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Sunday, 2 September 2018

On being argumentative in lieu of making an argument.

Integrity in Science — A Facebook Dialogue
Ann Gauger

Editor’s note: Our biologist colleague Ann Gauger had the following interaction on Facebook yesterday. Sadly typical, it speaks for itself.

Questioner: Has anyone in the Discovery Institute thought of doing any real experiments?

Ann Gauger: Read our papers. Plenty of real experiments there.

Q: Where, Ann Gauger?

Ann: [LINK]

Q: “Peer review” means different things in science and in intelligent design. Yet more deception.

Q: I don’t see any evidence of any science being done.

Another FB correspondent chimes in:

[LINK] Axe

[LINK Reeves

And this: [LINK] Gauger

Q: Who has reviewed these?

Ann: As you know, peer review is confidential and I do not know their names. They were qualified research scientists by no means all favorable to ID. More than that I cannot say. The process is rigorous. Many papers are turned down.

Q: From BIO-Complexity, Purpose and Scope:

Purpose: BIO-Complexity is a peer-reviewed scientific journal with a unique goal. It aims to be the leading forum for testing the scientific merit of the claim that intelligent design (ID) is a credible explanation for life. Because questions having to do with the role and origin of information in living systems are at the heart of the scientific controversy over ID, these topics — viewed from all angles and perspectives — are central to the journal’s scope.

To achieve its aim, BIO-Complexity is founded on the principle of critical exchange that makes science work. Specifically, the journal enlists editors and reviewers with scientific expertise in relevant fields who hold a wide range of views on the merit of ID, but who agree on the importance of science for resolving controversies of this kind. Our editors use expert peer review, guided by their own judgement, to decide whether submitted work merits consideration and critique. BIO-Complexity aims not merely to publish work that meets this standard, but also to provide expert critical commentary on it.
Ann: We have reviewers on both sides of the question and invite anyone who is qualified to participate. As you can guess, the problem is that we are being stonewalled. We have applied to mainstream journals and they are returned without being considered.

Q: Ann Gauger, you just said you don’t know who reviews what.

Ann: I don’t. But if you’ve been through peer review yourself, you know that you can get a general feel for the attitude of the reviewer without knowing their name.

Q: “Reviewing” and publishing in a specially protected environment basically makes them whiff of cargo-cult science and deliberate attempts at deception. They won’t win any scientific case that way, just take in gullible lay people.

Ann: Let me respond to your accusation of cargo-cult science and deception. BIO-Complexity is not the first journal to be founded to advance a controversial case. From Wikipedia [regarding the journal Nature] — you can check their sources:

“…Nature was conceived, born, and raised to serve polemic purpose.”[12] Many of the early editions of Nature consisted of articles written by members of a group that called itself the X Club, a group of scientists known for having liberal, progressive, and somewhat controversial scientific beliefs relative to the time period.[12] Initiated by Thomas Henry Huxley, the group consisted of such important scientists as Joseph Dalton Hooker, Herbert Spencer, and John Tyndall, along with another five scientists and mathematicians; these scientists were all avid supporters of Darwin’s theory of evolution as common descent, a theory which, during the latter-half of the 19th century, received a great deal of criticism among more conservative groups of scientists.[14] Perhaps it was in part its scientific liberality that made Nature a longer-lasting success than its predecessors. John Maddox, editor of Nature from 1966 to 1973 as well as from 1980 to 1995, suggested at a celebratory dinner for the journal’s centennial edition that perhaps it was the journalistic qualities of Nature that drew readers in; “journalism” Maddox states, “is a way of creating a sense of community among people who would otherwise be isolated from each other. This is what Lockyer’s journal did from the start.”[15] In addition, Maddox mentions that the financial backing of the journal in its first years by the Macmillan family also allowed the journal to flourish and develop more freely than scientific journals before it.[15].”
So spare me your sneers.

Q: Ann Gauger, I expect their eventual success rested on the veracity of their science.

