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... Logical reasoning is a cornerstone of scientific inquiry, enabling researchers to construct valid arguments, evaluate concepts and definitions, and systematically interpret evidence [6,19,21,26,[37][38][39][40][41][42][43]. Through logical reasoning, arguments can be assessed for soundness and consistency, flaws in reasoning can be identified, and coherent frameworks for understanding complex phenomena developed [5,6,21,26,37,42]. ...
... Logical reasoning is a cornerstone of scientific inquiry, enabling researchers to construct valid arguments, evaluate concepts and definitions, and systematically interpret evidence [6,19,21,26,[37][38][39][40][41][42][43]. Through logical reasoning, arguments can be assessed for soundness and consistency, flaws in reasoning can be identified, and coherent frameworks for understanding complex phenomena developed [5,6,21,26,37,42]. This structured approach typically involves applying deductive and inductive methods (Table 1) to distinguish valid arguments from invalid ones [6,37,38,44,45], ensuring that conclusions are derived from objective, logically consistent criteria rather than subjective biases [6,18,19,21,26,[37][38][39][42][43][44][45]. ...
... Through logical reasoning, arguments can be assessed for soundness and consistency, flaws in reasoning can be identified, and coherent frameworks for understanding complex phenomena developed [5,6,21,26,37,42]. This structured approach typically involves applying deductive and inductive methods (Table 1) to distinguish valid arguments from invalid ones [6,37,38,44,45], ensuring that conclusions are derived from objective, logically consistent criteria rather than subjective biases [6,18,19,21,26,[37][38][39][42][43][44][45]. ...
... It has long been known that the following theories are mutually interpretable: 1. SOL + DI 2. SOL + HP 3. second-order arithmetic. That (2) interprets (3) is essentially due to Frege [2,3]. That (1) interprets (3) is essentially due to Dedekind [4]. ...
... In this paper we give a more fine-grained characterization of the logical strength of HP, as measured by deductive implications rather than interpretability. 3 Our main result is that HP is not deductively conservative over SOL + DI (Theorem 12). In other words, HP proves additional theorems in the language of pure second-order logic that are not provable from SOL + DI alone. ...
... This system includes full second-order comprehension. See Section 2.1 for details.2 Second-order arithmetic is a powerful theory of arithmetic that seems capable of formalizing almost any ordinary mathematical theorem only involving countable objects and structures.3 This is a more fine-grained characterization, because everything that is deductively implied by a theory is interpretable in that theory, but not vice versa. ...
It has long been known that in the context of axiomatic second-order logic (SOL), Hume’s Principle (HP) is mutually interpretable with “the universe is Dedekind infinite” (DI). In this paper, we offer a more fine-grained analysis of the logical strength of HP, measured by deductive implications rather than interpretability. Our main result is that HP is not deductively conservative over SOL + DI. That is, SOL + HP proves additional theorems in the language of pure second-order logic that are not provable from SOL + DI alone. Arguably, then, HP is not just a pure axiom of infinity, but rather it carries additional logical content. On the other hand, we show that HP is Π11 conservative over SOL + DI, and that HP is conservative over SOL + DI + “the universe is well ordered” (WO). Next, we show that SOL + HP does not prove any of the simplest and most natural versions of the axiom of choice, including WO and weaker principles. Lastly, we discuss other axioms of infinity. We show that HP does not prove the Splitting or Pairing principles (axioms of infinity stronger than DI).
... If what I said above is reasonably plausible, then it is unlikely that regarding the degree of importance Frege lumps all theorems in Table 3 [27] and in Table 2 [28] together. It is perfectly conceivable that he considers them across the board indispensable in pursuit of his foundational project. ...
... In other words, it is the combination of these features-truth, generality, self-evidence and rational indubitableness, unprovability in a theory T , and at the same time the possession of significant cognitive value-that in Frege's view is the quintessence of axioms in general and supposed to approximate the classical Euclidean conception. 28 In contrast to a geometrical axiom, a primitive truth of logic that is selected as an axiom of a theory T must-and this requirement already looms in Begriffsschrift and in Grundlagen-satisfy the stronger condition of utmost generality, that is, its validity must extend to all areas of conceptual thought. When Frege introduced the concept-script version of Basic Law V he was probably aware that he was facing a conflict between the second and the third requirement. ...
... If we assume that in the scenario he committed himself to playing by the book he would, if certain conditions obtained, be heading to new shores, despite the supposed similarity between the two approaches that I just mentioned. In the logicist project that Frege pursues in [27] and [28], he gets by with value-ranges as the target objects of both his definition of the cardinal numbers and the envisaged definition of the real numbers as Relations on Relations. 50 If we pay attention to a remark that Frege makes at the beginning of [27, Section 9], we see that he had also planned to define the complex numbers as special valueranges. ...
... It is often claimed that the theory of function levels introduced by Frege [8] in his two-volume magnum opus Grundgesetze der Arithmetik anticipates the hierarchy of types that underlies the simple theory of types developed by Church [3]. 1 The claim is roughly that a ground type of objects and a type of functions is present in the ideography. I believe the anticipation is articulated more clearly in the words of Martin-Löf: ...
... Like any other rules of inference in the ideography, universal generalization operates on judgments as the premises and conclusions of the rule. 8 Therefore, Frege implies that we have a judgment in the premise of this rule. Despite Frege's implicit assumption, a second look reveals that the account of judgment outlined in Grundgesetze §5 is only applicable in the context of sentences, not Roman markers. ...
... This also means that instead of f (x) : o (x : ι), in constructive type theory we require f (x) : U (x : ι), where U is a universe type, a type whose terms are themselves types. Of course, I am not suggesting that Frege would be prepared to accept (8). I doubt he could even make sense of it as a rule of inference in his wildest dreams. ...
