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Logic denotes the intellectual discipline that the Greeks invented as a means of ensuring that the reasons that they thought up to explain the phenomena of the world actually matched the truth of Reality. The early philosophers had set themselves the task of discovering the ultimate causes of phenomena and they quickly discovered that human language did not give them a perfect instrument for discerning Truth. They saw that we need to supplement language with an awareness of the various fallacies that can flaw and invalidate our arguments. Such an awareness of the pathologies of Reason eventually led to an understanding that we have two fundamentally different ways by which we can reason ourselves to increased knowledge. We have induction, the preferred method of scientific reasoning, and deduction, the preferred method of mathematical reasoning, and, in a manner of speaking, each comes to us as the reverse of the other. In devising the Map of Physics we want to use deduction as our method of scientific reasoning, but it well behooves us to look at the method that has given us most of our knowledge of physics as well.
Induction gives us a method of obtaining general rules governing phenomena from observing many specific examples of those phenomena. Almost all of our scientific knowledge has come from the use of induction and the people who identify with the school of philosophy called Empiricism take it as the only reliable way to obtain true knowledge of Reality. Though Francis Bacon described scientific induction in his science-fiction story "The New Atlantis" around 1600, Isaac Newton provided the best description of induction in his Principia as "Rules of Reasoning in Philosophy". With appropriate commentary, those rules comprise:
"Rule I: We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances." Newton explained further that, "to this purpose the philosophers say that Nature does nothing in vain, and more is in vain when less will serve; for Nature is pleased with simplicity, and affects not the pomp of superfluous causes."
We know this rule better as Occam's Razor or the law of parsimony. It requires of us that from all the explanations that we can devise for a given phenomenon we must choose the simplest, though we must also bear in mind Einstein's admonition not to make our theories too simple.
"Rule II: Therefore to the same natural effects we must, as far as possible, assign the same causes."
Newton gave several examples to illustrate his meaning and one of them also lets me add a cautionary note. To the light emanating from cooking fires and from the sun, Newton said, we must assign the same cause. We can accept that statement as true to Reality if we identify the cause as heat agitating atoms and making them shake off sprays of photons, but as for the cause of the heat, in the former case we identify it as chemical combustion and in the latter case as thermonuclear fusion.
"Rule III: The qualities of bodies, which admit neither intension nor remission of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be esteemed the universal qualities of all bodies whatsoever."
Newton wrote this rule a little too broadly. He should have referred to "bodies of a specifiable class". We should interpret his comment on "neither intension nor remission of degrees" to mean that induction applies only to the quality per se and not to any quantitative aspect of it. Thus from our observations of birds we may induce the conclusion that all birds have feathers, but we may not induce any conclusion as to how many feathers birds have.
"Rule IV: In experimental philosophy we are to look upon propositions collected by general induction from phenomena as accurately or very nearly true, notwithstanding any contrary hypotheses that may be imagined, till such as other phenomena occur, by which they may either be made more accurate, or liable to exceptions."
In other words, if you have a proposition that you have induced from the results of experiments or from observations of Nature and if that proposition disagrees with some hypothesis you have thought up, then you must accept the proposition as true to Reality and discard the hypothesis as false to Reality. Only information from a further experiment or from a further observation of Nature can falsify your proposition or oblige you to modify it. In modern science we embody the idea implied in that statement in a rule that goes beyond Newton's list.
Rule V: We accept no theory of natural phenomena as valid unless it has a form that makes it potentially falsifiable.
One of the powers that we find inherent in the human mind, pattern recognition, seeks meaningful pictures in everything that comes before it. Part of the foundation underlying human intelligence, that power also acts as a source of considerable self-deception: we often see patterns in percepts that have no authentic connection (think of the constellations, pictures that we impute to the distribution of the stars that we see). To ensure against the survival of false patterns as theories of science, scientists take as an obligation the writing of their theories in ways that make clear what observations of fact could show those theories to be false to Reality.
