Tuesday, April 8, 2008

One of Several Ways to Prove the Negative

Suppose I tell you that “Screeds exist.”

Then you ask some questions and it turns out that what I mean by “screed” is something that is bleen, croom, and weeq.

Then you ask some more questions about the terms bleen, croom, and weeq. It turns out that those terms mean, “reptile,” “married,” and “bachelor,” respectively. So here’s the disproof:

1. Suppose that X is a screed. Then it would follow that:

2. X is bachelor, and

3. X is a reptile.

4. Bachelors are unmarried, adult human males. So,

5. X is human (by 4) and X is not human (by 3)


6. X is unmarried (by 4) and X is not married (by 3—reptiles can’t be married.)

7. Contradictions are impossible. Nothing can both have a property and not have it.

8. Nothing contradictory can exist.

9. Therefore, screeds cannot exist.

We just proved a negative. What’s the problem, exactly? Why is it that the urban myth that “you can’t prove a negative” persists, and persists, and persists?

For centuries, nonbelievers have been giving deductive proofs for the impossibility of God that demonstrate that there is no God using a strategy like this. But rather than actually consider any of those attempted disproofs, the widespread practice is to simply declare “Everyone knows that you can’t prove a negative.” That’s complete nonsense. We can and do prove negatives of all sorts—ask any mathematician. How do you think they conclude that some piece of mathematical reasoning is flawed. If I present you with a complicated logical formula like this one: ~(~a --> ~b) --> ((~a --> b) --> a)) do you think you can simply declare that it is true because “You can’t prove a negative”? It turns out that this formula is contradictory so we can prove that it must be false.

So if the concept of God is logically contradictory, which many people have argued, then we can prove the negative. For a recent collection of articles purporting to do just that, see Martin and Monnier’s anthology, The Impossibility of God.


anithri said...

I'm not great at this, but I think you haven't proved that screed doesn't exist, I think that you've proved that the proposition or that your assumptions are flawed is absurd (internally inconsistent).


I am in no way qualified to make this claim.

I guess I'm off to do some research.

Interesting subject though... been reading Saclzi's blog today?

anithri said...

And now I feel a bit foolish, given that when I came back I happened to notice that you were posting from experience, not in the exploratory way i took the post in the first place.

So given that I was wrong anyway...

It seems like "Can't Prove a negative" might be coming from a confused way of looking at "Failing to prove a negative does not affirm the positive". This seems to be a pretty widespread belief, any insight onto why that might be?

Matt McCormick said...

Hi Scott. Thanks for the comments. I'm not sure what "posting from experience" means. Bottom line is that we can be deductively certain that no thing exists that fits description that is logically contradictory. No square circles exist, no married bachelors, no four sided triangles, etc.

You're right that there is a widespread mistake of moving from the premise "X hasn't been proven to be false," to concluding "Therefore, it is justified to beleive that X is true." This fallacy is perhaps one of the single most common mistakes I see discussions with thousands of students and others about God. I think people's eagerness to infer the latter from the former is one of several confusions that can't be weeded out of one's thinking without a lot of hard work and dilligent self-analysis. If people don't want to see that error, there's not a lot that can be done to fix it. Nevertheless, I have written about it several times. See these previous posts, all linked on the left hand side of the main page:

Reasonable Belief, Proof, and Uncertainty

Proving the Negative

Possible, Possible, Possible: Overdrawing the God Account

Everything to the Glory of God

Confusing Possible with Probable and Having a Right to Believe


David B. Ellis said...

Reptiles can't be married?

More accurately: no known variety of reptiles has the cognitive capability to engage in a marriage contract.....but then when we find those intelligent descendents of velociraptors on the Lost Island that could all change.

Anonymous said...

I've heard a similar argument expressed as "Absence of evidence is not evidence of absence." And I thoroughly disagree with this argument as well. Nothing is literally 100% certain, so all "facts" are contingent upon additional evidence for or against them. It seems as if most people understand this for every day facts, but when the subject changes to god, their criteria for proof becomes unreasonable. You can no more prove that there is or isn't a god as you can a unicorn, yet most people don't claim to be agnostic of unicorns.

Up until the time that some evidence is found, absence of evidence IS evidence of absence. That is proof enough of a negative for me.

Jon said...

Surely we can be 100% certain about a great many things:
1)There is this thing that is called "I" that has perceptions.
3)2+2 does not equal 523 billion.

Anonymous said...

