’s demon, is (...) a better criterion for free will. If human action is fundamentally unpredictable, then we have free will. Scientists have studied a form of fundamental unpredictability, known as undecidability. The features that make a system capable of undecidable dynamics have been identified: program-data duality; potential to access an infinite computational medium; and the ability to implement negation. Humans have all three of these features, so we very likely are fundamentally unpredictable, so we have free will. (shrink)

This paper examines the practice of using predictive systems to lengthen the prison sentences of convicted persons when the systems forecast a higher likelihood of re-offense or re-arrest. There has been much critical discussion of technologies used for sentencing, including questions of bias and opacity. However, there hasn’t been a discussion of whether this use of predictive systems makes sense in the first place. We argue that it does not by showing that there is no plausible theory of punishment (...) that supports it. (shrink)

Based on the notion of a trope, this paper gives a novel analysis of identificational sentences such as 'this is Mary','this is a beautiful woman', 'this looks like Mary', or 'this is the same lump of clay, but not the same statue as that'.

In Tractatus, Wittgenstein held that there are null sentences – prominently including logical truths and the truths of mathematics. He says that such sentences are without sense (sinnlos), that they say nothing; he also denies that they are nonsensical (unsinning). Surely it is what a sentence says which is true or false. So if a sentence says nothing, how can it be true or false? The paper discusses the issue.

In this paper, we investigate the expressiveness of the variety of propositional interval neighborhood logics , we establish their decidability on linearly ordered domains and some important subclasses, and we prove the undecidability of a number of extensions of PNL with additional modalities over interval relations. All together, we show that PNL form a quite expressive and nearly maximal decidable fragment of Halpern–Shoham’s interval logic HS.

A suprise may occur if we use a similar strategy to the Liar's paradox to mathematically formalize "This sentence does not contain the symbol X".

Karen Barad’s Meeting the Universe Halfway relies on mutually incompatible grounding gestures, one of which describes the relationality of an always already material-discursive reality, while the other seeks to ground this relation one-sidedly in matter. These two materialisms derive from the gesture she borrows from the New Materialist (and other related) fields, which posits her work as an advance over the history of “representationalism” and “social constructivism.” In turn, this one-sided materialism produces a skewed reading of the quantum mechanical phenomena (...) with which she engages. Her attempt to create an ontological (not epistemological) interpretation of quantum mechanics proves deconstructible. Instead, a science of undecidability or science of quant à helps us to understand debates among scientists and philosophers over the completeness or incompleteness of quantum mechanics and its epistemological or ontological status – by demonstrating that these questions will necessarily remain unresolved. (shrink)

The influential proposal that the analytical component of a theory is captured by its ‘Carnap sentence’ is critically scrutinized. A counterexample which makes the suggestion problematic is presented.

En este ensayo planteo una dificultad que encuentro en la última propuesta de Quine sobre las oraciones observacionales. Argumento que esta dificultad impide que tales oraciones cumplan el rol que él les asigna en su filosofía y socavan su empirismo ilustrado. Luego exploro tentativamente un resquicio que encuentro en la propia filosofía quineana que eventualmente podría evitar los problemas derivados de dicha dificultad. El precio de seguir el camino apuntado por ese resquicio es, sin embargo, una cierta reinterpretación del espíritu (...) empirista de la filosofía de Quine. In this paper I set out a problem I find in Quine's last proposal on observation sentences. I argue that this problem prevents those sentences to play the role he ascribes to them in his philosophy and undermines his enlightened empiricism. Then I tentatively explore an option I find in Quine's own philosophy which can eventually avoid the issues raised by that problem. The price to pay in following this option, though, is a certain reinterpretation of the empiricist spirit of Quine's philosophy. Apresenta-se uma dificuldade na última proposta de Quine sobre as orações observacionais. Argumenta-se que essa dificuldade impede que essas orações cumpram o papel que ele concede a elas em sua filosofia e debilitam seu empirismo ilustrado. Explora-se tentativamente um resquício na própria filosofia quineana que eventualmente poderia evitar os problemas derivados dessa dificuldade. O preço de seguir o caminho indicado por esse resquício é, contudo, certa interpretação do espírito empirista da filosofia de Quine. (shrink)

