Proving negatives - Revision history

Mxmsj: /* A. Examples of Provable Negatives */

2025-08-19T12:26:05Z

A. Examples of Provable Negatives

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	: There seem to be ten valid categorical syllogisms which lead to negative conclusions. These ten examples are quoted from [http://www.philosophypages.com/lg/e08b.htm "Establishing Validity."] The names were given by medieval logicians. In each name, the vowels indicate the sequence of categoricals used in the argument. So, for example, "Baroco" has the sequence AOO.		: There seem to be ten valid categorical syllogisms which lead to negative conclusions. These ten examples are quoted from [http://www.philosophypages.com/lg/e08b.htm "Establishing Validity."] The names were given by medieval logicians. In each name, the vowels indicate the sequence of categoricals used in the argument. So, for example, "Baroco" has the sequence AOO.

			<div style="margin-left:2em">
	A syllogism of the form AOO-2 was called "'''Baroco'''":		A syllogism of the form AOO-2 was called "'''Baroco'''":

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	Some M are S.		Some M are S.
	Therefore, Some S are not P.		Therefore, Some S are not P.
			</div>
			That makes ten examples of proving a negative in formal logic.
	</div>		</div>

Mxmsj: /* A. Examples of Provable Negatives */

2025-08-19T12:22:07Z

A. Examples of Provable Negatives

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	: ". . . if either premise is negative, the conclusion must also be negative. For similar reasons, no affirmative conclusion about class inclusion can follow if either premise is a negative proposition about class exclusion. A violation results in the fallacy of drawing an affirmative conclusion from negative premises."<ref>[http://www.philosophypages.com/lg/e08b.htm "Establishing Validity."]</ref>		: ". . . if either premise is negative, the conclusion must also be negative. For similar reasons, no affirmative conclusion about class inclusion can follow if either premise is a negative proposition about class exclusion. A violation results in the fallacy of drawing an affirmative conclusion from negative premises."<ref>[http://www.philosophypages.com/lg/e08b.htm "Establishing Validity."]</ref>

			<div style="margin-left:4em">
	: There seem to be ten valid categorical syllogisms which lead to negative conclusions. These ten examples are quoted from [http://www.philosophypages.com/lg/e08b.htm "Establishing Validity."] The names were given by medieval logicians. In each name, the vowels indicate the sequence of categoricals used in the argument. So, for example, "Baroco" has the sequence AOO.		: There seem to be ten valid categorical syllogisms which lead to negative conclusions. These ten examples are quoted from [http://www.philosophypages.com/lg/e08b.htm "Establishing Validity."] The names were given by medieval logicians. In each name, the vowels indicate the sequence of categoricals used in the argument. So, for example, "Baroco" has the sequence AOO.

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	Some M are S.		Some M are S.
	Therefore, Some S are not P.		Therefore, Some S are not P.
			</div>

	'''"There is no such thing as a perfect voting system; every kind has one flaw or another" '''(Partha Dasgupta and Eric Maskin, "The Fairest Vote of All" [''Scientific American, ''March 2004, 94).		'''"There is no such thing as a perfect voting system; every kind has one flaw or another" '''(Partha Dasgupta and Eric Maskin, "The Fairest Vote of All" [''Scientific American, ''March 2004, 94).

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2022-12-13T08:58:32Z

Text replacement - "â€“" to "-"

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	* [[Carl Sagan's "Baloney Detector Kit."]]		* [[Carl Sagan's "Baloney Detector Kit."]]
	* [https://en.wikipedia.org/wiki/Evidence_of_absence#Proving_a_negative ''Wikipedia,'' "Proving a negative."]		* [https://en.wikipedia.org/wiki/Evidence_of_absence#Proving_a_negative ''Wikipedia,'' "Proving a negative."]
	* Hales, Steven D. (2005). [http://departments.bloomu.edu/philosophy/pages/content/hales/articlepdf/proveanegative.pdf "Thinking Tools: You can Prove a Negative"] (''Think'' 4 (4): ~~109â€“112~~).		* Hales, Steven D. (2005). [http://departments.bloomu.edu/philosophy/pages/content/hales/articlepdf/proveanegative.pdf "Thinking Tools: You can Prove a Negative"] (''Think'' 4 (4): 109-112).
	* [https://www.logicalfallacies.org/ Logical Fallacies website.]		* [https://www.logicalfallacies.org/ Logical Fallacies website.]

