The Structure of Frege's Thoughts

Fregean thoughts (i.e. the senses of assertoric sentences) are structured entities because they are composed of simpler senses that are somehow ordered and interconnected. The constituent senses form a unity because some of them are ‘saturated’ and some ‘unsaturated’. This paper shows that Frege's explanation of the structure of thoughts, which is based on the ‘saturated/unsaturated’ distinction, is by no means sufficient because it permits what I call ‘wild analyses’, which have certain unwelcome consequences. Wild analyses are made possible because any ‘unsaturated’ sense that is a mode of presentation of a concept together with any ‘saturated’ sense forms a thought. The reason is that any concept can be applied to any object (which is presented by a ‘saturated’ sense). This stems from the fact that Frege was willing to admit only total functions. It is also briefly suggested what should be done to block wild analyses.     

Davidson,correspondence truth and the frege-Gödel—church argument

This paper argues for a conditional claim concerning a famous argument—developed by Church in elucidation of some remarks by Frege to the effect that the bedeutung of a sentence is the sentence’s truth-value—the Frege–Gödel–Church argument, or FGC for short. The point we make is this :if, and just to the extent that, Arthur Smullyan’s argument against Quine's use of FGC is sound, then essentially the same rejoinder disposes also of Davidson's use of FGC against ‘correspondence’ theories of truth. We thus dispute a contention by Professor Davidson that it is coherent to accept that Smullyan’s rejoinder takes away the force of Quine’s version of FGC, while still consistently using FGC to establish that if true sentences (or utterances) correspond to anything, they all correspond to the same thing. We show that the differences between the cases discussed by Smullyan and Davidson’s version of FGC on which Davidson relies for his contention are irrelevant to the point under dispute     

A New–old Characterisation of Logical Knowledge

We seek means of distinguishing logical knowledge from other kinds of knowledge, especially mathematics. The attempt is restricted to classical two-valued logic and assumes that the basic notion in logic is the proposition. First, we explain the distinction between the parts and the moments of a whole, and theories of ‘sortal terms’, two theories that will feature prominently. Second, we propose that logic comprises four ‘momental sectors’: the propositional and the functional calculi, the calculus of asserted propositions, and rules for (in)valid deduction, inference or substitution. Third, we elaborate on two neglected features of logic: the various modes of negating some part(s) of a proposition R, not only its ‘external’ negation not-R; and the assertion of R in the pair of propositions ‘it is (un)true that R’ belonging to the neglected logic of asserted propositions, which is usually left unstated. We also address the overlooked task of testing the asserted truth-value of R. Fourth, we locate logic among other foundational studies: set theory and other theories of collections, metamathematics, axiomatisation, definitions, model theory, and abstract and operator algebras. Fifth, we test this characterisation in two important contexts: the formulation of some logical paradoxes, especially the propositional ones; and indirect proof-methods, especially that by contradiction. The outcomes differ for asserted propositions from those for unasserted ones. Finally, we reflect upon self-referring self-reference, and on the relationships between logical and mathematical knowledge. A subject index is appended.     

Formality of logic and Frege’s Begriffsschrift

This paper challenges a standard interpretation according to which Frege’s conception of logic (early and late) is at odds with the contemporary one, because on the latter’s view logic is formal, while on Frege’s view it is not, given that logic’s subject matter is reality’s most general features. I argue that Frege – in Begriffsschrift – retained the idea that logic is formal; Frege sees logic as providing the ‘logical cement’ that ties up together the contentful concepts of specific sciences, not the most general truths. Finally, I discuss how Frege conceives of the application of Begriffsschrift, and of its status as a ‘lingua characteristica’.     

非严格弗雷格自由逻辑

莱曼(S.Lehmann)的1994年的论文从三值逻辑角度处理语言中的空词项。弗雷格认为句子的所指是真值,包含空词项的句子缺乏所指,而只有涵义,那么这样的句子不真也不假。莱曼的文章刻画了弗雷格这一思想,但有两点不尽人意之处:其一、对量词进行特殊处理,使得量词公式是二值的;其二,把包含空词项的复合句一概归于第三值。针对上述两点,本文以一度衍推系统为基础,在语义上,以偏解释函数为基础构造了一个非严格弗雷格中性自由逻辑表列系统,使得量词公式是三值的,同时使得一部分包含空词项的复合公式或真或假。     

Existential Graphs: What a Diagrammatic Logic of Cognition Might Look Like

This paper examines the contemporary philosophical and cognitive relevance of Charles Peirce's diagrammatic logic of existential graphs (EGs), the ‘moving pictures of thought’. The first part brings to the fore some hitherto unknown details about the reception of EGs in the early 1900s that took place amidst the emergence of modern conceptions of symbolic logic. In the second part, philosophical aspects of EGs and their contributions to contemporary logical theory are pointed out, including the relationship between iconic logic and images, the problem of the meaning of logical constants, the cognitive economy of iconic logic, the failure of the Frege–Russell thesis, and the failure of the Language of Thought hypothesis.     