Contrary to nouns and verbs that either do not include a co-positing of parts, including nouns and some verbs, or if they are, their parts do not significate separately, a sentence (λόγος) is a ‘significant portion of speech by co-positing, its parts signify something separately, though not as a positive or negative judgment but as utterance.’ (OI ., I, 4, 16b26-28). Therefore, every utterance in language that i) includes parts, ii) its signification is based on the co-positing of its parts, (...) iii) each of its parts is separately significant but iv) the signification of its parts is the signification of an utterance and not a judgment, is a sentence. This definition distinguishes a sentence from: a) Nouns: because they do not have parts; b) Verbs: because they either do not have parts or if they have, their parts do not significate separately; c) Those co-positings of words whose significance is not based on their co-positing. Thus a co-positing like ‘wall tree house’ is not a sentence because it has no significance on the basis of its co-positing; d) Those co-positings that the signification of at least some of their parts is the signification of judgments. Thus, ‘Socrates is from Athens and is a philosopher’ is not a sentence because at least one of its parts, e.g. ‘Socrates is from Athens’ signifies a judgment. Those sentences that either of truth or falsity belongs to them are propositions (ἀπόφανσις or λόγος ἀπόφαντικὸς). (OI., I, 4, 17a2-3) It is this and only this kind of sentence that is the subject of study in philosophy. (OI., I, 4, 17a5-7) The primary forms of a proposition are, firstly, an affirmation (κατάφασις) and then a negation (ἀπόφασις). (OI., I, 4, 17a8-9) To have an affirmation or a denial and, thus, a proposition, what firstly is necessary is a verb or the tense of a verb (OI, I, 4, 17a9-12; I, 10, 19b12-13; OI, I, 4, 16b28-30) and a noun (OI, I, 10, 19b10-12). Aristotle draws a contrast between single and plural proposition: a) Single proposition: a proposition that ‘indicate a single fact, or the conjunction of the parts of which results in unity.’ (OI., I, 5, 17a15-16) b) Plural propositions: a proposition that ‘are separate and many in number, which indicate many facts, or whose parts have no conjunction.’ (OI., I, 5, 17a16-17) He also differentiates between simple and composite propositions (OI, I, 5, 17a20-22): a) Simple proposition: ‘that which asserts or denies something of something’; b) Composite propositions: ‘that which is compounded of simple propositions.’ A simple (ἁπλῆ) proposition is a statement about the belonging or not belonging of something (OI, I, 5, 17a22-24). Affirmation and denial are indeed kinds of simple proposition, one affirming the belonging and the other not belonging. An affirmation or denial may be single or non-single. A single affirmation or denial ‘signifies some one fact about some one subject. This singularity is destroyed neither by the universality of subject nor by the universal character of the statement. The only thing that destroys this singularity is that one part signifies more than one thing, which makes the proposition more than one proposition (OI, I, 8, 18a13-27). In other words, neither ‘predication of one thing of many subjects’ nor ‘many things of the same subject’ results in a unitary proposition. (OI., II, 11, 20b13-16) There is also another cause that might destroy the unity of predicate and thus the unity of proposition, namely when one of the parts of the predicate is implicit in the other. Thus, e.g. animal-man does not form a unity because the notion ‘animal’ is implicit in ‘man.’ (OI., II, 11, 21a16-18) The following are characteristics of propositions: 1. That about which an affirmation signifies something is a noun, whether definite or indefinite. (OI., I, 10, 19b5-12) Moreover, there can be no affirmation or denial without a verb. (OI., I, 10, 19b12-19) Therefore, a proposition must involve at least a noun and a verb. 2. Contrary propositions: since both belonging and not belonging of something to something is possible, we are able to make a negative proposition out of an affirmative one and vice versa. (OI., I, 5, 17a26-31) Therefore, ‘every affirmation has an opposite denial’ and vice versa. (OI., I, 6, 17a31-33) The contradictory of each proposition is made by changing the positivity or negativity of the verb ‘to be’ to its opposite but not by changing the subject to its opposite. Thus, the contradictory of ‘man is white’ is ‘man is not white’ and not ‘not-man is white.’ (OI., II, 11, 21a38-b5) What happens in the mentioned wrong form of contradictory propositions is that they change the subject while the denial must be the contrary of the affirmation about the same subject. (cf. OI., II, 14, 24b1-6) This is made plain in propositions involving a verb other than ‘to be’. What must be changed in these propositions is the verb and not the noun or the subject. Thus, the contradictory of ‘Socrates runs’ is ‘Socrates does not run’ and not ‘not-Socrates runs.’ 3. Conversion possible: ‘Out of conversion (μετατιθεμένου) of the name and the verb, the same affirmation and denial is generated. (OI., I, 10, 20b10-12) In other words, the two affirmations or denials ‘signify the same thing.’ (OI, I, 10, 20b1-2) . (shrink)

