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T.E.Mieczyslaw Omyla |
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Torun 21 IX 1987 r.) |
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1. The ontology of situations is a theory
in a so called first order language,
i. e.
language containing only one kind of
variables.
2. The ontology is formulated in a language
containing only logical symbols, i.
e. variables
and logical constants.
3. The theory describes only necessary facts
from a universe of situations earlier
presupposed.
I borrowed the term ontology of situations
from the title of [8], but this term
is understood
in a little different way here. If
the alphabet
of given language L does contain variables
which run over the universe of situations
UL , then no explicit theorems can
be stated
in it in a formalized way referring
to the
universe of situations associated with
the
language. To make it possible the alphabet
of this language should contain, among
others,
the following symbols: the identity
connective,
variables running over a given universe
of
situations and quantifiers binding
these
variables. According to Suszko - the
author
of non-Fregean logic - the variables
running
over the universe of situations are
fundamentally
different from other kinds of variables,
because the former are sentential formulas
and only sentential variables can be
substituted
for them. Nominal variables cannot
run over
the universe of situations. They run
over
the set of their reification equivalents.
These remarks lead to the following
notion
of a language of sentential logic:
L is a
language of sentential logic iff L
is a set
of formulas built, in the usual way,
of three
kinds symbols: statement letters, connectives,
and formator binding statement letters
(for
example quantifiers, description operator
and so on). To a language of sentential
logic
two interpretations usually apply:
(i) referential, when statement letters are
variables running over a certain universe,
(ii) substitutional, when statement letters
are schematic letters representing
sentences
of some languages or are abbreviations
of
sentences. Here are some examples of
referential
interpretations of a language of sentential
logic:
1. The Fregean interpretation of the classical
sentential logic. Sentential variables
run
over a two-elements universe: Truth,
and
Falsehood.
2. Lukasiewicz's and Post's interpretation
with sentential variables run over
a set
of logical values.
3. Wittgenstein's and Suszko's interpretation
with sentential variables running over
a
universe of situations.
4. Kripke's and Kit Fine's interpretation
with sentential variables running over
a
set of possible worlds. An important
example
of a language of sentential logic is
the
language of non-Fregean sentential
logic.
The formulas of the language are built,
in
the usual way, by means of the following
symbols: sentential variables: p, q,
r,...,
truth-functional connectives: Ø (negation),
Ù (conjunction), Ú (alternative), Þ
(implication),
Û (equivalence), identity connective:
º and
quantifiers: ", $ binding sentential
variables. Non - Fregean logic was
created
by Suszko under the influence of Wittgenstein's
Tractatus Logico-Philosophicus in 1968.
In
honour of Wittgenstein Suszko named
languages
in which the logic is defined , W-
languages.
In the alphabet of these languages
there
are, among other symbols, two kinds
of variables:
one running over a universum of situations
and the other one running over a unversum
of objects. Thus, the language of non-Fregean
sentential logic is such a fragment
of W-languages
which does not contain any nominl formulas.
If to the axioms of non-Fregean logic
the
axiom:
(p º q) º(p Û q) is added (called by Suszko
the ontological version of the Fregean
axiom)
then the truth-functional logic is
obtained
with two equivalence connectives, which
are
only graphically different but interchangeable
in any context without altering the
theorems
of any theory containing Fregean axiom..
The logical bivalence of non-Fregean
logic
is expressedby theorems: Ø( p Û Ø p)
p ÚØ
p
From them and the Fregean axiom it follows
that in classical logic variables run
over
two-elements sets which may be identified
with the set of logical values. In
the language
of non-Fregean sentential logic it
is possible
to formulate theorems related to any
universe
situations but in the language of the
classical
sentential logic only theorems releted
to
two-elements Boolean algebra can be
formulated.
Semantics for non-Fregean logic was
created
by S. L. Bloom and R. Suszko in their
papers
[1],[2] and [7]. According to the semantics
the models for the language of non-Fregean
sentential logic are structures: M
=(U, F)
where: U is a generalization of SCI-
algebra
on the given set U, and F is suitable
subset
of U. In any model M the logical constants
have the intended interpretation, therefore
any model for the language of sentential
logic will be treated as a formal representation
of a certain universe of situations
with
a distinguished set of of facts in
it. In
order to simplify the formulations
a generalized
algebra of any model M for non-Fregean
sentential
will be called the algebra of situations,
and the set F will be called the set
of all
facts occuring in this model. Algebra
of
situations is the same as what Suszko
named
semimodels. Let M be
be the family of all models for the language
of non-Fregean sentential logic L determined
by the algebra of situations U. Models
of
the class K are determined by the same
set
of situations. Moreover they are articulated
in the same way in the language L,
and differ
at most the set of facts realized in
them.
We denote: Tr(M) - the set of all formulas
true in M of the language of non-Fregean
sentential logic, Val(U) =Ç {TR(M):
M Î K}
Definition. T is an ontology of situations in the language
of non-Fregean sentential logic L iff T is
a theory in L and there exists an algebra
of situations U such that T Ì Val( U). Symbolically:
T is OSL Û T ÎTH(L) and T Ì Val(U) where:
OSL - denotes the set of all ontology of
situations in the language of non-Fregean
sentential logic L, TH(L) - the set of all
theories in the language L. Three direct
corollaries of the definition:
1. Cn (Æ) is the smallest ontology of situations,
i. e. Cn (Æ) is an ontology situations
and
moreover every ontology of situations
includes
Cn (Æ).
2. If X is a set of equalities i. e. X =
{ aºb: a, b ÎL} such that Gn(X) is
consistent
then Cn(Gn(X)) is a certain ontology
of situations,(where
Gn(X) is the set of all generalizations
of
all formulas from the set X).
3. In the language of classical sentential
logic there is only one ontology of
situations,
i. e. the set of all formulas true
in two-elements
Boolean algebra, i. e. the set of all
tautologies
of the classical sentential logic.
References
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