In my post on present value, I promised to explain how to turn a series of payments into a present value. This is the promised follow-up post.
I am 27 years old. I recently bought a life insurance policy with a face value of $100,000. This policy will last my whole life - in other words, no matter when I die, the payout happens. It cost me roughly $10,000 in today's money. If this is surprising to you, or you think the insurance company got a bad deal, then read this.
Everyone makes choices about whether they'd rather have something now, or something else later. Almost no one understands the economic concepts that describes these tradeoffs. they're called "present value" and "discount rate."
I will start by describing some simple examples that use these concepts, without using the jargon. Then I will explain what these all have in common. I'm not going to explain how to use these in real-life situations, but if you're interested, please let me know in the comments and I'll write a follow-up post.
Return on Investment
I'll start with a simplified example, with made-up numbers. Abby has a bank account with a bunch of money in it earning 2% guaranteed interest per year. She also owns a bond that would pay out $1,000 if she cashes it out now, or $1,030 if she cashes it out in a year. Should she cash it out now, or a year later?
Let's say that in any case she wouldn't use the money until a year from now. Then if she cashes out the bond now, she can immediately deposit the money, and in a year, she'll have $1,020. But that's less than the $1,030 she'd get if she held onto the bond for a year.
On the other hand, suppose she wants to use the money right now. Then if she cashes out the bond now, she has an immediate $1,000 to spend. On the other hand, let's say she holds onto the bond, and withdraws $1,000 from her bank account. Then in a year, she has $1,020 less in her account than she would have, but an extra $1,030 from the bond, putting her $10 ahead of the first strategy. So in this case too she should hold onto the bond for another year.
It should be easy to see that if the bond only returned $1,010 in a year, Abby comes out ahead by cashing out now, again regardless of whether she wants to use the money now or later. Because the bond gives her a lower return on investment (1%) than her savings account does (2%).
Then suppose the bond pays out $1,030 in a year, but her bank account offers 4% interest this year. Then Abby also comes out ahead by cashing out now, because the bond's return (3%) is less than the interest she gets on her bank account.
Cost of Funds
Brian doesn't have any savings - he a student. But he has a good credit rating and is able to borrow at 5% interest per year, and is allowed to pay off his loans at any time.
He is deciding whether to rent a textbook for $100, or buy it for $150 and sell it back used to his school's bookstore in a year for $55.
If Brian rents his textbook, then after a year, he will owe $105, including interest, and have no textbook. On the other hand, if he buys his textbook, then after a year, he will owe $157.50. He can then sell his textbook back to the bookstore for $55, use that to pay down his debt, and owe only $102.50. So buying the textbook is a better deal.
Suppose instead Brian can only borrow at 10% interest. Then if Brian rents his textbook, after a year, he will owe $110. On the other hand, if he buys his textbook, then after a year, he will owe $165-$55=$110. So he should be indifferent between the two alternatives.
If Brian has to pay 15% interest, then if he rents his textbook, after a year he owes $115, but if he buys, then after a year he owes $125, so he comes out ahead by renting.
On the other hand, suppose at the 5% rate of interest, Brian can only collect $50 for his textbook after a year. Then instead of owing $102.50 at the end of a year, he'd owe $107.50, more than the $105 he'd owe if he rented, so in that case renting again becomes more advantageous.
In each of the above examples, a future amount of money was related to a present amount of money, by either how much money you'd have if you used the current money in the best way available (either investing or paying off debt), or how much money you would have to have now, to produce the future money. The first is called the "future value" of money, and the second is called the "present value" of money.
When Abby is choosing between $1,000 now and $1,030 in a year, the "future value" of $1,000 is how much money she'd have at the end of a year if she put the money in her bank account yielding 2%. To get this, you multiply by (100%+2%=1.00+0.02=1.02): $1,000 * 1.02 = $1,020. This is less than the one-year future value of $1,030 in a year, which is of course $1,030.
The "present value" of the year-later $1,030 is the amount Abby would need today to produce that amount in a year. To calculate the value a year in the past, you simply do the opposite of what you did when calculating the value a year in the future: you simply divide by (100%+2%=102%=1.02), to get $1,030/1.02=$1009.80, more than the present value of $1,000 today (which is of course $1,000).
Another way to show this is algebraically:
Now let's look at the first example involving Brian. Brian is comparing making a single payment today, with making a payment today plus receiving a payment in a year.
Since Brian has to pay 5% interest on money he borrows, the future value of the textbook rental expense is how much Brian will owe in a year if he borrows the money, or $100*1.05=$105. The future value of the purchase price of the textbook is $150*1.05=$157.50, and the future value of the $55 Brian will receive for his textbook in a year is just $55. So the net future value of Brian's textbook expenses if he buys is $157.50-$55.00=$102.50, less than the $105 future value of the rental fee.