It is often claimed that the theory of function levels proposed by Frege in Grundgesetze der Arithmetik anticipates the hierarchy of types that underlies Church's simple theory of types. This claim roughly states that Frege presupposes a type of functions in the sense of simple type theory in the expository language of Grundgesetze. However, this view makes it hard to accommodate function names of two arguments and view functions as incomplete entities. I propose and defend an alternative interpretation of first-level function names in Grundgesetze into simple type-theoretic open terms rather than into closed terms of a function type. This interpretation offers a still unhistorical but more faithful type-theoretic approximation of Frege's theory of levels and can be naturally extended to accommodate second-level functions. It is made possible by two key observations that Frege's Roman markers behave essentially like open terms and that Frege lacks a clear criterion for distinguishing between Roman markers and function names.
... [25, pp. 132-133; translation modified to fit the Polish original 2 and its German translation 3 ] Since definitions were to be formulated as theses of the theory, the content of the rules would vary according to which logical symbols were primitive in its language. A definition introducing a new sign into a language would be formally correct (Polish "formalnie poprawna") or methodologically correct (German "methodologisch korrekt") if and only if it had the structure specified by the rules of definition of the language. ...
... It is not clear whether the change is substantive and if so why Tarski made it. If he was using the word 'non-creativity' with reference to the sense, to be discussed later, in which Frege [3] inveighed against creative definitions, then the change is not substantive. If he was using the word 'non-creativity' in the sense used by Łukasiewicz [12], Ajdukiewicz [1] and Suppes [14], then the change is substantive. ...
... This construal extends the condition of consistency beyond consistency as normally understood, since a theory to which a definition of an individual constant or term-forming functional constant is added without prior proof that exactly one object has the properties specified in the definiens may be consistent in the sense of not permitting derivation of a contradiction -namely, if unique satisfaction of the conditions described in the definiens is provable (just not actually proved). Frege [3] called definitions "creative" that conjured objects into existence by a definition of an individual constant or term-forming functional constant that was not preceded by a proof of unique satisfaction of the defining condition, and he inveighed against such creativity. Tarski thus accepted non-creativity in Frege's sense as a requirement of a formally correct definition but treated it as covered by the condition of consistency. ...
In his 1933 monograph on the concept of truth, Alfred Tarski claimed that his definition of truth satisfied “the usual conditions of methodological correctness”, which in a 1935 article he identified as consistency and back-translatability. Following the rules of defining for an axiomatized theory was supposed to ensure satisfaction of the two conditions. But Tarski neither explained the two conditions nor supplied rules of defining for any axiomatized theory. We can make explicit what Tarski understood by consistency and back-translatability, with the help of (1) an account by Ajdukiewicz (1936) of the criteria underlying the practice of articulating rules of defining for axiomatized theories and (2) a critique by Frege (1903) of definitions that conjure an object into existence as that which satisfies a specified condition without first proving that exactly one object does so. I show that satisfaction of the conditions of consistency and back-translatability as thus explained is guaranteed by the rules of defining articulated by Leśniewski (1931) for an axiomatized system of propositional logic. I then construct analogous rules of defining for the theory within which Tarski developed his definition of truth. Tarski’s 32 definitions in this theory occasionally violate these rules, but the violations are easily repaired. I argue that the Leśniewski-Ajdukiewicz theory of formal correctness of definitions within which Tarski worked is superior in some respects to the widely accepted analogous theory articulated by Suppes (1957).
... SeeWedin (1990). 9Frege (1893Frege ( /1903 andWhitehead/Russell (1910/1913. ...
... The point goes back toFrege (1893Frege ( /1903, vol. 2, §56, though Frege himself saw it as a reason to require that all vagueness be banned from the scope of logical theorizing. ...
The papers in this volume present some of the most recent results of the work about contradictions in philosophical logic and metaphysics; examine the history of contradiction in crucial phases of philosophical thought; consider the relevance of contradictions for political and philosophical actuality. From this consideration a common question emerges: the question of the irreducibility, reality and productive force of (some) contradictions.
... The later Frege assumed a different notion of content that contains abstract and objective presentations of individual objects, concepts and relations. A second difference is that he now conceives of the notion of an object in a semantic sense as the referent of a singular term, and conceives of concepts and relations in his technical and formal senses as functions from objects or n-tuples of objects to truth-values (Frege 1892b(Frege , 1893. ...
... 43 However, that is not what Frege explicitly says about such sentences. According to him, a Latin letter indicates certain objects in an undetermined way (Frege 1893(Frege , §8, 11, 1906b(Frege , 217, 1976. We could say that Latin letters are indicators for satisfying objects or substitutable names of a specific type. ...
In this paper we give a detailed comparison of the key elements of Frege's formal language of thought and apparently similar views in Stoic formal logic. That is, we compare their views on the following topics: connectives, negation, simple sentences, propositional content , predicates and their incompleteness, and quantifications. We show that in most of these cases the similarities between Frege's views and the Stoic views are only superficial. Frege's views are far more systematic, better developed and can in no case directly or fully be traced back to Stoic ideas. Furthermore, we show that Prantl, pace Bobzien 2021, is in most of these cases not a reliable source with respect to Stoic logic and that it is very unlikely that Frege's views on logic were influenced by Prantl's interpretation of Stoic formal logic. ARTICLE HISTORY
... Mit anderen Worten: Die Frage, ob eine Theorie der Subjektivität in Übereinstimmung mit dem Antipsychologismus anzubieten ist, ist diesem zwar inhärent, nicht aber die Formulierung dieser Frage als Frage nach dem Übergang vom Subjektiven zum Objektiven, denn sie macht nur unter der Voraussetzung des IP Sinn, verliert aber ihren Sinn, wenn man dieses Prinzip fallen lässt. Genau dies geschieht zuerst bei Frege [17] und dann bei Husserl [18], der einerseits den Begriff der abstrakten, für die Subjektivität absolut transzendenten Objekte einführt, andererseits aber den Zugang des realen psychologischen Subjekts zu diesen Objekten garantiert, indem er die Idee der Intentionalität verfeinert und die These verneint, dass das Subjekt nur einen direkten und unmittelbaren Zugang zu seinen eigenen Vorstellungen hat. In dieser entscheidenden Bewegung wird, und zwar nicht zufällig, sondern aus einer inneren systematischen Notwendigkeit heraus, die Kritik von Frege und Husserl an Lotze, vor allem im Hinblick auf seine Pflege des IP, eine grundlegende Rolle spielen, auch wenn in beiden Fällen der entscheidende Einfluss des Autors des "Mikrokosmus" [18][19][20] nicht unerkannt bleiben kann. ...
