Reasoning: An Introduction to Logic, Sets, and Functions

Subsection 4.1.3 The Building Blocks of Statements

Just as in Boolean logic:

• A statement is a logical expression that has a truth value (True, abbreviated T or False, abbreviated F).

• We’ll build complex logical expressions by starting with some simple building blocks and then combining them.

Two key differences between predicate logic and Boolean logic are:

• In Boolean logic, our building blocks were complete logical statements. For example, we gave the name W to the claim, “The sidewalks are wet.” In predicate logic, we’ll start with objects, like Smokey, and then build up our claims about them.

• In Boolean logic there was no way to generalize and make a single claim about many different objects (except by explicitly conjoining statements about each of them as individuals). In predicate logic, the quantifiers ∀ and ∃ will let us make such general claims in a single statement.

Now we’re ready to start with the building blocks of predicate logic. We’ll use:

• Objects drawn from our universe of discourse.

• Variables that may take on as their values any of the objects that we can talk about.

• Functions that apply to objects and return objects.

• Predicates, which are a special kind of function. A predicate must return a Boolean value (i.e., T or F). The job of objects and functions is to provide fodder to predicates. Predicates enable us to build logical expressions that we can reason with.