Serious actualists take it that all properties are existence entailing. I present a simple puzzle about sentence tokens which seems to show that serious actualism is false. I then consider the most promising response to the puzzle. This is the idea that the serious actualist should take ordinary property-talk to contain an implicit existential presupposition. I argue that this approach does not work: it fails to generalise appropriately to all sentence types and tokens. In particular, it fails to capture the (...) right distinctions we ought to make between what I call _typographical sentence types_—an interesting and previously undiscussed class of fine-grained sentence types which are partially individuated by their typography, or how they look when written out. (shrink)

This study proposes a framework by which grammatically and syntactically sound sentences are classified through the perceptual measurement in noise of multilinguals and monolinguals, using an objective measure called SPERI and an interpretivist measure called SPIn, with results evaluated using Shortlist models and the BLINCS model. Hereby filling a knowledge gap on the perception of sentences that combine in varying levels of contextual meaning, linguistic load and cognitive load, this study used sentence clustering methods to find limitations of (...) the proposed framework in determining an absolute and accurate prediction of performance between sentences in the proposed different categories, with factors such as sentence predictability and word frequency taking precedence. There were unintended findings including a relationship between the number of languages spoken and performance, proficiency in other languages decreasing performance despite being an English Native, and how mistakes by multilinguals were more semantically and phonetically influenced than monolinguals. (shrink)

This paper presents an exploration into Alan Turing's Halting Problem, providing a solution using ordinal arithmetic and trigonometric recursion. It builds on the principles of Turing machines, proposing a function H that correctly decides the expected behavior of Q at each recursive depth.

I discuss what I call practical Moore sentences: sentences like ‘You must close your door, but I don’t know whether you will’, which combine an order together with an avowal of agnosticism about whether the order will be obeyed. I show that practical Moore sentences are generally infelicitous. But this infelicity is surprising: it seems like there should be nothing wrong with giving someone an order while acknowledging that you do not know whether it will obeyed. I (...) suggest that this infelicity points to a striking psychological fact, with potentially broad ramifications concerning the structure of norms of speech acts: namely, when giving an order, we must act as if we believe we will be obeyed. (shrink)

This paper replies to Himmelreich's ‘The Paraphrase Argument Against Collective Actions’ [2017], which presents three putative counterexamples to the multiple agents analysis of plural action sentences. The paper shows that the argument from the first example, the discursive dilemma, fails because it relies crucially on a simplification of the target analysis, and that the others don't bear on the question because they turn out on examination to be about individual rather than group action sentences.

