scholarly journals Arithmetic Formulated in a Logic of Meaning Containment

2021 ◽  
Vol 18 (5) ◽  
pp. 447-472
Author(s):  
Ross Brady
Keyword(s):  
Relevant Logic ◽  
Peano Arithmetic ◽  
Recursive Functions ◽  
Primitive Recursive

We assess Meyer’s formalization of arithmetic in his [21], based on the strong relevant logic R and compare this with arithmetic based on a suitable logic of meaning containment, which was developed in Brady [7]. We argue in favour of the latter as it better captures the key logical concepts of meaning and truth in arithmetic. We also contrast the two approaches to classical recapture, again favouring our approach in [7]. We then consider our previous development of Peano arithmetic including primitive recursive functions, finally extending this work to that of general recursion.

Whither relevant arithmetic?

1992 ◽  
Vol 57 (3) ◽  
pp. 824-831 ◽  
Author(s):  
Harvey Friedman ◽  
Robert K. Meyer
Keyword(s):  
Relevant Logic ◽  
Peano Arithmetic ◽  
Central Question ◽  
Proof Methods ◽  
Classical Proof ◽  
Natural Way

AbstractBased on the relevant logic R, the system R# was proposed as a relevant Peano arithmetic. R# has many nice properties: the most conspicuous theorems of classical Peano arithmetic PA are readily provable therein; it is readily and effectively shown to be nontrivial; it incorporates both intuitionist and classical proof methods. But it is shown here that R# is properly weaker than PA, in the sense that there is a strictly positive theorem QRF of PA which is unprovable in R#. The reason is interesting: if PA is slightly weakened to a subtheory P+, it admits the complex ring C as a model; thus QRF is chosen to be a theorem of PA but false in C. Inasmuch as all strictly positive theorems of R# are already theorems of P+, this nonconservativity result shows that QRF is also a nontheorem of R#. As a consequence, Ackermann's rule γ is inadmissible in R#. Accordingly, an extension of R# which retains its good features is desired. The system R##, got by adding an omega-rule, is such an extension. Central question: is there an effectively axiomatizable system intermediate between R# and R#, which does formalize arithmetic on relevant principles, but also admits γ in a natural way?

A Hierarchy on the Class of Primitive Recursive Ordinal Functions

1976 ◽  
Vol 28 (6) ◽  
pp. 1205-1209
Author(s):  
Stanley H. Stahl
Keyword(s):  
Natural Numbers ◽  
Recursive Functions ◽  
Set Functions ◽  
Primitive Recursive ◽  
Class Of Functions ◽  
Recursive Set

The class of primitive recursive ordinal functions (PR) has been studied recently by numerous recursion theorists and set theorists (see, for example, Platek [3] and Jensen-Karp [2]). These investigations have been part of an inquiry concerning a larger class of functions; in Platek's case, the class of ordinal recursive functions and in the case of Jensen and Karp, the class of primitive recursive set functions. In [4] I began to study PR in depth and this paper is a report on an attractive analogy between PR and its progenitor, the class of primitive recursive functions on the natural numbers (Prim. Rec).

A Macro Program for the Primitive Recursive Functions

1991 ◽  
Vol 37 (8) ◽  
pp. 121-124
Author(s):  
Hilbert Levitz ◽  
Warren Nichols ◽  
Robert F. Smith
Keyword(s):  
Recursive Functions ◽  
Primitive Recursive

The relation of A to Prov ˹A˺ in the Lindenbaum sentence algebra

1973 ◽  
Vol 38 (2) ◽  
pp. 295-298 ◽  
Author(s):  
C. F. Kent
Keyword(s):  
Boolean Algebra ◽  
Normal Forms ◽  
Order Relation ◽  
Mathematical Induction ◽  
Axiomatic Theory ◽  
Recursive Functions ◽  
Primitive Recursive

Let U be a consistent axiomatic theory containing Robinson's Q [TMRUT, p. 51]. In order for the results below to be of interest, U must be powerful enough to carry out certain arguments involving versions of the “derivability conditions,” DC(i) to DC(iii) below, of [HBGM, p. 285], [F60, Theorem 4.7], or [L55]. Thus it must contain, at least, mathematical induction for formulas whose prenex normal forms contain at most existential quantifiers. For convenience, U is assumed also to contain symbols for primitive recursive functions and relations, and their defining equations. One of these is used to form the standard provability predicate, Prov ˹A˺, “there exists a number which is the Gödel number of a proof of A.” Upper corners denote numerals for Gödel numbers for the enclosed sentences, and parentheses are often omitted in their presence.This paper contains some results concerning the relation between the sentence A, and the sentence Prov ˹A˺ in the Lindenbaum Sentence Algebra (LSA) for U, the Boolean algebra induced by the pre-order relation A ≤ B ⇔ ⊦A → B. Half of the answer is provided by a theorem of Löb [L55], which states that ⊦Prov ˹A˺ → A ⇔ ⊦A. Hence, in the presence of DC(iii), below, it is never true that Prov ˹A˺ < A in the LSA. However, there is a large and interesting set of sentences, denoted here by Γ, for which A < Prov ⌜A⌝.

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