Logic · Quality Thinking

A foolish consistency is the hobgoblin of little minds, adored by little statesmen and philosophers and divines.
Ralph Waldo Emerson

tangent logo  The Blind Men and the Elephant
by John Godfrey Saxe

It was six men of Indostan
To learning much inclined,
Who went to see the Elephant
(Though all of them were blind),
That each by observation
Might satisfy his mind

The First approached the Elephant,
And happening to fall
Against his broad and sturdy side,
At once began to bawl:
"God bless me! but the Elephant
Is very like a wall!"

The Second, feeling of the tusk,
Cried, "Ho! what have we here
So very round and smooth and sharp?
To me ’tis mighty clear
This wonder of an Elephant
Is very like a spear!"

The Third approached the animal,
And happening to take
The squirming trunk within his hands,
Thus boldly up and spake:
"I see," quoth he, "the Elephant
Is very like a snake!"

The Fourth reached out an eager hand,
And felt about the knee.
"What most this wondrous beast is like
Is mighty plain," quoth he;
" ’Tis clear enough the Elephant
Is very like a tree!"

The Fifth, who chanced to touch the ear,
Said: "E’en the blindest man
Can tell what this resembles most;
Deny the fact who can
This marvel of an Elephant
Is very like a fan!

The Sixth no sooner had begun
About the beast to grope,
Than, seizing on the swinging tail
That fell within his scope,
"I see," quoth he, "the Elephant
Is very like a rope!"

And so these men of Indostan
Disputed loud and long,
Each in his own opinion
Exceeding stiff and strong,
Though each was partly in the right,
They all were in the wrong!

So oft in theologic wars,
The disputants, I ween,
Rail on in utter ignorance
Of what each other mean,
And prate about an Elephant
Not one of them has seen!

tangent logo  The Cockroach Syndrome

Patriotism is the last refuge of a scoundrel.
Sam Johnson

We can easily forgive a child who is afraid of the dark; the real tragedy of life is when men are afraid of the light.

When someone turns on a light (or tries to examine suspicious assertions more closely), they scurry for a dark hiding place. Be skeptical of things that only "work" when no one is looking.

cockroach syndrome

tangent logo  Explanation or Excuse?

  • begin with impartial observations or supportable facts and work toward conclusions in a logically consistent way. The starting point doesn’t require absolute certainty, but should be based on the best available information.
  • usually have some predictive value - if the initial situation or proposed cause arises later, a similar outcome should be likely.
  • consider all relevant information.


  • usually start with an actual outcome or desired conclusion and work backward to a cause (usually unconnected with the person offering the excuse). Any convenient or contrived reasoning may be used with little regard for quality or consistency.
  • don’t often work "forwards" - next time there may be a totally different outcome with no reasonable explanation for the change.
  • often ignore information that could invalidate the excuse (the cockroach syndrome), or avoid the actual starting point by only backtracking to an intermediate step that shifts blame to someone else.

tangent logo  But It’s Only a Theory

So is gravity. Gravity happens to be a very strong theory supported by substantial evidence but even after extensive research, some fundamental details remain elusive. In science, a hypothesis may be little more than a hunch and is often the starting point of an investigation. Once the research is done and the results are verified, the conclusions are combined with everything else that’s known with any certainty to form a theory. As observations and experimental evidence accumulate, the theory becomes more certain. Incomplete evidence becomes stronger in the absence of contradictory evidence and reasonable alternate explanations.

There’s always room for doubt, but take care not to fabricate special circumstances just to make a conclusion seem wrong or less useful. What are you trying to accomplish by doubting? Is there a specific fact or assumption that could be tested, or do you just want an excuse to believe something else?

tangent logo  Are -- better at -- than -- ?

Facts are stubborn, but statistics are more pliable.
Get your facts first, then you can distort them as you please.
Mark Twain

There are three kinds of lies: lies, damn lies, and statistics.
Benjamin Disraeli

It’s not always easy to find someone who is exactly average. Unlike Garrison Keillor’s Lake Wobegone, where "all the children are above average", if someone is above average, at least one other person in the group must be below average. If 3 people in the same group score 91, 94, and 10 on a test, their average score is 65. 2/3 are well above average and have a high score. None are close to average. 3 people in another group might take the same test and score 85, 80, and 84 for an average of 83. All are good scores and close to average. Which group is better? If you could choose one person, would you choose based on the group they’re in, or based on their personal score? If you could choose 3 people, would you choose all 3 from the same group? When would you need to know which group is better?

tangent logo  Is Anything Possible?

