Introduction

Welcome to Tim Harding’s blog of writings and talks about logic, rationality, philosophy and skepticism. There are also some reblogs of some of Tim’s favourite posts by other writers, plus some of his favourite quotations and videos This blog has a Facebook connection at The Logical Place.

There are over 2,300 posts here about all sorts of topics – please have a good look around before leaving.

If you are looking for an article about Skepticism, Science and Scientism published in The Skeptic magazine titled ”A Step Too Far?’, it is available here.

If you are looking for an article about the Birth of Experimental Science published in The Skeptic magazine titled ‘Out of the Dark’, it is available here.

If you are looking for an article about the Dark Ages published in The Skeptic magazine titled ‘In the Dark’, it is available here.

If you are looking for an article about the Traditional Chinese Medicine vs. Endangered Species published in The Skeptic magazine titled ‘Bad Medicine’, it is available here.

If you are looking for an article about the rejection of expertise published in The Skeptic magazine titled ‘Who needs to Know?’, it is available here.

If you are looking for an article about Charles Darwin published in The Skeptic magazine titled ‘Darwin’s Missing Link“, it is available here.

If you are looking for an article about the Astronomical Renaissance published in The Skeptic magazine titled ‘Rebirth of the Universe‘, it is available here.

If you are looking for an article about DNA and GM foods published in The Skeptic magazine titled ‘The Good Oil‘, it is available here.

If you are looking for an article about animal welfare published in The Skeptic magazine titled ‘Creature Features‘, it is available here.

If you would like to submit a comment about anything written here, please read our comments policy.

Follow me on Academia.edu

Copyright notice: © All rights reserved. Except for personal use or as permitted under the Australian Copyright Act, no part of this website may be reproduced, stored in a retrieval system, communicated or transmitted in any form or by any means without prior written permission (except as an authorised reblog). All inquiries should be made to the copyright owner, Tim Harding at tim.harding@yandoo.com, or as attributed on individual blog posts.

If you find the information on this blog useful, you might like to consider supporting us. Make a Donation Button

3 Comments

Filed under Uncategorized

What is logic?

The word ‘logic‘ is not easy to define, because it has slightly different meanings in various applications ranging from philosophy, to mathematics to computer science. In philosophy, logic’s main concern is with the validity or cogency of arguments. The essential difference between informal logic and formal logic is that informal logic uses natural language, whereas formal logic (also known as symbolic logic) is more complex and uses mathematical symbols to overcome the frequent ambiguity or imprecision of natural language.

So what is an argument? In everyday life, we use the word ‘argument’ to mean a verbal dispute or disagreement (which is actually a clash between two or more arguments put forward by different people). This is not the way this word is usually used in philosophical logic, where arguments are those statements a person makes in the attempt to convince someone of something, or present reasons for accepting a given conclusion. In this sense, an argument consist of statements or propositions, called its premises, from which a conclusion is claimed to follow (in the case of a deductive argument) or be inferred (in the case of an inductive argument). Deductive conclusions usually begin with a word like ‘therefore’, ‘thus’, ‘so’ or ‘it follows that’.

A good argument is one that has two virtues: good form and all true premises. Arguments can be either deductiveinductive  or abductive. A deductive argument with valid form and true premises is said to be sound. An inductive argument based on strong evidence is said to be cogent. The term ‘good argument’ covers all three of these types of arguments.

Deductive arguments

A valid argument is a deductive argument where the conclusion necessarily follows from the premises, because of the logical structure of the argument. That is, if the premises are true, then the conclusion must also be true. Conversely, an invalid argument is one where the conclusion does not logically follow from the premises. However, the validity or invalidity of arguments must be clearly distinguished from the truth or falsity of its premises. It is possible for the conclusion of a valid argument to be true, even though one or more of its premises are false. For example, consider the following argument:

Premise 1: Napoleon was German
Premise 2: All Germans are Europeans
Conclusion: Therefore, Napoleon was European

The conclusion that Napoleon was European is true, even though Premise 1 is false. This argument is valid because of its logical structure, not because its premises and conclusion are all true (which they are not). Even if the premises and conclusion were all true, it wouldn’t necessarily mean that the argument was valid. If an argument has true premises and its form is valid, then its conclusion must be true.

