Results for 'arithmetical truth'

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  1. On the Arithmetical Truth of Self‐Referential Sentences.Kaave Lajevardi & Saeed Salehi - 2019 - Theoria 85 (1):8-17.
    We take an argument of Gödel's from his ground‐breaking 1931 paper, generalize it, and examine its validity. The argument in question is this: "the sentence G says about itself that it is not provable, and G is indeed not provable; therefore, G is true".
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  2. The Modal Status of Contextually A Priori Arithmetical Truths.Markus Pantsar - 2016 - In Francesca Boccuni & Andrea Sereni (eds.), Objectivity, Realism, and Proof. FilMat Studies in the Philosophy of Mathematics. Cham, Switzerland: Springer International Publishing. pp. 67-79.
    In Pantsar (2014), an outline for an empirically feasible epistemological theory of arithmetic is presented. According to that theory, arithmetical knowledge is based on biological primitives but in the resulting empirical context develops an essentially a priori character. Such contextual a priori theory of arithmetical knowledge can explain two of the three characteristics that are usually associated with mathematical knowledge: that it appears to be a priori and objective. In this paper it is argued that it can also (...)
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  3. The Modal Status of Contextually A Priori Arithmetical Truths.Markus Pantsar - 2016 - In Francesca Boccuni & Andrea Sereni (eds.), Objectivity, Realism, and Proof. FilMat Studies in the Philosophy of Mathematics. Cham, Switzerland: Springer International Publishing.
    In Pantsar, an outline for an empirically feasible epistemological theory of arithmetic is presented. According to that theory, arithmetical knowledge is based on biological primitives but in the resulting empirical context develops an essentially a priori character. Such contextual a priori theory of arithmetical knowledge can explain two of the three characteristics that are usually associated with mathematical knowledge: that it appears to be a priori and objective. In this paper it is argued that it can also explain (...)
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  4. Arithmetic with Satisfaction.James Cain - 1995 - Notre Dame Journal of Formal Logic 36 (2):299-303.
    A language in which we can express arithmetic and which contains its own satisfaction predicate (in the style of Kripke's theory of truth) can be formulated using just two nonlogical primitives: (the successor function) and Sat (a satisfaction predicate).
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  5. Truth and Existence.Jan Heylen & Leon Horsten - 2017 - Thought: A Journal of Philosophy 6 (1):106-114.
    Halbach has argued that Tarski biconditionals are not ontologically conservative over classical logic, but his argument is undermined by the fact that he cannot include a theory of arithmetic, which functions as a theory of syntax. This article is an improvement on Halbach's argument. By adding the Tarski biconditionals to inclusive negative free logic and the universal closure of minimal arithmetic, which is by itself an ontologically neutral combination, one can prove that at least one thing exists. The result can (...)
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  6. The Truth Assignments That Differentiate Human Reasoning From Mechanistic Reasoning: The Evidence-Based Argument for Lucas' Goedelian Thesis.Bhupinder Singh Anand - 2016 - Cognitive Systems Research 40:35-45.
    We consider the argument that Tarski's classic definitions permit an intelligence---whether human or mechanistic---to admit finitary evidence-based definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways: (1) in terms of classical algorithmic verifiabilty; and (2) in terms of finitary algorithmic computability. We then show that the two definitions correspond to two distinctly different assignments of satisfaction and (...)
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  7. Truth, Proof and Gödelian Arguments: A Defence of Tarskian Truth in Mathematics.Markus Pantsar - 2009 - Dissertation, University of Helsinki
    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 (...)
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  8. Carnapian Modal and Epistemic Arithmetic.Heylen Jan - 2009 - In Carrara Massimiliano & Morato Vittorio (eds.), Language, Knowledge, and Metaphysics. Selected papers from the First SIFA Graduate Conference. College Publications. pp. 97-121.