It depends on what you mean by possible. We can say it’s possible to flip a coin 100,000,000 times and get heads each time. In this case, all we really mean is that it’s not impossible. It doesn’t violate any known laws of science, but do you really want to spend your time trying to confirm this possibility? A coin toss is often considered a fair way to randomly choose between two options. This is based on knowledge that the coin is balanced well enough to avoid a preferred outcome, and by observing many coin tosses by many people, and counting the number of heads and tails over time. Without doing any calculations, we can flip a coin many times in succession and observe that the number of heads and tails isn’t always the same. This suggests that it is possible to get 100,000,000 heads in a row, even though the probability is very small. This randomness is generally accepted, in part because it’s been observed so often, and the observations are consistent. These observations make it easier to accept the possibility of 100,000,000 successive heads even though it’s not likely anyone will ever see it happen.

Someone might say it’s possible that people are manipulated by a mind-control ray from aliens orbiting the earth in a flying saucer. Like the possibility of getting 100,000,000 heads in a row, this doesn’t seem impossible. Life elsewhere in the universe doesn’t violate any known laws of science, and beings from another planet might develop the technology for remote mind control and space travel. In addition to the "not impossible" test, we should also consider relevant observations and ask a few basic questions:

  • Has the presence of flying saucers been convincingly documented?
  • How does the behavior of those supposedly subjected to the mind control ray differ from the behavior from others?
  • Do the brains of those supposedly controlled by the ray have any observable physical, electrical or chemical abnormalities?
  • Was there ever a time when the aliens weren’t using their mind control ray, and if so, were the behaviors attributed to the ray ever observed then?
  • Do the aliens ever stop using the ray, and if so, does anyone’s behavior change?
  • Can the ray be detected or measured?
  • Are there any other possible explanations for the observed behavior?

The answers to the above questions don’t necessarily make it impossible, but might substantially reduce the probability. Just as you probably don’t want to devote your life to tossing a coin until you get 100,000,000 heads in a row, is it productive or useful to spend time wondering about mind control rays from flying saucers?

If you admit that one thing is possible, will you also admit that the alternatives are also possible? Perhaps people here on earth are manipulating the minds of the aliens, making the aliens think they are successfully using a mind-control ray. Should we prefer one possibility over all others?

tangent logo  Will It Ever be Wrong?

People sometimes feel confident or complacent "knowing" that they’re right, but logically sound reasoning requires the possibility that new observations or necessary assumptions might prove the conclusion wrong. A thorough, impartial evaluation of all supporting evidence is important, but it’s often useful to also ask, "What would persuade me to change my mind?" If the answer is, "Nothing will ever change my mind", the credibility of the conclusion is questionable and implies a refusal to consider all available evidence (the cockroach syndrome), and could be the result of making excuses rather than impartially seeking the best possible answer. Another useful question to ask someone with a belief or opinion that seems to be permanently entrenched is, "Was there ever a time you believed something different?". If so, what changed? If not, how and when did this belief or opinion originate?

Clinging to an insupportable conclusion may appear to minimize inconvenience or disruption of an established belief, but also slows progress, as it usually has no practical value or useful consequences. The truth (or best possible answer given the currently available information) usually provides a solid foundation to build on, and may help support or refute other conclusions. Remember the old advice, "Never argue with a fool". They usually don’t respond to reason, and often drag you down to their level - after a few minutes, it’s hard to tell who’s the bigger fool.

tangent logo  Will your sanity check bounce?

Reality is that which, when you stop believing in it, doesn’t go away.
A little experience often upsets a lot of theory.

The price of wisdom is eternal thought.
Frank Birch

It’s often tempting to generalize based on one or two specific instances, and sometimes that’s all you have to go on, but usually that’s just the beginning. Once you think you have a valid conclusion, try working it "backwards" to see if you can get back to the specific observations that got you started. Some people refer to this as a "sanity check" or a "reality check" because it scrutinizes the conclusion to make sure it’s still consistent with all actual observations.