... Given Second-Order Comprehension, V engenders trouble for Frege. It'll be important for our purposes to revisit why, following roughly Frege's own insightful presentation of the problem in the appendix of volume II of the Grundgesetze (Frege, 1903). ...
Call the combination of Unrestricted Composition and Composition as Identity Identity Universalism. I examine in detail some important features of this view in connection with Russell’s paradox and Cantor’s theorem, which have been of interest in the recent literature. I begin by showing that Identity Universalism entails an ontological explosion principle about plurals and mereology that’s exactly analogous to Frege’s infamous Basic Law V. I identify how Identity Universalism, unlike Frege’s system, nonetheless manages to avoid falling prey to a corresponding version of Russell’s paradox. I then argue that although Identity Universalism manages to forestall the letter of the paradox, it nonetheless remains in direct conflict with the Cantorian insight about cardinality at the basis of the paradox, which in this setting is the plural version of Cantor’s theorem. But I offer reasons why identity universalists needn’t be overly concerned about this. On the one hand, the same features of the view that preclude a derivation of the paradox also preclude a derivation of the theorem. On the other, identity universalists may help themselves to resources from recent discussions in fundamental metaphysics to articulate their understanding of reality, which may help allay concerns about their manifest anti-Cantorianism.
... Similarly, it was possible to investigate which variables could play a dominant role in the model (local/global sensitivity analysis) or which were the optimal parameters of a model given information affected by uncertainties (inverse sensitivity analysis). The together with the contemporary philosophers Gottlob Frege and Ludwig Wittgenstein, studied natural language for a long time in relation to its ambiguity and its ability to represent reality [20,21], laying the foundations for the Logical Atomism, an important philosophical view according to which reality is thought of as decomposition into non-reducible propositions, i.e., atomic facts, to which the principles of classical logic can be applied [22]. Both Russell and Wittgenstein came to consider vagueness as a ''degree'' to be attributed to reality based on the number of ''differences'' that the various systems derived from its representation (through language) can have. ...
The concept of uncertainty has always been important in the field of mathematical modeling. In particular, the growing application of Machine Learning and Deep Learning methods in many scientific fields has led to the implementation and use of new uncertainty quantification techniques aimed at distinguishing between reliable and unreliable predictions. However, the novelty of this discipline and the plethora of articles produced, ranging from theoretical results to purely applied experiments, has resulted in a very fragmented and cluttered literature. In this review, we have attempted to combine the well-established mathematical background of the Bayesian framework with the practical aspect of modern state-of-the-art emerging techniques in order to meet the urgent need for clarity on key concepts related to uncertainty quantification. First, we introduced the different sources of uncertainty, ranging from epistemic/reducible to aleatoric/irreducible, providing both a rigorous mathematical derivation and several examples to facilitate understanding. The review then details some of the most important techniques for uncertainty quantification. These methods are compared in terms of their advantages and drawbacks and classified in terms of their intrusiveness, in order to provide the practitioner with a useful vademecum for selecting the optimal model depending on the application context.
... Therefore Dedekind (1888), Peano (1889) and Frege (1893), again independently, offered three alternative constructions of the system of all natural numbers as the canonical infinite system of objects. So around 1890 a second shift of the problem of foundations of the differential and integral calculus occurred. ...
How mathematics confronts its paradoxes Paradoxes in mathematics show surprisingly many common features. The paper analyzes the historical development of the language of the particular mathematical theory (i.e., algebra, calculus, and predicate logic, respectively) and argues that the paradoxes occur at a particular phase of the historical development of the language. It argues that the paradoxes exhibit the expressive boundaries of the language of mathematics.
... 15 I employ Frege's symbol "∞ 1 " from Grundlagen for the number Endlos and dispense with the diagonal stroke crossing "0" and "1" which in Grundgesetze Frege uses to distinguish the cardinal numbers 0 and 1 from the numbers 0 and 1; cf. Frege (1903), §157. Likewise for the sake of convenience, I use "N" instead of Frege's special concept-script sign for the cardinality operator. ...
In this three-part essay, I investigate Frege’s platonist and anti-creationist position in Grundgesetze der Arithmetik and to some extent also in Die Grundlagen der Arithmetik. In section 1, I analyze his arithmetical and logical platonism in Grundgesetze. I argue that the reference-fixing strategy for value-range names – and indirectly also for numerical singular terms – that Frege pursues in Grundgesetze I gives rise to a conflict with the supposed mind- and language-independent existence of numbers and logical objects in general. In sections 2 and 3, I discuss the non-creativity of Frege’s definitions in Grundgesetze and the case of what I call weakly creative definitions. In Part II of this essay, I first deal with Stolz’s and Dedekind’s (intended) creation of numbers. In what follows, I focus on Grundgesetze II, §146, where Frege considers a potential creationist charge in relation to the stipulation that he makes in Grundgesetze I, §3 with the purpose of partially fixing the references of value-range names. I place equal emphasis on the related twin stipulations that he makes in Grundgesetze I, §10. In §10, Frege identifies the truth-values with their unit classes in order to fix the references of value-range names (almost) completely. He does so in a piecemeal fashion. Although in Grundgesetze II, §146 Frege refers also to Grundgesetze I, §9 and §10 in this connection, he does not explain why he thinks that the transsortal identifications in §10 and also the stipulation that he makes in §9 regarding the value-range notation may give rise to a creationist charge in addition to or in connection with the stipulation in §3, and if so, how he would have responded to it. The two main issues that I discuss in Part II are: (a) Has Frege created value-ranges in general in Grundgesetze I, §3? (b) Has he created the unit classes of the True and the False in §10? In part III, I discuss, inter alia, the question of whether in developing the whole wealth of objects and functions that arithmetic deals with from the primitive functions of Grundgesetze by applying the formation rules Frege creates special value-ranges and special functions. This procedure is fundamentally different from the reference-fixing strategy regarding value-range names that Frege pursues in Grundgesetze I, §3, §10–§12. It is just another aspect of his anti-creationism. In Grundgesetze II, §147, Frege makes a concession to an imagined creationist opponent which might suggest that he was fully convinced neither of the defensibility of his anti-creationist position regarding the syntactic development of the subject matter of arithmetic nor of his actual defence in §146 of the non-creativity of the introduction of value-ranges via logical abstraction in Grundgesetze I, §3 and the twin stipulations in §10. I argue that not only in Grundgesetze II, §146 but also in Grundgesetze II, §147 Frege falls short of defending his anti-creationist position. I further argue that on the face of it his creationist rival gains the upper hand in the envisioned debate in more than one respect.