The present value of the renting option, $100 today, is of course $100. The present value of the textbook's price today is also the same as the price, $150. The present value of getting $55 in a year is the amount of debt he'd have to pay off now, to owe $55 less in a year: $55/1.05=$52.38. So the present value of the cost of buying and selling back later is $150-$52.38=$97.62, less than the $100 textbook rental fee. So the buying option costs less, in present value terms, as well.
The key here is that by converting each value, whether positive or negative, into the equivalent value for a single time period - whether the present or the future - we end up with numbers that can be directly added and subtracted to find out which amount is higher on net.
You may have noticed that in Abby's case we were using the rate at which she could expect return on her savings to equate future and present amounts, but in Brian's case we looked at the interest rate he'd have to pay to borrow money. These might seem like quite different things, but in finance, there's little difference between spending saved money and borrowing money; in both cases money in the future is worth more than money in the present, and we assume a fixed conversion factor. Instead of calling it a cost of borrowing sometimes and an expected return on investment at other times, economics abstracts this into the more general term "discount rate", which is basically the extra share you can demand if you get your money in a year instead of today, or the share of your money you should expect to give up if you get your money today instead of a year from now.
This is related to the economic concept of "opportunity cost," which I will cover in a future post.
I will also cover how to deal with a series of future payments in a future post - and in the process show you that if you believe in discount rates, the future isn't as big a deal as it seems.
Which means, of course, that this is the first post in a series.
Can God create a rock so big that He can't pick it up? To understand the problem, we need to understand set theory. But I don't really want to talk about Russell's paradox quite yet - a big problem with set theory as it's taught is that it doesn't respond to a felt need, it's just plopped down at the beginning of a calculus or logic textbook without explanation. Here's a bunch of self-evident stuff! Go calculate what the union of the intersections is!
I'm not going to tell you how to do set theory here. You can look that up if you want. I'm just going to try to explain a little bit about why it matters, why you should be interested in it, and how to apply some set-theory-ish rules of thumb to your own thoughts.
Think about the difference between these two arguments:
The king of Freedonia is Phillip III.
The husband of Mary Teller is Phillip III.
Therefore, the king of Freedonia is the husband of Mary Teller.
Milk is white.
Snow is white.
Therefore, milk is snow.
The second argument looks just like the first one - but the first one works and the second one doesn't. Why?
Well, I've deliberately made it tricky by using the verb "is" in each case. "Is" is one of those tricky verbs whose meaning is very context dependent. Here's a more precise formulation of the arguments:
The king of Freedonia is the same as Phillip III.
The husband of Mary Teller is the same as Phillip III.
Therefore, the king of Freedonia is the same as the husband of Mary Teller.
Milk is one of the things that are always white.
Snow is one of the things that are always white.
Therefore, milk ??? snow.
it's not even clear which spurious consequence is supposed to follow from the second argument anymore. Is this a specious proof that milk and snow are identical, or that all milk is snow, or that all snow is milk, or just that some things are both milk and snow?
Here's another paired example:
A shark is an aquatic animal.
An aquatic animal is a living thing.
Therefore, a shark is a living thing.
A knife is an item in my silverware drawer.
An item in my silverware drawer is a spoon.
Therefore, a knife is a spoon.
And with more specific wording:
Every shark is an aquatic animal.
Every aquatic animal is a living thing.
Therefore, every shark is a living thing.
At least one knife is an item in my silverware drawer.
At least one item in my silverware drawer is a spoon.
Or better yet:
There exists at least one item that is both a knife and in my silverware drawer.
There exists at least one item that is both in my silverware drawer and a spoon.
Set theory is a way to force yourself to use statements more explicit than "X is Y", to prevent you from accidentally equivocating and "proving" that knives are spoons. Since math is all about proving possibly counterintuitive things, this is kind of important in math. But it's also important whenever you're making explicit compounded arguments of the (A, B, THEREFORE C) style.
In set theory you never say "X is Y." You instead are always talking about whether something is a member of a set. For now, think of a set as nothing more specific than a collection of things. There's a problem with this, but I'll get to it later.
You can say that something is a member of a set, or that if something is a member of one set, then it must be a member of another, or that there is at least one thing that is both a member of set A and a member of set B, etc. You can also negate these things - you can say that there are no things that are both members of set A and set B. Think about these sentences, and how to make them more precise:
- A mouse is in this cage.
- A mouse is an animal.
- This mouse is Pinky.
- Pinky is in this cage.
