Journal of Formalized Mathematics
Volume 15, 2003
University of Bialystok
Copyright (c) 2003
Association of Mizar Users
Intuitionistic Propositional Calculus in the Extended Framework with Modal Operator. Part I

Takao Inoue

The Iida Technical High School, Nagano
Summary.

In this paper, we develop intuitionistic
propositional calculus IPC in the extended language
with single modal operator. The formulation that
we adopt in this paper is very useful not only
to formalize the calculus but also to do a number
of logics with essentially propositional character.
In addition, it is much simpler than the past
formalization for modal logic. In the first section,
we give the mentioned formulation which the author
heavily owes to the formalism of Adam Grabowski's [4].
After the theoretical
development of the logic, we prove a number of valid
formulas of IPC in the sections 24. The last two
sections are devoted to present classical propositional calculus
and modal calculus S4 in our framework, as a preparation
for future study. In the forthcoming Part II of this paper,
we shall prove, among others, a number of intuitionistically
valid formulas with negation.
The terminology and notation used in this paper have been
introduced in the following articles
[5]
[7]
[6]
[8]
[3]
[1]
[2]

Intuitionistic Propositional Calculus IPC in the Extended
Language with Modal Operator

Formulas Provable in IPC: Implication

Formulas Provable in IPC: Conjunction

Formulas Provable in IPC: Disjunction

Classical Propositional Calculus CPC

Modal Calculus S4
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 [4]
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 [8]
Edmund Woronowicz.
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Journal of Formalized Mathematics,
1, 1989.
Received April 3, 2003
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