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2022-12-10T10:18:07Z

Text replacement - "--" to " — "

Show changes

Mxmsj: Text replacement - "Ã©" to "é"

2022-12-10T09:47:36Z

Text replacement - "Ã©" to "é"

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	'''"There is no general analytical solution for the three-body problem given by algebraic expressions and integrals."'''		'''"There is no general analytical solution for the three-body problem given by algebraic expressions and integrals."'''

	"In 1887, mathematicians Ernst Bruns and Henri ~~PoincarÃ©~~ showed that there is no general analytical solution for the three-body problem given by algebraic expressions and integrals. The motion of three bodies is generally non-repeating, except in special cases."<ref>[http://en.wikipedia.org/wiki/Three-body_problem ''Wikipedia,'' "Three-Body Problem."]</ref>		"In 1887, mathematicians Ernst Bruns and Henri Poincaré showed that there is no general analytical solution for the three-body problem given by algebraic expressions and integrals. The motion of three bodies is generally non-repeating, except in special cases."<ref>[http://en.wikipedia.org/wiki/Three-body_problem ''Wikipedia,'' "Three-Body Problem."]</ref>

	==== Don't Even Try to Contradict These Examples ====		==== Don't Even Try to Contradict These Examples ====

Mxmsj: /* IV. Let the Equivocation Begin */

2022-02-09T07:28:03Z

IV. Let the Equivocation Begin

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	You cannot breathe 100% nitrogen and survive for more than a couple of minutes at most..		You cannot breathe 100% nitrogen and survive for more than a couple of minutes at most..

	You cannot ~~cross~~ a ten-foot chasm in three easy steps.		You cannot walk across a ten-foot chasm in three easy steps.

	You cannot turn lead into gold by breaking and/or forming any series of chemical bonds.		You cannot turn lead into gold by breaking and/or forming any series of chemical bonds.
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	'''H. "But everybody says it. Why shouldn't I?"'''		'''H. "But everybody says it. Why shouldn't I?"'''
	<div style="margin-left:4em">		<div style="margin-left:4em">
	The argument from authority is the weakest form of argument, according to Aristotle (in appealing to the authority of Aristotle to repudiate appeals to authority, I am, of course, being self-referentially inconsistent!). The argument from "science says" or "all reputable authorities agree" or "everybody thinks" (''argumentum ad populum'') is the weakest form of the argument from authority. Unless you are gifted with ESP, the only way to determine what "science says" or "all authorities think" or what "everybody thinks" is to do a survey using the instruments of sociology.		The argument from authority is the weakest form of argument, according to Aristotle (in appealing to the authority of Aristotle to repudiate appeals to authority, I am, of course, being self-referentially inconsistent!). The argument from "science says" or "all reputable authorities agree" or "everybody thinks" (''argumentum ad populum'') is the weakest form of the argument from authority. Unless you are gifted with ESP, the only way to determine what "science says" or "all authorities think" or what "everybody thinks" is to do a survey using the instruments provided by sociology.

	I predict that the results of such a survey, if conducted fairly, would show a diversity of opinion. It is an unassailable truth that '''"people disagree."''' If you disagree with this proposition, you prove my point. My impression is that, by far, the vast majority of people who invoke this principle do so as a matter of "cultural conditioning." They are adopting a slogan without thinking critically about what it means and what it would take to establish it or to show that it is untenable. I don't mind adopting a minority position in this and in other cases. I do not subscribe to the view that truth and falsehood are to be determined by taking public opinion polls. I am accustomed to the fact that many people disagree with my view of reality.		I predict that the results of such a survey, if conducted fairly, would show a diversity of opinion. It is an unassailable truth that '''"people disagree."''' If you disagree with this proposition, you prove my point. My impression is that, by far, the vast majority of people who invoke this principle do so as a matter of "cultural conditioning." They are adopting a slogan without thinking critically about what it means and what it would take to establish it or to show that it is untenable. I don't mind adopting a minority position in this and in other cases. I do not subscribe to the view that truth and falsehood are to be determined by taking public opinion polls. I am accustomed to the fact that many people disagree with my view of reality.
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	<div style="margin-left:4em">		<div style="margin-left:4em">

	The burden of proof falls on the person who makes ~~the~~ claim. "No negative hypothesis can be proved" is a hypothesis in need of proof. It ought not to be given any more weight than the evidence for it allows. Theories, unlike persons, are not innocent until proven guilty. They are unusable as theorems until they are proven to be true. I've done my level best to show why I reject the hypothesis as untenable, both in terms of abstract logic and in terms of matters of fact.		The burden of proof falls on the person who makes a claim. "No negative hypothesis can be proved" is a hypothesis in need of proof. It ought not to be given any more weight than the evidence for it allows. Theories, unlike persons, are not innocent until proven guilty. They are unusable as theorems until they are proven to be true. I've done my level best to show why I reject the hypothesis as untenable, both in terms of abstract logic and in terms of matters of fact.
	</div>		</div>

