K. Menger. The algebra of functions: past, present, future. Rendiconti di matematica, vol. 20 (1961), pp. 409–430. - Karl Menger. Function algebra and propositional calculus. Self-organizing systems 1962, edited by Marshall C. Yovits, George T. Jacobi, and Gordon D. Goldstein, Spartan Books, Washington, D.C., 1962, pp. 525–532. - Karl Menger and Martin Schultz. Postulates for the substitutive algebra of the 2-place functors in the 2-valued calculus of propositions. Notre Dame journal of formal logic, vol. 4 no. 3 (1963), pp. 188–192. - Robert E. Seall. Truth-valued fluents and qualitative laws. Philosophy of science, vol. 30 (1963), pp. 36–10.

1966 ◽  
Vol 31 (2) ◽  
pp. 272-272 ◽  
Author(s):  
Bruce Lercher
Keyword(s):  
Philosophy Of Science ◽  
Propositional Calculus ◽  
Function Algebra ◽  
Formal Logic ◽  
Algebra Of Functions ◽  
Self Organizing

Alternative postulate sets for Lewis's S5

1956 ◽  
Vol 21 (4) ◽  
pp. 347-349
Author(s):  
E. J. Lemmon
Keyword(s):  
Propositional Calculus ◽  
Formal Logic ◽  
Propositional Formula ◽  
Modal Operator ◽  
Special Rules ◽  
Definition Of ◽  
Derived Rules

Professor Prior (Formal logic (1955), pp. 305–307) gives alternative postulate sets for Lewis's S5, one substantially Lewis's own set (p. 305, 6.1) and another, considerably simpler, due to Prior himself (pp. 306–307, 6.5). This note aims at showing that the systems based on these two postulate sets are in fact equivalent. We label the system based on Lewis's postulates ‘S5’, as usual, and that on Prior's postulates ‘P’.The form of the postulate set for P considered here is as follows. We add to the full propositional calculus one definition:Df. M: M = NLN;and two special rules:L1: ⊦Cαβ → ⊦CLαβ;L2: ⊦Cαβ → ⊦CαLβ, if α is fully modalized (a propositional formula is fully modalized, if and only if all occurrences of propositional variables in it lie within the scope of a modal operator).Prior's text (pp. 201–205) indicates sufficiently how it could be shown that any thesis of S5 is a thesis of P and that the rules and definitions of S5 are obtainable as derived rules of P. We turn, therefore, to the problem of showing that any thesis of P is a thesis of S5 and that the rules and definition of P are obtainable as derived rules of S5.

A note on nominalism and recursive functions

1949 ◽  
Vol 14 (1) ◽  
pp. 27-31 ◽  
Author(s):  
R. M. Martin
Keyword(s):  
General Theory ◽  
Philosophy Of Science ◽  
Recursive Function ◽  
Homogeneous System ◽  
Formal Logic ◽  
Abstract Objects ◽  
Natural Numbers ◽  
General Recursive Function ◽  
Recursive Functions ◽  
Definition Of

The purpose of this note is (i) to point out an important similarity between the nominalistic system discussed by Quine in his recent paper On universals and the system of logic (the system н) developed by the author in A homogeneous system for formal logic, (ii) to offer certain corrections to the latter, and (iii) to show that that system (н) is adequate for the general theory of ancestrale and for the definition of any general recursive function of natural numbers.Nominalism as a thesis in the philosophy of science, according to Quine, is the view that it is possible to construct a language adequate for the purposes of science, which in no wise admits classes, properties, relations, or other abstract objects as values for variables.

Gunnell on “Deductivism,” the “Logic” of Science and Scientific Explanation: A Riposte

1969 ◽  
Vol 63 (4) ◽  
pp. 1251-1258
Author(s):  
A. James Gregor
Keyword(s):  
Philosophy Of Science ◽  
Scientific Inquiry ◽  
Scientific Explanation ◽  
Logical Structure ◽  
Logical Empiricism ◽  
Formal Logic ◽  
Object Language ◽  
Logic Of Science ◽  
Social Scientific ◽  
Formal Representations

It is impossible to tender a reply to Professor Gunnell's essay, “Deduction, Explanation and Social Scientific Inquiry,” that would be both brief and adequate. It would be impossible to be brief because Gunnell conjures up a tagraggery of issues, none of which he seems prepared and/or disposed to resolve. But no matter how extensive a reply might be, it would still be impossible to conceive it as adequate for I am not sure that I, or anyone else, can determine precisely what he means to say in the essay before us. It is impossible for me to determine with any specificity whatsoever, for example, what it means to say:Logical empiricism as an approach to the philosophy of science has been concerned with developing formal representations or reconstructions of the logical structure of scientific explanation and with a meta-logical analysis of the language applied to science. In this view there is a very strict correlation between the philosophy of science and formal logic.I haven't the foggiest notion what a “metalogical analysis of the language applied to science” might mean. I had always understood “metalogic” to refer to discourse employing logic as its object language—just as metamathematics would be a language employed in the analysis of mathematics as an object language.

On the quantificational logic of intuitionistic set theory

1986 ◽  
Vol 99 (1) ◽  
pp. 5-10 ◽  
Author(s):  
Harvey M. Friedman ◽  
Andrej Ščedrov
Keyword(s):  
Set Theory ◽  
Propositional Logic ◽  
Propositional Calculus ◽  
Formal Logic ◽  
Predicate Calculus ◽  
Intuitionistic Propositional Calculus ◽  
Quantificational Logic ◽  
Law Of Excluded Middle ◽  
Excluded Middle ◽  
Intuitionistic Set Theory

Formal propositional logic describing the laws of constructive (intuitionistic) reasoning was first proposed in 1930 by Heyting. It is obtained from classical pro-positional calculus by deleting the Law of Excluded Middle, and it is usually referred to as Heyting's (intuitionistic) propositional calculus ([9], §§23, 19) (we write HPP in short). Formal logic involving predicates and quantifiers based on HPP is called Heyting's (intuitionistic) predicate calculus ([9], §§31, 19) (we write HPR in short).

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