Chapter Seven, Tutorial Three
English (and most all natural languages) have complicated quantifier procedures. Fortunately, much of this complication can be untangled when symbolizing by utilizing the existential and universal forms ("Some P are Q" and "All P are Q" respectively).
Lets continue to take a universe of discourse to be people at O.U. Among these people are two students: Jenny and Tom.
We may use 'j' and 't' respectively as PL names for the two. Let 'Mx' stand for "x is a male", 'Sx' for "x is a student", and 'Fxy' stand for "x is a friend of y".
1) Every male student is a friend of Jenny.
The "every" indicates a universal quantification. It's pretty easy to see how to fit this into the universal form mold:
(^x)( x is a male student > x is a friend of Jenny )
Then we need only symbolize the antecedent and consequent. But to say "x is a male student" is to say that x is both male and a student: 'Mx&Sx'. So, 1)'s symbolization is:
(^x)( (Mx&Sx) > Fxj )
2) Each male friend of Tom is a friend of Jenny's.
Here too we need to fit the universal mold:
(^x)( x is a male friend of Tom > x is a friend of Jenny )
To say "x is a male friend of Tom" is to say that x is male and a friend of Tom: '(Mx&Fxt)'. So, 2)'s symbolization is
(^x)( (Mx&Fxt) > Fxj )
3) A male friend of Tom is a friend of Jenny's.
This is like 2) but is of existential form. So, to fit the mold, we have:
(%x)( x is a male friend of Tom & x is a friend of Jenny )
Which is symbolized as
(%x)( (Mx&Fxt) & Fxj )
Finally, let 'i' be a name of myself (the author) and consider
4) A friend of Tom is a friend of mine.
Notice that this could easily be taken as either of universal or existential form. It might mean any friend of Tom is my friend too, or just that some is. So, 4) is ambiguous and can be symbolized as either
Symbolization I of 4: (^x)( Fxt > Fxi )
Symbolization II of 4: (%x)( Fxt & Fxi )Keep in mind that symbolizations I and II mean quite different things corresponding to two different readings of 4).
Now, you try a harder one: which of the following is a good symbolization for "Jenny is a friend of any friend of Tom"?