# Paper Review: The Paradoxes of Time Travel

We’ve all heard of the classic grandfather paradox of time travel: you go back in time and kill your grandfather before he sired your father. But then what happens? We can reason about it in something like the following way. You kill your grandfather, so your father never existed, so you never existed. But then who killed your grandfather? If you didn’t exist, then no one did, and then you should exist, since your grandfather lived, but then you do kill him…

You get the picture. This seems paradoxical, like there is no kind of stable outcome. This thought experiment is an example of a more general concern dealing with how time travel related to causality and, arguably, free will. It is because of such concerns that some people take time travel to be impossible.

David Lewis, however, doesn’t think this kind of argument against time travel works. This is a pretty classic argument in favour of the possibility of time travel. Let’s take a look at it.

***The original paper can be found here.***

To start off with Lewis makes a distinction between two ways in which we might think time travel works. These are the two-dimensional and one-dimensional views. We usually think of time as a kind of line with two directions, one pointing to the future and one to the past. However, as part of one strategy to make sense of time travel, we might try to think of time instead as two-dimensional–a plane. Let’s see how Lewis puts it:

It is tempting to reply that there must be two independent time dimensions; that for time travel to be possible, time must be not a line but a plane. Then a pair of events may have two unequal separations if they are separated more in one of the time dimensions than in the other. The lives of common people occupy straight diagonal lines across the plane of time, sloping at a rate of exactly one hour of time-1 per hour of time-2. The life of the time traveler occupies a bent path, of varying slope.

p. 145

This is presumably so we can make sense of cases where the time traveler spends one hour traveling, but ends up more or less than 1 hour in the future (or past). The time traveler moves from one point on the plane to another point on the plane, moving along both dimensions. We could utilize this second dimension to account for the travel time of the time traveler.

Lewis doesn’t like this proposal:

On closer inspection, however, this account seems not to give us time travel as we know it from the stories. When the traveler revisits the days of his childhood, will his playmates be there to meet him? No; he has not reached the part of the plane of time where they are. He is no longer separated from them along one of the two dimensions of time, but he is still separated from them along the other. I do not say that two-dimensional time is impossible, or that there is no way to square it with the usual conception of what time travel would be like. Nevertheless I shall say no more about two-dimensional time. Let us set it aside, and see how time travel is possible even in one-dimensional time.

p. 145

The two-dimensional view both doesn’t easily conform our intuitions about what time travel should mean, and it is a little more interesting to think about how one-dimensional time travel might work. So for the rest of the paper Lewis considers only the one-dimensional view. I follow him in this post.

The paper is already very readable, and in fact rather poetically written, so I encourage you to read it for all the details. For this post I will focus on the specific grandfather case Lewis considers, and look at the logical structure of his resolution of the paradox.

I like the particular reason of the paradox that Lewis considers since it is so colourful, so I’ll include it here:

Consider Tim. He detests his grandfather, whose success in the munitions trade built the family fortune that paid for Tim’s time machine. Tim would like nothing so much as to kill Grandfather, but alas he is too late. Grandfather died in his bed in 1957, while Tim was a young boy. But when Tim has built his time machine and traveled to 1920, suddenly he realizes that he is not too late after all. He buys a rifle; he spends long hours in target practice; he shadows Grandfather to learn the route of his daily walk to the munitions works; he rents a room along the route; and there he lurks, one winter day in 1921, rifle loaded, hate in his heart, as Grandfather walks closer, closer,. . . .

p. 148

So we have Tim the time traveler, with a vendetta against his grandfather, and the means to act on his hatred. Very fun.

Okay. So here is how Lewis thinks the usual arguments against time travel to work.

Fact 1: Tim cannot kill his grandfather. Since we have this one-dimensional view of time travel up and running, either “the events of 1921 timelessly do include Tim’s killing of Grandfather, or else they timelessly don’t” (p. 149). In other words, since Tim’s grandfather is alive in the “original” 1921, if Tim killed him in the “new” 1921, then Grandfather would be both alive and not alive in 1921, since on the one-dimensional view, the “new” and the “old” 1921 are the same thing. So Tim cannot kill his grandfather.

Fact 2: Tim can kill his grandfather. He clearly has the means and training to do so, as we see in the picture. As Lewis writes, “Tim is as much able to kill Grandfather as anyone ever is to kill anyone” (p. 148)

But now we see the issue clearly. Time travel was the thing that gave Tim the ability to kill his grandfather. But this leads to a contradiction: Tim both can and cannot kill his grandfather. Thus, time travel is impossible.

Should we give up on our dreams to build a time machine? Lewis says no.

Lewis argues that the contradiction is only apparent, and relies on an equivocation of two different uses of the word “can”. Once we disambiguate what “can” means in the two different statements, the contradiction is revealed as only apparent.

In order to make the story consistent, Tim must not in fact end up killing his grandfather. Let us see how we understand the claim “Tim can kill his grandfather” works then.

Lewis thinks that when we use the word “can” in the sense of something can happen, we mean that the thing happening is consistent (he uses the term “compossible”) with a certain sense of facts. So I guess if we were being very precise, then, we could put a subscript on each of our “can” claims that indicated the set of facts with respect to which the claim was being made. I like this idea as it would make clearer what our exactly we are claiming when we say a particular event can or cannot happen.

Now we being to see Lewis’ strategy. When we say “Tim can kill his grandfather” and when we say “Tim cannot kill his grandfather” our “can”s really have different subscripts (not the way Lewis would put it, but I think this is a useful way to think of it).

When we say he can, we mean with respect to the facts we usually mean when we say someone can do something. For example, most people think I can, if I want, stay in my apartment all day tomorrow, even if I don’t in fact do not say (which I won’t). We do this because we make the “can” claim with respect to a particular set of facts that are useful for my purposes. Whenever I consider my options, that I can do this, or I can do that, and imagine what the world likes in those different cases, I am reasoning about whether or not an event is compossible with a set of facts.

So, when we say Tim can kill his grandfather, this “can” is with respect to the same kind of set of facts that we use when we say I can stay home all day tomorrow. Perhaps it is true as a matter of fact that I will die tonight, and if we included that fact in our set of facts then it would not be true that I can stay home tomorrow, since I wouldn’t exist. However, this does not mean that with respect to one set of facts, the ones that I usually use when I reason about the possibility of my own actions, I can stay home tomorrow.

When we say Tim cannot kill his grandfather, the set of facts we are using is more of an “all things considered” kind of set of facts, in which are facts like “Tim’s grandfather does not die in 1921”. With respect to this set of facts, it is true that Tim cannot kill his grandfather.

But this reveals that Lewis is right, and there is not contradiction between Fact 1 and Fact 2. This is because, if we are more careful and do not equivocate our use of “can”, they are not logically incompatible.

I think this is a nice argument. The little pragmatic spin I put on it also lines up with how I think we do a lot of our counterfactual reasoning, and in particular can shed light on how we think about reasoning about our own possible actions.

So it seems that, lucky for Tim’s grandfather, even if time travel is possible, he need not fear his time traveling grandson.