Do predictions have a truth value?Posted: 11 February 2012 | |
An old philosophical question is whether a prediction, by which I here mean a statement about the future such as
P: ‘it will rain on Sydney Town Hall at 1pm tomorrow’
is true or false at the time it is made, or whether it is neither, and only becomes true or false when the event to which it refers occurs, or fails to occur by the predicted time.
I think whether you accept that a prediction currently has a truth value depends on whether you subscribe to an A or B theory of time.
The ‘A theory of time’ holds that the future does not exist but is in the process of being created. Under this interpretation most predictions have no truth value. There will be some exceptions, which are things that have to be true, such as: “at 1pm AEST on 1 Jan 2013 it will either be raining on Sydney Town Hall or it won’t”. But I don’t think they’ll be very interesting. This one about rain in a given spacetime location is just a particular instance of a logical rule of inference – the ‘law of excluded middle’. It is true even though the events to which it refers have not yet happened because, no matter how the future unfolds, the prediction will be true.
The A theory does not seem to me to be compatible with a belief in absolute determinism – that is, that there is only one possible future, which will inevitably unfold under entirely deterministic laws of nature – the ‘clockwork world’. Under the most widely-accepted interpretations of quantum mechanics there is an irreducible randomness at the quantum level that makes determinism impossible. There is a minority view held by some physicists that future discoveries will be made that dissolve the apparent quantum randomness by revealing underlying deterministic processes. A-theorists would have to eschew such a view though, as a future that is completely determined by the present state of the world seems as real as the past.
Under the B theory of time, all of spacetime exists as a four-dimensional block in which separate events may be earlier, later or contemporaneous with one another (under Einstein’s Theory of Relativity, the ordering of events is not uniquely determined. For two events u and v, u may be earlier than v in one reference frame and later than v in another), and the passing of time is just our subjective experience, as the trajectory of our life is embedded in that block. Under this theory, every proposition about events in the spacetime has a truth value, seen from a timeless framework that transcends the block, including propositions about spacetime events later than the time the proposition is stated. The passing of time only changes our ability to know the truth value, not its existence.
The rest of this essay considers the question in the context of the A theory.
A common argument from those that hold predictions do have a truth value is that a prediction is just a proposition about the future, and propositions are either true or false. If one accepts this then there is nothing more to say. But whether we accept it depends crucially on the definition of proposition. For a start, one would have to disallow as propositions all statements that are incoherent, such as “nibble, squawk, omnipotence“, “the colour of my aura is five” or “one of Jim’s legs is both the same“. But doesn’t coherent simply mean that the statement has a truth value (that is certainly the problem with these three examples)? If so, it becomes far from obvious that a prediction is a proposition, because it is incoherent unless it has a truth value. So one has to suppose the conclusion – that the prediction has a truth value at the time it is made – in order to prove it.
Alternatively, if we do not require that all propositions are coherent then the second part of the argument fails: a prediction being a proposition does not entail that it has a truth value.
One can argue that a prediction does not have a truth value by asserting that the statement ‘it will rain tomorrow’ has no meaning – it is as incoherent as “nibble squawk omnipotence” This seems a very strong and surprising claim, but it is not easy to see a rebuttal. It is hard to explain what the meaning of the statement is without simply restating it, at best substituting some synonyms for the words therein.
Why do people so often say things like ‘it will rain tomorrow’ then, if they have no meaning? I think the answer is that people generally don’t, or if they do they don’t mean what they say. What they do say, if they’re careful about their words, is:
P1: “I believe that it will rain on Sydney Town Hall at 1pm tomorrow“
The crucial difference from P is that P1 is a statement about the present not the future. It is describing a current state of the speaker’s mind, saying that it holds a certain belief. This doesn’t help much with the dilemma though, because we can ask ‘but what is it that the speaker believes?’ We can rewrite P1 as:
P2: “I believe that P“
and it looks like we’re back where we started.
P3: ‘Most of the possible states to which I believe the planet Earth can evolve at 1pm tomorrow, starting from its current state, include rain falling on Sydney Town Hall‘
P3 is a statement about the present, as it is a statement about the now-existing set of solutions to a set of mathematical equations that represent:
- the current state of the world, and
- the laws of nature under which states can evolve, as the speaker understands them
- quantum randomness, and
- the speaker’s necessarily incomplete understanding of the laws of nature
We can sidestep the second of these sources of uncertainty by considering the statement:
P4: ‘Most of the possible states to which the planet Earth can evolve at 1pm tomorrow, starting from its current state, include rain falling on Sydney Town Hall‘
In contrast to P3, P4 may well be false if the speaker has an erroneous understanding of the laws of nature, or the current state of the world. That is because P4 is a statement about the laws of nature and the current state of the world, rather than about the speaker’s beliefs about those things.
Finally, the residual (quantum) uncertainty can be removed from P4 only if the laws of nature are, contrary to current mainstream scientific opinion, deterministic. Only then can the following statement be coherent:
P5: ‘The state to which the planet Earth will evolve at 1pm tomorrow, starting from its current state, includes rain falling on Sydney Town Hall‘
As noted earlier though, determinism appears incompatible with the A theory, so this statement only makes sense within a B theory, under which statements about the future are generally regarded as having a truth value.
Let’s summarise. What do A and B theorists make of the following statements, made at spacetime location (x,t)?
- Event E happens at spacetime location (x’,t’).
For a B-theorist this statement has a truth value, which is determined solely by whether E happens at (x’,t’), regardless of the value of t. For an A-theorist the statement does not have a truth value if t<t’.
- I believe that event E happens at spacetime location (x’,t’).
This has a truth value for both A and B theorists, but in both cases the truth value depends solely on the speaker’s beliefs at time t, not on whether E happens at (x’,t’).
- Given the state of the spacetime in the spatial neighbourhood of (x,t), event E probably happens at spacetime location (x’,t’).
This has a truth value for both A and B theorists, which is determined by the states to which the state of the world can evolve starting at (x,t) under the non-deterministic laws of nature. This is a purely mathematical statement, not about reality. For a B-theorist, this statement can be true even if E does not happen at (x’t’).