@Jon - OK I'll grant you the math, but also clarify that math is an abstract concept, while I was referring to more concrete "facts" that exist in the real world, not just our brains. I did not make that clear.

As for the self that has perceptions, can you prove to me that you are a perceiving self and not just an automaton? How do you know you aren't the only perceiving self, with the rest of us automatons? Yes, I realize that is almost an absurd question, but it underlines what I mean by being literally 100% sure of something. At some point, almost everything we take as "fact" has at least a miniscule "leap of faith" to it.

Lippard said...

I've written and linked to some arguments about proving negatives here.

Jon said...

What I gave was just a partial list of the 100% certainties. It is not necessary that I am able to give certainties to all questions, only some questions and be able to give alot of them - enough to be considered a great many. Of this we can surely do right? Now in the God arguments if it can be shown that there are incompatible qualities, then certain certainties can be shown. For example, if we consider the existence of an omni-God to have the quality of omni-benevolence, then that being cannot be evil. Correct?

anithri said...

"posting from experience..." I just meant that as an Associate Professor, you are quite qualified to speak with authority.

And if I might draw on your expertise, could you recommend a good book or lecture series on Logic or Logical Argumentation that someone might enjoy reading if he didn't have to read it. As interesting subject as it is, life is too short to read most textbooks for fun.

Robert said...

It seems to me that there is an assumption in your proof, tu wit:4. Bachelors are unmarried, adult human males. This assumes a closed class that is finite...but is it? Unmarried human male adolescents would by definition be batchelors also...the term 'batchelor' cannot defined solely as adult, therefore it would seem that the proof does not hold based on #4 committing the Fallacy of Accident.

Anonymous said...

Well, Robert, unfortunately Merriam-Webster disagrees with you.

"3 a: an unmarried man"

1 a (1): an individual human; especially : an adult male human

I guess, sometimes, you don't get to randomly redefine words.

Robert said...

Dear Anon...my original post referred to the line: ‘4. Bachelors are unmarried, adult human males’, emphasis on “adult”...what about a 17 yr. old, never married male adolescent? Is he not a bachelor? If not, then what? Additionally, The Oxford English Dictionary, which is perhaps somewhat more rapidly updated than M-W includes the term: “bachelor girl: an independent unmarried young woman”. The other meanings are: “ a male bird or mammal that is prevented from breeding by a dominant male and 2: a person who holds a first degree from a university or other academic institution”.
If the wording originally were “All unmarried human males are bachelors”, there is no problem from the ‘adult’ standpoint. In the original sentence, the term “male”, on the other hand, is definitely a problem in the second meaning, and certainly with regard to a bachelor girl...or the animals --the point is that precision must be observed with not only the logical portion of the argument, but also the verbiage. The author creates a finite set “ (All) bachelors are unmarried, adult human males.” If one exception may be found that does not meet the stated criterion, then the Fallacy of Accident, also known as a dicto simpliciter ad dictum secundum quid , or denying the exception has been committed. The objection stands.

Anonymous said...

"what about a 17 yr. old, never married male adolescent?"

No. Not sure why the definition of Bachelor is so difficult for you to grasp. If the definition is an adult, not a male.

Additionally, as far as I can tell your "Bachelor girl" is also irrelevant, if it were in fact including women the original "bachelor" would not need the additional 'girl' in order to clarify a female.

The other entries for bachelor are obviously red-herrings, when a word is used in a specific context with a purposeful meaning it's rather irrelevant that the word can mean something else in other contexts with other references. It seems to me you'd just as readily argue against "The Redsocks pitcher has thrown seven strikes this inning" because obviously containers which hold water cannot throw a ball. It's absurd.

So, quite frankly, the only thing you've asked is any unmarried male a bachelor? Simply no, by definition. It seems you just wish to incorrect classify things to make a counterexample.

The only point of contention is under which age does one become an adult - which is completely irrelevant because regardless of which age is chosen the proof still works.

Robert said...

Anon...I took the liberty of checking with Merriam-Webster online and found the following:
"Main Entry: Bachelor-noun
Etymology: Middle English bacheler, from Anglo-French
Date: 14th century

1: a young knight who follows the banner of another 2: a person who has received what is usually the lowest degree conferred by a 4-year college, university, or professional school "bachelor of arts"; also : the degree itself received a bachelor of laws 3 a: an unmarried man b: a male animal (as a fur seal) without a mate during breeding time"...

