Predicate Logic

 

 

Goals for Unit IV

At the beginning of Unit III, I posed some lingering problems that still needed to be worked out. We actually dealt with one pretty swiftly, demonstrating that Stoic valid argument forms really were always valid. I reproduce that list here:

 

 

With regards to #1, our goal will be to unify Aristotle's categorical logic with Stoic truth-functional logic. Only then will we realize whose logic is more fundamental. With regards to #2, we will develop a mechanism through which we can distinguish between the Aristotelian existential viewpoint and the Boolean hypothetical viewpoint. Lastly, as we complete our survey of the history of first-order logic, we will discover whether or not logicism is true.

All this, however, begins with PL (short for predicate language), the logical language featured in this unit.¹

 

Important Concepts

 

 

 

 

Symbolizing atomic sentences

Let's begin mastering PL. As you learned in the Important Concepts above, PL breaks down natural language into just two components: singular terms and general terms. And yet, all propositional content can be expressed with just these two ingredients. In other words, propositions are simply a combination of one or more singular terms and one or more general terms. Let's begin with an atomic sentence, i.e., a sentence with a singular term as a subject and one (or more) general term(s) in the predicate.

Take as an example, "The tallest building in Chicago has more than 90 floors.” Recall that singular terms can be either proper names or definite descriptions. In this example, the singular term "the tallest building in Chicago" is a definite description, since there is only one thing in the world that "the tallest building in Chicago" could refer to. The second part of the sentence "____ has more than 90 floors" is a general term. This is because this predicate could be applied to many buildings in the world, such as the Empire State Building and Shanghai Tower (both of which have over a hundred floors). And so this atomic sentence is composed of one singular term ("the tallest building in Chicago") and one general term ("____ has more than 90 floors").

How would we symbolize this? Here are the steps:

1. Replace the singular term(s), i.e., the subject part, with an individual constant. Remember that you can use any lower case letter a-v, with or without a subscript; and don't forget that x, y, and z are reserved for variables).
2. Replace the general term(s), i.e., the predicate part, with a predicate letter. You can use any capital letter A-Z, with or without subscript.

 

 

Here are some examples

Symbolizing truth-functional compounds

Scaling up, we can see that it is easy to join atomic sentences together to form truth-functional compounds. For example, say we wanted to translate the following sentence in PL: "Steve has a backpack, and Mariela is under 6 foot." Well, we've already translated each of the atomic sentences within this conjunction above. Now we just need to join them with the corresponding logical operator, which is an ampersand:

Bs & Um

Food for thought...

 

How would you symbolize the sentence “Yuri is Russian”?

"Ry"?

 

 

Translating to PL (Singular Terms)

The following practice problems are taken from Herrick (2013; p. 476). Also, be sure to watch the Tablet Time! below before starting.

1. Grandpa Munster banks at the blood bank. 
2. Wimpy likes hamburgers and Popeye likes spinach. 
3. If Moe is happy, then Curly is happy and Larry is happy. 
4. It is not the case that if Frege lectures on logic then Russell will attend. 
5. The first man on the moon was an American astronaut. 

 

 

Tablet Time!

 

 

 

 

New symbols

The universal and existential quantifiers of PL are perhaps the most intimidating aspect of the language. However, by standardizing the way that we "read" these, we can go a long way towards mastering their use.

The universal quantifier: (∀x)

(∀x) should be understood as “For all x...” To take an example, the sentence “Everything is good” is symbolized as: (∀x)Gx. This could be read in two ways. Using the predicate letter "G" to stand in for the general term "____ is good", the first way to read this is “For all x, x is G.” Another way to read this is “Every x is such that x is G." Either way is equivalent. The sentences literally say that the predicate “____ is good” is true of everything in the universe.

Here are two more examples:

• “Everything is not good" is translated as (∀x)~Gx
  • This is read as "For all x, it is not the case that x is good."
• “All dogs have fleas” is translated as (∀x)(Dx ⊃ Fx)
  • This is read as "For all x, if x is a dog, then x has fleas."

The existential quantifier: (∃x)

(∃x) should be read as “There exists at least one x such that...” To take an example, “Some things are green” is symbolized as: (∃x)Gx. This is read as “There exists at least one x such that x is G.” It literally states that there is at least one thing in the universe of which “____ is green” is true.

Here are two more examples:

• “Some cars are noisy” is translated as (∃x)(Cx & Nx).
  • This is read as "There exists an x such that x is car and x is noisy."
• “Some students do not do their homework” is translated as (∃x)(Sx & ~Hx).
  • This is read as "There exists an x such that x is a student and x does not do their homework."

 

Storytime!

Mergers and acquisitions

As logicians were developing first-order logic, and simultaneously re-discovering Stoic logic (see O'Toole and Jennings 2004), they asked a question regarding primacy: which is the more fundamental logic—Aristotle's categorical logic or Stoic truth-functional logic?

In addressing this question, we should first note that both types of logic are "contained" within PL. Beginning with Stoic logic, convince yourself that any wff of TL counts as a wff of PL, but not every wff of PL counts as a wff of TL. This is what we mean when we say that PL is said to “contain” TL. But TL is simply the language that we learned to understand Stoic truth-functional logic and their main method of assessment for validity: pattern-recognition. This means that, isomorphically, Stoic logic is contained within PL. Next, note it's also the case that any categorical proposition can be expressed within PL. Enjoy the slideshow below:

 

 

So, categorical propositions and any wff of TL are expressible in PL. Hence, Aristotle’s categorical logic and Stoic truth-functional logic are both “contained” within PL. This means, in other words, that the two forms of logic have been unified.

Now that the general notion of containment is understood, the question is as follows: who acquired who? There are two ways of understanding this. First, note that the expressions of categorical logic can be expressed in PL without much effort. However, the expressive power of PL cannot in any way be matched by the categorical approach. That is, a propositional-approach to reasoning can express the categorical approach, but not vice versa. Aristotle's categorical approach, as we've noted before, has some expressive ambiguities. Just to give one example, recall that there was the problem of logical relativity. Clearly, the truth-functional approach subsumes the categorical approach.

Here's one more way of thinking about it. What type of logic does PL look more like: Stoic pattern-recognition or Aristotelian categories? Clearly, PL is a lot like the language that we used when learning Stoic pattern-recognition (TL). It looks like breaking down reasoning to the level of complete declarative sentences (and focusing on truth-functional compounds using logical operators) is the more fundamental approach. Historians of logic William and Martha Kneale summarize it this way:

Down goes Aristotle...

 

 

FYI

Homework— The Logic Book (6e), Chapter 7

• Read section 7.1 (p. 262-267) and 7.2 (p.268-274).
• Finish in-class practice problems (from the Do Stuff section)!

 

Footnotes

1. If you're wondering about the theme behind the images for this unit, I'll fill you in on the secret: it's about perspective.