Ann: And so we will let time tell.

Q: Not to mention their intellectual integrity and honesty.

Ann: No — read the history of the X Club. And I thought our conversation was honest and above ad hominems.

Q: Questioning honesty is not ad-hominem. It relates directly to the trustworthiness of what they are saying. Criticising an organisation or an individual for being naughty in a way that is unrelated to what they are arguing — that would be ad-hom.

Ann: Ad hominem attacks can take the form of overtly attacking somebody, or more subtly casting doubt on their character or personal attributes as a way to discredit …

Q: Ann Gauger, as I said, the exception is when we question their honesty and integrity in making statements.

“X is a habitual liar, so you cannot trust what he says about anything.” This is not ad-hom.

“X is smelly, so you cannot trust what he says about anything.” This is ad-hom.

Ann: Casting doubt on the integrity of an individual does not address the argument, it dismisses it, just as you dismiss our papers because of where they were published. Why not read them and judge them on their own merits?

And why not take my arguments as truthful, not deceptive, based on the evidence, not any prejudgment or label?

How are we to get an adequate hearing in the face of such prejudice?

Q: Ann Gauger, your history is against you. Starting with the Wedge document. My “prejudice” is based on years of investigating the claims of creationist/ID enthusiasts, and by learning about their underhanded tactics for infiltrating academe. You have brought all this upon yourselves.

Ann: Well, apparently, you know something about our history that I don’t. Infiltrating academe? What does that mean?

Ann: It took me a while to find this, but your accusation of infiltration reminded me of a quote from a book I had read:

[T]he process of conversion involved much hard lobbying by Huxley, Hooker and other naturalists who jumped to the defense of the theory. Huxley’s real triumph was in gradually extending the influence of those who shared his aversion to the design argument within the community of professional scientists. This influence was at last beginning to expand, and Huxley networked endlessly to ensure that people sympathetic to his position got the jobs that were opening up in the universities and elsewhere. It became unfashionable for a scientist to make open appeals to the supernatural, even if he (and they were still almost all men) believed in a Creator.

(Peter J. Bowler, Monkey Trials and Gorilla Sermons: Evolution and Christianity from Darwin to Intelligent Design (Cambridge, Massachusetts: Harvard University Press, 2007), pp. 38-39)
Q: Ann Gauger, do you have any evidence that Huxley was underhanded? Dishonest?

Ann: There’s more to the quote that I don’t have available to me now. Do you have such evidence?

Q: Nope.

Ann: And with regard to ID?

Q: I’ve already mentioned the Wedge document. It starts there.

[What’s the so-called “Wedge document” he keeps mentioning? See here . –Editor]

Ann: One more piece, this time about Darwin[LINK]:

Darwin’s picture of himself, and our dominant picture of him, as a pure scientist, slow at writing, poor at arguing, concerned only with the truth or with the judgment of a small group of experts and indifferent to public opinion, is a picture taken in the earlier part of his career and is irreconcilable with his motives and actions in the years of public controversy. Darwin’s self-image ignores his extensive efforts to proselytize, to simplify the bases for believing in evolution, and to broaden his base of support. Darwin’s failure to integrate his rhetorical action with his self-understanding as a scientist has powerfully reinforced a positivist interpretation of his achievement as merely the inevitable consequence of a superior account.

Finally the study has implications for our understanding of the ethics of scientific rhetoric. Given that science and civilization exist in an interdependent relationship, how far is it legitimate for science to mask its opposition to dominant cultural values? To what extent does the ethic of science demand truthfulness, when the price of truth may be a temporary, or perhaps even sustained, reaction against a particular scientific theory? In his use of Gray’s essays, does Darwin’s encouragement of the public to draw a theological conclusion beneficial to his persuasive aim, but counter to his personal convictions, make him cynical or dishonest? Or should we see him as acting to create a cultural environment beneficial to science and perhaps in time conducive to a more enlightened understanding of religion? It seems that however we interpret Dar win’s rhetorical strategies and tactics, we are left with the discomfiting realization that Darwin the historical truth bearer was not always Darwin the historical truth teller. The ethical obligation peculiar to rhetoric is to give truth effective advocacy.
Ann: The people I work with are men of integrity and honesty. I am not aware of any deception or misrepresentation on their part.