- Dallas's football team is heavier than the people in China.
- A dragon is not real.
- WEF wrestling is fake.
Here are some formulations that are a little more set theory-ish:
- There exists at least one thing that is both a member of the set (is a mouse) and a member of the set (things in this cage)
- Every member of the set (is a mouse) is a member of the set (is an animal).
- Every member of the set (this mouse) is a member of the set (Pinky). Also, every member of the set (Pinky) is a member of the set (this mouse).
(A pithier way to say that one is: Something is a member of the set (this mouse) if and only if it is a member of the set (Pinky). This is an "identity" relation.)
- Every member of the set (Pinky) is a member of the set (in this cage).
- The average of the weights of all the members of the set (members of Dallas's football team) is higher than the average of the weights of all the members of the set (the people in China).
(This one is tricky - the original statement is ambiguous, because it's worded as a statement about the set, but what exactly are we saying is heavier than what? Are we saying that each Dallas Cowboy is heavier than each person in China? Or that the Dallas Cowboys, weighed all together, are heavier than the people in China, weighed all together? Or that the average weight of a member of the first set is greater than that of a member of the second? It's important to be specific about things like this when talking about group characteristics.)
- There are no members of the set (dragons) that are members of the set (real things).
- Every member of the set (WEF wrestling matches) is a member of the set (fake things).
Do you get the pattern? You never simply talk about how something "is" or "is not" something else, only about whether a member of set A is never, sometimes, or always a member of set B, and whether an assertion is true or false.
This can be helpful in avoiding getting into stupid arguments. If someone says, "a mouse is an animal," do they mean that there is at least one mouse that is an animal, or that every mouse is an animal, or that something is a mouse if and only if it's an animal?
If they mean that there's at least one mouse that's an animal, then finding a mouse that's not an animal (like a computer mouse, or a robotic mouse) is not evidence against their point - all they have to do to prove it's true is find at least one mouse that is an animal. But if you phrase it explicitly like that, it's harder for them to equivocate and "prove" that a computer mouse is an animal.
Or maybe more realistically, if I "prove" that wiggins are thieves by showing you one wiggin who steals something (which only proves that there is at least one wiggin who is a thief), I might then pretend that you should draw the inference that some other wiggin is also a thief (which would only be valid if I had proved that every member of set "is a wiggin" is a member of set "is a thief").
If they mean that every mouse is an animal, then finding an example of a mouse that is not an animal is a counterexample, but finding an example of an animal that is not a mouse, like a dog, is a not counterexample. If they've shown to your satisfaction that all members of set "mouse" are members of set "animal", then you can go on and assume that's true for each new mouse you encounter - but it doesn't imply that all members of set "animal" are members of set "mouse".
Finally, if they show "if and only if," then you would have been able to prove them wrong just by showing them a dog. But if they convince you of this, then - and only then - you should accept the inference both ways.
It's easy to lose track of this when you say things like "mice are animals" or "wiggins are thieves", so it can be helpful to use set-theoretic language (which is almost as compact), like "MICE is a subset of ANIMALS."
OK, so what does this have to do with God's rocks? Well, sets are important, right? And we want to be correct when talking about important things - and sets help us be correct. So we want to describe sets using other sets. And talk about sets of sets!
Like you might want to talk about the properties of "sets that have no members." Or "sets that have a finite number of members." This is fine. But there are limits.
Let's walk through one of them - the rock paradox. It's usually stated as:
God is omnipotent. That means God can do any thing.
Making a rock so big that God can't pick it up is a thing.
Therefore, God can make a rock so big that God can't pick it up.
But picking up an arbitrary object that exists is also a thing.
Therefore God can pick up an arbitrary object that exists.
Now, let that arbitrary object be "a rock so big that God can't pick it up."
Then, God can pick up a rock so big that God can't pick it up.
Now, if the existence of such a rock were impossible, then this wouldn't be a problem. But we just said that God can make one.
But it's not really a rock so big that God can't pick it up, if God can pick it up.
Thus, the omnipotence of God implies a contradiction.
Therefore, there can be no omnipotent God.
The problem here seems to be using omnipotence in the definition of one of the powers. If you don't allow that, then there's no way to get the contradiction.
This brings up another set-theoretic principle: the "things" a set can be a collection of have to be well-defined, before we define any of the sets. So if we're talking about puppies, and we already know what puppies are, without using sets of puppies in the definition, then we can talk about sets of puppies. But we can't just define a collection of "puppies and sets of puppies," before we know what the sets of puppies are. And the sets of puppies can't themselves be defined until the puppies are defined.
So does the rock paradox follow this rule? No.
