In this blog post I'm going to talk about existential statements, the witnesses for these statements and whether there is value in existential statements without witnesses.
PART 1: How can we have existence without witness
An existential statement is of the form 'There exists X with property Y in domain Z ' which means that there is an (at least 1!) entity (call it X) that has a property (call it Y) in whatever domain or area that we're interested (call in Z). For example, 'there exists a bottle that is blue in the fridge'. The entity is the bottle, the property is blueness and the domain is the fridge.
Now, in practice people will make existential statements without explicitly following the format above. Consider statements like:
1A: 'There's a soulmate out there for you'
2A: 'Someone lives here'
One can consider these as elliptical constructions which means that the statement ommits certain words but are nonetheless understood in the context of the remaining elements. In other words, they can be considered short hand for a more expansive sentence. The equivalent expanded sentence might be:
1B: 'There exists a person who has the property of being your soulmate and is in the current world'
2B: 'There exists a person who has the property of living (in this building) and is in the current world'
A witness for an existential statement is an entity that satisfies the existential statement.
For example, a witness to 1A would be the persons actual soulmate. This person would validate the existential statement - it provides a proof that the statement is true.A witness to 1B would be the specific person who lives in the building.
Now ideally our existential statements all come alongside witnesses. I.e. whenever we say something exists, we can actually identify that specific thing that exists. As long as I have an entity, I can construct a existential statement in which the entity acts as the witness. The inverse is not however true - the fact that I can construct existential statements, even prove existential statements says nothing about whether I can identify the relevant witness.
Consider the following mathematical statement:
< 3 >: There exists irrational numbers A and B such that A^B is rational
Now consider the following argument (I will write R(X) to indicate the square root of an integer X):
R(2)^R(2) is either rational or irrational.
If is rational, then we have a witness to the existential statement, A = R(2) and B = R(2).
If it is irrational then let A = R(2)^R(2) and B = R(2).
Note that A^B = (R(2)^R(2))^R(2) = R(2)^2 = 2, which is rational. Therefore one of the above is a solution.
The point is that we know for sure that the existential statement is true - we have just proved it. However we are unable to determine with certainty a witness for the solution.
Now a keen observer might notice that the problem seems to be the first line of our proof - a claim that a number is either rational or irrational. This is an instance of the law excluded middle which states that it is always the case that (P OR not P). Whilst this law is true logically, is has the annoying habit of allowing us to provide existence without witness. In the example above we showed that the P case leads to a solution and the not P case leads to a solution, without identifying which one is the case. This lack of witness can be rather frustrating and has lead to an interesting area of mathematics known as constructivist mathematics, where one is unable to prove an existential statement without providing a witness. One way this is done is by restricting the use of the law of excluded middle.
PART 2: What value is there in existence without witness?
Our example with the irrational numbers showed that we can have existence without witness. In a lot of pure mathematics we probably don't care about picking out a specific witness that validates our proof. We will however construct a name for a witness (that we know exists) and allow the name to point to said witness. Let us call this a denoting witness.
This might seem a bit confusing, so let's consider an example. Suppose it is true that 'there exists a richest person in the world'. Now off the top of my head I might not know who this person is. However I might still make claims such as 'the richest person in the world should donate some of their money to help world hunger'. Essentially I'm using 'the richest person in the world' as a name that refers or denotes its witness and then making claims about this entity. The same is frequently done in mathematics where we might let X denote our witness to an existential claim. The reason this is useful is because we want to later use X (and it's proven properties) for further proof claims without going through laborious effort of actual identification.
It's clear that denoting witnesses have significant uses in mathematics, but I think this has arguably more important ramifications for metaphysics. In mathematics, we act within a supposedly objective world (there's contention here, but most people would argue that mathematics is mind-independant). In contrast it is often more difficult for us to determine to what extent our reality is objective.
Consider for example the following claim:
< 4 >: 'There is an objective morality'
This is an existential statement and a standard analysis is that it is elliptical for the following claims:
4i - Moral claims purport to facts and are true if they get those facts right
4ii - Some moral claims are true
4iii - The truth value of moral claims is determined mind independently.
Now, I think there are some very good arguments to think that other moral theories are weak and I've already argued about this in previous posts. So we have some reason to believe that < 4 > is true.
One can note however that < 4 > is to some extent uninformative - it tells us about the nature of morality without telling us what things are actually moral. It doesn't tell us whether murder is wrong, whether it is okay to lie or whether we should be helping the poor. Essentially, the next step to the existential claim is to ask for a witness - please give us some kind of thing that satisfies the conditions addressed above.
It might be considered that the lack of witness is a big problem for moral statements. Moral statements are strange and while most people have basic moral intuitions, this doesn't seem to be the most robust of epistemologies. Even as a moral objectivist, I would probably cede that it is difficult to produce a witness for our moral existential statement. Furthermore, some people believe that moral theories should be to some extent action guiding. To say that there is an objective morality but provide no way to identify what that morality is, is unhelpful as it doesn't say how we should behave.
In fact the question then becomes - does it provide any value at all? We haven't even proven the moral objectivst position and are already doubting what the point of doing so would be.
I argue that whilst having a witness would be fantastic, it is still very useful even if we don't.
Suppose we think < 4 > is true and let W be a denoting witness. What do we know about W? Well only that it exists and satisfies properties (4i, 4ii and 4iii), which actually amounts to quite a lot.
For example, if moral M satisfies 4i then that means that when people are talking about morality, they are actually referring to facts about the world. This is also means that if two people disagree about a moral claim, at least one of them is wrong. Suppose I make a moral claim, C and you make the opposite claim Not C. Now, C is true if gets the relevant facts about the world correct. However that means that Not C must be wrong as it would get the relevant facts about the world wrong.
Another interesting observation of W being true (even without a witness) is that it seems people can do things that are morally wrong without knowing it. One can act in a certain way, that goes against the moral facts of the world. In this sense, we could argue that the person is morally wrong even if they had no idea about it. This can have some interesting after effects - it means that people in the past could be morally wrong as they go against the moral standards of the world.
I might be wrong when judging the people of the past (or the people of different cultures) for being immoral, but the moral facts of the world don't care. You're either moral or not, depending on how your map to those facts. This therefore puts quite a lot of pressure on things like cultural relativism.
We might also consider what value 4ii has. As we lack a witness, I can't tell what moral claims are true. However, since I know that some moral claims are true, I know that it isn't the case that 'anything goes'. Things I do could be morally wrong. Knowing that I could be doing something bad is quite interesting as it enforces some restrictions on our behavior. A final comment on 4iii is that it doesn't matter as much what one thinks - it is the facts about reality that judge.
Finally, I know I often talk about morality so I do want to mention some other examples. One useful example to consider is physical reality. Many people think there is sufficient justification for a claim of scientific realism which is the view that the universe described by science is real regardless of how it may be interpreted. At the moment science provides an incomplete picture, but I think many scientific realists might endorse a stronger version - a claim that there is a 'true physics' that would fully describe the world. Note that this puts us an interesting situation - we might endorse the idea of a true physics, yet we obviously lack the witness to it. However by endorsing existence even if we lack witness, we can make some useful statements. For example we can reasonably conclude that people are crazy or hallucinating when they don't believe the same physical statements ('there is a chair') to others around them. We can consider some people irrational if they refuse to take a vaccine, especially if they believe in alternative methods that lack evidence.
I think I've written enough but I found this quite useful to jot some thoughts. People often argue with me about the importance of epistemic access about certain metaphysical claims I have made. What they are really asking for is a witness. Of course I'd love a witness - but I don't think they realise that existence if often enough.
