A logical measure function

1953 ◽  
Vol 18 (4) ◽  
pp. 289-308 ◽  
Author(s):  
John G. Kemeny
Keyword(s):  
Philosophy Of Science ◽  
Functional Calculus ◽  
Atomic Sentence ◽  
First Order ◽  
Measure Function

By a logical measure (for a given language) we mean a syntactically defined function which associates some value with each well-formed formula of the language. Various such logical measures have played a fundamental role in the development of Logic and the Philosophy of Science. The purpose of this paper is to define a logical measure which has much wider applications than measures so far studied.The new measure has two fundamental advantages, which will be referred to throughout the paper. First of all it can be applied to more (and richer) languages than the older measures. Most of the measures now in use are restricted to the first-order functional calculus, and frequently even to a first-order calculus with one-place predicates only. The measure to be defined will be applicable to richer languages as well, e.g., to functional calculi of all finite orders. But even as far as the first-order functional calculus is concerned, the new measure has a great advantage: We do not have to require that the atomic sentences of the calculus be independent. The previous measures depended in their construction on the requirement that “the basic statements of the language express independent facts.”By an atomic sentence we mean a well-formed formula formed by applying an h-place primitive predicate to h individuals (a well-formed formula no part of which is well-formed); by a permissible conjunction we mean a conjunction of atomic sentences and negations of other atomic sentences; and the requirement of independence is that an atomic sentence (or its negation) is logically implied by a permissible conjunction only if it (its negation) is one of the components of the conjunction.

On Theses of the First-Order Functional Calculus

1961 ◽  
Vol 7 (11-14) ◽  
pp. 175-184
Author(s):  
Juliusz Reichbach
Keyword(s):  
Functional Calculus ◽  
First Order

A derivation of number theory from ancestral theory

1952 ◽  
Vol 17 (3) ◽  
pp. 192-197 ◽  
Author(s):  
John Myhill
Keyword(s):  
Number Theory ◽  
Functional Calculus ◽  
Natural Numbers ◽  
Class Inclusion ◽  
First Order ◽  
Free Variables

Martin has shown that the notions of ancestral and class-inclusion are sufficient to develop the theory of natural numbers in a system containing variables of only one type.The purpose of the present paper is to show that an analogous construction is possible in a system containing, beyond the quantificational level, only the ancestral and the ordered pair.The formulae of our system comprise quantificational schemata and anything which can be obtained therefrom by writing pairs (e.g. (x; y), ((x; y); (x; (y; y))) etc.) for free variables, or by writing ancestral abstracts (e.g. (*xyFxy) etc.) for schematic letters, or both.The ancestral abstract (*xyFxy) means what is usually meant by ; and the formula (*xyFxy)zw answers to Martin's (zw; xy)(Fxy).The system presupposes a non-simple applied functional calculus of the first order, with a rule of substitution for predicate letters; over and above this it has three axioms for the ancestral and two for the ordered pair.

The completeness of the first-order functional calculus

1949 ◽  
Vol 14 (3) ◽  
pp. 159-166 ◽  
Author(s):  
Leon Henkin
Keyword(s):  
Functional Calculus ◽  
Propositional Calculus ◽  
Higher Order ◽  
New Method ◽  
Completeness Proof ◽  
New Approach ◽  
First Order ◽  
Formal Systems ◽  
The Subject ◽  
Image Position

Although several proofs have been published showing the completeness of the propositional calculus (cf. Quine (1)), for the first-order functional calculus only the original completeness proof of Gödel (2) and a variant due to Hilbert and Bernays have appeared. Aside from novelty and the fact that it requires less formal development of the system from the axioms, the new method of proof which is the subject of this paper possesses two advantages. In the first place an important property of formal systems which is associated with completeness can now be generalized to systems containing a non-denumerable infinity of primitive symbols. While this is not of especial interest when formal systems are considered as logics—i.e., as means for analyzing the structure of languages—it leads to interesting applications in the field of abstract algebra. In the second place the proof suggests a new approach to the problem of completeness for functional calculi of higher order. Both of these matters will be taken up in future papers.The system with which we shall deal here will contain as primitive symbolsand certain sets of symbols as follows:(i) propositional symbols (some of which may be classed as variables, others as constants), and among which the symbol “f” above is to be included as a constant;(ii) for each number n = 1, 2, … a set of functional symbols of degree n (which again may be separated into variables and constants); and(iii) individual symbols among which variables must be distinguished from constants. The set of variables must be infinite.

Completeness of the functional calculus of first order

Studia Logica ◽  
1955 ◽  
Vol 2 (1) ◽  
pp. 245-250 ◽  
Author(s):  
J. Reichbach
Keyword(s):  
Functional Calculus ◽  
First Order
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