    The subject of the first section is Carnapian modal logic. One of the things I will do there is to prove that certain description principles, viz. the ''self-predication principles'', i.e. the principles according to which a descriptive term satisfies its own descriptive condition, are theorems and that others are not. The second section will be devoted to Carnapian modal arithmetic. I will prove that, if the arithmetical theory contains the standard weak principle of induction, modal truth collapses to (...)
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  9. Truth-preserving and consequence-preserving deduction rules”,.John Corcoran - 2014 - Bulletin of Symbolic Logic 20 (1):130-1.
    A truth-preservation fallacy is using the concept of truth-preservation where some other concept is needed. For example, in certain contexts saying that consequences can be deduced from premises using truth-preserving deduction rules is a fallacy if it suggests that all truth-preserving rules are consequence-preserving. The arithmetic additive-associativity rule that yields 6 = (3 + (2 + 1)) from 6 = ((3 + 2) + 1) is truth-preserving but not consequence-preserving. As noted in James Gasser’s dissertation, (...)
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  10. Deflationary Truth and Pathologies.Cezary Cieśliński - 2010 - Journal of Philosophical Logic 39 (3):325-337.
    By a classical result of Kotlarski, Krajewski and Lachlan, pathological satisfaction classes can be constructed for countable, recursively saturated models of Peano arithmetic. In this paper we consider the question of whether the pathology can be eliminated; we ask in effect what generalities involving the notion of truth can be obtained in a deflationary truth theory (a theory of truth which is conservative over its base). It is shown that the answer depends on the notion of pathology (...)
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  11. Eternal truths and laws of nature.Dennis Des Chene - manuscript
    Are the laws of nature among the eternal truths that, according to Descartes, are created by God? The basis of those laws is the immutability of the divine will, which is not an eternal truth, but a divine attribute. On the other hand, the realization of those laws, and in particular, the quantitative consequences to be drawn from them, depend upon the eternal truths insofar as those truths include the foundations of geometry and arithmetic.
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  12. Frege, Dedekind, and the Modern Epistemology of Arithmetic.Markus Pantsar - 2016 - Acta Analytica 31 (3):297-318.
    In early analytic philosophy, one of the most central questions concerned the status of arithmetical objects. Frege argued against the popular conception that we arrive at natural numbers with a psychological process of abstraction. Instead, he wanted to show that arithmetical truths can be derived from the truths of logic, thus eliminating all psychological components. Meanwhile, Dedekind and Peano developed axiomatic systems of arithmetic. The differences between the logicist and axiomatic approaches turned out to be philosophical as well (...)
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  13. Self-reference and the languages of arithmetic.Richard Heck - 2007 - Philosophia Mathematica 15 (1):1-29.
    I here investigate the sense in which diagonalization allows one to construct sentences that are self-referential. Truly self-referential sentences cannot be constructed in the standard language of arithmetic: There is a simple theory of truth that is intuitively inconsistent but is consistent with Peano arithmetic, as standardly formulated. True self-reference is possible only if we expand the language to include function-symbols for all primitive recursive functions. This language is therefore the natural setting for investigations of self-reference.
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  14. The concept of truth in a finite universe.Panu Raatikainen - 2000 - Journal of Philosophical Logic 29 (6):617-633.
    The prospects and limitations of defining truth in a finite model in the same language whose truth one is considering are thoroughly examined. It is shown that in contradistinction to Tarski's undefinability theorem for arithmetic, it is in a definite sense possible in this case to define truth in the very language whose truth is in question.
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  15. Review of C. S. Jenkins, Grounding Concepts: An Empirical Basis for Arithmetical Knowledge[REVIEW]Neil Tennant - 2010 - Philosophia Mathematica 18 (3):360-367.
    This book is written so as to be ‘accessible to philosophers without a mathematical background’. The reviewer can assure the reader that this aim is achieved, even if only by focusing throughout on just one example of an arithmetical truth, namely ‘7+5=12’. This example’s familiarity will be reassuring; but its loneliness in this regard will not. Quantified propositions — even propositions of Goldbach type — are below the author’s radar.The author offers ‘a new kind of arithmetical epistemology’, (...)
