Chapter Seven, Tutorial Two
Natural language provides many ways to quantify over objects. Often these involve very complicated constructions. Fortunately, these complications are often founded on just two basic forms. In this tutorial we focus on these two forms, the "existential" and the "universal", and show how these give rise to categorical logic. In the next tutorial, we will see how the two forms can be seen as the basis for more complicated symbolizations.
The Existential Form
The existential form is not really new to you. Any English sentence like "Some prime number is even" or "Some students are freshmen" are simple examples of the form. In general, an English sentence is of the form if it fits this mold:
Some S are P.
where 'S' (the subject) and 'P' (the predicate of the expression) range over English predicates. The predicates stand for categories (like students and freshmen).
(Unfortunately, the term "predicate" is widely used in logic both for descriptive phrases in general and for P, the last clause of a categorical sentence like "Some S are P".)
Now, it's pretty easy to see how to symbolize sentences of this form. For instance,
Some whales are living in Ohio
may be symbolized as
assuming that 'Wx': "x is a whale", 'Ox': "x is living in Ohio".
Now, many English sentences are not exactly of this existential form but are close enough. An example we used in the last chapter was:
There is an even number less than three
This has the same meaning as "Some even number is less than three" so might be symbolizes as
Another familiar example
Someone will attend law school and need a loan.
is a simple stylistic variant of "Some persons who will attend law school are persons who will need a loan" and can be symbolized as
Once again, notice that all the English sentences highlighted on this page can be paraphrased as of the existential form:
Some S are P.
Each can be symbolized as of the form
(%x)(Sx & Px)
One may usefully force many English sentences into this form!
Now, given the symbolization used above, which of the following might symbolize "Some even number is prime"?