Using Deductive and Inductive Logic in Science

Posted on January 28, 2015 by Fallacy Man

There are several different types of logic, but probably the two most common are deductive and inductive. Both of these play a vital role in science, but we use them for different purposes. Therefore, it is my intention to explain the differences between these types of logic and when and how we use both of them in science.

Deductive logic
This is the most powerful form of reasoning. It is the type of logic that results in logical proofs. It goes from general concepts and/or specific observations to a focused conclusion. For example,

  1. The sum of the angles of any triangle equals 180 degrees (general concept)
  2. Angle A of triangle ABC = 45 degrees (specific observation)
  3. Angle B of triangle ABC = 90 degrees (specific observation)
  4. Therefore angle C of triangle ABC = 45 degrees (focused conclusion).

Notice that the conclusion is a certainty. This is the power of deductive logic, it tells you what absolutely must be true (assuming that your premises are true).

To give another famous example:

  1. All men are mortals (general concept)
  2. Socrates is a man (specific observation)
  3. Therefore, Socrates is a mortal (focused conclusion)

I used this example to bring up a very important point about deductive logic. Deductive syllogisms often include a premise that was arrived at via inductive logic, this will become important in the next section.

In science, deductive logic is typically what is used at to arrive at facts. In other words, we use it to determine the results of specific experiments. For example:

  1. I stomach flushed turtle A
  2. Its stomach contained the remains of a fish
  3. Therefore, turtle A ate a fish

So in short, deductive logic always gives a specific, focused conclusion and is used in science to determine facts and the outcomes of individual experiments.

Inductive logic

In contrast to deductive logic, inductive logic always results in a general conclusion and can be used to construct theories. It should be noted, that it is impossible to use deductive logic to arrive at a theory. Theories only come from inductive logic.

Inductive logic works somewhat backwards from deductive logic. It starts with specific observations and works towards a general conclusion (note: both types of logic start with observations and work to a conclusion). For example, think back to the Socrates example in the deductive logic section. How do we know that all men are mortals? Well, we know that from inductive logic,

  1. Every man that we have “tested” (observed) has been mortal (collection of specific observations)
  2. There is no reason to think that an immortal man could exist (logical statement)
  3. Therefore, all men are mortals (general conclusion)

Notice, an inductive conclusion is not as strong as a deductive conclusion, but it is still often very powerful. Technically speaking, it is true that I cannot be completely certain that there is not an immortal walking around pretending to be mortal, but there is simply no reason to think that such a person exists, so the conclusion is clearly valid. It should be noted, however, that not all inductive conclusions are equal. For example, if I said that, “I have liked every Christopher Nolen film so far, therefore, I like all Christopher Nolen films (present and future)” my conclusion is clearly dubious. There are so many variables involved in making a film that it is absurd to think that he will never make one that I don’t like. This is generally not the case in science. In science, we use inductive logic with as few variables as possible, and we generally support our conclusions with mathematical models. Also, the consistency of the physical universe adds an extra level of support to our inductive conclusions.

Additionally, because of the law of large numbers, the strength of an inductive conclusion increases as the number of observations used to form the conclusion increases. If I measure the rate of something once, it would be absurd to say that it always moves at that rate. If I measure it 100 times, however, it becomes more certain. If I measure it 1,000 times, it becomes even more certain.

Perhaps the greatest support of an inductive conclusion is, however, its ability to predict other events/make things work. Suppose that I build a device that would only work if the aforementioned rate was constant, and, when I turn the device on, it works. That would be extremely strong evidence that my inductive conclusion was correct. In fact, predictive power is the benchmark that we use to measure the strength and validity of theories.

This brings me to the restatement of a very important point about all scientific theories. They all rely on inductive logic. This is inherent in the nature of a theory (i.e., it is a general framework based on observations and used to explain other observations), but, something only gets promoted to the status of theory after it has been shown to have a high predictive power. For example:

  1. Every physical body with mass that we have tested has produced gravity and been acted upon by gravity.
  2. There is no logical reason to think that gravity wouldn’t be constant, and there are strong mathematical/logical reasons to think that gravity is a constant
  3. Numerous functional devices and calculations rely on the concept that gravity is constant
  4. Therefore all physical bodies with mass produce gravity and are acted upon by gravity (i.e., the universal theory of gravity).

Notice, technically, I cannot be 100% certain of the conclusion. I have not tested every physical body in the universe, but virtually everyone will agree that the conclusion is valid.