You may disagree with their motives or conclusions. But then as should be clear above, Huxley and Darwin were fighting their own cultural battle, and not always openly, with some deception involved. There is the story of how Darwin managed to get his theory admitted before the Royal Society at the same time as Wallace by lobbying with friends and Lyell. Your indignation over the Wedge document, which I have not read by the way, is mainly cultural, I suspect, and not because anyone was proposing underhanded science….

Ann: Gotta go. It’s been fun.

Saturday, 1 September 2018

Fear dumb people not smart machines?

Bill Dembski on the AI Boogeyman, and the Real AI Danger
Evolution News @DiscoveryCSC

On a new episode of ID the Future, Andrew McDiarmid reads an excerpt from a speech prepared by philosopher, mathematician, and trailblazing design theorist William Dembski for the launch of the 


Dr. Dembski asks whether we need to worry about an AI takeover. He says no, there’s no evidence that artificial intelligence (AI) could reach that level, or achieve consciousness, while on the other hand there is mounting evidence from both philosophy and the field of artificial intelligence technology that it cannot and will not.

“The real worry,” Dembski says, “isn’t that we’ll raise machines to our level, but that we’ll lower humanity to the level of machines.”

Big tobacco:A prelude to big marijuana?

Philip Morris v. Uruguay: Will investor-State arbitration send restrictions on tobacco marketing up in smoke?

How the consensus' gatekeepers got egg on their face.

James Tour on OOL science's circus.

On Origin of Life, Synthetic Chemist James Tour Delivers Chastisement to Jeremy England
David Klinghoffer | @d_klinghoffer  

As a postscript to Brian Miller’s  reply to MIT physicist Jeremy England, see this from the famed synthetic organic chemist James Tour, writing for the online journal Inference. InAn Open Letter to My Colleagues,” Tour sets out this way:

Life should not exist. This much we know from chemistry. In contrast to the ubiquity of life on earth, the lifelessness of other planets makes far better chemical sense. Synthetic chemists know what it takes to build just one molecular compound. The compound must be designed, the stereochemistry controlled. Yield optimization, purification, and characterization are needed. An elaborate supply is required to control synthesis from start to finish. None of this is easy. Few researchers from other disciplines understand how molecules are synthesized.

His colleagues are fooling themselves if they imagine otherwise. He gets around to England, not naming him except in a footnote, at the end:

If one understands the second law of thermodynamics, according to some physicists,15 “You [can] start with a random clump of atoms, and if you shine light on it for long enough, it should not be so surprising that you get a plant.”16

The quote, remarkably, is from Jeremy England in an interview with Natalie Wolchover for Quanta. Tour also cites England’s article “Statistical Physics of Self-Replication,” in the Journal of Chemical Physics, and one of the most absurdly titled God-bashing articles we’ve come across, “God is on the Ropes: The Brilliant New Science That Has Creationists and the Christian Right Terrified,” by Paul Rosenberg writing for Salon. Rosenberg quotes England from the same Quanta article, “[U]nder certain conditions, matter inexorably acquires the key physical attribute associated with life.” Oh, really, does it?

Tour goes on, referring to the notion that random atoms will become a plant if given plenty of light and plenty of time:

The interactions of light with small molecules is well understood. The experiment has been performed. The outcome is known. Regardless of the wavelength of the light, no plant ever forms.

We synthetic chemists should state the obvious. The appearance of life on earth is a mystery. We are nowhere near solving this problem. The proposals offered thus far to explain life’s origin make no scientific sense.