A CRITIQUE OF FREGE ON COMMON NOUNS

Frege analyzed the grammatical subject‐term ‘S’ in quantified subject‐predicate sentences, ‘q S are P’, as being logically predicative. This is in contrast to Aristotelian Logic, according to which it is a logical subject‐term, like the proper name ‘a’ in ‘a is P’– albeit a plural one, designating many particulars. I show that Frege’s arguments for his analysis are unsound, and explain how he was misled to his position by the mathematical concept of function. If common nouns in this grammatical subject position are indeed logical subject‐terms, this should require a thorough reevaluation of the adequacy of Frege’s predicate calculus as a tool for the analysis of the logic and semantics of natural language.     

Understanding Frege’s notion of presupposition

Why did Frege offer only proper names as examples of presupposition triggers? Some scholars claim that Frege simply did not care about the full range of presuppositional phenomena. This paper argues, in contrast, that he had good reasons for employing an extremely narrow notion of ‘Voraussetzung’. On Frege’s view, many devices that are now construed as presupposition triggers either express several thoughts at once or merely ‘illuminate’ a thought in a particular way. Fregean presuppositions, in contrast, are essentially tied to names.

     

超越弗雷格的“第三域”神话

<正>如达米特所指出的:"弗雷格关于思想及其构成涵义的看法是神话式的。这些恒久不变的实体居住在‘第三域’(the third realm),后者既不同于物理世界,也不同于任何经验主体的内心世界……。     

Freges Erläuterung des Urteils

According to Frege the notion of judgement cannot be defined, but must be explicated by referring to natural language. Frege uses a subjective and an objective notion of judgement: on the one hand a judgement is an inner mental process, on the other hand judging is explained as advancing from the thought to the truth-value. The subjective notion of judgement logically depends on the objective conception of a judgement, but nevertheless remains indispensable. To clarify the objective notion of judgement, the relation of thought and truth must be explicated by using Frege's theory of sense and reference. Only an intersubjective and action-guiding knowledge of the way in which the truth-value of a sentence is determined can make it possible to advance from the thought and the understanding of a sentence to the acknowledgement of its truth. Since by this advance a sentence receives a cognitive value, Frege's theory of sense and reference makes a contribution not only to semantics but also to epistemology.     

Traditional Logic,Modern Logic and Natural Language

In a recent paper Johan van Benthem reviews earlier work done by himself and colleagues on ‘natural logic’. His paper makes a number of challenging comments on the relationships between traditional logic, modern logic and natural logic. I respond to his challenge, by drawing what I think are the most significant lines dividing traditional logic from modern. The leading difference is in the way logic is expected to be used for checking arguments. For traditionals the checking is local, i.e. separately for each inference step. Between inference steps, several kinds of paraphrasing are allowed. Today we formalise globally: we choose a symbolisation that works for the entire argument, and thus we eliminate intuitive steps and changes of viewpoint during the argument. Frege and Peano recast the logical rules so as to make this possible. I comment also on the traditional assumption that logical processing takes place at the top syntactic level, and I question Johan’s view that natural logic is ‘natural’.     

Peter Winch on ‘Aristotelian’ and ‘Socratic’ Reasoning

Peter Winch often returned to questions about the nature of logic. In the context of his work on Wittgenstein and political philosophy in the 1990s, Winch described a contrast between ‘Aristotelian’ and ‘Socratic’ reasoning. Aristotelian conceptions of reasoning, attributed to Frege and Russell, would see logic as a formal science and rationality as consistency with pre‐existent rules of inference. The Socratic conception, attributed to Wittgenstein, understands rational argument as a form of socially embedded dialogue that involves moral relationships and a dimension of depth. Rational persuasion may also involve use of persuasive images and examples.     

语义预设的合理性辩护

弗雷格是第一个注意到命题的预设为假会影响到命题真值判定的哲学家。对这种因预设为假而形成的真值空隙,弗雷格、罗素和斯特劳森提出了不同的处理方式,由此奠定了预设的语义解释的基础。但预设的语义解释一直面临着种种质疑。这些质疑有说服力吗?应该放弃预设的语义解释吗?本文将对此进行分析,揭示这些质疑中所存在的问题,从而对预设的语义解释的合理性进行辩护。一、预设为假与真值空隙第一个把预设作为逻辑概念加以讨论的哲学家是弗雷格,其预设思想可以归纳为四点:第一,预     

Spinoza: une lecture d'Aristote

Does (affirmative) judgement have a logical dual, negative judgement? Whether there is such a logical dualism was hotly debated at the beginning of the twentieth century. Frege argued in ‘Negation’ (1918/9) that logic can dispense with negative judgement. Frege's arguments shaped the views of later generations of analytic philosophers, but they will not have convinced such opponents as Brentano or Windelband. These philosophers believed in negative judgement for psychological, not logical, reasons. Reinach's ‘On the Theory of Negative Judgement’ (1911) spoke to the concerns of these philosophers. While Frege took the distinction between affirmative and negative judgement to be logically redundant, Reinach argued that it is the result of confusing judgement with a different mental act. In this article, I present Reinach's arguments against the ‘old logical dualism’ in context, analyse them and discuss Reinach's innovative use of the notion of focus in the theory of judgement. Recently, there has been a revival of the view that sentential negation is grounded in a prior mental act of rejection. In the final section, I argue that Reinach's analysis of rejection poses a challenge for the revivalists.     

