R. M. Martin. A homogeneous system for formal logic. The journal of symbolic logic, vol. 8 (1943), pp. 1–23.

1943 ◽  
Vol 8 (2) ◽  
pp. 54-54
Author(s):  
J. C. C. McKinsey
Keyword(s):  
Homogeneous System ◽  
Formal Logic ◽  
Symbolic Logic

Disarming Stove’s Paradox: In Defence of Formal Logic

1998 ◽  
pp. 127-134
Author(s):  
R. Rodrigo Soberano
Keyword(s):  
Main Part ◽  
Formal Logic ◽  
Symbolic Logic ◽  
Logical Contradiction ◽  
False Conclusion

The argument (d) ("All arguments with true premises and false conclusions are invalid.") is an argument with true premises and false conclusion. Therefore "(d) is invalid" seems to be formally valid. Thus presumably formal logic has to admit it as valid. But then formal logic finds itself in a bind. For the above argument is problematic and even paradoxical since it involves an internal logical contradiction. The paradox, aptly termed "Stove's paradox," is fully realized by demonstrating with the help of symbolic logic the contradiction within the argument. Then as the main part of this essays shows, the paradox is attacked by exposing the paradox's genesis. It is shown that by appeal to some not so obvious logical considerations regarding sound linguistic construction and usage, the above argument could not have been legitimately construction. For its construction must have involved either equivocation or hiatus of meaningfulness in the use of the symbol (d).

A homogeneous system for formal logic

1943 ◽  
Vol 8 (1) ◽  
pp. 1-23 ◽  
Author(s):  
R. M. Martin
Keyword(s):  
Set Theory ◽  
General Type ◽  
Homogeneous System ◽  
Formal Logic ◽  
Strict Sense ◽  
Standard Methods ◽  
Logical Construction ◽  
Logical Type ◽  
Classical Mathematics ◽  
Axiomatic Set Theory

Two more or less standard methods exist for the systematic, logical construction of classical mathematics, the so-called theory of types, due in the main to Russell, and the Zermelo axiomatic set theory. In systems based upon either of these, the connective of membership, “ε”, plays a fundamental role. Usually although not always it figures as a primitive or undefined symbol.Following the familiar simplification of Russell's theory, let us mean by a logical type in the strict sense any one of the following: (i) the totality consisting exclusively of individuals, (ii) the totality consisting exclusively of classes whose members are exclusively individuals, (iii) the totality consisting exclusively of classes whose members are exclusively classes whose members in turn are exclusively individuals, and so on. Any entity from (ii) is said to be one type higher than any entity from (i), any entity from (iii), one type higher than any entity from (ii), and so on. In systems based upon this simplified theory of types, the only significant atomic formulae involving “ε” are those asserting the membership of an entity in an entity one type higher. Thus any expression of the form “(x∈y)” is meaningless except where “y” denotes an entity of just one type higher than the type of the entity denoted by “x” It is by means of general type restrictions of this kind that the Russell and other paradoxes are avoided.

scholarly journals Czy logika formalna ma sens?

2018 ◽  
Vol 13 (3) ◽  
pp. 49-55
Author(s):  
Ludomir Newelski
Keyword(s):  
Critical Thinking ◽  
Formal Logic ◽  
Symbolic Logic ◽  
Logical Inference ◽  
Semantic Inference ◽  
Definition Of ◽  
Rational Reasoning

Does formal logic make sense?This text is a commentary on the book Logika i argumentacja. Praktyczny kurs krytycznego myślenia Logic and Argumentation. A Practical Course In Critical Thinking by Professor Andrzej Kisielewicz. Prof. Kisielewicz argues there, among other things, that formal symbolic logic is inadequate to explain practical rational reasoning. This commentary defends formal logic in this respect. In particular, Prof. Kisielewicz proposes in his book a definition of practical logical inference. According to him, a conclusion follows from a given set of premises if there is no situation, where the premises hold, while the conclusion fails. In this commentary it is pointed out that this is a well-known notion of semantic inference in formal logic. It is also well-known that semantic and syntactic inference in logic are equivalent, i.e. equally strong.

Model-Completeness and Elementary Properties of Torsion Free Abelian Groups

1974 ◽  
Vol 26 (4) ◽  
pp. 829-840
Author(s):  
Elias Zakon
Keyword(s):  
High Power ◽  
Abelian Groups ◽  
Elementary Theory ◽  
Complete Classification ◽  
Formal Logic ◽  
Predicate Calculus ◽  
Symbolic Logic ◽  
Model Completeness ◽  
Free Abelian Groups ◽  
Torsion Free

The decidability of the elementary theory of abelian groups, and their complete classification by elementary properties (i.e. those formalizable in the lower predicate calculus (LPC) of formal logic), were established by W. Szmielew [13]. More general results were proved by Eklof and Fischer [2], and G. Sabbagh [12]. The rather formidable "high-power" techniques used in obtaining these remarkable results, and the length of the proofs (W. Szmielew's proof takes about 70 pages) triggered off several attempts at simplification. M. I. Kargapolov's proof [3] unfortunately turned out to be erroneous (cf. J. Mennicke's review in the Journal of Symbolic Logic, vol. 32, p. 535).

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.

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