Deductive logic is essentially about consistency. The rules of logic are not arbitrary, like the rules for a game of chess. They exist to avoid internal contradictions within an argument. For example, if we have an argument with the following premises:

Premise 1: Napoleon was either German or French
Premise 2: Napoleon was not German

The conclusion cannot logically be “Therefore, Napoleon was German” because that would directly contradict Premise 2. So the logical conclusion can only be: “Therefore, Napoleon was French”, not because we know that it happens to be true, but because it is the only possible conclusion if both the premises are true. This is admittedly a simple and self-evident example, but similar reasoning applies to more complex arguments where the rules of logic are not so self-evident. In summary, the rules of logic exist because breaking the rules would entail internal contradictions within the argument.

Inductive arguments

An inductive argument is one where the premises seek to supply strong evidence for (not absolute proof of) the truth of the conclusion. While the conclusion of a sound deductive argument is supposed to be certain, the conclusion of a cogent inductive argument is supposed to be probable, based upon the evidence given. An example of an inductive argument is: 

Premise 1: Almost all people are taller than 26 inches
Premise 2: George is a person
Conclusion: Therefore, George is almost certainly taller than 26 inches

Whilst an inductive argument based on strong evidence can be cogent, there is some dispute amongst philosophers as to the reliability of induction as a scientific method. For example, by the problem of induction, no number of confirming observations can verify a universal generalization, such as ‘All swans are white’, yet it is logically possible to falsify it by observing a single black swan.

Abductive arguments

Abduction may be described as an “inference to the best explanation”, and whilst not as reliable as deduction or induction, it can still be a useful form of reasoning. For example, a typical abductive reasoning process used by doctors in diagnosis might be: “this set of symptoms could be caused by illnesses X, Y or Z. If I ask some more questions or conduct some tests I can rule out X and Y, so it must be Z.

Incidentally, the doctor is the one who is doing the abduction here, not the patient. By accepting the doctor’s diagnosis, the patient is using inductive reasoning that the doctor has a sufficiently high probability of being right that it is rational to accept the diagnosis. This is actually an acceptable form of the Argument from Authority (only the deductive form is fallacious).

References:

Hodges, W. (1977) Logic – an introduction to elementary logic (2nd ed. 2001) Penguin, London.
Lemmon, E.J. (1987) Beginning Logic. Hackett Publishing Company, Indianapolis.

If you find the information on this blog useful, you might like to consider supporting us.

Make a Donation Button

18 Comments

Filed under Essays and talks

Reasoning

Rationality may be defined as as the quality of being consistent with or using reason, which is further defined as the mental ability to draw inferences or conclusions from premises (the ‘if – then’ connection). The application of reason is known as reasoning; the main categories of which are deductive and inductive reasoning. A deductive argument with valid form and true premises is said to be sound. An inductive argument based on strong evidence is said to be cogent. It is rational to accept the conclusions of arguments that are sound or cogent, unless and until they are effectively refuted.

A fallacy is an error of reasoning resulting in a misconception or false conclusion. A fallacious argument can be deductively invalid or one that has insufficient inductive strength. A deductively invalid argument is one where the conclusion does not logically follow from the premises. That is , the conclusion can be false even if the premises are true. An example of an inductively invalid argument is a conclusion that smoking does not cause cancer based on the anecdotal evidence of only one healthy smoker.

By accident or design, fallacies may exploit emotional triggers in the listener (e.g. appeal to emotion), or take advantage of social relationships between people (e.g. argument from authority). By definition, a belief arising from a logical fallacy is contrary to reason and is therefore irrational, even though a small number of such beliefs might possibly be true by coincidence.

