Translations

 

 

Important Concepts

 

As you learned in the Important Concepts above, today we are working with relational (or dyadic) predicates. Although this becomes intuitive with practice, initially the order of the individual constants and predicate letters might throw you off. The idea, however, is simple. In natural language, some predicates establish a relation between two (or more) singular terms (or variables), which means that, in PL, some predicate letters will have two (or more) individual constants (or variables) attached to them.

The following slideshow gives numerous examples that will begin to get you accustomed to using relational predicates.

 

 

Reflexive sentences

Be sure to be careful about reflexive sentences, i.e., sentences where the predicate establishes a relationship from an individual constant to that same individual constant. For example, consider the sentence “Narcissus loves himself”. If we let "Lxy" stand for “x loves y” and let "n" stand for “Narcissus”, the sentence would be symbolized as: Lnn. Take a moment to convince yourself that this is the case.

 

 

The following practice problems are taken from Herrick (2013; p. 511-12). See Tablet Time! section for help getting started.

  1. Pam is taller than Sue but Sue is older than Pam. 
  2. Archie Bunker does not like any liberal.
  3. Archie Bunker does not like every liberal.
  4. Wimpy, the hamburger man, respects himself. 
  5. Someone knows himself. 
  6. Everybody knows Bill Gates. 
  7. All elephants are larger than Nathan’s pet mouse. 
  8. Lorraine likes any horse. 
  9. Somebody knows Katie. 
  10. Matt is a friend of Elliot’s. 
  11. Sam dislikes somebody. 
  12. Sam dislikes everybody.

 

 

 

Mastering PL

Overlapping Quantifiers

In order to truly appreciate how expressive PL can be, we must take a look at instances of overlapping quantifiers. If one quantifier appears immediately to the right of another quantifier, then the scope of the left quantifier is that quantifier itself plus the scope of the right quantifier. For example, the sentence "(∀x)(∃y)Tyx" is a universally quantified sentence. This is because the universal quantifier has the largest scope. It applies to itself and to what is immediately to its right, namely the existential quantifier (which itself applies to the rest of the sentence).1

The sentence "(∃y)(∀x)Tyx" is an existentially quantified sentence. In this sentence, it is the existential quantifier that has the largest scope.

Here are some more examples of sentences of PL with overlapping quantifiers:

 

 

Universe of Discourse

One other characteristic of PL is that it can limit the range of objects that a variable can accept as values. Put more formally, the domain of a variable is the set of things the variable can take as values. When we specify the universe of discourse for a sentence containing a variable, we are stating the domain of the variable; i.e., we are specifying what it ranges over. If the domain of a variable is unrestricted (i.e., where literally any possible predicate is applicable to the variables in question), then we call this the universal domain. However, if the domain is confined to a set of things within the universe, we call this a restricted domain.

Restricting our domain allows us to simplify our expressions in PL. For example, say we wanted to translate the following sentence:

"All humans have moral rights."

Paraphrasing this as "For all x, if x is a human, then x has moral rights", our sentence in PL would be something like:

(∀x)(Hx ⊃ Mx)

However, suppose we stipulate that the domain is restricted only to persons; i.e., we stipulate that the variable x ranges only over persons. The result is that we can symbolize the sentence above as:

(∀x)Mx

This is because we no longer have to specify that the predicate letter H applies to the variable x since it is a given that H is true of all variables in this restricted domain. This convention will allow you to express more complex sentences with fewer symbols and in a way that is more human readable.

Putting it all together

Restricting domains can be combined with overlapping quantifiers to produce sentences featuring just a few symbols but that stand for propositions with complex semantic content. For example, consider the sentence “Someone knows someone.” If the universe of discourse is restricted to persons, we can translate this as:

(∃x)(∃y)Kxy

 

 

The following practice problems are taken from Herrick (2013; p. 518). See Tablet Time! section for help getting started.

Universe of Discourse: Human Beings

  1. There is a person who is universally respected. 
  2. There is a person who respects everybody. 
  3. Everybody respects someone or other. 
  4. Some people do not know anybody. 
  5. If everybody knows somebody, then somebody knows everybody. 
  6. If someone helps someone, then God is pleased. 
  7. Anyone who loves no one is to be pitied. 
  8. Someone is not known by anyone.

 

 

Food for thought...

Some students might wonder if the increased complexity of PL, when compared to TL, is justified—given that it seems to do a lot of what Stoic logic was able to do, such as make explicit certain patterns of reasoning and emphasize the truth-functionality of our argumentation. However, PL does feature a significant improvement over TL, and it has to do with relational predicates. Relational predicates are what allow PL to do what neither categorical nor Stoic truth-functional logic can: express n-place predicates. Some predicates, it turns out, relate to singular terms to each other. You hear it all the time. "Dan is taller than Pedro." "Trish graduated college before Micah." "Sally makes more money than her husband." In short, we use relational predicates all the time. Moreover, unsurprisingly, these are oftentimes a part of our arguments. And so we need to be able to express these in a logically precise way. TL doesn't do a very good job at this. In TL, these sentences would be something like "D", "T", and "S", respectively. But that doesn't tell you much of anything. Categorical logic, as studied by Aristotle, does even worse. You have invent all sorts of weird categories, like "persons who are taller than Pedro", "persons who graduated college before Micah", and "persons who make more money than their spouses". But PL can do all this with ease: "Tdp", "Gtm", and "Msh".

These are just two-place relational predicates, by the way. In other words, these are just predicates to which you attach two singular terms. You can have any number of singular terms attached, however. For example, here's a sentence with a three-place predicate: "Devin met Andy at the coffee shop." Here, the predicate is "_____ met _____ at _____", and the singular terms are "Devin", "Andy", and "coffee shop". Here's a sentence with a four-place predicate: "Devin met Andy at the coffee shop to talk about calculus". And on and on.

The point here is that PL is a quantum leap in our capacity to express highly-detailed propositions in a logically precise way. If you are a student of computer science, you may already recognize why this is so important. I'll keep it brief here. Just like the Industrial Age encouraged thinking in mechanical terms which served as an inspiration to Boole and DeMorgan, the increased precision of our logical languages in turn inspired visionaries like Alan Turing, Claude Shannon, and John von Neumann to build machines which we could "speak" to using a rigorous formal language (see Shenefelt and White 2013, chapter 9). The result? It's a brave new world.2

 

 

 

Tablet Time!

 

 

 

FYI

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

  • Do 7.3E #2 (p. 294-5).
    • Here is the portion of the student solutions manual for the relevant chapter.
  • Read section 7.4 (p. 296-310), in particular p. 300-310.

Related material—

 

Footnotes

1. Perceptive students might notice that this is much the same way as how the tilde works. For example, in the sentence "~~~K", it is the leftmost tilde that is the main operator, just like how in "(∀x)(∃y)Tyx" it is the leftmost quantifier that is the main operator.

2. One context in which you can see how logical languages like PL influenced researchers in the nascent field of computer programming is at the command line, where not only do you use truth-functional operators like "&&" and "||", but also issue commands that require two arguments, like a two-place predicate. For example, the mov command requires two arguments. For an intro to one of my favorite shells (i.e., the one I learned on), click here; see also Shotts (2019).