Jim Lippard

In his response, he writes that "Mr. Lippard has included the article below as a part of his website because, apparently, he believes that the criticisms directed at skeptics in the article are also representative of his views."

This is not an entirely accurate inference. The mere presence of something on my website is not necessarily an indication that I agree with it at all, let alone that it is "representative of [my] views." My website includes material that I completely disagree with, as well as material that I completely agree with. For example, I have a number of documents on my website simply because attempts have been made to censor them. I also have hosted documents which are critical of articles or essays which I have written, because I think they've made arguments worth being heard.

In the case of "Stupid Skeptic Tricks," I do agree--as does Debiak--that many of the points apply to some (but not all) skeptics. I must completely disagree with Debiak on the "proving a negative" point, however. It is simply false that "you can't prove a negative," as anyone who has taken an introductory symbolic logic course knows. (One common retort to this false claim is "Prove it!"). For more on this subject, I recommend Jeff Lowder's "Is a Sound Argument for the Nonexistence of a God Even Possible" and Richard Carrier's "Proving a Negative".

To expand a bit on the "proving a negative" issue: A more precise version of the claim is "You can't prove a universal negative proposition" or "You can't prove a negative existential proposition." (These are logically equivalent.) First, let me explain why the more precise version is on somewhat better footing than the less precise version, though still wrong.

It is important to distinguish "prove" in the logical/mathematical sense from "prove" in the more common use of language that would include legal proof, scientific "proof," or "proof" of any empirical claim. The former, logical/mathematical sense is the most precise sense and the most restricted sense. "Proof" in the logical sense is restricted to deductive consequences, yet even in that sense you can prove negative propositions, provided you have the right premises to start from.

There are lots of easy ways to prove simple negative propositions. If we have a premise of the form If P Then Q and a premise of the form Not-Q, then we can infer/prove Not-P using the deductive inference rule of modus tollens. If it were raining, my roof would be getting wet; since my roof is not getting wet, it's not raining. Further, every time you observe something in a particular state, you've also observed that it's not in any state that is a logical contradiction. If you can prove a positive, you can prove a whole bunch of negatives simultaneously. Likewise, if you can't prove a negative, you can't prove a positive, either. If you can't prove that a particular thing is not in every state that it's not, you can't prove that it is in the state that it's in. The evidence of the latter is evidence of the former, and vice versa. (Note that I'm leaving aside issues like wave-particle duality here.) This is logical, deductive proof--if the premises are true, the conclusion is a logically necessary consequence--it must also be true.

In the broader sense, as well, we can certainly prove negative propositions. Without making the statement about what can't be proved more precise (restricting it to universal negatives or negative existentials), the claim would entail that I can't prove, in the ordinary empirical sense, that there is no elephant (of the live animal sort) in the room with me. But clearly I can legitimately infer, based on what I know about the size of elephants and the space within this room, based on my observations, that there is no elephant in the room.

But even when we restrict the statement about what's provable to negative existential statements (there are no naturally pink elephants) or their equivalent universal negatives (all elephants are naturally not-pink), this is still something that can be proven in a number of ways, either by showing how it is logically contradictory to known facts (perhaps some fact about skin pigmentation in mammals) or by exhaustive enumeration over the domain or by the (deductive) process of mathematical induction. Again, we can make deductive inferences which yield proven negative existential statements or their equivalent universal negatives.

The sense of "prove" is weaker when we can't exhaustively enumerate over the domain of the bound variable in the proposition--where we use induction rather than deduction. This is really the idea behind the claim that "you can't prove a negative"--that we don't have the resources or ability to exhaustively enumerate all examples over the entire universe. But notice that this is an issue whether the proposition is positive or negative, and that all positive propositions have equivalent negative propositions, and vice versa. Also notice that, if the scope of the domain is sufficiently small, proof can be quite easy.

Let's take our domain to be swans, and look at the property of being purple. Which of these statements is supposed to be impossible to prove? (1) All swans are purple. (2) Not all swans are purple. (3) There is a purple swan. (4) There is no purple swan. Now, statements (1) and (2) are easy to disprove and prove, respectively--each requires only a single non-purple swan to demonstrate. The former is a universal positive statement, the latter is a negated universal. Statement (3), a positive existential statement, requires only a single purple swan to prove, but takes a lot of enumeration to disprove. Statement (4), a negative existential, is a statement of the type that is supposed to be impossible to prove, and it clearly requires the most work to demonstrate, but only a single purple swan to disprove. (This statement is equivalent to the universal negative, "All swans are not-purple.")