Funny you should mention it, it does actually have pretty much the same definition as the OED, except that you left out the parts you didn’t want to consider (I can understand the ‘young knight’ deletion). I draw your attention to #2 in particular: “A PERSON”. A person may be female, and they have been known to hold bachelor degrees.
A set has been created, tu wit: Bachelors are unmarried, adult human males

In setting up the Fallacy of Accident, it is but necessary to show the existence of ONE exception, and the statement is rendered false. The sentence would pass muster if it had said:” All unmarried human males are bachelors”, it allows for exceptions. The original sentence is false. The posting stands.

Anonymous said...

One of things which annoy me the most is when a religious person says "and you can't prove Gods NOT real." In which I say "you can't prove the invisible pink elephant that lives in my house isn't real either."

Now if I told you I saw invisible pink elephants living in my house you would say I was nuts.

When a religious person says they believe in an invisible God, then they have FAITH.

Anonymous said...

I'm not reading the comments, so I apologize if I repeat anything. One concern I have in both examples of negative proofs is that they depend on (i) analytic truths in the former and (ii) logical proof in the second. The logic does not make the world, since I can just as well make up any statement following modus ponens which could be absolute nonsense. The test is whether it is an actual truth. Therefore, one can logical prove or disprove God, given their system of logic, premises, etc. The question for that is whether it is anything of the natural world.

With (i) the issue is, like for (ii), whether analytic truths are of the actual world. We can say an unmarried bachelor is an analytic truth, but it's dependent on the linguistic system. That doesn't seem to get us any further than the logic since ultimately it appears we're just talking about abstract ideas for which the language is attempting to articulate. If that is the case, an analytic truth is like saying we have a logical tautology. We can even apply it to the real world, e.g., "Joe is a bachelor, therefore Joe is an unmarried man," but is that truth one of the world of merely a construction of logic applied to the world (ad hoc)? Joe, in this example, is only a bachelor or an unmarried man, inasmuch as we perceive him as such. It's dependent on many things we assume to hold true for the logical relation to hold true, but it need not be any more apart of the world than imaginary numbers need to be apart of the world simply because Maxwell's electromagnetism uses them as a means of conveying the electric and magnetic forces. Do we say imaginary numbers exist in the world because of that? Surely not, but it does show we can apply them in the real world.

Therefore, my concern is whether or not you can prove a negative outside of the realm of a deductive system of logic for which we can state such crisp and clear boundaries such as tautologies and mathematical truths. I'm one to motion for inductive truths in reality and proving a negative in that regard is a whole new ball game from the easy analytic or logical truths as demonstrated in the examples you provided.

Anonymous said...

One last thing to add to the conclusion, the kind of "proving the negative" we would derive from an inductive proof would be that of confidence (many question if induction provides proofs at all, however). I cannot prove or disprove the God of the Bible, but I can show it to be a very unlikely truth, given the way reality (nature) works (i.e., from our given knowledge set of the world). This kind of "proving the negative" is far weaker than the kind of proof we look for in the deductive logics as presented in the blog, but they seem to be how we make inferences in reality. Likewise, it also shows that we need a high confidence in believing in a God as well, given our knowledge of the world. But how one would derive that kind of inference opens up a whole other can of worms.

Matt McCormick said...

Thanks for your comments Bryan Goodrich. Lots of interesting ideas. You seem to be downplaying the significance of the fact that many descriptions of God are logically incoherent as if that doesn't really tell us anything about the issue. Claiming that the logical problem is just one of definition within a linguistic system suggests that we've got something else to fall back on. Look, if a description of an entity turns out to be unintelligible and riddled with contradictions, as a long tradition of deductive atheologists have argued, it's not like that just a minor speed bump for the believer. That means that the thing that they are claiming to exist is logically impossible. Which other "linguistic system" do you propose we use to describe this fantasy? One where the law of non-contradiction doesn't apply and there's no difficulty asserting P and ~P or that circles really aren't circles and bachelors really aren't bachelors? Logic is the bedrock of any sort of intelligible discourse. People often talk as if we can just drop it at will like changing from English to French. No, contradictions are ruled out at the most fundamental level in all languages. At the very least, the substantial burden of proof is on any believer to explain how it is the thing they believe in exists, but it defies the very fabric of reality and logic, and how believing in it can be rational. Until then, the deductive atheologist is perfectly justified in rejecting such claims as incoherent.

As for inductive proofs: Certainly that's another way to build the argument. But notice I haven't said those aren't possible. Deductive disproofs are one of several approaches to the question.


Anonymous said...

