# Modal Environment for Boolean Speculations

preliminary report
• George Gargov
• Solomon Passy
• Tinko Tinchev
Chapter

## Abstract

The common form of a mathematical theorem consists in that “the truth of some properties for some objects is necessary and/or sufficient condition for other properties to hold for other objects”. To formalize this, one happens to resort to Kripke modal logic K which, having in the syntax the notions of ‘property’ and ‘necessity’, appears to provide a reliable metamathematical fundament. In this paper we challenge this reliability. We propose two different approaches each claiming better formal treatment of the state of affairs. The first approach is in formalizing the notion of ‘sufficiency’ (which remains beyond the capacities of K), and consequently of ‘sufficiency’ and ‘necessity’ in a joint context. The second is our older idea to formalize the notion of ‘object’ in the same modal spirit. Having ‘property, object, sufficiency, necessity’, we establish some basic results and profess to properly formalize the everyday metamathematical reason.

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© Plenum Press, New York 1987

## Authors and Affiliations

• George Gargov
• 1
• Solomon Passy
• 1
• Tinko Tinchev
• 1
1. 1.Faculty of Mathematics at Sofia UniversitySofiaBulgaria