Asymmetries in two opposite arguments from ignorance

Question

Joe claims: "There is no proof that unicorns exist, therefore unicorns do not exist".

Alice claims: "There is no proof that unicorns do not exist, therefore unicorns exist".

Bob claims: "There is no proof either way, therefore I'm agnostic about the existence of unicorns".

From a formal point of view, Joe and Alice's claims are both arguments from ignorance, hence they are unsound. Bob's argument is sound.

But we all intuitively know that Joe is right (or at least probably right) despite the apparent formal symmetry between Joe's claim and Alice's claim.

So, there must be something asymmetric that makes Joe's claim stronger. But what is it? Where does the asymmetry stem from?

Related term: Russel's teapot

Answer 1

We all intuitively know that Joe is right, but we don't know that formally. We are talking about formal logic here. Formally we just don't know, and, based merely on the fact we don't have either proof, we can't say anything about probabilities.

Both arguments are equally unsound and weak. However, using other evidence (for example that nobody has ever seen a unicorn), we can say something about the probability of Alice's and Joe's conclusions: given the evidence that nobody has ever seen a unicorn, it is likelier that Joe is right than that Alice is right. This does not say anything about the weakness of their arguments though, only about the truth of their conclusions.

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Supposing the arguments were asymmetrical and the one form is indeed stronger than the other, we should be able to formalise them:

• Joe: there is no evidence for P, therefore ¬ P.
• Alice: there is no evidence for ¬ P, therefore P.

If we're really to say that Joe's argument is formally stronger, it must be stronger for every P. For example, we may fill in "Barack Obama's existence":

• Joe: there is no evidence for Barack Obama's existence, therefore, Barack Obama does not exist
• Alice: there is no evidence against Barack Obama's existence, therefore, Barack Obama exists

In this case, few people would think Joe's claim is stronger.

Answer 2

Joe claims: "There is no proof that unicorn exist, therefore unicorns do not exist".

Alice claims: "There is no proof that unicorns do not exist, therefore unicorns exist".

Bob claims: "There is no proof either way, therefore I'm agnostic about the existence of unicorns".

In purely formal logic Joe's and Alice's claims are invalid.

In the real world, when the existence of something would very likely have caused evidence about it to be available, and no evidence is available, then that is a strong indicator of non-existence. This is what's implied by Bob's claim when it's interpreted as being about the real world. That's because you know that in the real world the existence of unicorns would likely have produced a lot of evidence about them. There would have been a whole industry providing unicorn pictures and stories etc. to fans of unicorns.

I.e. the feeling of different strengths of arguments stem from interpreting the claims not as pure logic where the statements are all that is, but as real-world claims, with associated facts and inferences.

To formalize this I think one would need to replace the word "evidence", which has different meanings in the real word and in pure logic, with e.g. "reports". Then one can say (P1) if unicorns exist then the probability of having seen at least one unicorn-existence-report would be >0.998, (P2) if unicorns don't exist the probability of having seen at least one unicorn-non-existence report would be <0.001. And with these (or other better) probabilities one can reason about the above Joe and Alice claims.

Answer 3

Self-answering the question. After years, I think I formulated a nice answer. The unicorn situation is not really symmetrical. I think this is an intuitive interpretation of Bayesian Inference.

Let's think of the world as an infinite series of rooms. The real world is not like that, but this approximation is sufficient.

In order to prove unicorns exist, we only need to find one room which has a unicorn in it.

In order to prove unicorns do not exist, we need to scan an infinite number of rooms. This is infeasible. The best we can do is scan some of the rooms. The more rooms we scan and discover to be unicorn-free, our confidence in "unicorns do not exist" increases. If we scan a huge amount of rooms, the confidence is extremely high, but we're never 100% sure. This is why scientists say "you cannot prove a theory". A theory is typically a claim that something is true in every "room". When the confidence is sufficiently high, some theories can be regarded as facts for all practical uses.

As a human, you've been here on Earth for quite some time, and you've scanned many rooms. You probably didn't find a unicorn yet. This is why "unicorns do not exist" can be regarded as fact. (This is also combined with other pieces of knowledge about mammals and how the world works, as other answers have explained).

If a 1-second old baby is fully rational, and that baby has not scanned any rooms yet, then its most rational bet is: There is a 50% unicorns exist, and a 50% they do not. This is why formally, given no further info, both arguments are equal.

So why is the method proving a claim so different from proving its negation? That's because "unicorns exist" in fact means "there is at least one room with a unicorn", whilst "unicorns do not exist" means "all rooms have no unicorns". Are these negations? Logically, yes. But to fully de-mystify symantics and language, note there are actually 4 possible claims here:

1. Unicorns exist in at least 1 room. Easy to prove. Impossible to disprove.
2. Unicorns do not exist in any room. Easy to disprove. Impossible to prove.
3. Unicorns do not exist in at least 1 room. Easy to prove. Impossible to disprove. Already proven.
4. Unicorns exist in every room. Easy to disprove. Impossible to prove. Already disproven.