Beyond our planet, all the others that have been probed are lifeless, a result in accord with our chemical expectations. The laws of physics and chemistry’s Periodic Table are universal, suggesting that life based upon amino acids, nucleotides, saccharides and lipids is an anomaly. Life should not exist anywhere in our universe. Life should not even exist on the surface of the earth.17


It’s somehow more satisfying that England isn’t identified in the body of the article, but only in a footnote. That is a memorable instance of a senior scientist quietly taking a junior colleague out behind the woodshed. For more on the general subject, see Tour’s slashing 2016 lecture,  The Origin of Life: An Inside Story.”

Darwinism's quest for a free lunch hits yet another dead end?

Conservation of Information and Coevolution: New BIO-Complexity Article by Ewert and Marks
Brian Miller  

In a  previous article I described Winston Ewert, Robert Marks, and William Dembski’s book Introduction to Evolutionary Informatics which identifies the limitations of evolutionary algorithms to find solutions for complex problems. The book demonstrates that this class of programs is only capable of achieving non-trivial results unless information about desired outcomes is programmed into them. For instance, a program designed to find the best strategy for playing checkers must have detailed information about the game programmed into its search method. It could not develop a strategy to play chess without altering the underlying algorithm to include new chess-related information.

This constraint is a direct result of No Free Lunch (NFL) theorems and the related law of  Conservation of InformationAttempts have been make to overcome this challenge by appealing to what are termed coevolutionary searches. However, as Ewert and Marks demonstrate in a new article for the journal BIO-Complexitythese algorithms do not escape the NFL barrier, so they are no more efficient on average than random searches.

Evolutionary algorithms typically follow a standard set of steps. They generate trial solutions to a problem, and then assign each trial some “fitness” value. These values are then used to determine how the next iteration of trials is generated for testing. The process continues until a target is found.

Ewert and Marks use the illustration of generating recipes for making pancakes. In this example, a trial recipe represents a list of the specific amounts of each ingredient and details of the cooking, such as burner setting and times. The assigned value corresponds to the prepared pancake’s taste, and it determines how new recipes are generated. The process continues until a pancake is created that meets some taste standard. The set of values associated with all possible trials is described as a fitness landscape, and the search algorithm must navigate its terrain looking for targets. For standard algorithms, all information needed to assign a fitness value is known. In contrast, for coevolutionary algorithms information is not fully known, so incomplete or “subjacent” information must often be used. That’s because the fitness landscape continuously changes due to the presence of other organisms or other contingent factors.

This difference can be illustrated in terms of an examples from biological evolution. A standard evolutionary algorithm would correspond to an organism having a specific “fitness” which remains fairly constant in most situations. For instance, that of a desert plant might relate to such innate abilities as conserving water and processing sunlight. A computer program could model the plant’s fitness using related variables, such as the plant’s mass and the amount of chlorophyll produced in its leaves. These variables would fully determine the assigned fitness value. In contrast, a coevolutonary process would correspond to the fitness changing over time due to such factors as interactions with other species and details of the physical environment. For instance, chemicals in the skin could provide greater or lesser protection from different predators, and the shape of the plant could prove more or less helpful in different settings.

To model this increased complexity, the algorithms generate a query matrix where a row is assigned to each candidate solution, and each column corresponds to a different factor affecting fitness such as interactions with a particular species. Many cells in this matrix are often not known, whether due to computational or other practical limitations, so various methods are employed to assign each trial solution (row) an aggregate value based on the limited knowledge. The algorithm then proceeds as with traditional models. Many have claimed that coevolutionary programs can find solutions to a wide variety of problems more quickly on average than random searches, thus overcoming the restrictions of the NFL theorems. In other words, they eliminate the need for programmers to provide problem-specific “active information” to guide searches.

To test this claim, Ewert and Marks measured the performance of various coevolutionary algorithms for a variety of problems. They found that they could at best match the performance of traditional “full-search” methods, and they typically performed worse. Their article also describes how claims to the contrary were based on research that focused on solving very simple problems or that designed experiments in such a way as to provide hidden information to assist in finding targets. Therefore, coevolutionary processes cannot overcome NFL limitations, as they also require problem-specific information to perform properly.

