Luis VILLEGAS-FORERO, Janusz MACIASZEK

118 LUIS VILLEGAS-FORERO, JANUSZ MACIASZEK
of reference 4 . We will represent this notion by introducing type q
(syncategorematically) into the typology, in order to define types and
categories of q-functions called from now on “quasi-intensions”, or
simply q-intensions.
We define Tarski ontological system O as a sextuple , where Τ is an abstract Ty-
pology of entities, F is a First Order Tarski Frame, (K τ,F ) τ∈Τ is a
family of Categories of entities, typified by Τ and generated by F,
is an Algebra of ontological operations ,
(R j ) j∈J is a family of type-change Rules, and is a replace-
ment system.
1.1. Typology
Let the Typology Τ, the set of types of entities, be the smallest
set such that
1. e, t ∈ Τ, where e is the type of individuals and t, the type of truth
values.
2. If , ∈ Τ, then < , > ∈ Τ, where < , > is the type of functions
from entities of type to entities of type .
3. If ∈ Τ then ∈ Τ, where q is the syncategorematic type of
infinite sequences of individuals 5 and is the type of functions
from infinite sequences to things of type .
The types can be divided into extensional and q-intensional ones.
Τ ext , the set of extensional types, is the smallest subset of Τ such that:
(1) e, t ∈ Τ ext . (2) if , ∈ Τ ext , then < , > ∈ Τ ext . The set, Τ q , of
q-intensional types is therefore Τ ⎯Τ ext , and Τpq, the set of types of
proper q-intensions, is the smallest subset of Τq such that if ∈ Τ, then
∈ Τpq. In any case, is the type of q-intensions corresponding
to type entities. We can define as basic types of q-intensions
the type (q-propositions, and the type (q-individual concepts
).
Among the types defined we can distinguish between the so-called
relational and the q-relational types. Τ rel , the set of relational types is
4 They play an analogous role to possible worlds of the modal ontology
from in Carnapian tradition.
5 q will play a role analogous to that of Montague's [21] s, which corresponds
to possible worlds.