This paper shows that Russell’s theory of descriptions gives the wrong se-mantics for definite descriptions occurring in questions and imperatives. Depending on how that theory is applied, it either assigns nonsense to per-fectly meaningful questions and assertions or it assigns meanings that di-verge from the actual semantics of such sentences, even after all pragmatic and contextual variables are allowed for. Given that Russell’s theory is wrong for questions and assertions, it must be wrong for assertoric state-ments; for the semantics (...) of ‘the phi’ obviously doesn’t vary depending on whether it occurs in a question or an assertion or a command. (shrink)

Semanticists face substitution challenges even outside of contexts commonly recognized as opaque. Jennifer M. Saul has drawn attention to pairs of simple sentences - her term for sentences lacking a that-clause operator - of which the following are typical: -/- (1) Clark Kent went into the phone booth, and Superman came out. (1*) Clark Kent went into the phone booth, and Clark Kent came out. -/- (2) Superman is more successful with women than Clark Kent. (2*) Superman is (...) more successful with women than Superman. -/- She challenges us to explain why the upper and lower sentences in each pair differ, or at least appear to differ, in their truth-values and hence truth-conditions. This appearance of substitution failure is inherently puzzling. Moreover, it is taken by Saul to generate a dilemma for anyone hostile to direct reference accounts of that-clause constructions. Direct reference theorists regard the appearance of substitution failure in that-clause contexts as mere appearance, to be dealt with pragmatically rather than semantically. -/- Critics of such accounts need to say something about simple-sentence cases. If they choose to allow that intuitions of substitution failure can be over-ridden and explained away pragmatically in simple-sentence cases but not in that-clause cases, they lay themselves open to the charge of operating a double standard. But if they do not choose this option, they must offer a semantic explanation of apparent substitution failure in simple-sentence cases - no easy task, it turns out. Other respondents to Saul's challenge have sought to provide elaborate semantic treatments. In contrast, this paper proposes a far simpler pragmatic explanation of intuitions of substitution failure in simple sentences, an explanation that deploys no more resources than are to be found in Grice's 'Logic and Conversation'. Ironically, this proposal turns out to be incompatible with a direct reference perspective. So if it is, as I maintain, the most plausible treatment of simple-sentence cases available, Saul's initial thought gets turned around 180 degrees: the phenomenon she has drawn attention to ends up representing a challenge to supporters of direct reference theories. (shrink)

Spinoza distinguishes between causation that is external, as in A causing B where A is external to B, and causation that is internal, where C causes itself (causa sui), without any involvement of anything external to C. External causation is easy to understand, but self causation is not. This note explores an approach to self-causation based upon Gödelian undecidability and draws upon ideas from an earlier study of Gödel’s proof and the quantum measurement problem (Zwick, 1978).

The Sleeping Beauty problem remains controversial with disagreement between so-called Halfers and Thirders, although the Thirders appear to be leading these days. I analyze three popular arguments for the Thirder position, including the long-run frequency argument, Egla’s ‘symmetry’ argument, and new-information arguments, and find problems with each. The long-run frequency argument is almost unequivocally thought to strongly support Thirders, but in formalizing the argument for an arbitrary number of repetitions, I show that the expected proportion of Heads-Awakenings for a single-trial (...) experiment is unambiguously 1/2. My criticisms of Elga’s symmetry argument and the new-information arguments point to subtle misalignments between the narrative/causal description of thought-experiments and the mathematical probability expressions and theory we use to describe these narratives. I end with distinguishing two varieties of possibility—a dynamic forward type and static historical type—that help clarify the Sleeping Beauty problem, nullify the main criticism against Lewis’s Halfer argument, and have applicability to probability theory in general. (shrink)

Judges are typically tasked to consider sentencing benefits but not costs. Previous research finds that both laypeople and prosecutors discount the costs of incarceration when forming sentencing attitudes, raising important questions about whether professional judges show the same bias during sentencing. To test this, we used a vignette-based experiment in which Minnesota state judges reviewed a case summary about an aggravated robbery and imposed a hypothetical sentence. Using random assignment, half the participants received additional information about plausible negative consequences of (...) incarceration. As predicted, our results revealed a mitigating effect of cost exposure on prison sentence term lengths. Critically, these findings support the conclusion that policies that increase transparency in sentencing costs could reduce sentence lengths, which has important economic and social ramifications. (shrink)