  • Does it work in every possible situation, or do you have to make exceptions in some cases?
  • Can it reliably predict the outcome of all similar situations, or does it only work sometimes?
  • Are there unintended consequences or contradictions with related concepts or observations?

If there are exceptions, can you find a way to predict which future situations will follow the "rule" and which will be an exception? If your sanity check bounces, further refinements might be needed, or you might have to throw it out and start over.

tangent logo  Don’t Win the Wrong Argument

A fanatic is one who can’t change his mind and won’t change the subject.
Winston Churchill

Hain’t we got all the fools in town on our side? And hain’t that a big enough majority in any town?
Mark Twain

It might seem tempting to bolster a case with personal attacks, or by redirecting the discussion, but usually that’s just trying to win the wrong argument.

Vilifying those on the other side of a dispute may be simpler than intelligently discussing complex issues, but even a fool can repeat a noble truth. Accusing someone of foolishness might incline others to believe that the person in question is indeed a fool, but doesn’t necessarily mean that the persons words or actions were foolish in themselves.

The professor’s neighbor and self-proclaimed philosopher, Mediocrates, claims that roofers have the best/worst job in the world. They get lots of time off because they can’t work in the rain, but they don’t make much money because when it’s not raining, people don’t need roofs. Such observations could easily be the source of an endless argument. One person might explain all the great things that could be done with lots of free time, while an opponent goes on about the hardships endured by those with low incomes. They could each produce ever lengthening lists of people who enjoy/endure the situation, and specific accomplishments/suffering resulting from such a job. Each claims victory by overwhelming evidence, but nothing gets resolved. They each argue about only one aspect of the job, and make no effort to address the others concerns.

tangent logo  Calculating Credibility

There are so many claims about so many subjects, it’s sometimes difficult to know who or what to believe. The professor knows a bit about judging credibility, and Okum knows a crock of baloney when he finds it. Here are some suggestions.

Start with a thorough, impartial evaluation of all available evidence:

  • What qualifications or abilities are required to do such an evaluation?
  • Who determines those qualifications and why?
  • Who has those qualifications?
  • What percentage of qualified persons are for, against, undecided or neutral?

Is there a possible ulterior motive such as:

  • ego or emotional attachment
  • greed
  • stubbornness or arguing just for the sake of argument
  • personal comfort or convenience
  • a desire to bait or emotionally inflame others
  • just making excuses

Are the claims only relevant to one special case or can it be generalized to other things? If a position is based on cost or safety, is the same degree of safety or cost-consciousness expected from everything, or just this one issue?

If there is significant evidence for more than one position (or insufficient evidence for reasonable certainty):

  • What are the possible consequences of being wrong?
  • What are the expected early warning signs of failure?
  • How easily could errors be detected and corrected?

Are reasonable objections accounted for or just dismissed or ignored by changing the subject (the cockroach syndrome)? Does explaining away objections require adding more assumptions or greater complexity?

What circumstances or conditions led to the conclusion? Was it paid for, and if so, by whom?

The asymptote test: Is the conclusion reevaluated as new information becomes available?

tangent logo  Investigating the Investigators

Thorough investigations can provide valuable lessons about what happened, how or why it happened, and what might help to better control such situations in the future. Sometimes the results of these investigations contradict earlier accounts, which might warrant an investigation of the investigators. Assuming the investigation was impartial, some potentially useful questions are:
  • Did the later investigation consider all information previously available?
  • Did the later investigation have additional information not previously available?
  • Were earlier accounts or decisions reasonable under the circumstances when based on information available at the time?
  • Was there bias by any investigators which caused them to overlook or excuse certain things?

It could turn out the earlier actions or decisions were reasonable under the circumstances, and those involved would agree that with more information or resources, things would have been different. Improving the outcome of future similar situations might require access to better information or resources.

If the investigation reveals mistakes or poor decisions, perhaps better training or people with different skills would produce a better result next time.

If bias was suspected, was any evidence found such as ignored or dismissed observations?

