Solomon Feferman. Degrees of unsolvability associated with classes of formalized theories. The journal of symbolic logic, vol. 22 (1957), pp. 161–175. - J. R. Shoenfield. Degrees of formal systems. The journal of symbolic logic, vol. 23 no. 4 (for 1958, pub. 1959), pp. 389–392.

1962 ◽  
Vol 27 (1) ◽  
pp. 85-86
Author(s):  
Hartley Rogers
Keyword(s):  
Symbolic Logic ◽  
Formal Systems ◽  
Degrees Of Unsolvability

Degrees of unsolvability associated with classes of formalized theories

1957 ◽  
Vol 22 (2) ◽  
pp. 161-175 ◽  
Author(s):  
Solomon Feferman
Keyword(s):  
Formal System ◽  
Point Of View ◽  
Wide Sense ◽  
Recursively Enumerable Set ◽  
Formal Systems ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
The Difference ◽  
Degrees Of Unsolvability ◽  
Order Predicate Calculus

In his well-known paper [11], Post founded a general theory of recursively enumerable sets, which had its metamathematical source in questions about the decision problem for deducibility in formal systems. However, in centering attention on the notion of degree of unsolvability, Post set a course for his theory which has rarely returned to this source. Among exceptions to this tendency we may mention, as being closest to the problems considered here, the work of Kleene in [8] pp. 298–316, of Myhill in [10], and of Uspenskij in [15]. It is the purpose of this paper to make some further contributions towards bridging this gap.From a certain point of view, it may be argued that there is no real separation between metamathematics and the theory of recursively enumerable sets. For, if the notion of formal system is construed in a sufficiently wide sense, by taking as ‘axioms’ certain effectively found members of a set of ‘formal objects’ and as ‘proofs’ certain effectively found sequences of these objects, then the set of ‘provable statements’ of such a system may be identified, via Gödel's numbering technique, with a recursively enumerable set; and conversely, each recursively enumerable set is identified in this manner with some formal system (cf. [8] pp. 299–300 and 306). However, the pertinence of Post's theory is no longer clear when we turn to systems formalized within the more conventional framework of the first-order predicate calculus. It is just this restriction which serves to clarify the difference in spirit of the two disciplines.

Leibniz's interpretation of his logical calculi

1954 ◽  
Vol 19 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Nicholas Rescher
Keyword(s):  
Formal Logic ◽  
Point Of View ◽  
Symbolic Logic ◽  
Logical Calculus ◽  
Formal Systems ◽  
Logical Calculi

The historical researches of Louis Couturat saved the logical work of Leibniz from the oblivion of neglect and forgetfulness. They revealed that Leibniz developed in succession several versions of a “logical calculus” (calculus ratiocinator or calculus universalis). In consequence of Couturat's investigations it has become well known that Leibniz's development of these logical calculi adumbrated the notion of a logistic system; and for these foreshadowings of the logistic treatment of formal logic Leibniz is rightly regarded as the father of symbolic logic.It is clear from what has been said that it is scarcely possible to overestimate the debt which the contemporary student of Leibniz's logic owes to Couturat. This gratitude must, however, be accompanied by the realization that Couturat's own theory of logic is gravely defective. Couturat was persuaded that the extensional point of view in logic is the only one which is correct, an opinion now quite antiquated, and shared by no one. This prejudice of Couturat's marred his exposition of Leibniz's logic. It led him to battle with windmills: he viewed the logic of Leibniz as rife with shortcomings stemming from an intensional approach.The task of this paper is a re-examination of Leibniz's logic. It will consider without prejudgment how Leibniz conceived of the major formal systems he developed as logical calculi – that is, these systems will be studied with a view to the interpretation or interpretations which Leibniz himself intends for them. The aim is to undo some of the damage which Couturat's preconception has done to the just understanding of Leibniz's logic and to the proper evaluation of his contribution.

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