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	</div>		</div>

	'''M. "Your brief is too long. I can't read the whole thing."'''		'''M. "TLDR! Your brief is too long. I can't read the whole thing."'''
	<div style="margin-left:4em">		<div style="margin-left:4em">
	Sorry about that. I have trouble reading the whole thing, too. It wasn't this long or complicated when I got started on it. The nature of proof is a topic that has intrigued me since I took Thomas Aquinas as my confirmation sponsor in the spring of 1965, and my interest in "theological epistemology" (a term popularized, if not coined, by Avery Dulles, SJ) has been the foundation of my work in philosophy and theology. I think that explains why this note has grown so complex since the first draft in 2003.		Sorry about that. I have trouble reading the whole thing, too. It wasn't this long or complicated when I got started on it. The nature of proof is a topic that has intrigued me since I took Thomas Aquinas as my confirmation sponsor in the spring of 1965, and my interest in "theological epistemology" (a term popularized, if not coined, by Avery Dulles, SJ) has been the foundation of my work in philosophy and theology. I think that explains why this note has grown so complex since the first draft in 2003.

Mxmsj at 07:35, 5 May 2021

2021-05-05T07:35:30Z

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	: "This sentence is unprovably true."		: "This sentence is unprovably true."
	:::: ~~GÃ¶del~~ (1931--age 25)		:::: Gödel (1931--age 25)
	\|}		\|}
	</div>		</div>
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	:: Davis and Hersh reproduce a page from PM with the comment, "Russell and Whitehead were pioneers in a program to reduce mathematics to logic. Here, after 362 pages, the arithmetic proposition 1 + 1 = 2 is established."<ref>(D&H, 334).</ref>		:: Davis and Hersh reproduce a page from PM with the comment, "Russell and Whitehead were pioneers in a program to reduce mathematics to logic. Here, after 362 pages, the arithmetic proposition 1 + 1 = 2 is established."<ref>(D&H, 334).</ref>

	:: "As it happened, certainty was not to be had, even at this price. In 1930 ~~GÃ¶del~~'s incompleteness theorems showed that the Hilbert system was unattainable--that '''any consistent formal system strong enough to contain elementary arithmetic would be unable to prove its own consistency'''. The search for secure foundations has never recovered from this defeat" (D&H, 337). [After reading Torkel Franzén's book, ''Gödel's Theorem: An Incomplete Guide to Its Use and Abuse,'' I see that I may have to modify or remove these last three paragraphs. Mathematicians disagree about the MEANING of Gödel's Theorem for mathematics and about the changes in the philosophy of mathematics that should take place because of the theorem. Stay tuned for another revision, but probably not any time soon. My claim that "mathematics as a whole has not been proved to be true" is almost certainly a mathematically meaningless proposition.]		:: "As it happened, certainty was not to be had, even at this price. In 1930 Gödel's incompleteness theorems showed that the Hilbert system was unattainable--that '''any consistent formal system strong enough to contain elementary arithmetic would be unable to prove its own consistency'''. The search for secure foundations has never recovered from this defeat" (D&H, 337). [After reading Torkel Franzén's book, ''Gödel's Theorem: An Incomplete Guide to Its Use and Abuse,'' I see that I may have to modify or remove these last three paragraphs. Mathematicians disagree about the MEANING of Gödel's Theorem for mathematics and about the changes in the philosophy of mathematics that should take place because of the theorem. Stay tuned for another revision, but probably not any time soon. My claim that "mathematics as a whole has not been proved to be true" is almost certainly a mathematically meaningless proposition.]

	'''Now let's talk about proving negatives.'''		'''Now let's talk about proving negatives.'''
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	: This was true the day I took our silver Oldsmobile to a muffler shop shortly after the muffler fell off. Both the mechanic and I could see for ourselves that the muffler was not on the car. Fortunately, the man was able to remedy the situation at a very reasonable price and this statement has ceased to be true. But for a little while, it was a very provable negative. I had other evidence besides visual to help me reach the proper conclusion at the time that the observations were made. So I may now write, '''"The muffler was not on my car between 12:00 noon and 1:00 PM on July 29, 2003."''' (I've given the bill to the man in charge of cars and the credit card record to the Treasurer, so I don't have the time of day right at my fingertips.)		: This was true the day I took our silver Oldsmobile to a muffler shop shortly after the muffler fell off. Both the mechanic and I could see for ourselves that the muffler was not on the car. Fortunately, the man was able to remedy the situation at a very reasonable price and this statement has ceased to be true. But for a little while, it was a very provable negative. I had other evidence besides visual to help me reach the proper conclusion at the time that the observations were made. So I may now write, '''"The muffler was not on my car between 12:00 noon and 1:00 PM on July 29, 2003."''' (I've given the bill to the man in charge of cars and the credit card record to the Treasurer, so I don't have the time of day right at my fingertips.)