These results have direct implications for Darwinian evolution. Biologists often claim that coevolutionary interactions between different species or species and the environment can alter the underlying fitness landscape in such a way as to drive evolutionary changes. A classic example is the proposed coevolution of bees and flowering plantsMany plants need insects as carriers for their pollen. As a result, the presence of insects places selective pressure on the plants to produce smells, colors, and nectar to attract them. In turn, the presence of the plants places selective pressures on the insects to move toward the plants’ signals to obtain the food supply. In addition, insects are selected for thicker hairs on their legs to capture more pollen. They can then fertilize more plants resulting in a greater food supply. Such scenarios may sound plausible, but they only result in trivial adjustments to preexisting structures.

In contrast, complex innovations, such as new body plans, radical innovations which, in turn, require large amounts of new information.The environment is often claimed to provide this new information, which is believed to be hidden in the fitness landscapes coupled to coevolutionary interactions. However, this research by Ewert and Marks directly challenges the claim. The search space corresponding to biological forms is vastly greater than what could be searched through random mutations in the offspring of any species. And natural selection cannot help without being provided large amounts of information on where new forms reside. For coevolutionary processes are no more efficient at solving problems than traditional evolutionary algorithms, and the latter are no more efficient on average than random searches.

Friday, 31 August 2018

Cashing in on the gospel?

Winged navigators v. Darwin (again).

Migrating Birds Can Find Their Flocks After Many Miles and Days Apart

Carsten Egevang’s study of Arctic terns, highlighted in the Illustra Media documentary Flight: The Genius of Birds, stunned the ornithological world. It was the first time geolocators logged the exact routes of the world travelers, whose annual tour takes them from pole to pole. As geolocators (also called loggers) improve, more species can be studied, increasing our knowledge of migratory behavior. Now, for the first time, a new flight trace shows surprising social interactions en route for a bird that migrates between Europe and Africa.

The European bee-eater (Merops apiaster) is a flashy bird adorned in brilliant shades of rusty red, orange, yellow, and cyan with black highlights on the face. Males and females are similarly colored. Weighing just 2 ounces at a length of 10 to 11 inches, this gregarious bird with a cheerful song covers a lot of territory, looking for bees, wasps, dragonflies, and other insects to eat. Each bird can consume 250 bees a day, but this causes little impact on bee populations (less than 1 percent, according to Wikipedia). The birds catch their prey in flight, but remove the stingers first by banging the insects against hard surfaces before gulping them down. 

From Germany to Angola

The migration of M. apiaster takes it between Germany in summer and Angola in winter, a distance of 14,000 km (8,700 miles). This species migrates in small flocks usually ranging from 5 to 39 individuals. But without being able to follow a flock with ultralight aircraft, it is “it has previously been impossible to monitor spatiotemporal group dynamics in small migrating birds.”

Researchers from the Swiss Ornithological Institute, reporting in Current Biology, outfitted 77 Merops birds in 2015, and 92 more in 2016 with loggers. The instruments recorded ambient light for geolocation and atmospheric pressure for altitude. Ten birds were recovered in 2016, and 19 in 2017. The scientists wanted to investigate the following question:

From zebras to monarch butterflies, migratory species undertake some of the most extreme feats of endurance known in the animal kingdom. With the advent of novel tracking technologies, we are gradually completing the picture of where and when they travel. However, without being able to directly observe migration, we have very little knowledge of who might migrate with whom.

From the data, the scientists were able to piece together the routes of each individual, and infer social interactions within the flock. To the researchers’ surprise, some birds could separate from the main flock in small groups, fly thousands of miles apart on another route, and then re-join the main flock up to 5 days later, even though the sub-flocks probably encountered different environmental conditions along the way.

