The anatytic conception of truth and the foundations of arithmetic

2000 ◽  
Vol 65 (1) ◽  
pp. 33-102 ◽  
Author(s):  
Peter Apostoli
Keyword(s):  
Philosophy Of Mathematics ◽  
Logical Truth ◽  
Numerical Identity ◽  
Hume’S Principle ◽  
Contextual Definition ◽  
Definition Of ◽  
Axiomatic Set Theory ◽  
In Virtue Of ◽  
Second Order Logic ◽  
Foundations Of Arithmetic

Until very recently, it was thought that there couldn't be any current interest in logicism as a philosophy of mathematics. Indeed, there is an old argument one often finds that logicism is a simple nonstarter just in virtue of the fact that if it were a logical truth that there are infinitely many natural numbers, then this would be in conflict with the existence of finite models. It is certainly true that from the perspective of model theory, arithmetic cannot be a part of logic. However, it is equally true that model theory's reliance on a background of axiomatic set theory renders it unable to match Frege's Theorem, the derivation within second order logic of the infinity of the number series from the contextual “definition” of the cardinality operator. Called “Hume's Principle” by Boolos, the contextual definition of the cardinality operator is presented in Section 63 of Grundlagen, as the statement that, for any concepts F and G,the number of F s = the number of G sif, and only if,F is equinumerous with G.The philosophical interest in Frege's Theorem derives from the thesis, defended for example by Crispin Wright, that Hume's principle expresses our pre-analytic conception of assertions of numerical identity. However, Boolos cites the very fact that Hume's principle has only infinite models as grounds for denying that it is logically true: For Boolos, Hume's principle is simply a disguised axiom of infinity.

No Class

2014 ◽  
Author(s):  
Scott Soames
Keyword(s):  
Philosophy Of Mathematics ◽  
Philosophical Logic ◽  
Substitutional Quantification ◽  
Class Theory ◽  
Contextual Definition ◽  
Definition Of

This chapter explores Russell’s “no class theory,” originally expressed by his contextual definition of classes in Principia Mathematica. In recent years, some Russell scholars have trumpeted the virtues of the interpretation of Russell’s quantification as substitutional, among which is the sense it makes of the “no-class theory.” Such an interpretation does make some sense of Russell’s philosophical remarks about that theory, about the significance of his logicist reduction, and about the ability of the reduction to serve as a model for similar reductions outside the philosophy of mathematics. However this substitutional interpretation is not sufficient, since it is inconsistent with important aspects of Russell’s philosophical logic and is technically inadequate to support his logicist reduction. In short, if substitutional quantification is the source of the “no class theory,” then the theory is not vindicated, but refuted.

scholarly journals Reactivity in social scientific experiments: what is it and how is it different (and worse) than a Placebo effect?

2021 ◽  
Vol 11 (2) ◽  
Author(s):  
María Jiménez-Buedo
Keyword(s):  
Experimental Data ◽  
Conceptual Framework ◽  
Placebo Effect ◽  
Placebo Effects ◽  
Causal Inferences ◽  
Scientific Experiments ◽  
Social Scientific ◽  
The Many ◽  
Definition Of ◽  
In Virtue Of

AbstractReactivity, or the phenomenon by which subjects tend to modify their behavior in virtue of their being studied upon, is often cited as one of the most important difficulties involved in social scientific experiments, and yet, there is to date a persistent conceptual muddle when dealing with the many dimensions of reactivity. This paper offers a conceptual framework for reactivity that draws on an interventionist approach to causality. The framework allows us to offer an unambiguous definition of reactivity and distinguishes it from placebo effects. Further, it allows us to distinguish between benign and malignant forms of the phenomenon, depending on whether reactivity constitutes a danger to the validity of the causal inferences drawn from experimental data.

scholarly journals Carnap’s Structuralist Thesis

2020 ◽  
pp. 383-420
Author(s):  
Georg Schiemer
Keyword(s):  
Present Article ◽  
Structural Properties ◽  
Philosophy Of Mathematics ◽  
Mathematical Structure ◽  
Mathematical Structuralism ◽  
Definition Of ◽  
Philosophical Debates ◽  
Implicit Definitions

This chapter investigates Carnap’s structuralism in his philosophy of mathematics of the 1920s and early 1930s. His approach to mathematics is based on a genuinely structuralist thesis, namely that axiomatic theories describe abstract structures or the structural properties of their objects. The aim in the present article is twofold: first, to show that Carnap, in his contributions to mathematics from the time, proposed three different (but interrelated) ways to characterize the notion of mathematical structure, namely in terms of (i) implicit definitions, (ii) logical constructions, and (iii) definitions by abstraction. The second aim is to re-evaluate Carnap’s early contributions to the philosophy of mathematics in light of contemporary mathematical structuralism. Specifically, the chapter discusses two connections between his structuralist thesis and current philosophical debates on structural abstraction and the on the definition of structural properties.

