The “universe of discourse” (the “UD”) is the set of things that we are talking about. When Avril Lavigne asks “why does everything have to be so complicated?”, the truth of her implied claim, that everything is so complicated, lies in the UD. If the UD is the set of all social interactions then she might be right. If the UD is the set of all things in general, I would probably disagree with her; the instructions on a box of pizza pops, for instance, are not complicated. Some things are related to other things by relations. The relation “bigger than” relates (among other things to other things) me to my brother: I am bigger than my brother. Certain relations have certain properties. “Bigger than” has the property of being transitive: if x is bigger than y, and y is bigger than z, then x is bigger than z. Not all relations have the property of transitivity. If x loves y and y loves z, it does not follow that x loves z (often what actually follows is the opposite). Identity (x is identical to y) is a relation with a lot of properties. It is transitive: if x is identical to y and y is identical to z then x is identical to z. It is also symmetric: if x is identical to y then y is identical to x (note that symmetry does not hold with “I am bigger than my brother”, but it would with “I am the sibling of my brother”). Identity is also reflexive: x is identical to x. Most relations aren’t reflexive, mostly because most relations cannot relate a thing to itself. “Bigger than” is not reflexive. “The sibling of” is also not. Nonetheless, reflexivity at first seems like a rather unremarkable property (actually if you’re a normal person, all of this probably seems rather unremarkable). That’s why I was puzzled by the following passage in my logic textbook:
Before reading on, I gave it a little thought, and it turned out that I could not understand why those relations should not be reflexive. I am certainly the same age as me, my book is in the same place as itself, etc. Reading on, however:
In an unrestricted UD it is rather hard to find other reflexive relations. For example, a little thought should show that none of the following expresses a reflexive relation in an unrestricted universe of discourse: x is the same age as y x is the same height as y x is in the same place as y
At first that seems like cheating. Obviously the number 48 does not have an age, that’s why I didn’t even think about including the number 48 in the things that I was considering when I was searching for something that is not the same age as itself. My brain had automatically excluded numbers from the universe of discourse when thinking about age. But then again, the whole point of that little passage was that the UD is unrestricted; we are considering the set of all things. It just turns out that my brain isn’t totally primed to be dealing with an unrestricted UD. All of that got me thinking that it is interesting how my brain restricted the UD for me, without me even asking it to, and indeed in spite of my active attempts to consider an unrestricted UD. I guess pretty much all thinking occurs within a restricted UD unless we make a concerted effort otherwise. It’s interesting that people manage to align their UD’s without expressly talking about it (ie. Avril Lavigne doesn’t start off her song with “the universe of discourse is the set of all ways that boys interact with girls in high school), and yet whenever someone talks about “everything” and “anything”, we don’t get completely lost. One time I drove my car over a median. I took it to my newly-immigrated Eastern-European mechanic, who, after about five minutes looking under the car, stood up and said “I’ll tell you what is the problem. Everything is gone.” I did not even question for a moment that the UD was the set of all things that belong on the undercarriage of my ‘93 Ford Escort station wagon. I knew exactly what he meant. I think that a life time of talking about “everything” and “everyone” and “anything” and “anyone” has made it very difficult for my brain to consciously go from one universe of discourse to another, and even harder for it to think about a completely unrestricted UD. Incidentally, the book does not mention another property in an unrestricted UD, and I am at a loss trying to come up with one.
Since the number 48 is not of any age, it is not the same age as itself nor the same height as itself. So, too, neither the number 93 nor the set of human beings is in any place.
Symbolize the following sentence in PL:
All people who read this blog have noticed that I haven’t posted in ages.
Rxy: x reads y
Px: x is a person
Nxyz: x has noticed y has not posted in z
b: this blog
i: Colin Fraser
For all x such that x is a person and x reads this blog, x has noticed that Colin Fraser has not posted in ages
Note that as a universally quantified sentence, this claim does not carry existential import.
I’ll blog again after finals.