... 15 I employ Frege's symbol "∞ 1 " from Grundlagen for the number Endlos and dispense with the diagonal stroke crossing "0" and "1" which in Grundgesetze Frege uses to distinguish the cardinal numbers 0 and 1 from the numbers 0 and 1; cf. Frege (1903), §157. Likewise for the sake of convenience, I use "N" instead of Frege's special concept-script sign for the cardinality operator. ...
... 15 I employ Frege's symbol "∞ 1 " from Grundlagen for the number Endlos and dispense with the diagonal stroke crossing "0" and "1" which in Grundgesetze Frege uses to distinguish the cardinal numbers 0 and 1 from the numbers 0 and 1; cf. Frege (1903), §157. Likewise for the sake of convenience, I use "N" instead of Frege's special concept-script sign for the cardinality operator. ...
In this three-part essay, I investigate Frege’s platonist and anti-creationist position in Grundgesetze der Arithmetik and to some extent also in Die Grundlagen der Arithmetik. In Sect. 1.1, I analyze his arithmetical and logical platonism in Grundgesetze. I argue that the reference-fixing strategy for value-range names—and indirectly also for numerical singular terms—that Frege pursues in Grundgesetze I gives rise to a conflict with the supposed mind- and language-independent existence of numbers and logical objects in general. In Sect. 1.2 and 1.3, I discuss the non-creativity of Frege’s definitions in Grundgesetze and the case of what I call weakly creative definitions. In Part II of this essay, I first deal with Stolz’s and Dedekind’s (intended) creation of numbers. In what follows, I focus on Grundgesetze II, §146, where Frege considers a potential creationist charge in relation to the stipulation that he makes in Grundgesetze I, §3 with the purpose of partially fixing the references of value-range names. I place equal emphasis on the related twin stipulations that he makes in Grundgesetze I, §10. In §10, Frege identifies the truth-values with their unit classes in order to fix the references of value-range names (almost) completely. He does so in a piecemeal fashion. Although in Grundgesetze II, §146 Frege refers also to Grundgesetze I, §9 and §10 in this connection, he does not explain why he thinks that the transsortal identifications in §10 and also the stipulation that he makes in §9 regarding the value-range notation may give rise to a creationist charge in addition to or in connection with the stipulation in §3, and if so, how he would have responded to it. The two main issues that I discuss in Part II are: (a) Has Frege created value-ranges in general in Grundgesetze I, §3? (b) Has he created the unit classes of the True and the False in §10? In Part III, I discuss, inter alia, the question of whether in developing the whole wealth of objects and functions that arithmetic deals with from the primitive functions of Grundgesetze by applying the formation rules Frege creates special value-ranges and special functions. This procedure is fundamentally different from the reference-fixing strategy regarding value-range names that Frege pursues in Grundgesetze I, §3, §10–12. It is just another aspect of his anti-creationism. In Grundgesetze II, §147, Frege makes a concession to an imagined creationist opponent which might suggest that he was fully convinced neither of the defensibility of his anti-creationist position regarding the syntactic development of the subject matter of arithmetic nor of his actual defence in §146 of the non-creativity of the introduction of value-ranges via logical abstraction in Grundgesetze I, §3 and the twin stipulations in §10. I argue that not only in Grundgesetze II, §146 but also in Grundgesetze II, §147 Frege falls short of defending his anti-creationist position. I further argue that on the face of it his creationist rival gains the upper hand in the envisioned debate in more than one respect.
... В исходной версии Записи в понятиях 1879 года (Frege, 2000a) суждение обозначает утверждение наличия некоторого положения дел Φ: «имеет место Φ». Позднее, в работе «Основные законы арифметики» 1893 года (Frege, 1966), Фреге существенно меняет свой подход, и суждение теперь понимается как акт утверждения истинности или ложности предложения. Рассел и Уайтхед в Principia Mathematica переводят Urteil на английский язык как assertion, также понимая под ним утверждение истинности или ложности. ...
Martin-Löf’s type theory stems simultaneously from Frege and Russell’s logic-ontological ideas and Husserl’s phenomenology. The article examines this intermediate status of type theory using as examples Martin-Löf ’s syntactical-semantic method and the role of evidence and canonical objects in his approach. Martin-Löf borrows the syntactical-semantic method from Frege and extends it drawing on Husserl’s theory of meaning. In type theory this method leads to the identity (isomorphism) of syntax and semantics (formal logic and formal ontology). Unlike traditional formal logic the type theory is an interpreted system from the very beginning. Being intuitionistic, Martin-Löf ’s theory is based on the notion of proof, not truth. From the meaning theory point of view, it is a variant of proof-theoretic semantics (Gentzen, Prawitz, Dummett) which understands meaning as an object constructed according to certain rules. So understood, the proof is based on evidence, which allows us to associate it with the theory of intentionality by Husserl. The article compares Martin-Löf ’s type theory with Husserl’s intentionality theory, especially with the latter’s noematic component. We may consider type-theoretical rules for constructing objects and operating with them as a concretization and formalization of the phenomenological notion of noema. Both are explications of the more general concept of meaning. The article discusses the interrelation between notions of sense and meaning (Sinn, Bedeutung) in Frege, Husserl and Martin-Löf. This reveals the uncertainty of Martin-Löf ’s position in relation to meaning theories of Frege and Husserl.