If those involved simply refuse to accept the results of the investigation, perhaps they’re just making excuses.

tangent logo  Burdensome Proof

A lie can travel half way around the world while the truth is putting on its shoes.
A truth is not hard to kill and a lie told well is immortal.
Never let the truth get in the way of a good story.
Mark Twain

The first to present his case seems right, till another comes forward to question him.
Proverbs 18:17

Give me a grain of truth and I will mix it up with a great mass of falsehood so that no chemist will ever be able to separate them.
John Wilkes

Accusations are easily made and often difficult to disprove. If the accusation is hasty, frivolous, or poorly supported, the accused shouldn’t have to spend time and effort disproving the charge, especially if the accuser can just come back with yet another casual accusation. Sometimes, those accused simply respond with accusations of their own.

In the name of "fairness" or "equal time" it might seem tempting to simply accept the accusation and then ask the accused to disprove it. There’s nothing fair or equal about it. An accuser could make a casual accusation with little effort and no evidence, while the accused might need to invest substantial time and effort gathering evidence and presenting a convincing defense. If the accusation is a generality, such as "this person lies", or "this person is a thief", ask for specific examples.

Before seriously considering an accusation, demand more from accusers:

  • Did they ask the accused for evidence that might disprove the accusation?
  • Did they consider all evidence suggested by the accused?
  • Can they summarize the defense of the accused?
  • Can they convincingly explain why any defense is either wrong or irrelevant?

This shifts the burden to the accuser. Now the accused may be able to quickly respond by asking questions like:

  • Did the accuser talk to ... ?
  • Did the accuser read ... ?
  • How does the accuser explain ... ?

Now go back and ask the accuser to summarize the additional information suggested by the accused and explain why it’s wrong or irrelevant. If the accusation has little merit, accusers may give up once they realize that the burden of proof is on them.

tangent logo  Can God make a rock so big even he couldn’t move it?

This question can be divided into 2 parts:
  1. Can God make a really big rock?
  2. Can God not move a really big rock?

The answer to either question could be "yes" or "no" so we have 4 possible combinations:

  1. God can make a really big rock and also move a really big rock.
  2. God can make a really big rock, but not move a really big rock.
  3. God can’t make a really big rock, but could move a really big rock.
  4. God can’t make a really big rock, and can’t move a really big rock.

If the answers to question A and B are both "yes", then the answer to the entire question is "yes". If the answer to either question A or question B is "no" then the answer to the entire question is "no". Since question B is asked as a negative (Can God can not move a really big rock) only answer option 2 makes the answer to the original question "yes". The following table shows all possible answers to the original question.

Can God make a rock so big he can’t move it?
God can move a really big rock.
God can not move a really big rock.
God can make a really big rock.
God can not make a really big rock.

tangent logo  Squirrels, trees and circles

Imaging walking through the woods and seeing a squirrel on an oak tree. The squirrel also sees you, and immediately scurries to the opposite side of the tree. You circle the tree trying to catch another glimpse of the squirrel, but the squirrel is also circling the tree to stay on the opposite side. You are going around the tree, the squirrel is also going around the tree, but are you going around the squirrel?

This answer doesn’t depend on logic so much as on careful definitions. You’re walking in a circle, and the tree and squirrel are always inside the circle, but you and the squirrel would always be facing each other if the tree weren’t in the way. The word "around" could be understood either way, and the exact meaning determines the answer to the question. If you understand "around" to mean that if the tree weren’t there, you would see the squirrel from all sides (front, back, right, left) while walking "around", the answer is different than if you understand "around" to mean that the squirrel is always inside the circle you’re making with your footprints.

tangent logo  Absence of Proof and Proof of Nonexistence

Absence of proof could mean many things. It might seem tempting to assume that absence of proof (or a limited understanding) implies impossibility or opens the door for any type of speculation, but it really depends on what else is known. A good place to start is to consider the plausibility of the original claim. There’s no proof of a Loch Ness Monster, yet it’s not impossible. We might consider several things to estimate plausibility:
  • Loch Ness is difficult to explore.
  • New species are occasionally discovered.
  • There is an incredible diversity of life on earth.
  • Descriptions from claimed sightings are similar to known creatures, either living or extinct.

So far, the existence of a Loch Ness Monster doesn’t violate any known natural laws or principles, however the probability goes down with each new serious investigation:

  • Something that big isn’t easily missed.
  • Reported descriptions are most like known air-breathing creatures, which increases the probability of surface sightings.
  • Animals reproduce and die, yet no remains or partial remains of dead monsters have ever been found.