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  16. An epistemology for the Platonist? Platonism, Field’s Dilemma, and Judgment-Dependent Truth.Tommaso Piazza - 2011 - Grazer Philosophische Studien 83 (1):67-92.
    According to Hartry Field, the mathematical Platonist is hostage of a dilemma. Faced with the request of explaining the mathematicians’ reliability, one option could be to maintain that the mathematicians are reliably responsive to a realm populated with mathematical entities; alternatively, one might try to contend that the mathematical realm conceptually depends on, and for this reason is reliably reflected by, the mathematicians’ (best) opinions; however, both alternatives are actually unavailable to the Platonist: the first one because it is in (...)
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  17. Typed and Untyped Disquotational Truth.Cezary Cieśliński - 2015 - In T. Achourioti, H. Galinon, J. Martínez Fernández & K. Fujimoto (eds.), Unifying the Philosophy of Truth. Dordrecht: Imprint: Springer.
    We present an overview of typed and untyped disquotational truth theories with the emphasis on their (non)conservativity over the base theory of syntax. Two types of conservativity are discussed: syntactic and semantic. We observe in particular that TB—one of the most basic disquotational theories—is not semantically conservative over its base; we show also that an untyped disquotational theory PTB is a syntactically conservative extension of Peano Arithmetic.
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  18. Three Kantian Strands in Frege’s View of Arithmetic.Gilead Bar-Elli - 2014 - Journal for the History of Analytical Philosophy 2 (7).
    On the background of explaining their different notions of analyticity, their different views on definitions, and some aspects of Frege’s notion of sense, three important Kantian strands that interweave into Frege’s view are exposed. First, Frege’s remarkable view that arithmetic, though analytic, contains truths that “extend our knowledge”, and by Kant’s use of the term, should be regarded synthetic. Secondly, that our arithmetical (and logical) knowledge depends on a sort of a capacity to recognize and identify objects, which are (...)
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  19. A simple theory containing its own truth predicate.Nicholas Shackel - 2018 - South American Journal of Logic 4 (1):121-131.
    Tarski's indefinability theorem shows us that truth is not definable in arithmetic. The requirement to define truth for a language in a stronger language (if contradiction is to be avoided) lapses for particularly weak languages. A weaker language, however, is not necessary for that lapse. It also lapses for an adequately weak theory. It turns out that the set of G{\"o}del numbers of sentences true in arithmetic modulo $n$ is definable in arithmetic modulo $n$.
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    A defense of Isaacson’s thesis, or how to make sense of the boundaries of finite mathematics.Pablo Dopico - 2024 - Synthese 203 (2):1-22.
    Daniel Isaacson has advanced an epistemic notion of arithmetical truth according to which the latter is the set of truths that we grasp on the basis of our understanding of the structure of natural numbers alone. Isaacson’s thesis is then the claim that Peano Arithmetic (PA) is the theory of finite mathematics, in the sense that it proves all and only arithmetical truths thus understood. In this paper, we raise a challenge for the thesis and show how (...)
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  21. Cut elimination for systems of transparent truth with restricted initial sequents.Carlo Nicolai - manuscript
    The paper studies a cluster of systems for fully disquotational truth based on the restriction of initial sequents. Unlike well-known alternative approaches, such systems display both a simple and intuitive model theory and remarkable proof-theoretic properties. We start by showing that, due to a strong form of invertibility of the truth rules, cut is eliminable in the systems via a standard strategy supplemented by a suitable measure of the number of applications of truth rules to formulas in (...)
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  22. On the weak Kleene scheme in Kripke's theory of truth.James Cain & Zlatan Damnjanovic - 1991 - Journal of Symbolic Logic 56 (4):1452-1468.
