[Edit: it turns out that something quite similar to this is argued for in this paper by Nicholas Shackel]
The Grim Reaper Paradox (GRP) comes in various different forms. Sometimes it is about the divisibility of time, and sometimes it is about whether the past (or future) is finite. Even when we fix on which of these issues it is aimed at, there are also lots of different ways it can be cashed out. It can be reapers swinging their scythes, or placing point particles on planes, etc. Much of the discussion can be on how these details are to be understood.
Here I want to highlight what the GRP is at the most abstract level. It might be that once we think about it from this rarefied perspective, without the complications about what exactly the details are supposed to be, that we can see the paradox more clearly.
- The Schema
Any GRP has a logical form, which I shall refer to as the schema. Let’s just have as a toy example, the following:
The past has no beginning. There is an eternal machine such that each day at midnight, it checks to see if it has printed out anything yet from its printer. If it has, then it hibernates for the rest of the day. If it has not printed anything out yet, it immediately prints out the date and then hibernates for the rest of the day.
This is enough to generate our paradox. If it had not already printed anything out, this means that yesterday it would have run the same check and printed out the date. So it can’t be that the machine finds nothing printed out today. But that applies also to yesterday too, and every previous day. So although it can’t be that no date is printed out, no date could be printed on the paper.
The way to conceptualise this abstractly is as follows. There is a rule that characterises this example (and all the others). It is a universal condition that applies at all times. That condition says that some proposition p (which might be that a reaper kills Fred, or places a point particle on a plane, or that a machine prints out a date, etc) happens at a time if and only if p does not happen at any earlier time:
For all t (p at t iff for all t’ (if t'<t, then ~p at t’))
It says that p is true at t if and only if p is not true at any earlier time.
The schema on its own is not unsatisfiable. That is to say, if there are only finitely many times, then the schema can be true. In particular, the schema is true if there are only finitely many times and p is true at the first time. At that first time, p is true at t, and on the other side of the biconditional, the nested conditional (if t'<t, then ~p at t) has a false antecedent, and as it is in the scope of a universal quantifier it is vacuously true. So both sides of the schema are true. At all other times, p is not true at t (so the left side is false), and on the other side of the biconditional we have a condition that says that ~p is true at all earlier times, which is false because (as we just went through), p is true at the first time. So in all cases, the biconditional holds.
But if there is no first time, then we run into the familiar problem. If p is true at some time t, then the right side of the biconditional says that no earlier time, t’, could have p true at t’. But then take t’. It is also the case of that time that no earlier time, t”, has p true at it either. So given the left side of the biconditional, p is true at t’. Contradiction.
If p is not true at any time, then the left side of the condition is false for some arbitrary time t. But if p is not true at any time, then its not true at any time t’ earlier than t, which makes the right side of the biconditional true, which in turn implies p is true at t. Contradiction.
Now we have a purely logical version of the argument, freed from any distractions about reapers, or point particles, or eternal machines. The GRP really just says:
- There is no first time t
- For all t (p at t iff for all t’ (if t'<t, then ~p at t’))
As we have just seen, you can’t have both of these together. That is the GRP.