The above objections might be explained by small numbers of very shy monsters that die instantly and immediately sink to the bottom where they quickly decompose, but now we’re speculating about increasingly unlikely circumstances. These secondary speculations must be subjected to the same scrutiny as the original claim. Of course there’s always room for doubt when the only argument against existence is a lack of reliable observations or an inability to explain certain details which in themselves are not impossible. Besides, people love a good mystery, and the discovery of such a creature would be very exciting.

Proof of nonexistence is much trickier to deal with. In general, you can’t prove nonexistence in any absolute way since one can always fabricate explanations for the lack of direct evidence or observations. We can sometimes take an indirect approach and consider what else might be true if the subject of investigation really did exist. We could assume for instance that if there really were a Loch Ness Monster, it would eat, reproduce and die. This leads to several conclusions:

  • There must be sufficient food in Loch Ness to support something that big.
  • If it breathes air, it must surface at regular intervals.
  • If it reproduces, there must be at least 2.
  • When it dies, there will be remains.

If we could show that none of those things were true, that would be very strong evidence of nonexistence.

tangent logo  Extraordinary Claims, Extraordinary Proof

It’s often said that extraordinary claims require extraordinary proof. If your best friend tells you that a neighbor has a pet pig, you probably know that some people do have pet pigs and might believe your best friend without any further evidence. If someone tells you that a neighbor has a pet elephant, you’d probably be more skeptical, especially if you live in a city. You might think that perhaps it’s a toy elephant, or something other than a real live, eating, trumpeting elephant. If you’re told that a neighbor has a pet unicorn, you should be really skeptical. Even if you actually see a living creature with something sticking out of it’s forehead, you should still suspect a hoax. You might want to hear from an impartial veterinarian, see the results of a DNA analysis, or perhaps even observe the creature reproduce in a controlled environment and verify that the offspring really are unicorns.

tangent logo  Keep it Simple

Things should be made as simple as possible -- but no simpler.
Albert Einstein

If it ain’t broke, don’t fix it.

Sometimes a cigar is just a cigar.
Sigmund Freud

Okum likes to ask, "What’s the problem you’re trying to solve?". Once you really understand that, don’t fabricate a complex solution when a simple approach will get the results you want. If you see a picture of something on the surface of Mars that resembles a primate-like face, explore the simple solutions before jumping to conclusions about intelligent life on Mars. You might start by considering the size and contours of the "face", to see if they’re consistent with the results of geologic or atmospheric forces such as wind, water, or crust movement similar to those on earth. Even a short hike in the mountains of earth often reveals natural formations that resemble something else. Most 3-year-olds can "find" all sorts of things in earthly cloud formations. It’s just a result of normal brain functions and an active imagination. If the simple solution checks out, be very careful before seeking a more complex solution such as intelligent life on Mars. Attributing an apparently natural feature to living creators quickly leads to some important questions with no obvious answers:

  • Why didn’t entities with the technology to make such a large "face" leave other traces?
  • Who was it made for? If was made as a form of interplanetary message, they were advanced enough to believe they might not be alone and probably would have left other evidence or signals.
  • What other technology would they probably have, any why isn’t there any evidence?
  • Why didn’t they make anything else?
  • How likely is it that they would resemble us?

tangent logo  Don’t Read Everything You Believe

I respect faith, but doubt is what gets you an education.
Wilson Mizner

The price of wisdom is eternal thought.
Frank Birch

It’s natural to associate with like-minded people, and having your opinions and conclusions validated by friends, speakers or authors who share your beliefs can be very gratifying.  It can also make people so comfortable that they stop asking questions and no longer explore alternatives.  Don’t overlook the importance of an occasional, well-reasoned dissenting opinion.

tangent logo  A Second Opinion

The first to present his case seems right, till another comes forward to question him.
Proverbs 18:17

How reliable is a first (and only) opinion? Many puzzles and brainteasers begin with a plausible situation, proceeding step by step, only to end with a seemingly irreconcilable contradiction. Zeno’s Paradox is a famous example which concludes that motion is impossible by arguing that to reach any point, you must first pass the half-way point, the half-way point to the half-way point, etc. ultimately passing through an infinite number of intermediate points. Since passing through an infinite number of points would take forever, motion itself is therefore impossible, or at best an illusion. The "proof" that 1 equals 2 is another example. An isolated argument may seem reasonable at first, but is it really solid?