Unity and the Frege–Geach problem

The problem of the unity of the proposition asks what binds together the constituents of a proposition into a fully formed proposition that provides truth conditions for the assertoric sentence that expresses it, rather than merely a set of objects. Hanks’ solution is to reject the traditional distinction between content and force. If his theory is successful, then there is a plausible extension of it that readily solves the Frege–Geach problem for normative propositions. Unfortunately Hanks’ theory isn’t successful, but it does point to significant connections between expressivism, unity, and embedding.     

Lotze and frege: The dating of the ’Kernsätze’ ∗

Michael Dummett has shown that the fragment ‘17 Kernsätze zur Logik’ is evidence that Frege knew Lotze's Logik Dummett’s dating of this fragment prior to 1879, however, must be rejected.The present paper shows that there are other articles of Frege’s which bear clear traces of Lotze's LogikFirst of all, the expressions Vorstellungsverlauf from ‘Über die wissenschaftliche Berechtigung einer Begriffsschrift’, and veranlassenden Ursachen, from ‘Logik’, certainly are borrowed from Lotze.Second, there are links between ‘Booles rechnende Logik und die Begriffsschrift’ and Lotze's Logik. Furthermore, it is shown that Frege’s ‘Kernsätze’, the ‘Dialog mit Pünjer über Existenz’, and his ‘Logik’ are intimately connected.All of this indicates that these texts were written in roughly the same period, namely the early 1880s.Conclusive evidence for this is that the terms Vorstellung and Vorstellungsverbindung are used indiscriminately in both a psychological and a logical sense in the ‘Begriffsschrift’, a fact which contradicts the ‘Kernsätze’     

Logical Foundations and Kant's Principles of Formal Logic

The abstract status of Kant's account of his ‘general logic’ is explained in comparison with Gödel's general definition of a formal logical system and reflections on ‘abstract’ (‘absolute’) concepts. Thereafter, an informal reconstruction of Kant's general logic is given from the aspect of the principles of contradiction, of sufficient reason, and of excluded middle. It is shown that Kant's composition of logic consists in a gradual strengthening of logical principles, starting from a weak principle of contradiction that tolerates a sort of contradictions in predication, and then proceeding to the (constructive) principle of sufficient reason, and to a classical-like logic, which includes the principle of excluded middle. A first-order formalisation is applied to this reconstruction, which reveals implicit modalities in Kant's account of logic, and confirms the implementability of Kant's logic into a sound and complete formal system.     

The Role of Structural Reasoning in the Genesis of Graph Theory

The concept of quantity (Größe) plays a key role in Frege's theory of real numbers. Typically enough, he refers to this theory as ‘theory of quantity’ (‘Größenlehre’) in the second volume of his opus magnum Grundgesetze der Arithmetik (Frege 1903). In this essay, I deal, in a critical way, with Frege's treatment of the concept of quantity and his approach to analysis from the beginning of his academic career until Frege 1903. I begin with a few introductory remarks. In Section 2, I first analyze Frege's use of the term ‘source of knowledge’ (‘Erkenntnisquelle’) with particular emphasis on the logical source of knowledge. The analysis includes a brief comparison between Frege and Kant's conceptions of logic and the logical source of knowledge. In a second step, I examine Frege's theory of quantity in Rechnungsmethoden, die sich auf eine Erweiterung des Größenbegriffes gründen (Frege 1874). Section 3 contains a couple of critical observations on Frege's comments on Hankel's theory of real numbers in Die Grundlagen der Arithmetik (Frege 1884). In Section 4, I consider Frege's discussion of the concept of quantity in Frege 1903. Section 5 is devoted to Cantor's theory of irrational numbers and the critique deployed by Frege. In Section 6, I return to Frege's own constructive treatment of analysis in Frege 1903 and succinctly describe what I take to be the quintessence of his account.     

On the Reduction of Necessity to Essence

In his influential paper ‘‘Essence and Modality’’, Kit Fine argues that no account of essence framed in terms of metaphysical necessity is possible, and that it is rather metaphysical necessity which is to be understood in terms of essence. On his account, the concept of essence is primitive, and for a proposition to be metaphysically necessary is for it to be true in virtue of the nature of all things. Fine also proposes a reduction of conceptual and logical necessity in the same vein: a conceptual necessity is a proposition true in virtue of the nature of all concepts, and a logical necessity a proposition true in virtue of the nature of all logical concepts. I argue that the plausibility of Fine's view crucially requires that certain apparent explanatory links between essentialist facts be admitted and accounted for, and I make a suggestion about how this can be done. I then argue against the reductions of conceptual and logical necessity proposed by Fine and suggest alternative reductions, which remain nevertheless Finean in spirit.