Chapter Six, Tutorial Five
PL for Categorical Logic

Natural language provides many ways to quantify over objects. Often these involve very complicated constructions. Think about the complicated English quantification you needed to symbolize in the exercises of 6.4.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The Existential Form

The existential form is pretty easy to symbolize in PL.

For anyone skipping over 6.2 to 6.4: Any English sentence like "Some prime number is even" or "Some students are freshmen" are simple examples of the form. 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).

Now, it's pretty easy to see how to symbolize sentences of this form. For instance,

Some whales are living in Ohio

means that there is some thing or things that are whales and that are living in Ohio. This may be symbolized as

(%y)(Wy&Oy)

(Read this as "there is a y such that y is W and y is O".)

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

(%x)(Ex&Lx)

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

(%x)(Wx&Nx)


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"?

  1. (%x)(Ex=Px)
  2. (%x)(Ex&Px)
  3. (%x)(Ex>Px)
  4. (%x)(ExvPx)