I explained in a previous post that laws are synonymous with theories, allow me to demonstrate this by showing that the second law of thermodynamics was also arrived at using inductive logic.

  1. Every closed system that we have ever observed has increased in entropy
  2. There is no logical reason to think that a closed system could decrease in entropy, and there are strong mathematical/logical reasons to think that entropy must always increase
  3. Numerous devices/experiments only work because all closed systems increase in entropy
  4. Therefore, all closed systems increase in entropy (i.e., the second law of thermodynamics)

To summarize, scientists generally use deductive logic to determine the outcomes of specific experiments (sometimes inductive logic is also required depending on the nature of the experiment), and we use inductive logic to generalize from those experiments and form laws and theories. This is true for all laws/theories, whether we are talking about the laws of thermodynamics or the theory of gravity.

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Science and the Public Part 3: A Scientific Consensus is Based on Evidence, not Peer Pressure and Adherence to Dogma

Posted on January 28, 2015 by Fallacy Man

In this post, I am going to debunk an argument that is very commonly used by the anti-science movement. Namely, the argument that scientists merely go along with the accepted dogma of their field and either refuse to consider contrary evidence, or even if they realize that their position is flawed, they refuse to speak up for fear of being rejected by the scientific community. I have frequently heard claims such as, “many scientists realize that global warming isn’t true, but they keep quiet because if they spoke up they would be ridiculed by their peers and might lose their job.” This argument generally appears either in an attempt to persuade people not to trust scientists or as a response to the dilemma presented by the fact that there is enormously strong agreement among scientists on issues such as global warming, vaccines, evolution, etc. In either form, it is horribly, horribly flawed. In the former situation, it commits both the ad hominem and question begging fallacies, and in the later, it commits the ad hominem and ad hoc fallacies. These are the same logical fallacies committed by the conspiracy argument (see Part 2). These arguments attack the scientists, rather than their results (ad hominem). Further, there is no evidence to support them, and they wouldn’t be believed by anyone who wasn’t already convinced that the scientific results were false (question begging/ad hoc fallacy depending on context). So I could really stop right here. Anyone who says that you shouldn’t believe scientists because they are either involved in a conspiracy or are simply refusing to accept contrary evidence is not following the rules of logical analysis and you shouldn’t listen to a word they say. Nevertheless, I will explain the problems with this argument in more detail.

To avoid a strawman fallacy, I will begin with a quote from a conversation I recently had with an anti-scientist, “All [scientific education] does is indoctrinate people with pre-established beliefs and prevent them from truly thinking for themselves. That’s why it’s impossible to have an intelligent conversation with most ‘scientists.'”

Overlooking how completely insulting this is to those of us who spend our entire lives studying science, let’s consider the many problems with this statement. First, this statement (and every statement like it that I have ever seen) was made by someone with no scientific education. So in what way is this person qualified to comment on scientific education? Further, this statement makes it blatantly obvious that people in the anti-science movement have utterly no clue how science actually works because not one word of this statement is even remotely true. Someone who has never attended a science class, published a paper, gone to a professional conference, etc. is in no position to make any judgments about how science works.

A scientific education accomplishes multiple things, not the least of which is to impart a massive amount of background knowledge. Science is complicated, and there is an huge quantity of knowledge that is required to be able to properly analyze scientific results. If I were to stack together all of the scientific books/papers that I have studied, the stack would be taller than I am. Notice, this is in no way an “indoctrination,” rather, this is learning what is already known. All that any scientist does, is build on what has already been discovered, but you can’t do that unless you know what has already been discovered. Science cannot move forward unless we know what other scientists have found. This is one of the key problems with the anti-science movement, the bloggers and other anti-scientists have not acquired the necessary background knowledge. Time and time again when I read anti-science blogs, books, etc., they are wrong on the most basic scientific facts. No wonder they don’t accept vaccines, global warming, etc. when they don’t understand the concepts behind them!