    It is well known that the following features hold of AR + T under the strong Kleene scheme, regardless of the way the language is Gödel numbered: 1. There exist sentences that are neither paradoxical nor grounded. 2. There are 2ℵ0 fixed points. 3. In the minimal fixed point the weakly definable sets (i.e., sets definable as {n∣ A(n) is true in the minimal fixed point where A(x) is a formula of AR + T) are precisely the Π1 1 sets. (...)
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  23. Structure and Categoricity: Determinacy of Reference and Truth Value in the Philosophy of Mathematics.Tim Button & Sean Walsh - 2016 - Philosophia Mathematica 24 (3):283-307.
    This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.
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  24. After Non-Euclidean Geometry: Intuition, Truth and the Autonomy of Mathematics.Janet Folina - 2018 - Journal for the History of Analytical Philosophy 6 (3).
    The mathematical developments of the 19th century seemed to undermine Kant’s philosophy. Non-Euclidean geometries challenged Kant’s view that there is a spatial intuition rich enough to yield the truth of Euclidean geometry. Similarly, advancements in algebra challenged the view that temporal intuition provides a foundation for both it and arithmetic. Mathematics seemed increasingly detached from experience as well as its form; moreover, with advances in symbolic logic, mathematical inference also seemed independent of intuition. This paper considers various philosophical responses (...)
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  25. The Significance of Evidence-based Reasoning for Mathematics, Mathematics Education, Philosophy and the Natural Sciences.Bhupinder Singh Anand - forthcoming
    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 (...)
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  26. Descriptions and unknowability.Jan Heylen - 2010 - Analysis 70 (1):50-52.
    In a recent paper Horsten embarked on a journey along the limits of the domain of the unknowable. Rather than knowability simpliciter, he considered a priori knowability, and by the latter he meant absolute provability, i.e. provability that is not relativized to a formal system. He presented an argument for the conclusion that it is not absolutely provable that there is a natural number of which it is true but absolutely unprovable that it has a certain property. The argument depends (...)
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  27. Is Incompatibilism Compatible with Fregeanism?Nils Kürbis - 2018 - European Journal of Analytic Philosophy 14 (2):27-46.
    This paper considers whether incompatibilism, the view that negation is to be explained in terms of a primitive notion of incompatibility, and Fregeanism, the view that arithmetical truths are analytic according to Frege’s definition of that term in §3 of Foundations of Arithmetic, can both be upheld simultaneously. Both views are attractive on their own right, in particular for a certain empiricist mind-set. They promise to account for two philosophical puzzling phenomena: the problem of negative truth and the (...)
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  28. Counterpossibles in Science: The Case of Relative Computability.Matthias Jenny - 2018 - Noûs 52 (3):530-560.
    I develop a theory of counterfactuals about relative computability, i.e. counterfactuals such as 'If the validity problem were algorithmically decidable, then the halting problem would also be algorithmically decidable,' which is true, and 'If the validity problem were algorithmically decidable, then arithmetical truth would also be algorithmically decidable,' which is false. These counterfactuals are counterpossibles, i.e. they have metaphysically impossible antecedents. They thus pose a challenge to the orthodoxy about counterfactuals, which would treat them as uniformly true. What’s (...)
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  29. Do Goedel's incompleteness theorems set absolute limits on the ability of the brain to express and communicate mental concepts verifiably?Bhupinder Singh Anand - 2004 - Neuroquantology 2:60-100.
    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 (...)
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  30. Debunking Arguments: Mathematics, Logic, and Modal Security.Justin Clarke-Doane - 2017 - In Michael Ruse & Robert J. Richards (eds.), The Cambridge Handbook of Evolutionary Ethics. New York: Cambridge University Press.