I offer as a simple example a little theory that I devised when I was about five or six years old. I had been wondering about traffic lights and, in particular, about what makes them change from green to yellow to red and then back to green in a regular cycle. My theory originated in two observations: first that toggle switches mounted on walls are used to control fixed lights inside and outside houses and other buildings and second that a large gray concrete block of a building (presumably a power substation) had a large number of wires coming out of it (though not nearly as many as my little theory would require). From those observations I made the reasonable (for a child) inference that scattered around the city were large gray blocks of buildings that enclosed large volumes of empty space (rather wasteful, but a child doesn't think of such things). Covering all four walls on the inside of each building were trios of toggle switches that controlled the traffic lights in the area around the building and catwalks on which workers moved as they flipped the switches to change the lights.
Though not much of a theory, my idea did conform to one requirement of a proper scientific theory - it was falsifiable. All I had to do was to show that these strange buildings did not exist (yeah, just try to prove a negative) or make an observation that would show that a different method was used to control traffic lights. And indeed in time I learned about the clock-like mechanisms that move a rotating electrical contact over connections to the circuits that feed electricity to the lights. Those mechanisms would have to be contained in a box somewhere near the traffic lights that they controlled. Sure enough, near every intersection that has traffic lights I could see a gray metal box set on top of a pipe rising from the sidewalk. So my little theory died, but it had a properly scientific death.
That example illustrates the empirical-inductive Scientific Method that we all learned in grammar school and which has produced almost all of our current scientific knowledge of Reality. Rationalism, on the other hand, uses the axiomatic-deductive method to produce what we hope will be an accurate representation of Reality without our having to consult Reality through observation or experiment.
Deduction gives us a method of combining two known-to-be-true propositions to create a new, guaranteed-to-be-true proposition. That may seem like magic, but it works more like chemistry. Just as the properties of a new chemical compound pre-exist hidden in the elements that make up the compound, so does hidden truth pre-exist in the propositions whose combination brings it out into the open. Our application of that idea to science embodies our confidence that premises true to Reality lead through deduction to conclusions that are also (and necessarily so) true to Reality. If we extend that notion to its logical conclusion, we infer that, given the right axiom, we can deduce all of the laws of physics from that one axiom. The school of philosophy that accepts that idea and whose students thus believe that deduction gives us the only reliable way to obtain true knowledge of Reality bears the name Rationalism. This website provides an example of a Rationalist treatise (at least in embryonic form).
What philosophers teach as classical logic comprises primarily a study of deduction (the overwhelming emphasis on deduction in classical logic offers a strong explanation of why science, which we base almost exclusively on induction, took so long to develop). That comes from the fact that the old logicians wanted to devise methods of reasoning that would lead with certainty from true premises to equally true conclusions. They founded their discipline upon the bedrock notion (or axiom if you prefer the more technical term) that statements of fact can either be true to Reality or false to Reality but can never be both. Out of that law of the excluded middle the ancient logicians evolved their methods of formal reasoning.
The most famous formal method of reasoning that the Greek philosophers developed bears the name syllogism (from syn (with) + logos (statement), which means "statements taken together") and the most famous example of a syllogism comes from a retort that Aristotle allegedly made when some orator imprudently referred to "immortal Sokrates";
Sokrates is a man (Major premise)
All men are mortal (Minor premise)
Therefore, Sokrates is mortal. (Conclusion)
The Greeks intended that kind of verbal arithmetic to eliminate all ambiguity from human reasoning and, though it seems that the Greek logicians developed the syllogism first, it stands at the head of a long line of methods. A partial list of the methods of logic includes:
1. Syllogism; If A implies B and if B implies C, then A implies C. This makes classical logic look like some Štherial game of dominoes.
2. Modus Ponens; If A implies B and A is true, then B is true. For example, if (and only if) it rains, then the streets will be wet. It has rained, so the streets are wet.
If (and only if) the train is on time, then it will be in the station now. The train is on time, so it is in the station now.
3. Modus Tollens; If A implies B and B if false, then A is false. For example, if (and only if) the train is on time, then it will be in the station now. The train is not in the station now, so it is not on time.
If (and only if) it rains, then the streets will be wet. The streets are not wet, so it has not rained.
4. Reductio Ad Absurdum; If A implies B and A implies not-B, then A is false.
The principle of Relativity tells us that Reality must display the same phenomena for you as it does for me.
Hypothesis: moving objects expand in the directions perpendicular to their motion relative to an observer. (Proposition A)
i. Thus, a basketball hoop in your frame expands for me so that the basketball that I carry will pass through it. (Proposition B)
ii. Thus, a basketball in my frame expands for you so that it will not pass through the basketball hoop that you carry. (Proposition not-B)
Necessary Conclusion: moving objects do not expand in directions perpendicular to their motion relative to an observer.
