Truth Functional Expansions and

Interpretations of Quantified Formulas

 

Every quantified formula ranges over a universe of discourse, that is, a set of objects which either have or lack the properties or relations used in the formula.  Sometimes the universe of discourse for a quantified formula is the entire universe as we know it.  Other times the universe may be somewhat or excessively restricted.  Whenever the universe of discourse is restricted, the formula in question is taken to apply only to those objects in the universe of discourse--all other objects can be ignored.  When dealing with formulas taken from mathematics, we often limit the universe to, say, the natural numbers.  Doing so greatly simplifies the formula, it no longer being necessary to specify that an item being talked about is a natural number (as opposed, for instance, to a real number, an irrational number, a cat, an elephant, etc.).  Another way in which a universe of discourse can be restricted is to specify a finite set of objects (or elements) as the universe.  One might, for instance, limit the universe of discourse to three elements (objects) a, b, and c, without needing to say any more about those objects.  Any quantified formula will be true or false of that universe depending on whether the elements {a, b, c} have or lack the properties used in the formula in the manner specified by the formula.

Using a restricted universe of discourse to determine whether a given formula is true or false requires two additional bits of information:  a truth-functional expansion of the formula across the universe; and an interpretation of the extension of the predicates used in the formula with respect to the elements in the universe.  A truth-functional expansion of a formula tells you what properties the elements in the universe must have in order for the formula to be true of that universe.  Consider the formulas:

 

1.  (x)(Fx Gx)

2.  (x)(Fx Gx)

 

The truth-functional expansions of these formulas over a three element universe {a, b, c} produce:

 

1a.  [(Fa Ga) (Fb Gb)] (Fc Gc)

2a.  [(Fa Ga) v (Fb Gb)] v (Fc Gc)

 

which say, respectively, that if 'a' is an F then it is a G, and so on for 'b' and 'c', and either 'a' is both F and G, or 'b' is, or 'c' is.  We are not yet in a position, however, to determine whether either formula is true of the universe, we still need to know which elements are F and which are G.  That information is provided by an interpretation of the extension of the predicates.

Suppose that 'a' is both F and G, 'b' is not F but is G, and 'c' is F but not G.  We can represent this information in a table as follows:

F          G

a          +          +

b          -           +

c          +          -

in which case 1 is false in the universe {a, b, c} given the interpretation, because one conjunct of 1a is false.  2, however, is true in the universe {a, b, c}  given the interpretation because the first disjunct of 2a is true.

 

NOTE:  The truth-functional expansion of a universally quantified proposition is a conjunction of the instances, the truth-functional expansion of an existentially quantified proposition is a disjunction of the instances.


Use the following chart as an interpretation of the three element universe {a, b, c} and determine which, if any of the following formulas is true in the universe as interpreted.

  Element    F G H I
a + + +
b + + -
c + - +

                                                                

1)         (x)(Fx Gx) 

2)         (x)(Fx (Gx Hx))

3)         (x)(Fx (Gx v Hx))

4)         (x)(~Ix ~Fx)

5)         (x)(Fx (~Gx v ~Hx))

6)         (x)(Gx (Hx ~Ix))

7)         (x)Fx (y)~Gy

8)         (x)Fx (y)(Gy Iy)               Solutions

9)         (x)Fx (y)Hy