I do not really disagree with what you say. What I am getting at is that not all definitions of God (or anything for that matter) are so clean cut to say we can evaluate it by simple deductive logic. To prove a negative in such cases is rather easy. Like the reductio, all you need to do is show one value to be inconsistent, and you get the other. What if there are more possible truth values? As soon as you lose the excluded middle you have to show many more negatives to claim the positive. Certainly I am not saying ~A isn't a valid claim, but ~A may consist of X, Y and Z. Of course, all that requires is showing Z is absurd, Y is absurd and X is absurd, then we claim A. What if there's more? Apparently any n set of possible truth values needs n-1 reductio proofs! What if n is infinite? It's like a falsification problem. Can we disprove the claim that a box contains X? That's as easy as looking in the box and seeing if it is there. But what if the box is extremely big or infinitely large? Not only does it become an accessibility problem, but possibly impossible to do a "reductio" kind of proof. My point being that we need to be very creative to negatively do a negative proof (not saying it can't be done, depending on the system of mathematics you use to do it).

Back on point, I do not discount the fact that if God is clearly defined to be contradictory, we can easily identify that it cannot be true if the world is such to be logically sound. But does that reach the scope of all definitions of God, or even most? On what criteria are the definitions measured? We all like to say it is absurd to think unicorns or Santa Clause is real, but is there clearly something in their definitions that is contradictory? How do we prove the negative in this case? Searching an inaccessible or infinite box for how they do not fit into it? Unlikely, but that is why induction proves useful. If magic gnomes cannot be shown absurd by mere deduction (and definition), how can God be demonstrated to be as such any better?

Anonymous said...

If a mind perceives that a god is real, it is real within that mind.

Matt McCormick said...

Sure, you can form whatever ideas you want in your mind--unicorns, Bigfoot, a million dollars in your checking account, a hot supermodel girlfriend, but they aren't real. And fantasizing about them all you want or believing in them as fervently as you want won't change that.


Kailya said...

This is all very fun but it's an age old argument which has been solved a thousand times over and it's a natural fallacy. Let's examine a word from the real dictionary; "Run". Run has one of the longest definitions in the English language, and if we were to use this logic to evaluate each, then the word cannot exist because "to run" as in sports is not the same "to run" as in computer programming because one is the rapid movement of a human's legs and the other is the process of on off switches which by definition have no legs by which to run.

So to say that the word (sorry forgot it already) cannot be a term for a reptile and a term for a bachelor is also a logical fallacy since language has such massive variances such as slang, conjugations, etc.

I like reading your blog when I get the time, but, and this is not meant as a personal attack, it's old stuff. Nothing new and most of it already solved. I love debates, I love civil conversations, but that also requires knowning both sides of the topic. There are some really good arguments out there now that the great minds are hashing out, let's talk about some of those.

Kailya said...
This comment has been removed by the author.
Matt McCormick said...

Kailya, thanks for reading my blog and for posting. But some of your input isn't really helpful. First, if by "natural fallacy" you mean "naturalistic fallacy," you need to look that one up; you're really off the mark, I think. You've frequently accused me of rambling, being unknowledgeable, or otherwise expressed your frustration with my lack of insight. It's hardly fair to accuse me of not addressing the recent literature--I have regularly cited, quoted, mentioned, or responded to a long list of references from the last 20 years in the philosophical literature. If there's some vast literature out there that you think I"m missing, I'm all ears. Maybe it's all the shelves of Christian apologetic drivel at the Bible store you've got in mind. You're right, I don't read much of that stuff. After looking through lots of it and trying unsuccessfully to find clear, thoughtful arguments that weren't just blatant appeals to authority (look up that fallacy too) I have given up. William Lane Craig, Alvin Plantinga, William Alston, Peter Van Inwagen are all important and influential philosophers and Christians. You will find that I have responded to all of them frequently here.

Finally, you're point about the various definitions of words is interesting. But here's the problem. It sounds like you are suggesting that it will be impossible to every disprove any notion of God because the definition keeps changing or different people mean different things by it. Ok, fine. Let's take the one account of God that figures centrally in the Christian, Muslim, and Jewish traditions--the God that is all powerful, all knowing, and all good. I have gone through a long list of problems with that definition of God here, and there is a vast literature out there addressing problems in that account (that I don't think you're familiar with). That notion of God has not proven to be intelligible. If you've got some other account of a being that you think is worthy of the name and that we should all think about, then we can talk about that one too. But the problems with the mainstream account of God can't just be waved away with personal attacks and vague allusions as you're doing.