... However, the modern development of modal logic can be traced back to the work of philosopher and logician C.I. Lewis in the early twentieth century [19]. • Predicate Logic: Predicate logic has its origins in the work of Gottlob Frege in the late nineteenth century, who developed a formal system for representing logical relationships between different objects and properties [6]. The system was further developed by other logicians, such as Bertrand Russell and Alfred North Whitehead [27]. ...
The history of fuzzy sets can be traced back to the work of mathematician Lotfi A. Zadeh in the 1960s. Zadeh’s idea of “fuzzy sets” was a departure from the classical “crisp” set theory, which assumed that an element either belongs or does not belong to a set. Fuzzy set theory, on the other hand, allows for degrees of membership, which better captures the ambiguity and vagueness present in many real-world applications. Since its inception, fuzzy set theory has evolved and been applied in a wide range of fields, including engineering, economics, medicine, and more. Today, the fuzzy set theory continues to inspire new research and innovation in the field of artificial intelligence and beyond.
... It is modern defenders include the likes of (Tahko, 2014(Tahko, , 2021McSweeney, 2018McSweeney, , 2019Maddy, 2007Maddy, , 2012. Historic defenders include but are certainly not limited to Frege (1879Frege ( , 1893, the early Wittgenstein (1922) and Quine (1960Quine ( , 1981. ...
This paper outlines three broad ways one might think about logical correctness: the Realist approach, the One-Language approach and my own Neo-Carnapian view. Although the realist and one-language views have dominated the philosophy of logic in recent years, I argue against them, favouring of the Neo-Carnapian approach.
So far, the method of supervaluations has been mainly employed to define a non-gradable property of sentences, supertruth, in order to provide an analysis of truth. But it is also possible, and arguably at least as plausible, to define a gradable property of sentences along the same lines. This paper presents a supervaluationist semantics that is quantitative rather than qualitative. As will be shown, there are at least two distinct interpretations of the semantics — one alethic, the other epistemic — which can coherently be adopted to address key issues such as vagueness and future contingents.
In Section 2, I analyze Frege's principle of logical and notational parsimony in his opus magnum Grundgesetze der Arithmetik (vol I, 1893, vol. II, 1903). I argue inter alia that in order to carry out the proofs of the more important theorems of cardinal arithmetic and real analysis in Grundgesetze Frege's identification of the truth-values the True and the False with their unit classes in Grundgesetze I, §10 need not be raised to the lofty status of an axiom. Frege refrains from doing this but does not provide any reason for his restraint. In Section 3, I argue that he considered the primitive function-name "ξ = ζ" indispensable in pursuit of his logicist project. I close with remarks on the nature of identity. I suggest that there is no need to interpret identity in a non-standard fashion in order to render it logically palatable and scientifically respectable.
Abstractionism in the philosophy of mathematics aims at deriving large fragments of mathematics by combining abstraction principles (i.e. the abstract objects , are identical if, and only if, an equivalence relation holds between the entities ) with logic. Still, as highlighted in work on the semantics for relevant logics, there are different ways theories might be combined. In exactly what ways must logic and abstraction be combined in order to get interesting mathematics? In this paper, we investigate the matter by deriving the axioms of second-order Peano Arithmetic from Frege’s Basic Law V (the extension of F is identical with the extension of G if, and only if, F and G are extensionally equivalent) in the presence of a relevant higher-order logic. The results are interesting. Not only must we take on logic as true, and not only must we apply our logic to abstraction principles, but also we have to apply our theory of abstraction back to the logic in order to arrive at arithmetic. Thus, what Abstractionism gives us is not simply what we get from abstraction via logic, but also what we get from logic via abstraction.
This chapter compares Tarski’s paper on the establishment of scientific semantics to his other published summaries of his ideas and results on the concept of truth, explains what led Tarski to embark on the project of defining truth, and reveals the basis of Tarski’s choice of the title. It discusses each paragraph. Among other things, it explains what Tarski meant by ‘semantics’ and why he called his semantics ‘scientific’. It explains why Tarski took the structural description of a “formalized language” to include its rules of definition and points out how following them could guarantee that a definition was formally correct. It explains how Tarski’s conception of the material adequacy of a way of introducing a semantic concept into the meta-theory of a formalized language is rooted in a mentalistic, “intuitionist” conception of meaning. It documents how objections of logicians in Vienna to appealing to intuition explain Tarski’s studious avoidance of such appeals in the German-language (but not the Polish-language) version. It explains the background to Tarski’s two substantive objections to introducing a semantic concept axiomatically and points out the weakness of its two novel objections. It notes that Tarski does not even sketch a proof of his main thesis about the definability of semantic concepts.
In this paper we will look at Frege's work under a metaphysical lens aiming to identify crucial concepts to be employed in the treatment of objectivity. After presenting the principle of epistemological simplicity and its relation to objectivity through a discussion about naive metaphysics, we analyze how meaning and truth can be employed to understand apparent reality. On the central matter of the nature of truth, we will argue that, under a naive metaphysical perspective, its indefinability is a desirable feature and that it could be advantageous to study truth in terms of an internal and an external versions of judgment. We conclude with some brief examples of applications of these notions.