  • Was any relevant information ignored, dismissed, misunderstood or distorted?
  • Are unrelated things lumped together as if they’re the same?
  • Are further distinctions needed between similar facts or concepts?

Zeno’s Paradox ignores the fact that as you increase the number of intermediate points, the distance between the points (and the time to move between adjacent points) gets smaller, ultimately approaching zero as the number of points approaches infinity. The "1=2" argument sneaks in a division by 0.

A second opinion from someone reaching a different conclusion can help locate flaws in the original argument, but the second opinion should be scrutinized as carefully as the first.

  • Do they both start with the same facts or observations?
  • Do they justify alternatives, or just say the other is wrong?
  • Does one simply substitute one unknown for another?

Sometimes there just isn’t enough information to be certain and even experts will disagree.

tangent logo  None of the Above

"If we don’t use my idea right now, the situation will spin out of control with disastrous consequences!"

The intended implication is that only the proposed solution will prevent a catastrophe, but there are really 2 separate statements:

  • Without intervention, the situation may develop on it’s own with potentially unpleasant consequences.
  • There’s at least one possible solution.

These two issues could be considered separately. Is action really required right now, and if so, what kind of action would be most appropriate?

  • Will the proposed solution fix the problem permanently?
  • How likely are side effects?
  • Is it practical?
  • Will everyone support it?

If people disagree on significant aspects of the solution, more options would probably be helpful, especially if the proposed solutions are mutually exclusive.

"If I don’t get $3000 right away, my life is ruined!"

This might start an argument with one person demanding $3000 and another refusing to provide it. Exploring the matter further might suggest alternatives.

"Why do you need the money?"
"To buy a car."
"Why do you need a car right now?"
"So I can get to work."

Someone started out by asking for $3000, but really wants transportation to work. This new information opens up new possibilities:

  • Is another job available closer to home?
  • Would moving closer to work help?
  • Is another method of transportation available?
  • Would a cheaper car provide adequate transportation?

Finding alternatives isn’t always easy, but some possible questions are:

  • Is there an underlying problem (needing money to buy a car to get to work)?
  • Are there several aspects that could be considered separately?
  • Can any consequences be considered separately?
  • Is this problem similar to another that’s already been solved?

tangent logo  Are Logic and Truth Relative?

Truth and logic are subject to certain restrictions and preconditions. It’s often said that the shortest distance between two points is a straight line, but if you walk from your house to a neighbor’s house, you’re really walking on the surface of the earth (a sphere), so traveling a truly straight line might require burrowing through the earth. This might be the shortest distance, but is not the most efficient in terms of travel time or required effort. We might amend the old adage to say that it’s only practical on a flat surface. Even then we might want to further clarify what "flat" really means, since the limits of our perception might make us think we’re on a flat surface when we really aren’t. After all, the earth can sometimes appear flat to us unless we observe it from space, or in relationship to the sun, moon, or other objects apart from the earth.

Apart from carefully defined conditions, be very skeptical of any claims of variations in "truth". A good test of questionable claims would be to look for practical applications of the alternate "truth". If someone claims that "For me, 2+2=5. That’s my reality", put that claim to a practical test. You get a stack of $1 bills, and have the other person get a stack of $5 bills. You give the other person $2, then another $2, and in return you get $5. Keep the exchange going all day.

moon not made of cheese

tangent logo  Are All Opinions Equal?

There are many types of opinions. An opinion may be totally subjective, or objective and based on anything from wishful thinking to absolute knowledge. A subjective opinion might involve color or style of clothing, although there might be other considerations if one is dressing for a job interview or other situation where the impression given by the clothes might have practical consequences.

Objective opinions can be compared and evaluated based upon the underlying factual information, which constitutes the basis of the opinion. One might say for instance "I believe the moon is made of swiss cheese. That’s my opinion and I’m entitled to it." It is in fact an opinion, but has little value. If this person traveled to the moon, would they pack a lunch or just plan on scooping handfuls of cheese? This opinion has practical consequences, which can be used to judge the quality of the opinion.