Question

The two basic facts about the quantifiers you need to understand, and from which all of the logical properties of the quantifiers follow are:

Basic Fact 1: A universal quantifier (x) Fx is equivalent to an infinite conjunction: Fa & Fb & Fc & Fd & ........

where a, b, c, d, are the names of objects in the universe picked out by the 'x' in the universal quantifier '(x)'.

Basic Fact 2: An existential quantifier is equivalent to an infinite disjunction

Fa v Fb v Fc v Fd v ......

Expand in a two-element universe

(a) ~(x) ((Fx v Gy) v Ka)

Answer 1

Using a restricted universe of discourse to determine whether a given formula is true or false requires two additional bits of information: a truth fractional 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	+	-	+	-