Husserl's Philosophy of Mathematical Practice explores the applicability of the phenomenological method to philosophy of mathematical practice. The first section elaborates on Husserl's own understanding of the method of radical sense-investigation (Besinnung), with which he thought the mathematics of his time should be approached. The second section shows how Husserl himself practiced it, tracking both constructive and platonistic features in mathematical practice. Finally, the third section situates Husserlian phenomenology within the contemporary philosophy of mathematical practice, where the examined styles are more diverse. Husserl's phenomenology is presented as a method, not a fixed doctrine, applicable to study and unify philosophy of mathematical practice and the metaphysics implied in it. In so doing, this Element develops Husserl's philosophy of mathematical practice as a species of Kantian critical philosophy and asks after the conditions of possibility of social and self-critical mathematical practices.
This paper explores the criticism of psychological reductionism from the perspective of Husserl's phenomenology, with a focus on the relevance of these critiques for contemporary thought. We analyze Husserl's critique of psychologism, emphasizing his proposal for a rigorous, anti-reductionist philosophy. Additionally, the paper examines Hering’s contributions to the phenomenology of religious experience and their implications for modern theological crises. Through a comparative analysis of both thinkers, we aim to demonstrate how phenomenology provides valuable epistemological tools to overcome the challenges of subjectivism and relativism in the study of knowledge and religious phenomena.
Any attempt at the unitary reconstruction of science as a whole, or at least of a set of sciences presumed to be representative of it, pursues a certain ideal of unity of scientific knowledge. One is interested here in the ideal of logical unity (alias, logical universalism), aiming at the reconstruction of all the sciences concerned within the same logical framework, and in the more demanding ideal of a systematic unity (alias, systematic universalism), aiming at the reconstruction of all of these sciences within one and the same system. In his famous Wahrheitsbegriff of 1935 (postscript not included), Tarski, the inventor of semantics as a science, proposed a criterion, the so-called convention T, which any definition of the notion of truth should meet in order to be adequate, and he constructed a semantic definition of this notion meeting convention T for some explicitly given systems taken as examples, fitting into a logical framework which he considered universal, viz. the simple extensional theory of types. But, he also brought to light the limits of the exercise by demonstrating negative theorems, which implied that, for no system fitting into the logical framework and including it, is a definition of the notion of truth meeting convention T possible—be this within this system itself or, more broadly, within the chosen logical framework—, without the system in question falling, in one way or another, prey to the Liar. And, moreover, there was every reason to think that one could not be content with an axiomatic explication of the notion of truth. Under these conditions, taking for granted that the Tarskian explication of the notion of truth for science should be on the reconstruction agenda, it would be all over for logical universalism (and a fortiori for any universalism more demanding than it), and Tarski in fact gave up on it, though without saying so, in his postscript to the Wahrheitsbegriff. In the present article, it is preferred to go back to the program of the Wahrheitsbegriff with a new logical framework, viz. Zermelo-Fraenkel set theory with axiom of choice, and to challenge convention T in favor of new conventions on adequacy. It then turns out to be possible to save logical universalism and to take a few steps forward on the path to systematic universalism.
The lecture spells out the difference between the validity of inference (‐figure)s and validity applied to demonstrations (‘proof acts’). The latter notion is not an ordinary characterizing one; in Brentano's terminology it is a modifying one. A demonstration lacking validity is not a real demonstration, just as a false friend is no true friend. Throughout, the treatment makes crucial use of an epistemological perspective that is cast in the first person. Furthermore, the difference between (logical) consequence among propositions and the validity of inference from judgement to judgement is explained. Particular attention is paid to alleged issues of circularity in the definition of the validity of inference, and to the ‘explosion’ validity of inference from contradictory premisses. Drawing upon a version of the dialogical framework of Per Martin‐Löf, namely, ‘When I say Therefore , I give others my permission to assert the conclusion’, while stressing also the importance of the first person perspective, both difficulties can be neutralized.
La llamada “polémica en torno al psicologismo” (Psychologismusstreit) se compone en realidad de una serie de polémicas singulares que no poseen necesariamente vínculo factual entre sí. La discusión entre Herbart y Beneke es una de ellas, poseyendo particular importancia por encontrarse en el comienzo del conflicto. En ella el psicologista Beneke percibe con agudeza los puntos débiles de la posición herbartiana e inicia una línea de argumentación que, con variaciones, se mantendrá constante en el psicologismo y dará sentido a que el último acto del drama llamado Psychologismusstreit sea dado por la fenomenología husserliana.
Gottlob Frege rozlišuje mezi smyslem a významem výrazů, přičemž smyslem rozumí to, co je výrazem vyjadřováno, významem pak to, co je jím označováno. Toto rozlišení se týká jak singulárních, tak obecných výrazů, ovšem Fregovy rozbory týkající se významu obecných výrazů jsou všeobecně méně známy. Chci zdůraznit skutečnost, že Frege považuje za význam obecných výrazů pojem, který chápe jako tzv. nenasycenou funkci, nikoli jako množinu nebo zobrazení. To mu umožňuje zachovat některé podstatné intuice týkající se obecnin – můžeme např. mít obecný pojem (znát nějakou vlastnost či relaci), aniž bychom věděli, jaké předměty pod něj spadají. Zároveň se snažím ukázat, že není jasné, nakolik Frege skutečně zastával tezi, která mu bývá obvykle přičítána, totiž že identita pojmů je dána jejich koextenzivitou. Gottlob Frege distinguished between sense and reference of expressions. 'Sense' according to him is that what the expression means, 'reference' is that what is designated by it. This distinction concerns both singular and general terms but Frege’s analyses focused on general terms are much less known. It is crucial, I argue, to appreciate the importance of the fact that the reference of a general term is, according to Frege, a concept – identified with an unsaturated function in the Fregean sense, i.e. not a set, and not a mapping. In this way Frege can keep some important intuitions about universals – for instance that we can have a general concept (know a property or a relation) even if we don’t know which objects fall under it. I also try to show that it is not clear whether Frege really defended the thesis usually attributed to him that the identity of concepts is guaranteed solely by their coextensivity.