We present the first evidence of a migratory bird flying together with non-kin of different ages and sexes at all stages of the life cycle. In fact, 49% stay together throughout the annual cycle, never separating longer than 5 days at a time despite the ∼14,000-km journey. Of those that separated for longer, 89% reunited within less than a month with individuals they had previously spent time with, having flown up to 5,000 km apart. These birds were not only using the same non-breeding sites, but also displayed coordinated foraging behaviors — these are unlikely to result from chance encounters in response to the same environmental conditions alone. Better understanding of migratory group dynamics, using the presented methods, could help improve our understanding of collective decision making during large-scale movements.

Functional Advantages

The authors speculated about functional advantages for this divide-and-conquer strategy that they call “fission-fusion” events, but did not expect to see unrelated birds cooperating so well.

In conclusion, we find that birds from the same colony do not always follow the same migratory routes but will in fact join with birds from nearby colonies post-breeding to form groups that migrate together. Groups are generally stable during migration. However, if groups separate, they can reunite in the non-breeding grounds to form dynamic groups that repeatedly forage together, sometimes separating for 1–5 days at a time before migrating back to the breeding grounds together. Most surprisingly, these groups showed no age or sex structure and consisted of non-kin. Our research is the first to show such behavior between migratory non-breeding non-kin bird groups, displaying rare spatiotemporal group dynamics more often observed in mammals.

The wonder of this behavior can be appreciated by reversing roles. Imagine the birds running an experiment on humans. They remove all smartphones and electronic devices, outfit them with loggers the people can’t read, and send them through uncharted territory in Africa in small groups of mixed sex and age who don’t know each other, telling them to meet in Germany. Wouldn’t the birds be flabbergasted to watch small groups split off from the main group, hike for a thousand miles on a different route without communication, and then re-join the main group a week later? Even if the humans used ultralight aircraft, it would be astonishing to see them find one another. 

It’s not like the terrain has some bottleneck that forces the birds toward a particular narrow flight corridor. Somehow, these birds show a degree of social coordination only previously seen in mammals.

Fission-fusion can occur without individuals being able to “recognize” each other per se. The same individuals could encounter each other again and again at the same site as a result of migratory connectivity, simply because it is the only one available to them at a particular period. Under such circumstances, resource bottlenecks are likely driving group fusion, not social relationships. On the other hand, where resources occur broadly over a large area, animals must coordinate decisions to fuse into a long-term group — especially if they regularly fissure and must find each other again. Only species with high social cognition, such as elephants, dolphins, and bats, have been found to form long-term social bonds by coordinating decisions, despite separations imposed by migration. In birds, long-term social bonds, despite fission-fusion dynamics, have been observed between non-migratory non-kin, migratory kin, or migratory bonded pairs. Long-term social bonds, despite fission-fusion dynamics, are poorly understood in non-kin migratory birds.

Group Formation

What they’re saying is that unrelated birds that don’t migrate are known to form social bonds, and mating pairs can be expected to cooperate during migration. But why would unrelated birds of mixed sex and age do this? There are advantages to cooperation: “Group formation is pivotal in allowing individuals to interact, transfer information, and adapt to changing conditions,” they say. But it also comes with risks:

Migratory species are notable for their propensity to aggregate in large numbers. The stability of migratory groups over time can be important in determining survival, navigational accuracy, migratory speed, transfer of information, and new migratory behaviors. However, migrating with others is not without risk, as it can increase both disease prevalence and resource competition. Group size typically fluctuates over time and space, with individuals coming together and separating; hereafter termed “fission-fusion dynamics”) as they trade off the different benefits and costs of cooperation. Indeed, resource patches are distant, seasonal, and often unpredictable. One slow individual could, for instance, force the entire group to slow down and miss peaks in resource availability, creating conflict. Groups can therefore either compromise to remain together or spilt into subgroups, for example, of different migratory speeds.