The next major function of a scientific education is, in fact, to teach you to think for yourself. As a graduate student, I am constantly encouraged by all of my professors to question the accepted wisdom. Graduate students are required to take seminars which are specifically designed to make us question and think for ourselves, and we are repeatedly cautioned NOT to blindly follow what’s in the literature but to analyze the data for ourselves and compare it to other studies. Further, the whole point of conducting independent research (i.e. our theses) is to see whether or not we can think for ourselves and solve unique problems. Where anyone got the idea that a scientific education is an indoctrination is beyond me. Virtually all scientists got into science because we love to ask question and acquire new information, that’s the whole point of being a scientist. The statement that a scientific education renders us incapable of thinking for ourselves could not be more untrue. A scientific education is specifically designed to force you to think for yourself, and anyone who tells you otherwise doesn’t have a clue what they are talking about.

Finally, lets address the nonsense argument that evidence against vaccines, evolution, climate change, etc. is unpublishable because the scientific community refuses to accept it. My fellow graduate students and I have often laughed at this claim because it is so absurd. For example, if I had evidence that truly disproved evolution, I could publish in any biological journal that I wanted. I would have just made my career. I would have my pick of universities to work at, and I would probably get a Nobel Prize. No scientist in his right mind would sit on data like that. All of history’s great scientists have been great because they found evidence that was contrary to a common view. It is every scientists dream to find a groundbreaking result that alters the way that the entire scientific community thinks. So the notion that a lot of us actually know that vaccines don’t work or that the climate isn’t changing, and we are just sitting on that evidence for fear of reprisal is, quite frankly, idiotic. Once again, anyone who makes that claim clearly doesn’t understand how science works.

Look around you, everything from the computer you are sitting at to the headache medicine that you will surly take after reading my rant was brought to you courtesy of modern science. Science works. The anti-science movement is miss-informed and downright dangerous. If individuals want to personally live in ignorance, that is their problem, but we should not allow them to drag all of society back to the dark ages where logic is unheard of, leaches are used to cure illnesses, and magnetism is the result of supernatural forces.

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Science and the Public Part 2: Scientific Results Are Facts, Not Conspiracies

Posted on January 28, 2015 by Fallacy Man

As I explained in the first post of this series, there is widespread and unfounded disagreement between what scientists know to be true and what the general public chooses to believe. Many people choose to blindly reject the science behind vaccines, global climate change, evolution, etc., but this position presents an obvious dilemma. Namely, why is it that virtually every scientist in the world accepts these as fact? How is it that virtually the only people who reject vaccines, climate change, etc. are non-academics that have never done any actual research? When confronted with this issue, people in the anti-science movement usually take one of two approaches (or often a combination of the two). The first, which I will deal with here, is simply to claim that the scientists are actually involved in an enormous conspiracy. The second response, which I will deal with in a later post, is to claim that scientists are biased, and in order to keep getting funding and be accepted, they have to go along with scientific dogma, even though they know it isn’t correct.

First, let us consider the scope of these conspiracies. The plausibility of a conspiracy theory increases as the number of people involved decreases. Simple logic and common experiences tell us this. For sample, a secret is easier to keep when fewer people know it. When a massive group of people know it, it is harder to prevent someone from slipping. This is especially true for a secret whose story changes constantly. Consider a lie that has to be constantly built on as people ask questions. Everyone has to be in agreement on what the false answers will be in order to prevent inconsistencies, but this gets harder and harder to maintain as the number of people who are in on it increases. Just for example, if someone proposes that a small group of ten solders are involved in a conspiracy, that is plausible, it would not be difficult to keep the story consistent with only ten people involved, but, it is far less plausible to think that every solder in every branch of the military is involved in an elaborate conspiracy. It would be virtually impossible to maintain a constant story with that many people involved.

Now, let’s think about the notion that virtually all of the world’s scientists are involved in a conspiracy. We are talking about millions of people that would have to be in agreement. Further, science evolves constantly, making it necessary for continual contact among all members of the scientific community in order to maintain a consistent story. The amount of networking that would be have to be involved in this is ridiculous. It is utterly absurd to think that millions of people from all over the world are in on this.

The second, and perhaps greatest problem is a lack of motive. Conspiracy theorists often cite money as the motive (especially for vaccines where false claim is made that all of the research supporting vaccines is funded by pharmaceutical companies), but this argument suffers multiple flaws. First, only a small portion of the scientific community is funded by pharmaceutical companies, climate change focused agencies, etc. so what is in it for the rest of us? Why would we go along if we aren’t getting the money? There is utterly no motive for the rest of the scientific community.