5. Double Negation; If A is true, then not-A is false (what is A is not not-A). That seems too trivial to have any use, but an excellent example of its application appeared in one of the new episodes of The Twilight Zone that were shown on TV in 1987. The episode was called "Eye of Newton" and it concerned a professor of mathematics (played by Sherman Hemsley) who accidently conjures up a demon (played by Ron Glass). The professor must give the demon a task it cannot perform or suffer a fate worse than death. The professor determines that there is no place in the Universe or beyond to which the demon cannot go or from which he cannot return. The professor reasoned something like this:
The demon can go anywhere.
What can be done cannot not be done.
Therefore, the demon cannot not go anywhere.
The professor's soul-saving command was thus the delightful and delicious pun, "Get lost!"
If you think of logical argument as a kind of computer program for turning true statements into other true statements, then fallacies comprise the bugs that make the program go wrong and fail. The Greek logicians debugged their program long ago and they left us a list of twenty-four major bugs that we must guard against. If you see one of these fallacies in an argument, logic obliges you to dismiss that argument as invalid until the arguer corrects the fallacy. However, the use of logical fallacies is not completely illegitimate: as you will see from some of my examples, a substantial amount of comedy comes from deformed logic.
A: Fallacies of Shifting Meaning;
1. Ambiguity - an argument in which a term has a double meaning.
The tall alien stepped from his spaceship, raised his hand in greeting, and said,"We have come to serve Man."
(OK, is he a galactic missionary or does he represent some fast-food outlet like Kentaurus Fried Human?)
2. Accentus - an ambiguity that comes from emphasis on a word or phrase.
"He's still with us (meaning alive), so he's still with us (meaning on our side)."
3. Amphiboly - literally"a word thrown two ways at once".
"The accuracy of the scientists' examination is necessary to good science." (Are the scientists doing the examining or are they being examined?) Or consider an advertisement for "hot women's fashions". (Are they fashions for sexy women or are they ultra-popular fashions for women?)
4. Equivocation - a term is used one way in a premise and another way in another premise or the conclusion.
The secret of comedy is good timing.
Good timing is obtained from a quartz-based clock.
Therefore, the secret of comedy is obtained from a quartz-based clock.
B: Fallacies of Irrelevance;
5. Argumentum Ad Baculum - The Argument of the Fist uses a threat, either direct or indirect, to gain acceptance of the argument.
All forcible religious conversions embody this fallacy and, some would say, thereby invalidate the very religion they seek to promote.
6. Argumentum Ad Hominem - uses the character of the proponent as a premise promoting acceptance or rejection of their argument.
"Relativity is false because Einstein is a Jew." (Yes, the Nazis, not the brightest bulbs on the marquee, actually used that one.)
But this can get a little subtle at times. Consider an incident reported on page 8 of The Nation for 2006 Dec 11. In discussing the legacy of the recently deceased economist Milton Friedman, William Greider wrote:
"Art Hilgart, a retired industrial economist, recalls hearing Friedman lecture in 1991 and recommend the destruction of Medicare, welfare, the postal system, Social Security and public education. The audience was dumbfounded. Finally, a brave young woman asked what this would mean for poverty. "There is no poverty in America," Friedman instructed."
If we argue that we must discard Friedman's economic theories because he was a Jew, that is argumentum ad hominem and is, therefore, invalid.
If we argue that we must discard Friedman's economic theories because he was bald and wore dorky-looking glasses, that is argumentum ad hominem and is, therefore, invalid.
But if we argue that we must discard Friedman's economic theories because he was oblivious to a major fact of economic reality, that is not argumentum ad hominem. The existence or nonexistence of poverty in America must be taken into account in any theory that seeks to give an accurate and true accounting of the American economic system and someone who gets that fact wrong will, of necessity, produce a theory that is inaccurate and wrong. Even though it might be taken as a personal attack, the state of Friedman's knowledge of the American economy is relevant to any judgement of his work and thus is part of a valid argument.
OK, did you spot the argumentum ad hominem that I snuck in above? Certainly we must dismiss the reasoning of the Nazis because they used argumentum ad hominem (among other fallacies), but not because they were "not the brightest bulbs on the marquee".