Gottlob Frege rozlišuje mezi smyslem a významem výrazů, přičemž smyslem rozumí to, co je výrazem vyjadřováno, významem pak to, co je jím označováno. Toto rozlišení se týká jak singulárních, tak obecných výrazů, ovšem Fregovy rozbory týkající se významu obecných výrazů jsou všeobecně méně známy. Chci zdůraznit skutečnost, že Frege považuje za význam obecných výrazů pojem, který chápe jako tzv. nenasycenou funkci, nikoli jako množinu nebo zobrazení. To mu umožňuje zachovat některé podstatné intuice týkající se obecnin – můžeme např. mít obecný pojem (znát nějakou vlastnost či relaci), aniž bychom věděli, jaké předměty pod něj spadají. Zároveň se snažím ukázat, že není jasné, nakolik Frege skutečně zastával tezi, která mu bývá obvykle přičítána, totiž že identita pojmů je dána jejich koextenzivitou. Gottlob Frege distinguished between sense and reference of expressions. 'Sense' according to him is that what the expression means, 'reference' is that what is designated by it. This distinction concerns both singular and general terms but Frege’s analyses focused on general terms are much less known. It is crucial, I argue, to appreciate the importance of the fact that the reference of a general term is, according to Frege, a concept – identified with an unsaturated function in the Fregean sense, i.e. not a set, and not a mapping. In this way Frege can keep some important intuitions about universals – for instance that we can have a general concept (know a property or a relation) even if we don’t know which objects fall under it. I also try to show that it is not clear whether Frege really defended the thesis usually attributed to him that the identity of concepts is guaranteed solely by their coextensivity.
Tato práce se podrobně věnuje způsobu, jakým David Hilbert (1862–1943) pojal aritmetizaci geometrie v knize Grundlagen der Geometrie z roku 1899. Nejprve stručně představíme Hilbertovy předchůdce z téže doby, kteří buď po změnách v založení geometrie volali, nebo je již sami prostřednictvím axiomaticko-deduktivní metody zapracovali. Neopomeneme přitom, co dílu předcházelo v dřívějších Hilbertových přednáškách. Následně se pokusíme nastínit obsah prvních dvou kapitol knihy a vysvětlit dobové i věcné souvislosti, nutné k jejich pochopení. Představíme způsob implicitních definic základních pojmů a vztahů v axiomech, a dále Hilbertovo rozdělení axiomů do skupin, přičemž se zejména zaměříme na axiomy spojitosti v kontextu s otázkou o její bezespornosti. K tomu popíšeme konstrukci aritmetického modelu axiomů geometrie, který Hilbert pro důkaz bezespornosti používá. V závěru se pokusíme nastínit hlavní důvody, které Hilberta k napsání díla vedly, a některé klíčové důsledky jeho pojetí axiomatiky geometrie.
O presente artigo se propõe clarificar o lugar das “Investigações lógicas” husserlianas no Psicologismusstreit primariamente desde o ponto de vista histórico-filosófico, mas sem desconsiderar a perspectiva sistemática.
This paper sheds light on an epistemological dimension of Frege’s “On Sense and Reference.” Under my suggested reading of it, one of its aims is to suggest a picture about propositional knowledge and its production. According to this picture, judgment, which produces propositional knowledge, is identification of the truth-value True with the reference of a given sentence. The propositional knowledge that p, produced by the judgment that p, consists in the knowledge of the identity between the True and the reference of “p.” Judgment as such is a primitive kind of identification. It produces non-propositional knowledge of the identity of the True to which propositional knowledge is reduced.
In this essay, I critically analyze Wittgenstein's dispensation with " = " in a correct concept-script. I argue inter alia (a) that in the Tractatus the alleged pseudo-character of sentences containing " = " or =-sentences remains largely unexplained and propose how it could be explained; (b) that at least in some cases of replacing =-sentences with equivalent identity-sign free sentences the use of the notion of a translation seems inappropiate; (c) that in the Tractatus it remains unclear how identity of the object as that which is expressed by identity of the sign should be understood specifically; (d) that there are =-sentences which have no obvious equivalent in Wittgenstein's novel notation; (e) that Wittgenstein's adherence to (non-relational) identity, although he dispenses with " = ", is probably motivated by his desire to ensure that the expressive power of an identity-sign free concept-script of first-order is on a par with standard first-order logic containing " = ". In the concluding section, I critically discuss some claims in Lampert and Säbel (The Review of Symbolic Logic, 14, 1-21, 2021) and defend Wehmeier's account of pseudo-sentences in the Tractatus (2012) against the objections they raise.
ABSTRACT In this article, I discuss Frege’s conception of analyticity, the nature of abstraction in Die Grundlagen der Arithmetik and the proof of Hume’s Principle.
It is argued that standard music notation pictures Aristotle’s time (time, as Aristotle conceived of it) in a number of important respects, which concern its micro-structure. It is then argued that this allows us to see some features of Aristotle’s time more clearly. Most importantly, Aristotelian instants can be pictured by bar-lines. This allows us to see as how radically devoid of any content Aristotelian instants should be interpreted. Thus, attention to music notation may show why Aristotle was not a presentist.
Existe de Herbart a Husserl una evolución en el antipsicologismo en el sentido de ofrecer una concepción de subjetividad acorde con el mismo. Lotze constituye un momento intermedio en este proceso, en cuanto, por un lado, contra Herbart, da un paso decisivo para superar la idea naturalista de subjetividad, por otro, sin embargo, mantiene con Herbart la validez irrestricta del principio de inmanencia, principio éste que será objeto de la expresa crítica de Frege y Husserl. En este último, como es sabido, la superación definitiva del psicologismo se consuma como superación definitiva del naturalismo.
Non-factive representing is what makes room for truth and falsehood. In the ontologically central aspect of the verb it comes in two forms: allorepresenting (saying-that), and autorepresenting (taking-that). Each form relates thinkers to thinkables in its proprietary way. Autorepresenting invites a certain sort of misunderstanding. It may seem to call for enabling in a particular determinate way. Just here psychologism despite oneself may strike. Allorepresenting rests on capacities of a different sort. It relates itself, and thereby its author, to a thinkable in what seems at first sight a manifestly different way than does autrorepresenting, one perhaps less likely to invite psychologism. In doing this it throws light on autorepresenting which, if rightly understood, I suggest, may undo the compulsion to psychologise this last. Armed with this tool I suggest a better picture of what autorepresenting is.