Further, this argument presents a clear misunderstanding of how science works. When a researcher gets a one million dollar grant to study climate change, he doesn’t get to go out and by a new Ferrari. Rather, he gets to go buy a bunch of expensive equipment for his research. We don’t get to keep grant money. Look at the average income of a scientist. It’s not much. If we wanted a career that would make money, we could have done much better than science. We don’t do research to get rich, we do it for the love of knowledge.

Additionally, there is an inherent paradox here. The supposed money flow is circular. Supposedly, scientists are going along with vaccines and global warming to get money, but the committees of most granting agencies are composed of scientists (remember that these scientists also have to be in on the conspiracy). So why on earth, would the scientists on the committee who are not personally receiving one dime of the funding that they are passing out agree to fund research on something that they know is a load of crap!? Scientists don’t need to invent phenomena to get funding, there are plenty of legitimate research questions out there.

Finally, let’s consider the inherent absurdity in the notion that scientists are involved in a massive lie. Generally speaking, becoming a scientist requires four years of undergraduate studies, 6-8 years of graduate studies, and 1-2 years of postdocing. During this period, you will work all day, every day, and you will forgo sleep and free time in order to achieve your goal. Even after you finally get a job as a professor (which is the job held by most scientists), you will work a minimum of 60 hours a week at an often thankless job, and you will make very little money. The salaries of scientists are absolutely pathetic compared to those of lawyers, doctors, and other professions that require comparable amounts of training. So, why does anyone go through that? Why would we subject ourselves to all those years of hard work if we don’t even make much money? Quite simply, we do it for the love of knowledge. We do what we do because we love to ask questions, learn new facts, and share what we have learned. So, why would anyone, go through all of that training, all of those years of long, often miserable days just to throw away everything that they have learned and join some bizarre conspiracy that is intent on deceiving everyone? The very notion that all scientists are in a conspiracy is antithetical to everything that we started studying science for in the first place.

To sum up, the idea that scientists are involved in massive conspiracies is utterly ridiculous for the following reasons: first, this would require a massive and constant networking and agreement of millions of people working under different granting agencies, from thousands of universities, from every country of the world! Second, there is utterly no motive. It takes an incredible amount of work to become a scientist, and scientists don’t make much money. If we were really interested in money, we would never have gone into science, because very few people get rich from studying vaccines, climate change, etc. The cost/benefit ratio simply doesn’t play out here. In short, scientists do research for knowledge, not money.

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Science and the Public Part 1: Why You Shouldn’t Trust Blogs

Posted on January 28, 2015 by Fallacy Man

An enormous disparity exists between what scientists know to be true, and what the general public chooses to believe. This disparity exists largely because of the internet, and it is perpetuated by those who readily read and disperse blogs and unfactual websites. Allow me to begin by using an example to illustrate the absurdity of the situation. I have little knowledge about archeology. I’ve seen some History Channel specials, and I can quote all of the Indiana Jones movies by heart, but that’s about the extent of my knowledge. Now, suppose that I did some reading online, then came up to you and said, “I have absolutely no credentials in archeology, I’ve never done any field work, I’ve never had any training, I’ve never gone to a professional meeting, I’ve never published a paper, I haven’t read more than 10 peer-reviewed papers, but I figured out something that the entire archeological community is wrong about! In fact, I think that they are all in on an elaborate conspiracy to cover this up in order to get more funding.” Would you listen to me about my discovery? I highly doubt it. You’d probably laugh in my face, and with good reason! It would be the epitome of arrogance for me to claim to know more than the entire archeological community. I cannot think of a more audacious claim, yet this is exactly what happens every day on the internet. People with absolutely no credentials or experience claim to know more than the entire scientific community, and people believe them! If you wouldn’t believe me about my archeological epiphany, then you shouldn’t believe them when they claim that the entire scientific community is wrong about vaccines, global climate change, evolution, etc.?

The simple fact is this, we live in a world where information is more easily available than ever before. One needs only to press a few buttons on their phone to have the summation of all human knowledge at their fingertips. This blessing is, however, also a curse. The internet is not filtered. There are no quality control of peer-review mechanisms. As a result, for every factual website, there are 20 unfactual websites (my personal estimate). Any eloquent idiot can write a blog and deceive thousands of people, and most readers aren’t discerning enough to tell which websites are good and which websites are faulty. The internet is replete with blatantly false articles that are extremely well written, include impressive looking graphs, and cite peer-reviewed scientific papers. To the masses who are untrained in the sciences, these blogs/websites seem very convincing. People rarely bother to fact check them and see whether or not the sources are reliable, or if the papers that are being cited actually support the author’s claims, or if the author even has any credentials in the field that he is claiming to be an expert in.