    I discuss the structure of genealogical debunking arguments. I argue that they undermine our mathematical beliefs if they undermine our moral beliefs. The contrary appearance stems from a confusion of arithmetic truths with (first-order) logical truths, or from a confusion of reliability with justification. I conclude with a discussion of the cogency of debunking arguments, in light of the above. Their cogency depends on whether information can undermine all of our beliefs of a kind, F, without giving us direct reason (...)
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  31. Objects are (not) ...Friedrich Wilhelm Grafe - 2024 - Archive.Org.
    My goal in this paper is, to tentatively sketch and try defend some observations regarding the ontological dignity of object references, as they may be used from within in a formalized language. -/- Hence I try to explore, what properties objects are presupposed to have, in order to enter the universe of discourse of an interpreted formalized language. -/- First I review Frege′s analysis of the logical structure of truth value definite sentences of scientific colloquial language, to draw suggestions (...)
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  32. Neo-Logicism and Gödelian Incompleteness.Fabian Pregel - 2023 - Mind 131 (524):1055-1082.
    There is a long-standing gap in the literature as to whether Gödelian incompleteness constitutes a challenge for Neo-Logicism, and if so how serious it is. In this paper, I articulate and address the challenge in detail. The Neo-Logicist project is to demonstrate the analyticity of arithmetic by deriving all its truths from logical principles and suitable definitions. The specific concern raised by Gödel’s first incompleteness theorem is that no single sound system of logic syntactically implies all arithmetical truths. I (...)
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  33. Why did Fermat believe he had `a truly marvellous demonstration' of FLT?Bhupinder Singh Anand - manuscript
    Conventional wisdom dictates that proofs of mathematical propositions should be treated as necessary, and sufficient, for entailing `significant' mathematical truths only if the proofs are expressed in a---minimally, deemed consistent---formal mathematical theory in terms of: * Axioms/Axiom schemas * Rules of Deduction * Definitions * Lemmas * Theorems * Corollaries. Whilst Andrew Wiles' proof of Fermat's Last Theorem FLT, which appeals essentially to geometrical properties of real and complex numbers, can be treated as meeting this criteria, it nevertheless leaves two (...)
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  34. On What Ground Do Thin Objects Exist? In Search of the Cognitive Foundation of Number Concepts.Markus Pantsar - 2023 - Theoria 89 (3):298-313.
    Linnebo in 2018 argues that abstract objects like numbers are “thin” because they are only required to be referents of singular terms in abstraction principles, such as Hume's principle. As the specification of existence claims made by analytic truths (the abstraction principles), their existence does not make any substantial demands of the world; however, as Linnebo notes, there is a potential counter-argument concerning infinite regress against introducing objects this way. Against this, he argues that vicious regress is avoided in the (...)
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  35. Three Dogmas of First-Order Logic and some Evidence-based Consequences for Constructive Mathematics of differentiating between Hilbertian Theism, Brouwerian Atheism and Finitary Agnosticism.Bhupinder Singh Anand - manuscript
    We show how removing faith-based beliefs in current philosophies of classical and constructive mathematics admits formal, evidence-based, definitions of constructive mathematics; of a constructively well-defined logic of a formal mathematical language; and of a constructively well-defined model of such a language. -/- We argue that, from an evidence-based perspective, classical approaches which follow Hilbert's formal definitions of quantification can be labelled `theistic'; whilst constructive approaches based on Brouwer's philosophy of Intuitionism can be labelled `atheistic'. -/- We then adopt what may (...)
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  36. Formal Background for the Incompleteness and Undefinability Theorems.Richard Kimberly Heck - manuscript
    A teaching document I've used in my courses on truth and on incompleteness. Aimed at students who have a good grasp of basic logic, and decent math skills, it attempts to give them the background they need to understand a proper statement of the classic results due to Gödel and Tarski, and sketches their proofs. Topics covered include the notions of language and theory, the basics of formal syntax and arithmetization, formal arithmetic (Q and PA), representability, diagonalization, and the (...)
