In our study of sentential logic we learned how to symbolize some sentences so
as to be able to check certain arguments for validity. But sentential logic is
limited to the structures or forms generated by the presence of
truth-functional connectives. Consider, for example, the following very simple
argument:
Every whale is a mammal.
Willy is a whale.
So, Willy is a mammal.
This is clearly a valid argument but, as we have noted before, operating within
a system of sentential logic we could only symbolize such arguments in s some
way similar to the following:
A
B
So, C
We do not, in sentential logic, have any means of representing the common
elements in the internal structure of these sentences. This does not show that
there is something wrong with sentential logic; it merely indicates something
we have already noted, that sentential logic is limited. The above argument
regarding Willy, although valid, is not truth-functionally valid. When we
looked at categorical logic we did indeed "look inside" some sentences, but the
techniques we developed there were very limited.
What we will here do is to develop a more advanced branch of logic, often
called either "predicate logic" or "quantification theory," which terms we
shall use interchangeably. Our goal will be to present a complete system of
quantification theory, but one that will be restricted in certain ways. We will
emphasize only the more elementary parts of the system. Nonetheless we will,
with this system, be able to assess the validity of a much broader range of
arguments than we could before.
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Section 1: The Basics of Quantification TheoryThe structure of the language we will develop in this topic will be much more complex than that of the language used for sentential logic. This is to be expected, since we are now concerning ourselves with the internal complexity of the sentences we will be symbolizing.
Just as we did in sentential logic, we relate actual statements to our formal
language by way of symbolizing them. So we shall look at some examples
symbolizing sentences. Here we will focus on one-place predicates
Here we spoke of various properties of relations, in particular transitivity,
symmetry and reflexivity. We also introduced identity, a relation of extreme
importance in reasoning.
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Section 2: Further Questions of Interpretation.
In this section we note that the construction of worlds is sometimes helpful if we are to determine what a given quantificational sentence means. And we noted that quantificational logical truths are those that come true in every interpretation in a non-empty domain. Arguments are quantificationally valid if the conclusion is true in every interpretation in a non-empty domain in which all the premises are true. |
Covering Section 1 |
Covering Section 2 |