That last point is one of the biggest issues: blogs’ authors generally aren’t scientists. Scientific articles are written in the jargon of science, and unless you understand the intricacies of experimental design and the meaning of terms like “P value,” “type 1 error,” “standard deviation,” “ANOVA,” “Bonferroni correction,” “null hypothesis,” etc. there is simply no way that you truly can understand a scientific publication and make a valid evaluation of its results (note: this is not a statement of “scientists are better than everyone else,” rather this is a statement of, “science is complicated and requires years of careful study, and it’s absurd to think that someone who hasn’t received that training will understand it in adequate detail”). Time and time again, when I am debating someone who denies global warming or is anti-vaccine, they send me a link to some blog that cites a bunch of peer-reviewed papers that, according to the blog’s author, demonstrate that global warming isn’t true or vaccines don’t work. When I read the original papers, however, it becomes abundantly clear that the author of the blog doesn’t understand science/statistics and either misunderstood the papers’ or intentionally distorted their results.

Blogs/private websites basically boil down to this, someone who is not a scientist and doesn’t know how to interpret scientific results went ahead and tried to interpret scientific results, and now expects you to accept their interpretation. The amazing thing is, of course, that millions of people actually do believe something just because several blogs claimed that its true! Not only do they believe it, they post the link to their Facebook account, encouraging their friends to read this, “excellent blog on why we shouldn’t use vaccines” or “why evolution isn’t true” or “why climate change is just a natural cycle.” They are often willing to put themselves and everyone around them at risk all because of what they read on the internet!

The take home message here is basically this, do not trust the internet!!!. If a “scientific” blog/website is not written by someone who is credentialed in that field, don’t even both to read it. Your time would be better spent elsewhere. Second, even if they have credentials, view it cautiously, especially if they are making a claim that is contrary to the general consensus of scientists. There is an old saying that extraordinary claims require extraordinary evidence. If they are right, then you should have no problem finding information published by well established, reputable, scientific organizations (e.g. NOAA, CDC, USGS, etc.) that agrees with them (note: NaturalNews.com, Merecola.com, and Whale.to are not reputable sources). If you cannot find peer-reviewed, reputable sources that agree with your blogger, then ignore him/her. As I have said many times, it doesn’t matter what crackpot notion you choose to believe, you can find someone with an advanced degree that agrees with you. You can find M.D.s that argue that smoking doesn’t cause cancer, and Ph.D.s in physics that argue that the sun moves around the earth, but you would be crazy to believe them. Even so, finding one or two “scientists” that agree with you that the entire scientific community is wrong does not constitute a valid reason to reject scientific evidence, and it would be crazy for you to believe these people. Having an advanced degree does not inherently make you a scientist. If you want to honestly research a topic, start by avoiding any site that has a “.com” address, and stick to the reputable sources.

Note: it may seem hypocritical to write a blog about why you shouldn’t trust blogs, so I want to clarify a few points. First, I actually am a published scientist who participates in the peer-review system, which means that I have been trained on how to read and assess scientific papers. Second, despite my qualifications, I AM NOT suggesting that anyone blindly believe any information on my blog. I strongly encourage everyone to fact check everything here (as you should with all websites). All that I am saying, is that when you fact check, you have to do it against original sources of peer-reviewed information, not other blogs. Finally, many of my posts (such as this one) are logical arguments not expositions of scientific facts. You don’t fact check logical arguments, you analyze them on their own merits. Again, you should carefully consider them rather than blindly believing them, but there is no way, even in concept, to fact check a post such as this one.

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Basic Statistics Part 2: Correlation vs. Causation

Posted on January 28, 2015 by Fallacy Man

Updated with additional sources on 16-June-16

It is fairly widely known that correlation does not inherently indicate causation. In fact, inappropriately asserting causation is a logical fallacy known simply as a correlation fallacy. Nevertheless, there is a great deal of confusion around this topic, and many people use it selectively. For example, anti-vaccers are adamant that the correlation between in the introduction of vaccines and the decline in diseases is not valid evidence that vaccines work, yet they insist that the supposed correlation between autism and vaccines is 100% proof that vaccines are dangerous. Therefore, in this post I will endeavor to unravel the mysteries of correlation and causation.

