HELENA RASIOWA, 1917 - 1994
by
W. Bartol, E. Orłowska, A. Skowron
Algebraic Logic
The study of the relationship between logic and algebra, originated by the work of George Boole and continued by A.Lindenbaum, A.Tarski, M.H.Stone, R.McKinsey, A.Mostowski and L.Henkin - to mention only a few - was the main research subject of Helena Rasiowa. One of the crucial moments in the development of algebraic study of logic was the introduction of Lindenbaum's and Tarski's method for treating equivalence classes of formulas as elements of an abstract algebraic system. This provided a link between theories based on classical logic and Boolean algebras. An analogous correspondence between intuitionistic logic and pseudo-complemented lattices was established by Stone and Tarski, while the application of algebraic methods to modal logics was originated by McKinsey and Tarski (1944 - 1948). Another important idea leading to the emergence of the field of algebraic logic was the treatment of formulas as algebraic functions in certain algebras, initiated by Ł ukasiewicz and Post with their generalization of truth tables.
Helena Rasiowa became very active in these areas in the early fifties. Her research work on algebraic logic was aimed at finding a precise description for the mathematical structure of formalised logical systems. Her early papers contain several examples of algebras associated with logical systems as well as algebraic proofs of some of their properties. Jointly with Roman Sikorski she presented the first algebraic proof of the Gödel completeness theorem for classical predicate logic ([4]). Next, she algebraically proved analogous theorems for intuitionistic and modal logics ([5]). In the papers [9, 13, 14, 16, 18, 21, 22, 23, 24, 25, 26] she turned to other nonclassical logics, applying with success the algebraic tools she had developed. All these investigations led her to a synthesis of the results hitherto obtained; she developed a general framework for an algebraic presentation of propositional and first order logics. In particular, she established elegant formal techniques for associating classes of algebras with logical systems and for providing their algebraic semantics. In the following we shall briefly outline the basic elements of her method for an algebraization of logics.
Rasiowa's first monograph The Mathematics of Metamathematics
([31]) written jointly with R.Sikorski, contains a comprehensive survey
of algebraic theories of logical calculi. Algebraization is applied to
classical, intuitionistic, modal and positive logics. The book focuses
on metalogical theorems on the predicate calculi of these logics and on
an investigation of elementary theories based on them. The main idea
underlying the concept of associating algebraic systems with logical
calculi is the following: Let be a first order language such that
is its set of individual variables,
are the binary propositional
connectives of
,
is a unary propositional
connective and
are the
quantifiers of
Let
be a complete algebra such that
are binary operations,
is a unary operation and
and
are infinitary operations the
common domain of which is the class of all nonempty subsets of
By a realization of the language
in a nonempty set
and in the algebra
we mean any mapping
which assigns a function
to each function symbol
of
and a function
to any predicate symbol
of
The realization
is then extended to all the terms
and all the formulas of
Let
be a valuation of individual
variables in a set
Then we define:
{} for any
individual variable
{}
{}
{}
{}
{}
{}
{} where
and
for all
{} where
is as above.
The algebraic semantics of the classical predicate logic is
determined by the class of realizations of the language in Boolean algebras. We say that
a formula is true in a realization
whenever
for every valuation
where 1 is the unit element of
the underlying Boolean algebra. A formula
is valid in the classical
predicate logic if and only if
is true in all realizations in
any Boolean algebra. Similarly, the semantics of intuitionistic logic
is given by realizations in Heyting algebras, positive first order
logic is associated with realizations in the class of relatively
pseduocomplemented lattices. Modal logic, also considered in the book,
is semantically determined by the class of realizations in what is
known as topological Boolean algebras, i.e. Boolean algebras endowed
with a unary operation behaving like an interior operator.
In the next monograph An Algebraic Approach to Non-classical
Lo-gics ([49]) Helena Rasiowa develops a general algebraization
theory for propositional logical systems. Within the framework of this
theory she presents algebraizations of a number of non-classical logics
such as: classical and positive implicative logics, logics weaker than
positive implicative logic, minimal logic, positive logic with
semi-negation, constructive logic with strong negation and many-valued
Post logics. The fundamental idea behind algebraization is the
following: Let be a propositional logic, where
is a propositional language that
includes implication as a binary connective and
is a consequence operation
determined by the axioms and inference rules. As usual, for a set
of formulas we denote by
the least set of formulas which
contains the axioms and the set
and is closed with respect to
the inference rules. In particular,
is the set of all the theorems
of
An algebra associated with
is any algebra of a type similar
to that of the algebra of formulas of
and endowed with a distinguished
element 1. Elements of such an algebra are interpreted as truth values,
whereas 1 represents the value true. Let
be the set of propositional
variables of
By a valuation in an algebra
associated with
we mean any mapping
Every formula
of the language
induces a mapping
defined as follows:
{} for any
propositional variable
,
{} for every unary propositional
connective
of
{} for every binary connective
of
A formula is said to be true in
if and only if
for every
Any propositional logic determines thus a class of
-algebras, where an
-algebra is an algebra
associated with
such that for every formula
we have
for every
The completeness theorem states
that if
is consistent, then the set
is equal to the set of all
formulas which are true in each
-algebra.
Helena Rasiowa paid very much attention to many-valued Post logics.
She introduced several generalizations of the standard -valued Post logics; she
considered the propositional and predicate versions of
-valued Post logics and used them
in her reasoning on programs. The following section describes some of
these applications in more detail.