We give a brief historical sketch of the clash of dualism vs. naturalism and then analyze the argument that Gödel’s incompleteness theorems support dualism by implying human-machine non-equivalence. We prove that this implication is not valid. Instead, we give a correct implication.
In the context of the genealogy of classical logic, the philosophical foundations of its renewal and mathematization, which took place starting from the middle of the 19th century, are being considered. Attention is paid to the philosophical foundations and grounds for updating logic renewal and applying to it the basic principles of arithmetic. In terms of the genealogy of logic and the process of its reformation, the role of mathematical research relevant at that time, which became an important basis and prerequisite in this process, is analyzed. The role of G. Frege's activities in the direction of the mathematization of logic and the creation of systems for counting statements and predicates is analyzed and determined. Classical key differences between syllogistics and modern types of predicate logics are studied. On the basis of these provisions, conclusions are formed that allow us to understand the main philosophical foundations in the creation of mathematical logic, and a consistent genealogy of these foundations is built.
By playing a crucial role in settling open issues in the philosophical debate about
logical consequence, logical evidence has become the holy grail of inquirers investigating the domain of logic. However, despite its indispensable role in this endeavor, logical evidence has retained an aura of mystery. Indeed, there seems to be a great disharmony in conceiving the correct nature and scope of logical evidence among philosophers. In this paper, I examine four widespread conceptions of logical evidence to argue that all should be reconsidered. First, I argue that logical apriorists are more tolerant of logical evidence than empiricists. Second, I argue that evidence for logic should not be read out of natural language. Third, I argue that if logical intuitions are to count as logical evidence, then their evidential content must not be propositional. Finally, I argue that the empiricist proposal of treating experts’ judgments as evidence suffers from the same problems as the rationalist conception.
Para determinar o lugar que corresponde a determinado autor no processo denominado “polêmica em torno do psicologismo” (Psychologismusstreit), é necessário ir além de suas afirmações expressas sobre a relação entre psicologia e lógica e atender às suas afirmações sobre a subjetividade. Nas linhas que se seguem, este princípio geral é aplicado a Herbart, o que permitirá determinar o lugar que ocupa na polémica e esclarecer a relação em que se encontra com outros autores mais conhecidos e paradigmáticos da mesma, como é o caso de Frege.
Frege's fictitious names possess meaning but lack denotation. Both these names and the sentences containing them are deemed fictitious. Since any proper name can potentially refer to an imaginary entity, it is crucial to consider the speaker's intention. When making a statement, the speaker may refer to the real or the imaginary. In the latter case, the thought cannot be explicitly expressed, and consequently, denotation cannot be reached. In Frege's framework, fictional thoughts hold little significance for decision-making and actions. Therefore, we consistently seek to discern whether the discourse pertains to the real or the imaginary. To make this knowledge accessible, it must be incorporated into the content of a sentence, effectively becoming a thought. However, not every statement expresses a thought, even if it conforms to the structure of a sentence. I will now elucidate three intensionalization procedures that Frege proposes for constructing a sentence that expresses a thought, even if certain components within it lack denotation: the articulation of a naming relation, the formulation of a propositional attitude of intention, and the formulation of a propositional attitude that conveys a metafictional context. Through these methods, the speaker's intent to indicate a real or fictional object becomes a constituent of thought, i. e., the sense of the sentence. Fictions themselves become components of thought when they are found in an indirect context, wherein their sense serves as their denotation. When considered independently, the sense of a proper noun is an entity with a parameter that acquires a value in the specific situation where the name is employed by a particular speaker. Frege's foundational concepts are juxtaposed with certain aspects of Aristotle and Leibniz's doctrines.
Um dos feitos mais notáveis alcançados por Frege no âmbito do seu trabalho lógico-filosófico foi o de ter proposto uma visão inteiramente nova do conceito. Como aqui se mostrará, é esta visão do conceito, em que o mesmo é tomado como uma função de um tipo específico, que, por sua vez, se encontra subjacente aos sucessos e insucessos da Lógica de Frege e do seu programa logicista.
En este artículo, el autor analiza la relación entre dos destacados filósofos del siglo XX en Europa y Gran Bretaña: Ludwig Wittgenstein y Bertrand Russell. De acuerdo con una gran cantidad de correspondencia disponible hoy en día, podemos reconstruir no solo el entorno de pensamiento en Cambridge a principios y la primera mitad del siglo XX, sino también encontrar motivos subjetivos y personales para los cambios de relación entre pensadores, malentendidos entre a ellos. Tal tipo de reconstrucción biográfico-histórica no interfiere, sino que ayuda a comprender el origen, desarrollo y crítica de las ideas y teorías filosóficas. En este contexto, la relación personal entre Russell y Wittgenstein, su amistad durante más de 30 años y la ayuda mutua, tanto intelectual como empresarial, jugaron un papel importante en el desarrollo y formación de ambos filósofos como filósofos. Incluso los desacuerdos mutuos y la crítica de ideas a lo largo del tiempo les ayudaron a ver las deficiencias y, a veces, incluso el fracaso total de sus tesis y declaraciones. Primero, el autor describe en detalle el papel de Russell en la vida de Wittgenstein: primer encuentro e inspiración para hacer filosofía, apoyo en los estudios, asistencia en la publicación del primer libro de Wittgenstein, apoyo y facilitación para regresar a Cambridge en 1929, obtener el doctorado y recibiendo la beca del Trinity College. En segundo lugar, el autor considera los puntos básicos y fundamentales de los desacuerdos filosóficos entre dos filósofos y la crítica de Wittgenstein a las ideas de Russell.
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