[6.5.] “Existential Quantifiers.” (Ch.7:4)

 

($x), the existential quantifier, is used to indicate that something, i.e., at least one thing, has a given property:

 

($x)Hx             =                      Something is happy. (I.e., at least one thing is happy.)

($x)Sx              =                      Something is shiny.

($x)Px              =                      Something is perfect.

 

As with universal quantifiers, existential quantifiers can be combined with tildes:

 

~($x)Hx            =                     It is not the case that something is happy, i.e., nothing is happy.

($x)~Hx            =                     Something is unhappy.

 

[Important -- these two sentences are not equivalent!]

 

You can use the existential quantifier to predicate more than one property at a time:

 

($x)(Hx · Sx)    =                      Something is happy and shiny.

Some happy thing is shiny.

Some shiny thing is happy.1

 

($x)[(Hx · Sx) · Px]  =              Some happy things are shiny and perfect.

Some things are happy, shiny and perfect.

 

The existential quantifier signifies any quantity greater than none and less than all:

 

($x)(Sx · Hx)    =                      A few shiny things are happy.

There are several shiny, happy things.

A number of shiny things are happy.

Many shiny things are happy.

Lots of shiny things are happy.

 

NOTE: Just as it is most common for a universal quantifier directly to bind a conditional, it is most common for an existential quantifier directly to bind a conjunction.

 

See examples on p.175.

 

Exercise 7-5, p.176

·         do all of these; we’ll go through the odds next time

 

 

 

Stopping point for Wednesday March 5. For next time, do ex.7-5, and read ch.7:5-7 (pp.176-80)