These birds appear to have reached an optimal trade-off between benefits and risks. How could such complex behavior arise by mutation and selection? It seems a challenge too daunting for neo-Darwinism. Maybe that’s why the researchers don’t speculate about evolution. 

As wonderful as the observed group dynamics have been shown to be, there are other wonders even greater. Try to build a robot that can fly and power itself by capturing and consuming bees. While you’re at it, make it beautiful to look at and able to sing a beautiful song. Fit all that capability into two ounces, and give it software to find a target 9,000 miles away employing optimal group dynamics. Then endow it with the ability to make copies of itself. If you succeeded at all this, would you not be offended to hear some people say it “emerged” by chance?

On Darwinists' shifting of the goalposts.

A clash of titans. LXXV

Partisan politics is balkanising the U.S.?:Pros and cons.

On the law of noncontradiction.

re to Think About
IV. The Law of Excluded Middle
       One logical law that is easy to accept is the law of non-contradiction. This law can be expressed by the propositional formula ¬(p^¬p). Breaking the sentence down a little makes it easier to understand. p^¬p means that p is both true and false, which is a contradiction. So, negating this statement means that there can be no contradictions (hence, the name of the law). In other words, the law of non-contradiction tells us that a statement cannot be both true and false at the same time. This law is relatively uncontroversial, though there have been those who believe that it may fail in certain special cases. However, it does lead us to a logical principle that has historically been more controversial: the law of excluded middle.  
       The law of excluded middle can be expressed by the propositional formula p_¬p. It means that a statement is either true or false. Think of it as claiming that there is no middle ground between being true and being false. Every statement has to be one or the other. That’s why it’s called the law of excluded middle, because it excludes a middle ground between truth and falsity. So while the law of non-contradiction tells us that no statement can be both true and false, the law of excluded middle tells us that they must all be one or the other. Now, we can get to this law by considering what it means for the law of non-contradiction to be true. For the law of noncontradiction to be true, ¬(p^¬p) must be true. This means p^¬p must be false. Now, we must refer back the truth table definition for a conjunction. What does it take for p ^ ¬p to be false? It means that at least one of the conjuncts must be false. So, either p is false, or ¬p is false. Well, if p is false, then ¬p must be true. And if ¬p is false, then p must be true. So we are left with the disjunction p _ ¬p, which is exactly the formulation I gave of the law of excluded middle. So we have just derived the law of excluded middle from the law of non-contradiction.  
       What we just did was convert the negation of a conjunction into a disjunction,
by considering what it means for the conjunction to fail. The rule that lets us do this is known as De Morgan’s rule, after Augustus De Morgan. Formally speaking, it tells us that statements of the following two forms are equivalent: ¬(p^q) and ¬p_¬q. If you substitute ¬p for q in the first formula, you will have the law of non-contradiction. So you might want to try doing the derivation yourself. You will, however, need the rule that tells us that p is equivalent to ¬¬p. The point of this exercise, however, was to show that it is possible to derive the law of excluded middle from the law of non-contradiction. However, it is also possible to convince yourself of the truth of the law of excluded middle without the law of non-contradiction.  
       We can show using the method of truth tables that the disjunctive statement
p _ ¬p is always true. As a point of terminology, a statement that is always true, irrespective of the truth values of its components, is called a tautology. p _ ¬p is a tautology, since no matter what the truth value of p is, p_¬p is always true. We can see this illustrated in the truth table belowp    ¬p    p _ ¬p
T     F     T
F     T     T  
       You can try constructing a similar truth table to show that the law of non-contradiction is also a tautology. Its truth table is a bit more complicated, though. However, since the law of excluded middle is a tautology, it should hold no matter what the truth value of p is. In fact, it should be true no matter what statement we decide p should represent. So the law of excluded middle tells us that every statement whatsoever must be either true or false. At first, this might not seem like a very problematic claim. But before getting too comfortable with this idea, we might want to consider Bertrand Russell’s famous example: “The present King of France is bald.” Since the law of excluded middle tells us that every statement is either true or false, the sentence “The present King of France is bald” must be either true or false. Which is it?  