Leave a comment

Filed under Uncategorized

Tribal truth fallacy

During the 2020 US presidential election campaign, one of the starkest visual differences between Trump and Biden supporters was in the wearing of masks. Most Biden supporters appeared to wear masks, whereas most Trump supporters didn’t.

Even during Trump’s announcement of his nomination of Judge Amy Barrett to the US Supreme Court, very few people in the White House Rose Garden were wearing a mask. Worse still, some of the guests were seen hugging and kissing each other. 

As a result, the White House has become a COVID-19 ‘hotspot’ or super-spreader location, with seven attendees at the Justice Amy Barrett nomination announcement testing positive for coronavirus – even President Trump and the First Lady. More people caught the virus at the White House election ‘celebration night’. 130 Secret Service agents have also tested positive.

It is clear that at least some, if not most, of Trump supporters refuse to wear masks on political grounds. They seem to associate mask wearing with what they perceive to be ‘liberal’ pro-science attitudes. It is also possible that some Biden supporters might wear masks as a form of visual political opposition to the Trump supporters. In either case, this is irrational tribal behaviour. 

A similar phenomenon may be occurring in the climate change debate. Some beliefs against human causes of climate change may be genuinely (but mistakenly) held on the basis of personal interpretation of the evidence. But at least some of the far-right wing opposition is due to a perception of climate science being some sort of left-wing plot against fossil fuel industries.

The far left is not immune from such irrational tribal behaviour either. At least some of the opposition to GMOs and vaccines seems to be based on ideological opposition to large agribusinesses and the pharmaceutical industry, rather than on evidence-based health concerns.

Another example is where some atheists oppose the idea of free will simply because Christians believe in it.  (This is despite the fact that prominent atheists such as Professor Daniel Dennett also believe in free will).

The tribal truth fallacy lies not in the falsity of beliefs about mask wearing, climate change, GMOs, vaccines or free will per se; but in the basis for these beliefs being identification with one’s own tribe or opposition to a rival tribe.

1 Comment

Filed under Logical fallacies

Readers’ wildlife photos

Why Evolution Is True

Send in your photos, but make sure they’re good ones. Thanks!

Today was have some lovely landscapes from reader Bill Zorn. His captions are indented.

Sand Dunes, Colorado:

The Narrows, Utah:

Altamaha River, Georgia:

Monument Valley, Utah:

Jekyll Island, Georgia:

Jekyll Island, Georgia:

Linville Gorge, North Carolina:

Irwin Creek, Colorado:

Reeds, Georgia:

North Fork of the Virgin River, Utah:

These images were made using a Linhof Master Technika 2000 camera, Fuji Velvia film. I sent the film to a lab, and printed these on Ilfochrome. These are scans from the transparencies.

View original post

Leave a comment

Filed under Reblogs

Nobel Prize in Physics goes to three for showing that formation of black holes is predicted by relativity theory

Why Evolution Is True

This morning the Karolinska Institute awarded the 2020 Nobel Prize in Physics to two men and a woman—Roger Penrose, Reinhard Genzel, and Andrea Ghez—for work on black holes.  As the press release notes:

Three Laureates share this year’s Nobel Prize in Physics for their discoveries about one of the most exotic phenomena in the universe, the black hole. Roger Penrose showed that the general theory of relativity leads to the formation of black holes. Reinhard Genzel and Andrea Ghez discovered that an invisible and extremely heavy object governs the orbits of stars at the centre of our galaxy. A supermassive black hole is the only currently known explanation.

Penrose got half the prize, with Genzel and Ghez sharing the other 50%.

