[9.] Relational Predicate Logic

 

[9.1.] "Relational Predicates." (Ch.10:1)

 

 

[9.1.1.] Relational Properties and Property Constants.

 

All of the property constants we have used up to now have represented properties that characterize one individual at a time:

 

“Michael is tall”                        =          Tm

“Dumbo is an elephant”           =          Ed

“Someone is conceited”                       =          ($x)Cx

“All cows are mammals”          =          (x)(Cx É Mx)*

 

*Even in this example, the property being a cow and the property being a mammal are properties had by individual entities. Although there are many cows and many mammals, the property of being a cow and the property of being a mammal are possessed by one individual entity at a time.

 

But not all properties are like this. Some properties hold, not for a single individual, but between two individuals. For example:

·         being taller than (or, being shorter than)

·         being the mother of (or, being the child of, or father of, or brother of)

·         being louder than (or, being quieter than)

·         being on top off (or, being under, or being beside, or being behind)

 

And some properties hold among three or more individuals. For example:

·         being between (A is between B and C);

·         being given (A was given to B by C).

 

Such properties are relational properties.

 

We can define property constants so that they represent relational properties, e.g.,

 

Txy = x is taller than y

Sxy = x is shorter than y

Mxy = x is the mother of y

Cxy = x is the child of y

 

So if we let “a” stand for Andrew and “b” stand for Bill…

 

Tab      =          Andrew is taller than Bill.

 

And notice that the order of “a” and “b” makes a big difference. If “Txy” has been defined as “x is taller than y,” then

 

Tba      =          Bill is taller than Andew.

 

 

Another example:

 

Sxyz     =          x sat between y and z

 

So letting “c” stand for Cindy…

 

Sacb     =          Andrew sat between Cindy and Bill

Sbac     =          Bill sat between Andrew and Cindy

Scab     =          Cindy sat between Andrew and Bill

 

 

[9.1.2.] Relational Property Constants and Quantifiers:

 

Relational predicates can be integrated with quantifiers:

 

“Everyone knows Bill.”

 

To symbolize this sentence, use the following strategy:

 

1.      Ask to whom or what the sentence is attributing a property:

 

This sentence is attributing a property to all people, and this suggests that the symbolization will look like this:

 

(x)(Px É ...)    [Px: x is a person]

 

2.      Ask what property is being attributed:

 

In this case, the property that is being attributed is the property of knowing Bill.

 

The sentence means: for all x, if x is a person, then x knows Bill. It should thus be symbolized:

 

(x)(Px É Kxb)       [Kxy: x knows y]

 

 

Other examples [all with unrestricted domains of discourse]:

 

“Bill knows everyone.”                                      (x)(Px É Kbx)

“Bill knows someone.”                                       ($x)(Px · Kbx)

“Someone knows Bill.”                                      ($x)(Px · Kxb)

“If everyone knows Bill, then someone does.”     (x)(Px É Kxb) É ($x)(Px · Kxb)

“Bill doesn’t know anybody.”                             ~($x)(Px · Kbx) or (x)(Px É ~Kbx)

 

[the following have a domain of discourse restricted to all people]

 

“Bill knows everybody Andy knows”                  (x)(Kax É Kbx)               or   ~($x)(Kax · ~Kbx)

“Andy knows everybody Bill knows”                  (x)(Kbx É Kax)   or   ~($x)(Kbx · ~Kax)

“Andy knows somebody Bill doesn’t know”        ($x)(Kax · ~Kbx)

 

 

Exercise 10-1 (p.224)

·         do all problems for next time; we’ll review the odds next class

·         notice that the book uses the same letter to represent a one-place property and a two-place property in a single sentence (e.g. #3... Mxy = x is married to y, and Mx = x is married).

·         notice that in #20 there is a 4-place predicate!

 

 

[10.2.] “Symbolizations Containing Overlapping Quantifiers.” (Ch. 10:2)

 

Recall that the scope of a quantifier is the part of a sentence in which variables may be bound to that quantifier. For example...

 

(x)Fx                                        scope of “(x)” = “Fx”

(x)(Fx É Gx)                             scope of “(x)” = “(Fx É Gx)”

(x)(Fx É Gx) Ú Gx                    scope of “(x)” = “(Fx É Gx)”  [scope does not include “Gx”]

 

It is possible for an expression to contain two or more quantifiers the scopes of which overlap.

 

E.g. (these examples come from your textbook-- p.225)

 

[domain of discourse limited to persons]

 

Everyone loves everyone.          (x)(y)Lxy

 

I.e., for any two members of the DOD you happen to choose, one loves the other.

 

Someone loves someone            ($x)($y)Lxy

           

I.e. there is at least one individual in the DOD that loves at least one individual in the DOD.

 

Not everyone loves everyone.    ~(x)(y)Lxy

 

I.e., it is not the case that for any two members of the DOD you happen to choose, one loves the other.

 

No one loves anyone.                ~($x)($y)Lxy    or         (x)(y)~Lxy

 

I.e., it is not the case that there is at least one individual in the DOD that loves at least one individual in the DOD; i.e., for any two members of the DOD you happen to choose, it is not the case that one loves the other.

 

 

 

 

Exercise 10-2 (pp.227-28)

·         [WE DID ALL OF THESE IN CLASS]

 

 

Exercise 10-3, pp.229-30

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

·         look out for #17 -- it’s relatively difficult

·         there is an important error throughout this exercise: it needs a separate constant to stand for the property of being a good student; just assume that being good and being a student are, together, equivalent to being a good student, so that you don’t need a separate constant for “x is a good student”

 

 

Stopping point for Wednesday April 9. For next time, complete ex. 10-1 and 10-3, and read ch.10:4 (p.230).