1 and 2 are logical negations. 3 and 4 are also logical negations.

Answer 4

This feeling is accounted formally, at least in part, in a form of logic that has been constructed to represent 'Intuitionistic' mathematics.

Brower, the intial framer of intuitionism, became famous for the 'hairy ball' theorem in topology -- basically, that you cannot uniformly paint a sphere entirely with non-overlapping brush strokes along the surface of the sphere, there will always be an overlap or there will be points left that have to be filled in by the point of the brush at the end -- you can't comb a hairy ball. But he did so by 'reductio ad absurdum', by assuming there is no flaw at any point and getting a contradiction, in a way that does not identify the flawed point.

He felt he had done the world something of a disservice by answering the question. What was really needed was a 'constructive' answer that identified the point, or continued search for a solution. So he stepped back and tried to capture what would happen to mathematics if we took this dissatisfaction seriously.

So you are not alone in your feeling about the two assertions, there has been a school of math that feels that way too. We intuitively expect more out of an existential quantifier than the assertion of possibility. We are fairly comfortable with universal statements about entities which have no reason to believe exist being nominally true and letting them go. All unicorns are black, and all of them are also white. Until we meet an actual unicorn, we are fine with that to a large degree.

But for an existential statement, this is not satisfactory, we would really like a stronger sense of proof. Although assertions about situations in which there is no evidence in general are really neither true or false, unproven existential statements are less true, in an important way, than unproven generalizations.

This means that existential statements should require stronger evidence than the negations of universal statements, so De Morgan's laws, as they apply to quantifiers, are overly strong, and we need a different way of looking at negation, especially when it is applied over infinite or other merely potential sets.

Since Brower was a better mathematician than philosopher, his own internal study of intuition became less than compelling. (It also happened in Germany right before Nazism, so it got interpreted as a sort of Nationalism and anti-Semitism, since the deepest thinking about infinities, logical complexities, etc. near that time originated from Jewish, Polish and English investigators.)

But there are clearer formulations of this kind of thing that keep arising, and it seems this will not stop until some school of Intuitionistic or Construtivist Mathematics captures what Boolean logic takes out of more natural logic in a way that is adequately clear.

Answer 5

The soundness of an argument depends on whether the premises are "actually true" or not. This definition of soundness comes from the Internet Encyclopedia of Philosophy:

A deductive argument is sound if and only if it is both valid, and all of its premises are actually true. Otherwise, a deductive argument is unsound.

Soundness is about the actual truth of the premises, not whether the argument has the form of an argument from ignorance.

The OP wants to know if there is some assymmetry between the following two arguments:

Joe claims: "There is no proof that unicorns exist, therefore unicorns do not exist".

Alice claims: "There is no proof that unicorns do not exist, therefore unicorns exist".

Douglas Walton describes arguments from ignorance in terms of strength or weakness: (page 270)

In an inquiry, the argumentum ad ignorantiam for a particular proposition becomes stronger and stronger as the knowledge cumulated by the inquiry becomes more and more firmly established.

Consider the question: So, there must be something asymmetric that makes Joe's claim stronger. But what is it? Where does the asymmetry stem from?

There would be asymmetry between the two statements based on "cumulated knowledge". Joe's statement would be stronger. That gives Joe's argument a greater "degree of plausibility" than Alice's argument.

It should be noted that neither of the arguments are fallacies, according to Walton, unless they are "used in a context of dialog (of which there can be many types) as a tactic of deception to trick a speech partner in an exchange, or as an underlying, systematic, and serious type of error of reasoning." (page 270) From what little is provided of the context and the content of the arguments that seems unlikely.

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anonymous, "Validity and Soundness", Internet Encyclopedia of Philosophy https://www.iep.utm.edu/val-snd/

Walton, D. (1996). Arguments from ignorance. Penn State Press.

Answer 6

Joe's negative existential claim that there are no unicorns is true by default until a unicorn is proven to exist, which would falsify his negative existential claim (since only negative existential claims can be falsified).

Alice's positive existential claim that a unicorn does exist is falsified by default the second she mutters it until a unicorn is proven to exist, which would upon occurence falsify Bob's negative existential claim that they do not exist, since again, only negative existential claims can be falsified.

Bob is neither right nor wrong, since he has not expressed an opinion other than to say he does not know. However, he is correct in stating that he does not know for certain that unicorns exist, nor is he technically certain they do not exist because one or more could be hiding somewhere. This is assuming that he has not seen one and chosen to lie about it.

In summation, Joe starts out right until they find a unicorn simply by saying they don't exist, Alice is right when they find one, and Bob could never be right or wrong. This is, of course, assuming everyone is telling the truth.