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Click Here for New & Used College Textbooks at Discount Prices Click Here for College Education Information & Study Resources Shop Amazon Stores in the Radical Academy Bookstore Magazine Outlet Music Store Classical Music Store Video Store DVD Store Computer Store Camera & Photo Store Computer/Video Games Software Store Musical Instruments Outlet Store Cellular Phones Toys & Games Tools & Hardware Automotive Store Outdoor Living Consumer Electronics Home & Garden Kitchen & Housewares Baby Superstore Apparel & Accessories Gourmet Food Grocery Store Sporting Goods Jewelry & Watches Health & Personal Care Beauty Store		Posted December 9, 2005 Can You Prove A Negative? This is in response to an email from Bill W. in Minnesota who asks the following question: What precisely does the principle "You can't prove a negative" claim? Mathematicians prove negative propositions all the time. Thanks for your question, Bill, here is my response: The question of the moment is: "Do cats exist?" The proposition of the moment is: "Yes, cats exist." This is a statement affirming that, indeed, cats exist -- a positive existential proposition. How do we know it is a true proposition? Well, simply, because I and millions of others have observed cats, raised cats, kept them as pets, and run up hundred of dollars in expenses to feed them and care for them. I don't know anyone (well, there might be some absolute skeptic out there but we can ignore it) who denies that cats exist. The evidence is simply overwhelming. The question now is: "Do unicorns exist?" The new proposition is: "No, unicorns do not exist." This is a negative existential proposition. Is it true? How do we know? "Well," says the proponent, "no one has proved that unicorns exist." Does this constitute proof or evidence? Sorry, it does not and to argue in this manner is to commit an all-too-common fallacy, and it falls within the "you can't prove a negative" category. It is "possible," although highly unlikely, that unicorns exist. In other words, such a proposition cannot be dismissed as proven one way or the other. It is what I prefer to call an "open question." (But, believe me, I'm not going to go around searching for a unicorn just for sake of the argument; I don't think they exist, but I can't "prove" they do not.) Some years ago, the newspapers carried a story about an "extinct" fish which was caught off the coast of Africa. The science textbooks of the time had stated that this primitive fish had lived millions of years ago and was now extinct. The statement was definitive and made as a matter of "fact." Well, of course, the "fact" was not a fact at all. The science textbooks should have been a little more generous and stated something like "is now thought to be extinct" or "there currently is no evidence that this fish still lives," or whatever qualification would be appropriate. So we need to be very cautious about statements we make, particularly negative ones, those that deny the existence of something. (I have always suspected that the day I deny the existence of "flying saucers" is the day one will choose to land in my back yard!) Seriously, though, the fact that a proposition has not been conclusively proven to be true or false often establishes nothing but one's inability to prove or disprove it. To treat this inability as establishing the truth or falsity of the proposition is to use a fallacious type of argument known as the argument from ignorance (sometimes called the argumentum ad ignorantiam). The basic form of this fallacy is : "There is no evidence or proof that it is the case that X; therefore, it is not the case that X." As philosopher Dr. Mortimer Adler says: "An affirmative existential proposition can be proved, but a negative existential proposition -- one that denies the existence of some thing -- cannot be proved." I recently wrote a book review about a book by a trained psychologist (at least he told me he was) entitled Natural Atheism, wherein the author "proves" that God does not exist because no one has "proved" it. This is what I had to say in the review about this point: However, the most blatant unscientific and illogical proposition he [the author] submits is that it is possible to prove a negative; that is, it is possible to prove that God (or gods) does (or do) not exist. He says: "It is unnecessary to prove a negative, but if you can, then the case against the claim becomes even stronger, perhaps conclusive." And then he proposes to "prove" that God (or gods) does not exist. It is true that one doesn't "have" to prove a negative. But, I hate to inform this "scientist," one cannot "prove" a negative at all, which he assumes can be the case in this situation.   [The author] attempts to justify this "prove-a-negative" procedure by using the analogy of the courtroom where a defendant "proves" his innocence through an alibi. According to [the author], the statement "I am not guilty," which is a "negative" claim, can be "proved" by providing an alibi. Sorry to say, [the author] should have consulted an attorney before venturing into this legal and logical battleground. Any first-year student of logic is taught that the courtroom is not the science laboratory or the philosophical symposium. The judicial system has its own set of "rules of evidence" and "procedural protocols." The "logic" of ordinary science and philosophy does not apply in this context. This analogy is incorrect and misleading. [The author] fails miserably here with his courtroom example. It remains as true: One cannot prove a negative. And [the author] cannot "prove" that God does not exist. There are, in spite of what [the author] maintains, certain limits to logical disproof. Now, let me quote from Irving Copi's Introduction to Logic (Fifth Edition, p. 91), probably the most-widely used textbook in undergraduate logic courses and the one which I used when I taught college courses in logic: The fallacy of argumentum ad ignorantiam is illustrated by the argument that there must be ghosts because no one has ever been able to prove that there aren't any. The argumentum ad ignorantiam is committed whenever it is argued that a proposition is true simply on the basis that it has not been proved false, or that it is false because it has not been proved true. But our ignorance of how to prove or disprove a proposition clearly does not establish either the truth or the falsehood of that proposition. This fallacy often arises in connection with such matter as psychic phenomena, telepathy, and the like, where there is no clear-cut evidence either for or against. It is curious how many of the most enlightened people are prone to this fallacy, as witness the many students of science who affirm the falsehood of spiritualist and telepathic claims simply on the grounds that their truth has not been established.   Although this mode of argument is fallacious in most contexts, it should be pointed out that there is one special context in which it is not fallacious -- namely, in a court of law; for in a court of law the guiding principle is that a person is presumed innocent until proved guilty. The defense can legitimately claim that if the prosecution has not proved guilt, this warrants a verdict of not guilty. Since this claim is based upon the special legal principle mentioned, however, it is quite consistent with the fact that the argumentum ad ignoratiam is a fallacy in all other contexts. S. Morris Engel, in his excellent work, With Good Reason: An Introduction to Informal Fallacies, has this to say on the subject: The fallacy of appeal to ignorance is an argument that uses an opponent's inability to disprove a conclusion as proof of the conclusion's correctness. By shifting the burden of proof outside the argument onto the person hearing the argument, such an argument becomes irrelevant. One's inability to disprove a conclusion cannot by itself be regarded as proof that the conclusion is true.   The following two arguments attempt to shift the burden of proof:   a) There is intelligent life in outer space, for no one has been able to prove that there isn't. b) I know that every action we perform is predetermined because no has proved that we have free will.   Such fallacious arguments involve an appeal to the emotions in that one hopes to place opponents on the defensive, causing them to believe that the proposed conclusion must be true merely because they cannot prove otherwise. ... In logical argument, it is always the obligation of those who propose conclusions to provide the proof.   If the absence of evidence against a claim could be counted as proof for it, we could prove anything we liked... Regarding the second part of your inquiry: "Mathematicians prove negative propositions all the time." In regard to mathematics, we are not dealing with "real" objects, but "ideal" objects. The numeral "0" in mathematics, for instance, does not stand for a physical "non-being" in an existential proposition. The philosophical concept of "non-being" refers to nothing in actual existence, which means there is no "being" which defines it. We are dealing in philosophy and empirical science with "actual" realities, wherein in mathematics one is dealing with "ideal" realities (present only as pure intellectual ideation). Now, it is true that mathematical symbols, numerals, and notations can be "used" to "represent" actual realities or relations, but the mathematical "representation" is not the same as an actually existing thing or relation in the physical world. For classical realists, such as I am, mathematics is considered to be "second-degree" formal abstraction. The formal terms and relations of mathematics in their pure form do not exist outside the human mind. A "negative" in mathematics is not the same as a "negative" in science or philosophy. What we now call the physical, natural, or empirical sciences are considered "first-degree" formal abstraction because they are coextensive with that part of reality which falls directly under our senses -- the universe of changing bodies. There are no physical realities which are coextensive with mathematical terms -- the number "one" can theoretically refer to any existent. Philosophy, by the way, and especially metaphysics or ontology is considered "third-degree" abstraction, because here the intellect grasps "being" as "being," being isolated in all its intelligible purity as against its partial revelation in this or that particular being. I hope the above makes some sense and answers your inquiry to some extent. Best regards, Jonathan Dolhenty, Ph.D. P.S. The logic books mentioned above are available in The Radical Academy Bookstore in the Philosophy Section. 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