All that lies behind the more precise statement, "you can't prove a universal negative (or negative existential) statement" is that the most straight-forward, direct manner of proof--exhaustive enumeration--is not always available due to practical limitations. (Though if we restrict the scope of the domain to "swans in this room," such enumerative proof can be trivial.) But as we've already pointed out, that's not the only kind of proof there is.

Within the fields of mathematics and logic, we can use such proof methodologies as mathematical induction and reductio ad absurdum (or proof by contradiction) to deductively prove universal negatives or negative existentials over infinitely large domains, despite not having the time to individually test each instance. We can also derive such statements from other already-proved theorems. (A note on mathematical induction: despite the name, it is a form of deductive logical proof. This method of proof proceeds by showing that a statement is true for, say, n=0, and then showing that if the statement is true for n=k, the statement is also true for n=k+1. This methodology can be used to prove mathematical propositions such as (for a very simple example) that no number is greater than or equal to its successor.)

Looking at the empirical realm, we can still prove such statements, deductively, by inferring them from other things we know. To the extent we know the things we use as premises, we know, deductively, the conclusion. If we know that nothing can travel faster than the speed of light in a vacuum, then we can derive the negative existential statement that no alien spacecraft has traveled at faster than the speed of light. Likewise, if we can derive a contradiction from a positive existential statement, we have thereby proved its negation.

But in the broader sense of "proof," we can even go farther--we can "prove" in the sense of providing strong evidence such that denial is as irrational as solipsism or the belief that we are brains in a vat being fed false stimuli, based on statistical inferences. In that sense, we can prove that there are no swans that can write novels, for example.

It is important to note that if this kind of inference is ruled illegitimate, or doesn't count as "proof" in the broad sense, then we can't prove positive empirical statements, either--they, like negative empirical statements, are based on premises that are justified in terms of similar inductive inferences. Anything that requires reliance on sense perception, memory, or inference from past to future events is similarly dependent on such reasoning.

So both "you can't prove a negative" and "you can't prove a universal negative (or negative existential) statement" are false--it would be more accurate to say that "It can often be more difficult, as a matter of practicality, to prove an empirical universal negative (or negative existential) statement than to prove an empirical positive existential or a negative universal statement," or "It can be impossible, as a matter of physical limitations, to prove universal negative (or negative existential) statements by exhaustive enumeration of instances." But you must remember that exhaustive enumeration of instances is not the only method of proof we have.

July 21, 2004

[Bloomsberg University philosophy professor Steven D. Hales
published an article making similar points, in a less technical way,
in *Think* in Summer 2005 titled "You
*Can* Prove a Negative.".]

Note added May 15, 2008: I think I was perhaps somewhat too
dismissive of the practical (as opposed to logical) epistemological asymmetry between the different
types of statements above. This asymmetry is the subject of Nassim
Nicholas Taleb's book, *The
Black Swan: The Impact of the Highly Improbable*, which I
recommend. This is also discussed (specifically in reference to the
above essay) in the comments following The Two Percent Company's Rant,
"Only
God Can Prove a Negative, and There Is No God." In that
discussion, I think The Two Percent Company (like most skeptics) is
more concerned about Type II errors (accepting a false hypothesis)
than Type I errors (rejecting a true hypothesis), but either type of
error has the potential for running into a "black swan" in Taleb's
sense. Skeptics tend to allow their concern about Type II errors to
lead to rejecting data (what Charles Fort called "the damned"), while
believers tend to allow their concern about Type I errors into
accepting all manner of nonsensical explanations for data. (See my
Powerpoint presentation on "What
Skeptics can Learn from Forteans.")

Added November 25, 2010: Vincent Bugliosi's massive tome,
*Reclaiming History: The Assassination of President John F. Kennedy*
(2007, W.W. Norton and Company) contains a footnote on p. 973:

One frequently hears that "it is impossible to prove a negative." But this, of course, is pure myth. In some situations, as in the murder of President Kennedy, it is impossible, but in many situations in life it is very easy. For instance, in a criminal case where a defendant says he did not commit the robbery or burglary, or what have you, because he was somewhere else at the time, the prosecution routinely proves the negative (that he was not somewhere else) by establishing through witnesses, fingerprints, DNA, or sometimes even film, that he did commit the crime and was not where he said he was at the time it happened.On an even more obvious level, if someone were to say, "I have [or do not have] pancreatic cancer," medical tests can disprove this (i.e., prove the negative), if such be the case.

In the Kennedy case, I believe the absence of a conspiracy can be proved to a virtual certainty.