My Nobel Prize Contest (see here and here) is already a big flop this year, with nobody guessing even one person from each of…

View original post 85 more words

Leave a comment

Filed under Reblogs

Fall of an antivaxxer

The Doctor Who Fooled the World
Andrew Wakefield’s War on Vaccines
By Brian Deer
Scribe, A$35.00

(An edited version of these book reviews was published in The Skeptic magazine, September 2020, Vol 40 No 3)

The Doctor Who Fooled the World - Brian Deer

This is an important book. Whether it’s really about “the scientific deception of our time”, as the blurb on the back cover describes it, or the result of “the most extensive investigation by a reporter into an aspect of medicine ever undertaken”, as the author describes it, history will decide. But it has to be said that the subject of the book has caused immeasurable damage to the lives of many thousands, and possibly millions, of people.

Therefore we deemed it appropriate to run two reviews – the first by your editor, and the second by noted anti-anti-vaccination campaigner Peter Bowditch, who adds a personal perspective.

Brian Deer is an experienced journalist with a list of exposes of medical fraud and mispractice in his CV. But Wakefield is probably his magnum opus. He has spent 16 years following the saga of Andrew Wakefield, “the doctor without patients”. He wrote a number of revelatory articles on Wakefield’s progress for the Sunday Times, the publication that supported him throughout, both financially and legally, as well as many TV, radio and public appearances, and a significant expose in the British Medical Journal.

Readers of this publication will be aware of Andrew Wakefield’s role in the anti-vaccination movement – the search for autistic kids who could be linked to the measles/MMR vaccine, publication in The Lancet, Wakefield’s promotion of the ‘link’ which lead to a major decline in MMR vaccinations across the world and consequent increases in cases of measles and the damage that has caused.

Deer came to this story during the anti-vaccination campaign, with Wakefield a high-profile figure riding a wave of publicity and, frankly, adulation. From 2002 until the writing of this book in 2019, Deer has followed the vicissitudes of his subject, and played a key role in exposing a range of misconduct and duplicity that eventually led to the longest-ever inquiry by the UK General Medical Council (GMC). In January 2010, the GMC judged Wakefield to be “dishonest”, “unethical” and “callous”, and on 24 May 2010, Wakefield was struck off the UK medical register. Responding to Deer’s findings, The Lancet partially retracted Wakefield’s research in February 2004 and fully retracted it in February 2010 following the GMC findings. In 2011, Deer published his findings in the BMJ with an endorsement by the editors.

Wakefield, of course, became a tragic hero of the anti-vaccination movement, a martyr to the cause, now living in the US and still promoting the supposed autism link, despite the masses of evidence against him and his claims.

Needless to say, Wakefield does not come out of it looking rosy. In fact, Deer portrays Wakefield as an opportunist, a mediocre researcher who used his personal charisma as a tool to promote himself, and who cottoned on to a ‘good thing’ and milked it – and continues to milk it – for everything it’s got.

But Deer makes clear that, despite the book’s title, there were many others contributing to the failed theory and whose involvement made them just as guilty as Wakefield; it is just that Wakefield had the charisma and drive – if not the medical knowledge and skill – to push the case to a broader public. These others include scientific and medical associates, adulatory followers, politicians, parents, learned journals, lawyers (importantly) and, of course, a complicit media. Some of these have paid the price of their association with Wakefield.

But Deer’s coverage of the media is a bit surprising. This reviewer’s background is journalism, and seeing how the media promoted and boosted Wakefield’s scare tactics was always disappointing, to say the least! There would not have been a vaccine scare without some media putting the case in hyperbolic terms. Their role in the growth of the anti-vaccination movement is considerable and intrinsic to spreading misinformation and paranoia.

Therefore, it is interesting that Deer doesn’t spend more time on the media’s involvement. Certainly, he makes reference to a number of specific and highly partisan journalists, such as Lorraine Fraser of the Mail on Sunday, Jeremy Paxman and Susan Watts on the BBC’s Newsnight, and Matt Lauer of the NBC’s Today program, but his coverage of the media is as much about their attacks on him as it is their support for Wakefield, and in some cases, once Deer’s work had been publicised, trying to gazump him with a scoop.