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  37. Σ01 soundness isn’t enough: Number theoretic indeterminacy’s unsavory physical commitments.Sharon Berry - 2023 - British Journal for the Philosophy of Science 74 (2):469-484.
    It’s sometimes suggested that we can (in a sense) settle the truth-value of some statements in the language of number theory by stipulation, adopting either φ or ¬φ as an additional axiom. For example, in Clarke-Doane (2020b) and a series of recent APA presentations, Clarke-Doane suggests that any Σ01 sound expansion of our current arithmetical practice would express a truth. In this paper, I’ll argue that (given a certain popular assumption about the model-theoretic representability of languages like (...)
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  38. The inexpressibility of validity.Julien Murzi - 2014 - Analysis 74 (1):65-81.
    Tarski's Undefinability of Truth Theorem comes in two versions: that no consistent theory which interprets Robinson's Arithmetic (Q) can prove all instances of the T-Scheme and hence define truth; and that no such theory, if sound, can even express truth. In this note, I prove corresponding limitative results for validity. While Peano Arithmetic already has the resources to define a predicate expressing logical validity, as Jeff Ketland has recently pointed out (2012, Validity as a primitive. Analysis 72: (...)
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  39. The gödel paradox and Wittgenstein's reasons.Francesco Berto - 2009 - Philosophia Mathematica 17 (2):208-219.
    An interpretation of Wittgenstein’s much criticized remarks on Gödel’s First Incompleteness Theorem is provided in the light of paraconsistent arithmetic: in taking Gödel’s proof as a paradoxical derivation, Wittgenstein was drawing the consequences of his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. It is shown that the features of paraconsistent arithmetics (...)
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  40. Axiomatic Theories of Partial Ground I: The Base Theory.Johannes Korbmacher - 2018 - Journal of Philosophical Logic 47 (2):161-191.
    This is part one of a two-part paper, in which we develop an axiomatic theory of the relation of partial ground. The main novelty of the paper is the of use of a binary ground predicate rather than an operator to formalize ground. This allows us to connect theories of partial ground with axiomatic theories of truth. In this part of the paper, we develop an axiomatization of the relation of partial ground over the truths of arithmetic and show (...)
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  41. String theory.John Corcoran, William Frank & Michael Maloney - 1974 - Journal of Symbolic Logic 39 (4):625-637.
    For each positive n , two alternative axiomatizations of the theory of strings over n alphabetic characters are presented. One class of axiomatizations derives from Tarski's system of the Wahrheitsbegriff and uses the n characters and concatenation as primitives. The other class involves using n character-prefixing operators as primitives and derives from Hermes' Semiotik. All underlying logics are second order. It is shown that, for each n, the two theories are definitionally equivalent [or synonymous in the sense of deBouvere]. It (...)
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  42. L'etica moderna. Dalla Riforma a Nietzsche.Sergio Cremaschi - 2007 - Roma RM, Italia: Carocci.
    This book tells the story of modern ethics, namely the story of a discourse that, after the Renaissance, went through a methodological revolution giving birth to Grotius’s and Pufendorf’s new science of natural law, leaving room for two centuries of explorations of the possible developments and implications of this new paradigm, up to the crisis of the Eighties of the eighteenth century, a crisis that carried a kind of mitosis, the act of birth of both basic paradigms of the two (...)
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  43. Self-referential theories.Samuel A. Alexander - 2020 - Journal of Symbolic Logic 85 (4):1687-1716.
    We study the structure of families of theories in the language of arithmetic extended to allow these families to refer to one another and to themselves. If a theory contains schemata expressing its own truth and expressing a specific Turing index for itself, and contains some other mild axioms, then that theory is untrue. We exhibit some families of true self-referential theories that barely avoid this forbidden pattern.
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  44. ‘‘Quine’s Evolution from ‘Carnap’s Disciple’ to the Author of “Two Dogmas.Greg Frost-Arnold - 2011 - Hopos: The Journal of the International Society for the History of Philosophy of Science 1 (2):291-316.