Let’s start with basic definitions. Correlation is simply a relationship between two variables. Either they both increase together, both decrease together, or one increases as the other decreases. So, for example, among people under the age of 20, there is a correlation between age and height. As age increases, height also increases. In contrast, causation means that not only do the two variables change together, but the change in one variable is actually causing the change in the other variable. The height example is, in fact, a causal one. Being older means that you have had more time to grow. Thus, on average, an 18 year old will be taller than a 12 year old because the 18 year old has had more time to grow.

organic food autism corrleation logical fallacy

Correlation does not equal causation. Organic food sales and autism rates are tightly correlated, but that does not mean that organic food causes autism. Image via the Genetic Literacy Project

The problem is that simply being correlated does not mean that one variable causes the other. I’ll use a classic example to illustrate. There is a strong correlation between ice cream sales and drowning accidents. Both of them increase together and decrease together, yet it would clearly be absurd to claim that ice cream sales are causing people to drown. The reality is that both ice cream sales and drowning accidents are caused by a third variable: time of year. People consume more ice cream in the summer than in the winter and people spend more time in the water in the summer than in the winter (which leads to more drowning accidents). Thus, these factors are correlated, but not causally related. This is why the correlation between increased vaccines and increased autism rates is not evidence that vaccines cause autism. There are countless other factors that could be causing autism to increase. Amusingly, it turns out that the increase in the sale of organic food also correlates well with the increase in autism (note: the supposed increase in autism is largely artificial as it is mostly caused by a change in the definition of autism, i.e., people who would not have been considered autistic 20 years ago are considered autistic today, also, there is a great deal of evidence that vaccines do not cause autism).

Having now established that correlation does not automatically mean causation, we run into our second point of confusion. Many people are under the impression that you can never use correlation to show causation. In reality there are two circumstances in which you can use correlation to conclude that there is a causal relationship. The first, and most powerful, is by simply controlling everything except for the two variables you are interested in. This is why scientists carefully design controlled studies such that one variable (the experimental variable) is deliberately changed  and another variable (the response variable) is measured to see if it changes in response to the experimental variable, but all other variables are controlled so that they do not change. Under these circumstances, you can conclude that the correlation is causal because there are no other potential causes. If the only things that change are the two variables you are interested, then the changes in one variable must be caused by the changes in the other variable. This is why, for example, we can claim that there is a causal relationship between increased CO2 in our atmosphere and the increase in our planet’s temperature. We have carefully monitored output from the sun and the other major drivers of our planet’s climate and none of them correlate closely with the increase in temperature. In other words, without including CO2 , we can’t account for the current warming (Stott et al. 2001; Meehl et al. 2004; Allen et al. 2006; Lean and Rind 2008; Imbers et al. 2014).

The second way that we can infer causation is by using additional data. For example, a few months ago, I was debating with someone who was opposed to higher education, and I showed them a figure documenting the increase in salary that accompanied increasing levels of education. They, of course, accused me of a correlation fallacy, but there was a very obvious additional piece of information that demonstrated that the relationship was causal. Namely, the fact that the high paying jobs required higher education. This fact makes it abundantly clear that higher education was responsible for the higher salaries. Similarly, with global climate change we have additional data from laboratory trials that show that increased CO2 traps more heat, and we have data from satellites that show that less IR radiation (the energy that CO2 traps) is leaving the earth now than it was 30 years ago (Harries et al. 2001; Griggs and Harries. 2007). These additional pieces of evidence confirm that the relationship between the increase in CO2 and the increase in temperature is a causal one.

There are several key take home messages here. First, do not be duped by people who are trying to use correlation inappropriately to prove causation. Look carefully at their argument, and if there are other variables that were not controlled for, please tell them that they have committed a logical fallacy then proceed to ignore their argument. If, however, someone presents you with a carefully controlled study in which only the variables of interest have changed, you can then conclude that the relationship is causal. Similarly, if they can back up the relationship with independent facts which clearly demonstrate that one variable will cause the other, you can conclude that the relationship is causal. Finally, this once again demonstrates the superiority of carefully controlled studies over anecdotal evidence. Anecdotes have no controls whatsoever so you absolutely cannot conclude that your herb cured your cold or your vaccination caused your autism because there are too many other variables (note: these specific examples are technically post hoc ergo propter hoc fallacies).  Using carefully controlled studies is the one and only way to test the relationships among variables.

Other posts on statistics:

 

 

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