Formalization of functionally complete propositional calculus with the functor of implication as the only primitive term

Studia Logica ◽  
1989 ◽  
Vol 48 (4) ◽  
pp. 479-494 ◽  
Author(s):  
Czes?aw Lejewski
Keyword(s):  
Propositional Calculus ◽  
Primitive Term

Relation of Leśniewski's mereology to Boolean algebra

1974 ◽  
Vol 39 (4) ◽  
pp. 638-648 ◽  
Author(s):  
Robert E. Clay
Keyword(s):  
Boolean Algebra ◽  
Final Stage ◽  
Propositional Calculus ◽  
Ordinary Language ◽  
Complete Boolean Algebra ◽  
Primitive Term ◽  
Logical Foundation ◽  
Formal Relations

It has been stated by Tarski [5] and “proved” by Grzegorczyk [3] that: (A) The models of mereology and the models of complete Boolean algebra with zero deleted are identical.Proved has been put in quotes, not because Grzegorczyk's proof is faulty but because the system he describes as mereology is in fact not Leśniewski's mereology.Leśniewski's first attempt at describing the collective class, i.e. mereology, was done in ordinary language with no rigorous logical foundation. In describing the collective class, he needed to use the notion of distributive class. So as to clearly distinguish and expose the interplay between the two notions of class, he introduced his calculus of name (name being the distributive notion), which is also called ontology, since he used the primitive term “is.” At this stage, mereology included ontology. Then, in order to have a logically rigorous system, he developed as a basis, a propositional calculus with quantifiers and semantical categories (types), called protothetic. At this final stage, what is properly called mereology includes both protothetic and ontology.What Grzegorczyk describes as mereology is even weaker than Leśniewski's initial version. To quote from [3]: “In order to emphasize these formal relations let us consider the systems of axioms of mereology for another of its primitive terms, namely for the term “ingr” defined as follows:A ingr B.≡.A is a part B ∨ A is identical to B.The proposition “A ingr B” can be read “A is contained in B” or after Leśniewski, “A is ingredient of B”.”

Corrections for "On the lengths of proofs in the propositional calculus preliminary version"

1974 ◽  
Vol 6 (3) ◽  
pp. 15-22 ◽  
Author(s):  
Stephen Cook ◽  
Robert Reckhow
Keyword(s):  
Propositional Calculus ◽  
Preliminary Version

scholarly journals Preface

2011 ◽  
Vol 21 (4) ◽  
pp. 671-677 ◽  
Author(s):  
GÉRARD HUET
Keyword(s):  
Mathematical Logic ◽  
Computer Science ◽  
20Th Century ◽  
Theorem Proving ◽  
Propositional Calculus ◽  
Interactive Theorem Proving ◽  
Predicate Calculus ◽  
Special Issue ◽  
Mathematical Structures ◽  
Infinite Domains

This special issue of Mathematical Structures in Computer Science is devoted to the theme of ‘Interactive theorem proving and the formalisation of mathematics’.The formalisation of mathematics started at the turn of the 20th century when mathematical logic emerged from the work of Frege and his contemporaries with the invention of the formal notation for mathematical statements called predicate calculus. This notation allowed the formulation of abstract general statements over possibly infinite domains in a uniform way, and thus went well beyond propositional calculus, which goes back to Aristotle and only allowed tautologies over unquantified statements.

Complete Connectives for the 3-Valued Propositional Calculus

1966 ◽  
Vol s3-16 (1) ◽  
pp. 167-191 ◽  
Author(s):  
Roger F. Wheeler
Keyword(s):  
Propositional Calculus

A Lattice-Diagram for the Propositional Calculus

1962 ◽  
Vol 46 (356) ◽  
pp. 119 ◽  
Author(s):  
John Evenden
Keyword(s):  
Propositional Calculus

scholarly journals Metalogic of Intuitionistic Propositional Calculus

2010 ◽  
Vol 51 (4) ◽  
pp. 485-502 ◽  
Author(s):  
Alex Citkin
Keyword(s):  
Propositional Calculus ◽  
Intuitionistic Propositional Calculus

On simple, weak and strong models of propositional calculi

1973 ◽  
Vol 74 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Ronald Harrop
Keyword(s):  
Simple Model ◽  
Propositional Calculus ◽  
Finite Model ◽  
Model Property ◽  
Finite Model Property ◽  
Finite Structure ◽  
Propositional Calculi ◽  
Simple Models

In this paper we will be concerned primarily with weak, strong and simple models of a propositional calculus, simple models being structures of a certain type in which all provable formulae of the calculus are valid. It is shown that the finite model property defined in terms of simple models holds for all calculi. This leads to a new proof of the fact that there is no general effective method for testing, given a finite structure and a calculus, whether or not the structure is a simple model of the calculus.

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