"God is omnipotent" can be rephrased as:
For every ability X, let there be a set (entities that have ability X).
Every omnipotent being is a member of every such set.
God is an omnipotent being.
Therefore, for every ability X, God is a member of the set (entities that have ability X.)
Now, this works for abilities like "walk on water" or "use set-theoretic notation" or "make ten commands". Because those things are well-defined even if we don't know about God.
How about "make a rock so big that God can't pick it up." Is this well-defined before we start talking about sets of abilities? No, because the ability is defined by a reference to what God can do, and what God can do is defined by a particular set of abilities. So a collection of abilities that includes "make a rock so big that God can't pick it up" is simply not a well-defined collection that we can take sets of.
In fact, "make a rock so big that [someone] is not a member of set (entities that have the ability to pick up a rock of that size)" is never a first-order ability.
A set-theoretically valid definition of omnipotence would be something more like this:
Define some collection of "abilities," none of which reference other powers or omnipotence directly.
Define omnipotence as the set of all these abilities.
Now, maybe "make an arbitrarily large rock" is one of the powers. And maybe "pick up an arbitrarily large rock" is a power. But none of the powers refer to each other, or to sets of powers, no matter how indirectly. So "make a rock so big that God can't pick it up" isn't an ability.
We can then think of sets of abilities, like the set of rock-making and rock-picking-up. Omnipotence is the ability-set that is contains all abilities.
Now we need to use a concept called a "subset." A X is a subset of Y if every member of set X is also a member of set Y. For example, "Puppies" is a subset of "Animals," and "Animals" is also a subset of "Animals, but "Animals is not a subset of "Puppies."
So every ability-set is a subset of omnipotence.
Of course, that doesn't mean that no one can make a rock so big that someone else can't pick it up. Or even a rock so big that they themselves can't pick it up. But that's a statement about combinations of abilities and inabilities.
So what if you wanted to describe all the collections of abilities that don't include certain abilities? Well, that's a second-order set. Call it a schmet. So you might have a schmet of ability-sets that include walking on water, but not swimming. Or making a 32kg rock, but not picking it up.
Now let's get back to that paradox. Can God make a rock so big that He can't pick it up? How does that cash out when thinking about sets of abilities?
If someone can make a rock so big they can't pick it up, that means that their ability-set is a member of a certain schmet. In particular, it's the schmet that includes ability-sets where for some size X, they include the ability "can make a rock of size X", and also do not include any ability "can pick up a rock of up to size Y", for any Y>=X.
So the question is, is God's ability set (omnipotence) a member of that schmet? The answer is no: omnipotence is not a member of the schmet "can make a rock so big you can't pick it up."
There's no paradox, because a schmet is not an ability. Remember, we had to define all the abilities before defining any of the ability-sets, and we had to define the ability-sets before defining the schmets. So there can't be an ability that refers to a schmet! And omnipotence is an ability-set, so its definition can't refer to schmets either - it's just the ability set that includes all abilities.
If you look up Russell's paradox explained, you will find a similar exposition, except it's less fun because it isn't about God and rocks.
I did a little research and think I finally understand what zeugma and syllepsis are, and how they relate to each other.
Zeugma is any case where a single mention of a word is treated as a part of more than one clause of a sentence. Syllepsis is a type of zeugma where the word in question is used in contexts that require it to do different things.
So "He prefers dogs, she cats" is zeugma because "hate" gets re-used, but it is not syllepsis because "prefers" is doing the exact same thing in "He prefers dogs" and the implied "She prefers cats."
There are two types of syllepsis.
Grammatical syllepsis is where the difference is in verb form. So "I prefer dogs, she cats" would be grammatical syllepsis, because "she prefer cats" is ungrammatical - the implicit extra verb is "prefers", a different form of the same word.
Semantic syllepsis is where the difference is in meaning. So in "And he said as he hastened to put out the cat, the wine, his cigar and the lamps", "put out" can mean to expel, to retrieve from storage, to extinguish, and possibly to turn off (I am not sure whether that last sense would be anachronistic).
You can't have a syllepsis without a Zeugma because the different usages have to be attributed to the same single mention, which is the definition of zeugma. So if the line from the old song went, "And he said as he hastened to put out the cat, put out the wine, put out his cigar and put out the lamps", we would have a weird phrasing, but nothing disorienting - the casual listener might not even notice the repetition, instead hearing "put out [X]" as a whole phrase. It is where we have to attribute to each clause the same mention of the verb that our attention is called to its different meanings.
Zeugma in itself may be nothing more than parsimonious phrasing, and asylleptic zeugma may easily pass unnoticed. Syllepsis is a distinct type of zeugma because it is unexpected, and draws attention to itself.