    Recent scholarship indicates that Quine’s “Truth by Convention” does not present the radical critiques of analytic truth found fifteen years later in “Two Dogmas of Empiricism.” This prompts a historical question: what caused Quine’s radicalization? I argue that two crucial components of Quine’s development can be traced to the academic year 1940–1941, when he, Russell, Carnap, Tarski, Hempel, and Goodman were all at Harvard together. First, during those meetings, Quine recognizes that Carnap has abandoned the extensional, syntactic approach (...)
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  45. Julius Caesar and the Numbers.Nathan Salmón - 2018 - Philosophical Studies 175 (7):1631-1660.
    This article offers an interpretation of a controversial aspect of Frege’s The Foundations of Arithmetic, the so-called Julius Caesar problem. Frege raises the Caesar problem against proposed purely logical definitions for ‘0’, ‘successor’, and ‘number’, and also against a proposed definition for ‘direction’ as applied to lines in geometry. Dummett and other interpreters have seen in Frege’s criticism a demanding requirement on such definitions, often put by saying that such definitions must provide a criterion of identity of a certain kind. (...)
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  46. Gödelova věta a relace logického důsledku.Jaroslav Zouhar - 2010 - Teorie Vědy / Theory of Science 32 (1):59-95.
    In his proof of the first incompleteness theorem, Kurt Gödel provided a method of showing the truth of specific arithmetical statements on the condition that all the axioms of a certain formal theory of arithmetic are true. Furthermore, the statement whose truth is shown in this way cannot be proved in the theory in question. Thus it may seem that the relation of logical consequence is wider than the relation of derivability by a pre-defined set of rules. (...)
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  47. The Potential in Frege’s Theorem.Will Stafford - 2023 - Review of Symbolic Logic 16 (2):553-577.
    Is a logicist bound to the claim that as a matter of analytic truth there is an actual infinity of objects? If Hume’s Principle is analytic then in the standard setting the answer appears to be yes. Hodes’s work pointed to a way out by offering a modal picture in which only a potential infinity was posited. However, this project was abandoned due to apparent failures of cross-world predication. We re-explore this idea and discover that in the setting of (...)
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  48. Operators in the paradox of the knower.Patrick Grim - 1993 - Synthese 94 (3):409 - 428.
    Predicates are term-to-sentence devices, and operators are sentence-to-sentence devices. What Kaplan and Montague's Paradox of the Knower demonstrates is that necessity and other modalities cannot be treated as predicates, consistent with arithmetic; they must be treated as operators instead. Such is the current wisdom.A number of previous pieces have challenged such a view by showing that a predicative treatment of modalities neednot raise the Paradox of the Knower. This paper attempts to challenge the current wisdom in another way as well: (...)
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  49. Gottfried Wilhelm Leibniz.Lloyd Strickland - 2021 - Oxford Bibliographies 2.
    Gottfried Wilhelm Leibniz (1646–1716) was a universal genius, making original contributions to law, mathematics, philosophy, politics, languages, and many areas of science, including what we would now call physics, biology, chemistry, and geology. By profession he was a court counselor, librarian, and historian, and thus much of his intellectual activity had to be fit around his professional duties. Leibniz’s fame and reputation among his contemporaries rested largely on his innovations in the field of mathematics, in particular his discovery of the (...)
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  50. Steps Towards a Minimalist Account of Numbers.Thomas Schindler - 2021 - Mind 131 (523):863-891.
    This paper outlines an account of numbers based on the numerical equivalence schema, which consists of all sentences of the form ‘#x.Fx=n if and only if ∃nx Fx’, where # is the number-of operator and ∃n is defined in standard Russellian fashion. In the first part of the paper, I point out some analogies between the NES and the T-schema for truth. In light of these analogies, I formulate a minimalist account of numbers, based on the NES, which strongly (...)
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