Truth Functional Expansions of Formulas Using Relational Predicates

The truth functional expansion of a formula using relational predicates works just like a truth functional expansion of a formula using only monadic predicates.  Consider the 2 element universe {a,b}.

The truth functional expansion of

A.  ( x)( y)Lxy,

which says that everything in the universe bears L to something or other is:

[Laa v Lab] · [Lba v Lbb] (a bears L to either itself or b, and b bears L either to itself or to a)

The truth functional expansion of

B.  ( x)( y)Lxy,

which says that some one thing bears L to everything, is:

[Laa · Lab] v [Lba · Lbb]

Interpretations of Universes with Relational Predicates

Providing an interpretation of a universe using relational predicates is a bit more difficult than with monadic predicates.

We can provide an exhaustive list of all of the ordered sets of elements in the universe

R

aa

ba

bb

In this universe, as interpreted, both formula A and formula B are true:  everything bears L to something or other and some one thing, b, bears L to everything.

Graphic Representations of Interpretations of Universes

Alternatively, we can use a graphic representation of the extension of a predicate.  In the following diagrams, the circle represents the universe.  The numbers in the circles are the elements in the universe.  An arrow running from one number to another, 1à 2, means that 1 bears the relation in question to 2, but not that 2 bears the relation to 1 unless there is another arrow, 2à 1.

Universe I                                                                    Universe II

The formula in Universe I is FALSE because there is nothing in the universe that bears B to everything.  8 comes close, but since 8 does not bear B to itself, it does not bear B to everything.  The formula in Universe II is TRUE because the addition of the self-referential loop to 8 means that 8 now bears B to itself and everything else in the universe.

This approach can be expanded to incorporate different sorts of elements.  Instead of using just numbers as the elements in our universe, we can allow letters to stand for predicates.  Suppose S represents the students in the universe and P represents the professors and an arrow indicates that the first element has as its favorite the second (so S P indicates that P is S's favorite professor).  Consider the claim, one that professors hope is true, that every professor is some student's favorite.  Symbolically that gives us:

x(Px → ∃y(Sy Fxy))

In this universe, the formula is FALSE because P1 is no one's favorite professor.