Part Eight: introduction to predicate logic

 
 

Introduction

What is predicate logic? Here’s a quick answer that serves more as foreshadowing than as explanation: it’s a type of logic that breaks down sentences into smaller components - subjects and predicates. To unpack this, let’s reiterate the basics of sentential logic. 

Consider this sentence/proposition: “Uncle Tony is a war criminal.” Sentential logic treats the whole sentence as its atomic component. We could symbolize it simply as T. We could make the sentence more complex by appending other sentences to it via boolean operators, e.g. “Tony is a war criminal and he is a pescetarian.” In sentential logic, this complex sentence consists of two atoms - two whole sentences: “Uncle Tony is a war criminal” (call it T) and “Uncle Tony is pescetarian” (call it P). Both of these sentences, T and P, would be perfectly fine on their own. This is how we’ve been treating sentences so far - as atoms. 

In predicate logic, we divide sentences like T and C each into two smaller components: 1) a subject; and 2) a property that the subject has - the predicate.

Let’s give T and P predicate logic treatment. For each of these sentences, what’s the subject and what’s the predicate (property)? Uncle Tony is the subject in both sentences, but he has a different property ascribed to him in either sentence. In T, he has the property (predicate) of being a war criminal; in P, he has the property of being pescetarian. 

For this part of the tutorial, we’re going to focus on the syntax of predicate logic. Let’s get into it.

 

Cast of characters

 
 
tony-card-3-compressed.png
 
 

The basics

Symbolizing sentences

As mentioned in the introduction, there are two basic constituents to a simple sentence in predicate logic, the subject and the predicate. Let’s take the sentence “Uncle Tony is pescetarian”. We will symbolize it as Pt, where P is the predicate (being pescetarian) and the lowercase subscript t is Uncle Tony - the subject. The t is a non-logical symbol called an individual constant (to be contrasted with an individual variable): its function is to be interpreted as the name of some particular subject, in this case, Uncle Tony. They are thus arbitrary, but it's a good idea to pick the first letter in the name so that it's easier to remember.

The P - the predicate - can be read as "___ is pescetarian". When it has a name attached to it (such as Pt) you have a singular sentence. If it doesn't have a name attached, then it's not a sentence. When we want to talk about the predicate alone, without a particular name attached to it, we don't just write P; we have to put an individual variable as a placeholder, e.g. Px, to indicate that P is a property of something. The symbol given for the variable is also arbitrary, but mathematics conventions is always nice (so x, y, z are good candidates). Note that Px is not the same as a singular sentence, because it doesn't have a particular name attached to it. It cannot, on its own be true or false; only when you have a specific name/subject attached to it does it become a sentence with a true or false meaning. We will make use of these standalone predicates when we want to talk about something general, or where the subject is unspecified. Another word for predicates like Px is propositional function. We will use the terms predicate and propositional function interchangeably.

Examples

Let’s look at some examples. These will mostly be compound sentences. I’ll provide a key for the predicates here:

 
  • Dx = x has a dark past
  • Jx = x will know justice
  • Lx = x is locked away
  • Rx = x pays reparations
  • Ix = x is who he says he is
  • Sx = x sold Scatman John bootleg CDs in the early 2000s
  • Nx = x seems like a nice person
  • Fx = x made a fortune through morally questionable means
  • Tx = x can be trusted
 
Notice the two equivalent forms on point 5. Recall conditional exchange

Notice the two equivalent forms on point 5. Recall conditional exchange

 
 

Part Eight wrap-up

What we have learned so far

In this section of the tutorial, we dipped our toes into the lukewarm swamp that is predicate logic. We learned that predicate logic breaks down sentences into two basic components - subjects and predicates. This is unlike propositional logic, which simply treats a sentences as an atom. This will allow us to construct more low-level arguments and proofs.

Where are we going next?

There’s still a bit of exposition needed before we start working with predicate logic. So, in the next tutorial we’ll introduce some new methods and rules.