7. Argumentum Ad Ignorantum - claims that a proposition is true because it has not been proven false or vice versa. This makes credulity rather than scepticism the default in reasoning and the Greeks would have none of it (and neither should we).
"Well, nobody has proven that flying saucers don't exist!"
8. Argumentum Ad Populam - an appeal to the beliefs of the multitude.
"Everyone believes that the sun goes around the Earth, so it must be true."
9. Argumentum Ad Misericordium - an appeal to pity to gain acceptance of an argument.
I'm just a poor preacher who's trying to do his best." (OK, give him your sympathy, but not your assent.)
10. Argumentum Ad Vericundiam - an appeal to an authority on matters outside their realm of expertise.
"Nine out of ten American League pitchers say that the Godzillatron should be built at Coney Island." (Any resemblance between a fastball and a relativistic subatomic particle is less than superficial.)
C: Fallacies of Covert Insertion of Invalid Assumptions;
11. Petitio Principii (also known as begging the question and circular reasoning) - the conclusion to be proven is assumed in one of the premises.
Perhaps one of the best examples of a circular argument comes from George Berkeley's proof of the existence of God. Berkeley began by asserting that for anything to exist it must be perceived. He then asked his famous question "If a tree falls in the forest and no one is nearby to hear, has it made a sound?" We answer that it has done all of the things that we associate with the creation of a sound and so answer that it must have done so. But then Berkeley says that he agrees but also that if no one was nearby to hear the sound the sound could not have existed, so someone must have been nearby and that someone was God, who is always everywhere.
In that argument Berkeley gives us an improper Modus Ponens in which we set A = sound of tree falling in the forest. We have
If A exists, then A is perceived.
A exists, so A is perceived.
But we have specified that no material perceiver is present, so we must, according to Berkeley, infer the existence of a non-material perceiver, one that is omnipresent. But Berkeley's major premise makes this argument a petitio principii by tacitly assuming the existence of an omnipresent perceiver, for if existence necessitates being perceived, then all things that exist have a perceiver.
(Oh, but look at Section 4 following this list!)
12. Many Questions - demands a simple answer to a complex question.
"Do you still molest children, yes or no?"
13. Composition - assuming that the whole has a property because its parts have it.
The pieces of uranium in the nose of a United States Air Force rocket are inert, but you don't want to bet your city that the single mass they are meant to become will share that property.
14. Division - assuming that the various parts have a property because the whole has it.
The Los Angeles telephone directory will stop a small-caliber bullet, but you don't want to bet your maiden aunt's favorite china teacup that a single page from the directory will do the same.
15. Hypostatization - attributing to a thing properties that it does not actually possess, such as treating an abstraction as if it were an actual concrete object. In the form of attributing human motives to inanimate objects it's known as the pathetic fallacy.
"At this temperature the metal wants to become superconducting, but the magnetic field prevents the electrons from forming Cooper pairs." (Wanting is an attribute of animals, not of metals.) More properly we should say, "At this temperature the metal would become superconducting if the magnetic field did not prevent the electrons from forming Cooper pairs."
16. Ignoratio Elenchi - proves a proposition different from the one proposed (also known as missing the point).
To illustrate the proposition that alcohol consumption is unhealthy, the young preacher dropped a worm into a glass of booze, where the worm went into convulsions and died. One parishioner rejoiced loudly that, since he was a heavy drinker, the experiment assured him that he would never have worms.
17. Non Causa Pro Causa - an argument to reject a proposition because another proposition that seems to be a consequence of it (but actually is not) is false.
"The hypothesis that the Earth moves is to be rejected because we would feel any such motion and we feel nothing of the sort." (The concept of frictionless, vibration-free motion was unknown to the people who made that argument in the Sixteenth Century.)
We can also express that idea as an improper Modus Tollens:
If Earth moves, then we must feel the effects of motion.
We do not feel the effects of motion.
Therefore, Earth does not move.
18. Post Hoc, Ergo Propter Hoc - arguing that "A preceded B" necessitates "A caused B".
This fallacy is the basis for many gamblers' tales, such as the one about a man who won big money shortly after a pigeon defecated on his head and subsequently went around trying to get pigeons to defecate on him.
