Conjuring Propositions

Our last discussion ended after an overview of truth tables and a promise to look at quantifying a variable such as the x in continous equations. The implication, if we’ll recall, is propositional logic technology such as truth tables cannot be used to compute truth value solutions to non-propositional statements. Never fear, not all is lost. Let’s explore predicate logic and figure out how to conjure propositions out of predicate statements.

What is a Predicate?

Previously, we explored statements such as This is an essay on discrete mathematics and it was written by Dr. Jason M. Pittman. We defined such propositions because the assertion can be True or False. Further, we claimed x > 10  is not a proposition because we haven’t the foggiest idea what value x holds. By comparing the structure and definitions of the two, we find a hint as to what a predicate is in discrete math. We’ll pursue this line for a moment.

We know x as a variable. We wouldn’t be out of hand to refer to x as such. However, we can be more precise. In fact, we can draw a parallel to common grammatical structure and refer to x as a subject. Then, we’re left with greater than ten as a description of x. That my friends is a predicate.

Subjects and predicates can come in a variety of forms. Take the non-math statement, “Some computer y is powered on in the computer science laboratory”. Just like x, we don’t know which specific computer is represented by y. Consequently, we have to check each computer. That’s a good segue to our main point actually.

A Function to Turn Predicates to Propositions

Great- now we know what a predicate is compared to a proposition. Can we turn predicates into propositions though? You bet we can.

We can conjure a proposition from x > 10 by way of the function P(x). Put simply, the function assigns a value to our subject and allows us to compute the truth value of the predicate as a proposition. For example:

Let P(x) be x > 10 where P(100)

While this function may look technically challenging, all we really need to do is assign 100 to x and evaluate the new proposition. Thus x > 10 becomes 100 > 10 which is clearly True.

A small note: it would not be proper to refer to x > 10  as an equation although doing so is tempting. Equations compare values as is the case in something like x + 10 = 100.

Going Forward

So far we’ve covered the the basic building blocks of propositional and predicate logic. Before moving on to another topic, I’d like to dig deeper into analyzing and creating logic problems out of practical issues. Doing so will give us a powerful toolset to compute solutions going forward.