The Proof of Hume’s Principle

2019 ◽  
pp. 182-206 ◽  
Author(s):  
Robert C. May ◽  
Kai F. Wehmeier
Keyword(s):  
Initial Part ◽  
Formal Proofs ◽  
Hume’S Principle ◽  
The Core ◽  
Logical Status ◽  
Definition Of ◽  
The Right

Beginning in Grundgesetze §53, Frege presents proofs of a set of theorems known to encompass the Peano-Dedekind axioms for arithmetic. The initial part of Frege’s deductive development of arithmetic, to theorems (32) and (49), contains fully formal proofs that had merely been sketched out in Grundlagen. Theorems (32) and (49) are significant because they are the right-to-left and left-to-right directions respectively of what we call today “Hume’s Principle” (HP). The core observation that we explore is that in Grundgesetze, Frege does not prove Hume’s Principle, not at least if we take HP to be the principle he introduces, and then rejects, as a definition of number in Grundlagen. In order better to understand why Frege never considers HP as a biconditional principle in Grundgesetze, we explicate the theorems Frege actually proves in that work, clarify their conceptual and logical status within the overall derivation of arithmetic, and ask how the definitional content that Frege intuited in Hume’s Principle is reconstructed by the theorems that Frege does prove.

The Nature of Ususfructus

1946 ◽  
Vol 9 (2) ◽  
pp. 159-170
Author(s):  
Kopel Kagan
Keyword(s):  
Great Difficulty ◽  
Roman Law ◽  
Present Position ◽  
Modern Times ◽  
The Subject ◽  
The One ◽  
Definition Of ◽  
The Right ◽  
Opening Statements ◽  
In Virtue Of

No satisfactory definition of Dominium in Roman Law has yet been achieved. Amongst English writers Austin many years ago found great difficulty in this question while in modern times Professor Buckland has written ‘it is thus difficult to define Dominium precisely.’ Again, Poste, dealing with Gaius' discussion of dominium, says that his opening statements are ‘deplorably confused.’ These examples are enough to indicate the condition, of uncertainty which prevails. In my submission this uncertainty exists mainly because the conception of ususfructus has never yet been explained adequately. Of Possessio it has been said ‘the definition of Possessio to give the results outlined is a matter of great difficulty. No perfectly correct solution may be possible,’ and this statement is generally accepted as a correct assessment of the present position in juristic literature. But here, too, in my opinion, the reason is again connected with usufruct, for the possessio of the usufructuary has not yet been adequately determined. Gaius (2.93) tells us ‘usufructuarius vero usucapere non potest; primuum quod non possidet, sed habet ius utendi et fruendi.’ Ulpian holds that he had possessio in fact (‘Naturaliter videtur possidere is qui usum fructum habet’ D.41.2.12). On this subject Roby says ‘the fructuary was not strictly a possessor and therefore if he was deprived from enjoying he had not a claim to the original interdict de vi but in virtue of his quasi-possessio a special interdict was granted him.’ Austin saw difficulty in the whole problem of possessio. He wrote ‘by Savigny in his treatise on possessio it is remarked that the possessio of a right of usufruct … resembles the possessio of a thing, by the proprietor, or by an adverse possessor exercising rights of property over the thing. And that a disturbance of the one possession resembles the disturbance of the other. Now this must happen for the reason I have already stated:—namely, that the right of usufruct or user, like that of property, is indefinite in point of user. For what is possession (meaning legal possession not mere physical handling of the subject) but the exercise of a right ?’

The development of arithmetic in Frege's Grundgesetze der arithmetik

1993 ◽  
Vol 58 (2) ◽  
pp. 579-601 ◽  
Author(s):  
Richard G. Heck
Keyword(s):  
Cardinal Number ◽  
Formal Theory ◽  
Second Order ◽  
Order Logic ◽  
Hume’S Principle ◽  
Philosophical Significance ◽  
Philosophy Of Arithmetic ◽  
Second Order Logic

AbstractFrege's development of the theory of arithmetic in his Grundgesetze der Arithmetik has long been ignored, since the formal theory of the Grundgesetze is inconsistent. His derivations of the axioms of arithmetic from what is known as Hume's Principle do not, however, depend upon that axiom of the system—Axiom V—which is responsible for the inconsistency. On the contrary, Frege's proofs constitute a derivation of axioms for arithmetic from Hume's Principle, in (axiomatic) second-order logic. Moreover, though Frege does prove each of the now standard Dedekind-Peano axioms, his proofs are devoted primarily to the derivation of his own axioms for arithmetic, which are somewhat different (though of course equivalent). These axioms, which may be yet more intuitive than the Dedekind-Peano axioms, may be taken to be “The Basic Laws of Cardinal Number”, as Frege understood them.Though the axioms of arithmetic have been known to be derivable from Hume's Principle for about ten years now, it has not been widely recognized that Frege himself showed them so to be; nor has it been known that Frege made use of any axiomatization for arithmetic whatsoever. Grundgesetze is thus a work of much greater significance than has often been thought. First, Frege's use of the inconsistent Axiom V may invalidate certain of his claims regarding the philosophical significance of his work (viz., the establish may invalidate certain of his claims regarding the philosophical significance of his work (viz., the establishment of Logicism), but it should not be allowed to obscure his mathematical accomplishments and his contribution to our understanding of arithmetic. Second, Frege's knowledge that arithmetic is derivable from Hume's Principle raises important sorts of questions about his philosophy of arithmetic. For example, “Why did Frege not simply abandon Axiom V and take Hume's Principle as an axiom?”

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