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Jump to navigationJump to search→1.3.10.3. Propositions and Sentences: segment
their informal comprehension can be converted into an explicit subject matter,
their informal comprehension can be converted into an explicit subject matter,
one that depends on grasping an array of increasingly formalized concepts.
one that depends on grasping an array of increasingly formalized concepts.
| NB. In the following asciification of a pre-existing text,
| NB. In the following asciification of a pre-existing text,
of distinction, respectively, between the boolean values of the
of distinction, respectively, between the boolean values of the
entities in question.
entities in question.
In logical studies, one tends to be interested in all of the
In logical studies, one tends to be interested in all of the
the values of its putative arguments, is not as variable in its values as
the values of its putative arguments, is not as variable in its values as
the values of its variables are allowed to be.
the values of its variables are allowed to be.
The "indicator function" or the "characteristic function" of a set Q c X,
The "indicator function" or the "characteristic function" of a set Q c X,
subset of its designated universe, and even though one will eventually,
subset of its designated universe, and even though one will eventually,
most likely, want to know exactly what subset that is.
most likely, want to know exactly what subset that is.
According to the stated understandings, a proposition is a function that
According to the stated understandings, a proposition is a function that
[| f |] = f^(-1)(%1%) = {x in X : f(x) = %1%}.
[| f |] = f^(-1)(%1%) = {x in X : f(x) = %1%}.
The definition of a fiber, in either the general or the boolean case,
The definition of a fiber, in either the general or the boolean case,
that is identified by the sign "f(x)", and thus an object that might
that is identified by the sign "f(x)", and thus an object that might
as well be identified with the value f(x).
as well be identified with the value f(x).
The upshot of this whole discussion of evaluation is that it allows one to
The upshot of this whole discussion of evaluation is that it allows one to
idea of an "assertion" and notion of an "equation", and it allows one
idea of an "assertion" and notion of an "equation", and it allows one
to treat logical equality on a par with the other logical operations.
to treat logical equality on a par with the other logical operations.
As a purely informal aid to interpretation, I frequently use the letters
As a purely informal aid to interpretation, I frequently use the letters
since the apodosis is intended to do nothing more than to abbreviate and
since the apodosis is intended to do nothing more than to abbreviate and
to summarize what is already stated in the protasis.
to summarize what is already stated in the protasis.
A "boolean connection" of degree k, also known as a "boolean function"
A "boolean connection" of degree k, also known as a "boolean function"
any sign that is taken to denote an indicator function, or any sign whose
any sign that is taken to denote an indicator function, or any sign whose
object happens to be a function of the form f : X -> %B%.
object happens to be a function of the form f : X -> %B%.
An "expression" is a type of sign, for instance, a term or a sentence,
An "expression" is a type of sign, for instance, a term or a sentence,
Indeed, there are problems involved in this whole style of definition
Indeed, there are problems involved in this whole style of definition
that need to be discussed, and doing this requires a slight excursion.
that need to be discussed, and doing this requires a slight excursion.
</pre>
=====1.3.10.4. Empirical Types and Rational Types=====
1.3.10.4. Empirical Types and Rational Types
<pre>
In this Subsection, I want to examine the style of definition that I used
In this Subsection, I want to examine the style of definition that I used
to define a sentence as a type of sign, to adapt its application to other
to define a sentence as a type of sign, to adapt its application to other
