This module introduces you to basic ideas about the quantifiers. The are two basic facts about the quantifiers you need to understand and from which all of the logical properties of the quantifiers follow: Basic Fact 1: A universal quantifier (x) Fx is truth-functionally 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 truth-functionally equivalent to an infinite disjunction Fa v Fb v Fc v Fd v ...... where a, b, c, d, ... are the names of objects in the universe picked out by the 'x' in the existential quantifier (Ex) These two basic facts express the idea of an expansion of a predicate logic wff. Fa v Fb v Fc ...is the expansion of the predicate logic wff (x)Fx. When you expand a predicate logic wff, you drop the quantifier, and replace the bound variables by the names of objects in the universe over which the quantifier ranges. The 'x' in (x) and in (Ex) is not a free or bound variable. It (i) names the universe over which the quantifier ranges and (ii) names the bound variables used in the matrix to its right. In (x) Fx, the term Fx is the matrix and x in Fx is the bound variable.

1. Expand in a two-object universe (the objects are named 'a' and 'b')

(a) (x) ((Ax & Bd) v Ky)

(b) (x) ~ (Dx Cy)

(c) ~(Ex) (Dy & (Hx ~Ga))

Expanding a wff is not the same as reducing the wff to conjunctive normal form (or reducing it to disjunctive normal form). Read the text and my handout on how to do an expansion.

2. For the following wffs, indicate which variables are free and which are bound (you can use 'F' for free and 'B' for bound.) Either (i) draw a vertical line underneath each variable with the letters 'F' or 'B' at the bottom of each vertical line or (ii) color bound variables red and free variables green. Make sure you know which symbols are used for variables. Not every symbol in the language of predicate logic is used for variables. The reading (Scope, binding, and quantifier expansions) lists the symbols that are used for variables and the symbols that used for the names of objects (i.e., individual constants). Names of objects are not names of variables. The free/bound distinction only applies to variables. It does not apply to individual constants (i.e., names of objects)

(a) (Ex) (y) (z) ((Ayz Bzzy) v (Fxac Hzzu))

(b) (x) (y) Hyyy (z) (Fzy v Hxx)

(c) (Ez) (x) (y) (Axyzbbw v Bxycdvz)