       Since there is no present King of France, it would seem quite unusual to claim that this sentence is true. But if we accept the law of excluded middle, this leaves us only one option - namely, to claim that it is false. Now, at this point, we might choose to reject the law of excluded middle altogether, or contend that it simply does not hold in some cases. This is an interesting option toconsider, but then we might need to consider why the method of constructing truth tables tells us that the law of excluded middle holds, if it actually doesn’t. We would also have to consider why it is derivable from the principle of non-contradiction. After all, this sentence doesn’t pose a problem for the law of non-contradiction, since it’s not both true and false. So we’ll ignore this option for now.    
       Returning to the problem at hand, we must now consider the question of what it means for the sentence “The present King of France is bald” to be false. Perhaps it means “The present King of France is not bald.” But that would imply that there is a present King of France, and he is not bald. This is not a conclusion we want to reach. Russell, in his 1905 paper “On Denoting” presented his own solution to this problem, which comes in the form of a theory of definite descriptions. Under this theory, we can think of there being a hidden conjunctive structure in the sentence “The present King of France is bald.” What the sentence really says is that there is a present King of France, and he is bald. So the fact that there is no present King of France implies that this sentence is false, and we have the solution we need.    
       Russell’s solution clearly suggests that we can’t just extract the logical structure of a sentence from its grammatical structure. We have to take other things into account. If you’re interested in issues about the relationship between logic and language, you might want to take a class in philosophy of language. The other essay in this section, entitled “Logic and Natural Language”, covers some other issues in this area.    
       One method of proof that comes naturally from the law of excluded middle is a proof by contradiction, or reductio ad absurdum. In a proof by contradiction, we assume the negation of a statement and proceed to prove that the assumption leads us to a contradiction. A reductio ad absurdum sometimes shows that the assumption leads to an absurd conclusion which is not necessarily a contradiction. In both cases, the unsatisfactory result of negating our statement leads us to conclude that our statement is, in fact, true. How does this follow from the law of excluded middle? The law of excluded middle tells us that there are only two possibilities with respect to a statement p. Either p is true, or ¬p is true. In showing that the assumption of ¬p leads us to a contradictory conclusion, we eliminate the possibility that ¬p is true. So we are then forced to conclude that p is true, since the law of excluded middle is supposed to hold for any statement whatsoever.    
       Now, I’ve been a bit flippant in talking about statements. Statements can be about a lot of different things. The above discussion illustrated a problem with statements about things that don’t actually exist. I’m sure most people would agree that the designation “the present King of France” refers to something that doesn’t exist. But what about situations where it’s not so certain? One of the main metaphysical questions in the philosophy of mathematics is the question of whether or not mathematical objects actually exist. Think about the question of whether numbers exist. If they do exist, then what are they? After all, they’re not concrete things that we can reach out and touch. But if they don’t exist, then what’s going on in math?    
       Metaphysical worries have motivated certain people to argue that proofs by contradiction are not legitimate proofs in mathematics. Proponents of intuitionism and constructivism in mathematics place a significant emphasis on the construction of mathematical objects. One way to characterize this position is that in order to show that a mathematical object exists, it is necessary to construct it, or at the very least, provide a method for its construction. This is their answer to the metaphysical question. Suppose we had a mathematical proof in which we assumed an object did not exist, and proved that our assumption lead us to a contradiction. For an intuitionist or a constructivist, this proof would not be a sufficient demonstration that the object does exist. A sufficient demonstration would have to involve the construction of the object.    
       Even if questions about existence get too complicated, we can still ask the question “What mathematical objects can we legitimately talk about?” The intuitionist answer is that we can talk about those mathematical objects which we know can be constructed.    
       Simply speaking, intuitionistic logic is logic without the law of excluded middle. I have outlined some small part of the motivation behind developing such a system, but more details can be found in the work of L.E.J. Brouwer and Arend Heyting.