D: Fallacies of Improper Procedure;
19. Affirmation of the Consequent (improper Modus Ponens) - reasoning from the truth of a hypothetical statement and the truth of the consequent (the part that follows the implication sign) to the truth of the antecedent (the part that precedes the implication sign).
"If Koppernigk is right and Earth goes around the sun, then Venus will show phases like those of the moon. We see that Venus shows phases like those of the moon, so Koppernigk is right and Earth goes around the sun." (Galileo made that particular error in reasoning, failing to see that Venus could show moon-like phases for reasons other than the truth of the Copernican model of the solar system. Tycho Brahe's version of the Copernican model would do just as well.)
20. Denial of the Antecedent (improper Modus Tollens) - reasoning from the truth of a hypothetical statement and the falsity of its antecedent to the falsity of the consequent.
"If we see a parallax of the stars, then we will know that Earth moves. We do not see any stellar parallax, so Earth does not move." (That displays Ptolemy's reasoning and it is clearly wrong. The real reason nobody saw stellar parallaxes in classical times is that the stars are so far away from us that the parallaxes are too small to be detected with instruments and techniques available prior to the Nineteenth Century).
21. Non Sequitur - the conclusion is not a necessary consequence of the premises.
"Earthquakes are a force of Nature and mere humans cannot oppose the forces of Nature, so there is no way to prevent earthquakes." (Preventing earthquakes does not involve opposing the forces acting on Earth's tectonic plates, but rather could be achieved by lubricating the faults so that the plates slide freely one past another and don't build up the stresses whose sudden release generates an earthquake).
22. Secundum Quid - a proposition is used as a premise without attention to an obvious condition affecting its applicability.
If people here obey the traffic laws, then no one will run a red light with impunity.
A fire engine ran the red light with impunity.
Therefore, people here don't obey the traffic laws.
This is an improper Modus Tollens. It ignores the fact that emergency vehicles, such as fire engines, are granted exceptions to the traffic laws.
23. Undistributed Middle (also known as Sweeping Generalization) - a syllogism in which the middle term (the one common to both premises) is not distributed (that is, does not cover all members of the class of objects it denotes) in at least one premise.
"Confucius was a well-educated man. Well-educated men speak perfect Greek. Therefore, Confucius spoke perfect Greek." (Not all well-educated men speak perfect Greek; some, though, speak perfect Mandarin Chinese.)
24. Illicit Process (also known as Hasty Generalization) - a syllogism in which a term is distributed in the conclusion but not in the premises.
"Carpenters need wood. Wood is obtained by cutting down trees. Therefore, carpenters are going to cut down all the trees." (Well, they will cut down some trees, but surely not all of them.)
4. A Bit of Circular Reasoning
Consider the following chain of logical deduction based on Hans Christian ěrsted's discovery that an electric current exerts a force upon a magnet:
a. Modus Ponens:
If A exerts a force upon B, then B exerts a reciprocal force upon A (Newton's third law of motion).
Electric current exerts a force upon a magnet.
Therefore, a magnet exerts a force upon an electric current.
Electric current is electric charge in motion.
Objects in motion do not move the same for all observers.
Therefore, electric current does not move the same for all observers.
c. Modus Ponens:
If I put an electrostatic charge on a moving silk thread, I will have an electric current.
I put an electrostatic charge on a silk thread that's moving past a magnet.
Therefore, I have an electric current that's moving past a magnet.
The magnet exerts a force upon the moving electrified thread.
Whatever exerts a force upon an object causes that object to change its motion (Newton's first law of motion).
Therefore, the magnet causes the moving thread to change its motion.
e. Modus Ponens:
If the motion between us does not change, then we will both see the same change in the motion of any object.
Your motion matches the original motion of the silk thread (that is, the thread is stationary in your frame of reference), but does not change.
Therefore, you see the same change in the motion of the thread that I see.
f. Modus Tollens:
If the magnet does not make the thread change its motion, then some other object exerts a force upon the thread.
No other object exerts a force upon the thread in this imaginary experiment.
Therefore, the magnet alone exerts a force upon the thread.
Only an electric force can act on a stationary electric charge.
In your frame of reference the moving magnet acts on the stationary electric charge (the one on the thread).