	'''"~~Aether~~ does not exist."'''		'''"The aether does not exist."'''

	: Physicists in the 19th century assumed that every wave needed a medium; sound travels through the air the way waves travel through water. Since they knew that light acted as if it were a wave, they hypothesized that the wave would be carried by the "aether." Maxwell's equations showed that light propagates by the production electro-magnetic fields, which require no physical substratum. Careful observations of the speed of light showed that no effect of the motion of the earth through the aether could be detected.		: Physicists in the 19th century assumed that every wave needed a medium; sound travels through the air the way waves travel through water. Since they knew that light acted as if it were a wave, they hypothesized that the wave would be carried by the "aether." Maxwell's equations showed that light propagates by the production electro-magnetic fields, which require no physical substratum. Careful observations of the speed of light showed that no effect of the motion of the earth through the aether could be detected.

Mxmsj: /* Links */

2021-05-05T07:32:44Z

Links

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	* [http://www.fallacyfiles.org/index.html ''The Fallacy Files''] by Dr. Gary N. Curtis, Ph.D.		* [http://www.fallacyfiles.org/index.html ''The Fallacy Files''] by Dr. Gary N. Curtis, Ph.D.
	* [[Carl Sagan's "Baloney Detector Kit."]]		* [[Carl Sagan's "Baloney Detector Kit."]]
	* [~~http~~://en.wikipedia.org/wiki/Evidence_of_absence#~~.22You_can.27t_prove_a_negative.22~~ ''Wikipedia,'' "~~You Can't Prove~~ a ~~Negative~~."]		* [https://en.wikipedia.org/wiki/Evidence_of_absence#Proving_a_negative ''Wikipedia,'' "Proving a negative."]
	* Hales, Steven D. (2005). [http://departments.bloomu.edu/philosophy/pages/content/hales/articlepdf/proveanegative.pdf "Thinking Tools: You can Prove a Negative"] (''Think'' 4 (4): 109â€“112).		* Hales, Steven D. (2005). [http://departments.bloomu.edu/philosophy/pages/content/hales/articlepdf/proveanegative.pdf "Thinking Tools: You can Prove a Negative"] (''Think'' 4 (4): 109â€“112).
	* [https://www.logicalfallacies.org/ Logical Fallacies website.]		* [https://www.logicalfallacies.org/ Logical Fallacies website.]

Mxmsj: /* III. Disproof by Counter-Example */

2021-05-05T07:24:53Z

III. Disproof by Counter-Example

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	In deciding the issue, we want to '''weigh the evidence carefully'''. "'One unerring mark of the love of truth,' wrote John Locke in 1690, 'is not entertaining any proposition with greater assurance than the proofs it is built upon will warrant.'"<ref>Sagan (1996), 66.</ref> I strive to meet the standards set by "that remorseless taskmaster called the scientific method: Everything hinges on the matter of evidence."<ref>Sagan (1996), 69.</ref> I will alter my opinion if the evidence forces me to do so; if it does not compel me, I will not change my mind.		In deciding the issue, we want to '''weigh the evidence carefully'''. "'One unerring mark of the love of truth,' wrote John Locke in 1690, 'is not entertaining any proposition with greater assurance than the proofs it is built upon will warrant.'"<ref>Sagan (1996), 66.</ref> I strive to meet the standards set by "that remorseless taskmaster called the scientific method: Everything hinges on the matter of evidence."<ref>Sagan (1996), 69.</ref> I will alter my opinion if the evidence forces me to do so; if it does not compel me, I will not change my mind.

