scholarly journals “How did the serpent of inconsistency enter Frege’s paradise?”

2019 ◽  
pp. 411-436
Author(s):  
Crispin Wright
Keyword(s):  
Cardinal Number ◽  
Ordinal Number ◽  
Higher Order ◽  
Order Logic ◽  
Higher Order Logic ◽  
Indefinite Extensibility ◽  
Basic Law ◽  
Per Se

The paper explores the alleged connection between indefinite extensibility and the classic paradoxes of Russell, Burali-Forti, and Cantor. It is argued that while indefinite extensibility is not per se a source of paradox, there is a degenerate subspecies—reflexive indefinite extensibility—which is. The result is a threefold distinction in the roles played by indefinite extensibility in generating paradoxes for the notions of ordinal number, cardinal number, and set respectively. Ordinal number, intuitively understood, is a reflexively indefinitely extensible concept. Cardinal number is not. And Set becomes so only in the setting of impredicative higher-order logic—so that Frege’s Basic Law V is guilty at worst of partnership in crime, rather than the sole offender.

Definitions in Begriffsschrift and Grundgesetze

2019 ◽  
pp. 538-566
Author(s):  
Michael Kremer
Keyword(s):  
Higher Order ◽  
Order Logic ◽  
Higher Order Logic ◽  
Variable Binding ◽  
Basic Law ◽  
Value Range ◽  
New Variables

Frege’s definitions in Part III of Begriffsschrift introduce novel forms of variable-binding and quantification. Frege’s commentary, however, shows that he did not fully grasp the logical significance of his notation, treating the new variables as themselves somehow defined. In Grundgesetze, such issues are avoided by exploiting the value-range notation as a substitute for functional abstraction, relying on the inconsistent Basic Law V. In presenting Frege’s Theorem without appeal to Basic Law V, Richard Heck reinstates a generalized form of the notation of Begriffsschrift by using a higher-order logic employing generalized variable-binding to form names of higher-level functions. Heck and Robert May suggest that by the time of Grundgesetze Frege had achieved the necessary understanding of variable-binding to appreciate Heck’s notation. However, even the later Frege had only a piecemeal characterization of what we now see as falling together as variable-binding. For him, different forms of variable-binding do different logical work, marked by distinct ranges of variables. Eliminating the value-range notation in his definitions after the manner of Heck would require rethinking the role of variable-binding operators. This would not be philosophically cost-free for Frege, though there is some slight evidence that Frege may have begun to move in this direction toward the end of his career.

scholarly journals WHAT RUSSELL SHOULD HAVE SAID TO BURALI–FORTI

2017 ◽  
Vol 10 (4) ◽  
pp. 682-718 ◽  
Author(s):  
SALVATORE FLORIO ◽  
GRAHAM LEACH-KROUSE
Keyword(s):  
Set Theory ◽  
Current Literature ◽  
Concept Formation ◽  
Ordinal Number ◽  
Higher Order ◽  
Order Logic ◽  
Higher Order Logic ◽  
Class Theory

AbstractThe paradox that appears under Burali–Forti’s name in many textbooks of set theory is a clever piece of reasoning leading to an unproblematic theorem. The theorem asserts that the ordinals do not form a set. For such a set would be–absurdly–an ordinal greater than any ordinal in the set of all ordinals. In this article, we argue that the paradox of Burali–Forti is first and foremost a problem about concept formation by abstraction, not about sets. We contend, furthermore, that some hundred years after its discovery the paradox is still without any fully satisfactory resolution. A survey of the current literature reveals one key assumption of the paradox that has gone unquestioned, namely the assumption that ordinals are objects. Taking the lead from Russell’s no class theory, we interpret talk of ordinals as an efficient way of conveying higher-order logical truths. The resulting theory of ordinals is formally adequate to standard intuitions about ordinals, expresses a conception of ordinal number capable of resolving Burali–Forti’s paradox, and offers a novel contribution to the longstanding program of reducing mathematics to higher-order logic.

Frege’s Little Theorem and Frege’s Way Out

2019 ◽  
pp. 384-410
Author(s):  
Roy T. Cook
Keyword(s):  
Higher Order ◽  
Order Logic ◽  
Higher Order Logic ◽  
Basic Law

Shortly before the second volume of Basic Laws went to press, Frege appended an Afterword, analyzing the Russell paradox and suggesting an alternate formulation of basic law V. This attempted ‘way out’ has been characterized as “the wrong guess of a man in a hurry” and has been accused of leading to either paradoxes or absurd consequences. The former claim is wrong, and the latter, although correct, has until now been misunderstood. The bulk of the paper works though a new series of proofs—ones that pay closer attention to whether we are working in Frege’s original formalism or modern higher-order logic, and which do not consist merely of un-illuminating reductios—that highlight exactly where Frege’s attempted solution goes wrong. As a result, it becomes clear that Frege’s ‘way out’, although failing in the end to avoid Russellian problems, is far from merely a “guess” or the result of haste. On the contrary, the amended principle results from a natural application of theorems about, and insights into, abstraction provided by Frege himself in the Afterword. Ironically, however, natural generalizations of these theorems and insights (ones not noticed by Frege) point to precisely the problems that were later found to besiege the amended law V.

scholarly journals Closed Structure

2021 ◽  
Author(s):  
Peter Fritz ◽  
Harvey Lederman ◽  
Gabriel Uzquiano
Keyword(s):  
Formal Language ◽  
Syntactic Structure ◽  
Natural Extension ◽  
Propositional Attitudes ◽  
Higher Order ◽  
Order Logic ◽  
Bertrand Russell ◽  
Structured Propositions ◽  
Higher Order Logic ◽  
Closed Structure

AbstractAccording to the structured theory of propositions, if two sentences express the same proposition, then they have the same syntactic structure, with corresponding syntactic constituents expressing the same entities. A number of philosophers have recently focused attention on a powerful argument against this theory, based on a result by Bertrand Russell, which shows that the theory of structured propositions is inconsistent in higher order-logic. This paper explores a response to this argument, which involves restricting the scope of the claim that propositions are structured, so that it does not hold for all propositions whatsoever, but only for those which are expressible using closed sentences of a given formal language. We call this restricted principle Closed Structure, and show that it is consistent in classical higher-order logic. As a schematic principle, the strength of Closed Structure is dependent on the chosen language. For its consistency to be philosophically significant, it also needs to be consistent in every extension of the language which the theorist of structured propositions is apt to accept. But, we go on to show, Closed Structure is in fact inconsistent in a very natural extension of the standard language of higher-order logic, which adds resources for plural talk of propositions. We conclude that this particular strategy of restricting the scope of the claim that propositions are structured is not a compelling response to the argument based on Russell’s result, though we note that for some applications, for instance to propositional attitudes, a restricted thesis in the vicinity may hold some promise.

Adapting functional programs to higher order logic

2008 ◽  
Vol 21 (4) ◽  
pp. 377-409 ◽  
Author(s):  
Scott Owens ◽  
Konrad Slind
Keyword(s):  
Higher Order ◽  
Order Logic ◽  
Higher Order Logic
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