Therefore, a moving magnet exerts an electric force.
h. Modus Tollens:
An object that exerts a force must touch the object that it forces.
The magnet itself does not touch the silk thread.
Therefore, the magnet itself does not exert an electric force.
i. Modus Ponens:
If object A exerts a force on object B but does not touch B, then there must exist an object C that touches both A and B, through which A exerts its force on B.
The moving magnet exerts an electric force upon the thread without touching it.
Therefore, there exists an intermediate object (which we call a magnetic field) through which the magnet exerts its force upon the thread.
A moving magnet creates an electric field at a point in space.
A moving magnet carries its magnetic field with it.
Therefore, a moving magnetic field creates an electric field at a point in space.
A moving, nonuniform magnetic field generates an electric field of a single fixed strength at a succession of points that comprise a line that lies precisely parallel to and, point for point, an unchanging distance from the succession of points representing the motion of the magnet.
The sequential occupation of a succession of points by something that does not change constitutes motion of that thing.
Therefore, a moving magnetic field generates a moving electric field.
A moving magnetic field generates a moving electric field.
A moving electric field generates a moving magnetic field (converting ěrsted's discovery into forcefield terminology).
Therefore, a moving magnetic field generates a secondary moving magnetic field.
If a magnet and I are moving away from you and if I make the magnet move away from you even faster, then I will see the magnet generate an additional magnetic field that it did not generate before I accelerated it.
An event that occurs for me must also occur for you if there is no change in the situation between us.
Therefore, any increase in the speed of a moving magnetic field produces an increase in the strength of the secondary magnetic field that it generates.
The strength of a secondary magnetic field is increased by increases in the speed of the moving magnetic field that generates it.
We can make a magnetic field move at any speed we desire (not true, but no Nineteenth Century physicist would have known. It turns out to be just true enough, though, that the conclusions that we draw from it remain valid, though we would be obliged by our ultimate conclusion to come back and check it).
Therefore, we can make a magnetic field move fast enough to generate a secondary field equal to itself.
That which makes a thing remain equal to itself is all that's required to make that thing exist (because for a thing to exist is for that thing to remain equal to itself).
Motion at the specific speed cee (299,792.458 kilometers per second) makes a magnetic field remain equal to itself.
Therefore, a magnetic field can exist by itself (without the support of a material magnet) so long as it moves at the speed cee.
p. Reductio Ad Absurdum:
If a self-supporting magnetic field passes me at the speed cee, it will pass you at a different speed if you are moving relative to me.
i. It passes me at the speed cee, so its intensity does not change.
ii. It passes you at a speed greater or less than cee, so its intensity changes.
An event that does not occur for me (a change in the intensity) does not occur for you if the situation between us does not change, so our premise is false to Reality.
Therefore, if a self-supporting magnetic field passes me at the speed cee, it will not pass you at a different speed, even if you are moving relative to me.
That conclusion is clearly the second and more important of Einstein's postulates of Relativity. What makes that chain of logic extending from ěrsted's discovery to Einstein's postulate circular is the fact that magnetism, which I introduced in an axiom in the first link of the chain, is a relativistic effect. Magnetism is the physical manifestation of a modification of the law of electricity necessitated by the effects of relative motion. Thus I have rather subtly assumed in my initial premise the very thing that I set out to deduce - the theory of Special Relativity.
We must, of course, avoid fallacies in logic, but I believe that we will encounter some difficulty in eliminating this one. If the fundamental assumption of Rationalism is correct, then all the laws of Nature are components in one grand petitio principii. Whenever we see a connection between two laws of Nature, we are seeing not a straight line of logic, but rather a short segment of a giant circle of reasoning. If we do indeed create a circle of logic that truly encompasses all the laws of Nature, a circle that is thus a true and accurate description of the Universe, it will, nonetheless, be fallacious. What I'm implying in this series of essays is that a fully Rationalist physics does underlie the workings of the Universe. If I am right about that, then we may find ourselves in the awkward position of believing in an absurdity, a Reality whose rules constitute a logical fallacy. Logic will then oblige us to confront the question How do we correct that fallacy?
We resolve this particular fallacy by deducing the theory of Relativity from a separate set of propositions and applying it to the electric force to deduce the existence of the magnetic field. Resolving the larger fallacy of the Map of Physics itself will take a little more effort.
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