	'''The maxim in question cannot be established by any finite list of examples of negatives that are difficult or impossible to prove.''' The maxim covers all negatives of any kind whatsoever. It is an unrestricted proposition. The set of all possible negatives is probably infinite, since there is no reason why new negatives cannot be constructed at will. The resources are readily available to anyone who wishes to create a new negative hypothesis. No matter how ~~may~~ examples of unprovable negatives one lists, if it is a finite list, it is not exhaustive and cannot establish the universal negative. Davis and Hersh describe "another well-verified conjecture--known to be true in the first billion cases--which Littlewood proved is false eventually" (364).		'''The maxim in question cannot be established by any finite list of examples of negatives that are difficult or impossible to prove.''' The maxim covers all negatives of any kind whatsoever. It is an unrestricted proposition. The set of all possible negatives is probably infinite, since there is no reason why new negatives cannot be constructed at will. The resources are readily available to anyone who wishes to create a new negative hypothesis. No matter how many examples of unprovable negatives one lists, if it is a finite list, it is not exhaustive and cannot establish the universal negative. Davis and Hersh describe "another well-verified conjecture--known to be true in the first billion cases--which Littlewood proved is false eventually" (364).

	'''The maxim that "You cannot prove a negative" is defeated by finding one example of a provable negative.''' I've found that when I start giving examples of provable negatives, people start hopping on different horses from the one they rode in on. Suddenly, their definition of what "a negative" is and what "proof" is start to change and become elusive. It is very frustrating, which is why I've taken so much time to write this essay and map out my position. I'm sure I will collect more examples of equivocation as people read and comment on this. As I give examples, people will say "No, that's not what I mean by proof of a negative." Well, define your terms. I will then give a counter-example according to your definitions. If you can't define your terms, then you're not qualified to advance the maxim as part of reasoned discourse.		'''The maxim that "You cannot prove a negative" is defeated by finding one example of a provable negative.''' I've found that when I start giving examples of provable negatives, people start hopping on different horses from the one they rode in on. Suddenly, their definition of what "a negative" is and what "proof" is start to change and become elusive. It is very frustrating, which is why I've taken so much time to write this essay and map out my position. I'm sure I will collect more examples of equivocation as people read and comment on this. As I give examples, people will say "No, that's not what I mean by proof of a negative." Well, define your terms. I will then give a counter-example according to your definitions. If you can't define your terms, then you're not qualified to advance the maxim as part of reasoned discourse.

Mxmsj: /* II. Formal Logic Does Not Treat Negatives as a Special Case */

2021-05-05T07:21:48Z

II. Formal Logic Does Not Treat Negatives as a Special Case

← Older revision		Revision as of 09:21, 5 May 2021
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	:: Davis and Hersh reproduce a page from PM with the comment, "Russell and Whitehead were pioneers in a program to reduce mathematics to logic. Here, after 362 pages, the arithmetic proposition 1 + 1 = 2 is established."<ref>(D&H, 334).</ref>		:: Davis and Hersh reproduce a page from PM with the comment, "Russell and Whitehead were pioneers in a program to reduce mathematics to logic. Here, after 362 pages, the arithmetic proposition 1 + 1 = 2 is established."<ref>(D&H, 334).</ref>

	:: "As it happened, certainty was not to be had, even at this price. In 1930 GÃ¶del's incompleteness theorems showed that the Hilbert system was unattainable--that '''any consistent formal system strong enough to contain elementary arithmetic would be unable to prove its own consistency'''. The search for secure foundations has never recovered from this defeat" (D&H, 337). [After reading Torkel ~~FranzÃ©n~~'s book, ''~~GÃ¶del~~'s Theorem: An Incomplete Guide to Its Use and Abuse,'' I see that I may have to modify or remove these last three paragraphs. Mathematicians disagree about the MEANING of ~~Gödel~~'s Theorem for mathematics and about the changes in the philosophy of mathematics that should take place because of the theorem. Stay tuned for another revision, but probably not any time soon. My claim that "mathematics as a whole has not been proved to be true" is almost certainly a mathematically meaningless proposition.]		:: "As it happened, certainty was not to be had, even at this price. In 1930 GÃ¶del's incompleteness theorems showed that the Hilbert system was unattainable--that '''any consistent formal system strong enough to contain elementary arithmetic would be unable to prove its own consistency'''. The search for secure foundations has never recovered from this defeat" (D&H, 337). [After reading Torkel Franzén's book, ''Gödel's Theorem: An Incomplete Guide to Its Use and Abuse,'' I see that I may have to modify or remove these last three paragraphs. Mathematicians disagree about the MEANING of Gödel's Theorem for mathematics and about the changes in the philosophy of mathematics that should take place because of the theorem. Stay tuned for another revision, but probably not any time soon. My claim that "mathematics as a whole has not been proved to be true" is almost certainly a mathematically meaningless proposition.]

	'''Now let's talk about proving negatives.'''		'''Now let's talk about proving negatives.'''