With that in mind, Deer’s book covers a lot of his investigation in addition to what he is investigating. This adds an element akin to a detective thriller, which takes the book along at a very readable pace. Overall, media coverage notwithstanding, this is an excellent book. The depth and detail are spot on, from well-explained scientific and medical protocols and procedures to well-told human interest elements (the parents’ responses ranging from suspect support to desperate self-blame).

It is highly likely that this is the definitive version of the Wakefield and Co saga. Now for one on the evolution of the anti-vaccination movement to complement it.

PS: The story is bookended by a couple of Australians. In the beginning was John Walker-Smith, a gastroenterologist who was instrumental in testing the children who would eventually become the basis of The Lancet paper (and who only just missed out on suffering the same professional fate as Wakefield), and at the tail end we have Elle ‘the Body’ Macpherson, super-model and consort of the superstar Wakefield. One wonders whether everyone got their just desserts.

– Reviewed by Tim Mendham

In 1996 I was commissioned to write a book about the Internet. It was to explain to people who didn’t know anything about it or the technology behind it or what it could be good and bad for. There was some hysteria about the possibility of a flood of pornography filling our lounge rooms so I actually had to research porn (it was boring!) to answer the inevitable questions in interviews. I also looked for other forms of bad information because it was obvious even then that there would be dubious information coming down the tubes. One of the bad things I found was a group of websites spreading fear about vaccinations. I commented at the time that none of the pornography I was forced to watch was as offensive as some of these sites.

In 1999 I started paying more attention to the anti-vaccination sites and it wasn’t long before I was sneeringly told that a paper by a Dr Andrew Wakefield had been published in The Lancet (the world’s second-most prestigious and influential medical journal) which proved that the MMR vaccine caused autism. As I had experience of people citing unlikely research results in the hope that nobody would check, I read the paper for myself (I had access to the medical library at Westmead Hospital) and it proved no such thing – it only suggested there might be a link. There were several red flags on the paper, one of which was that the editors of The Lancet felt the need to include an editorial statement implying the clichés “further research is needed” and “the science is not settled”.

The biggest red flag for me came from something I had been taught about research methodology at university – the sample of subjects looked too good to be true. It seemed highly unlikely that the parents of the children had independently and randomly sought out a doctor (who didn’t see patients!) at a small and relatively unknown London hospital. I mentioned my concerns in a conference presentation in 2001. The most charitable view was that there had been some cherry-picking going on mixed with some confirmation bias. The peer review process can’t always detect outright fraud, so this was a case of “the benefit of the doubt”.

But fraud it certainly was.

Journalist Brian Deer had been investigating suspicious matters around the pharmaceutical industry for some years, and in 2003 he was approached by an editor at the Sunday Times and asked to apply his investigative skills to the Wakefield story, which by then had started to have a serious effect on vaccination levels and public health. There was enough information and doubt from within the medical profession itself to suggest that the public didn’t know all the things it should have known, and it wasn’t long before the facts started coming out – that Wakefield was paid a large amount of money to find what he wanted to find, that he had applied for a patent on a measles vaccine that would have made him very wealthy if it replaced the current vaccine, that the subjects of the study had not been randomly chosen but had been supplied by a lawyer, Roger Barr, who intended taking legal action against vaccine manufacturers, that Barr had used a loophole in the regulations to stripmine the Legal Aid system for tens of millions of pounds (shared with Wakefield), that Wakefield and Barr both had close associations with prominent anti-vaccination campaigners, that the laboratory doing the tests for measles DNA had less credibility than a school science project … the list went on.

In 2010 Wakefield’s registration as a medical practitioner was cancelled and The Lancet retracted the 1998 paper. It took too many years, but we thought that at last it might all be over. We were wrong.

Brian Deer (described as “a lying dog of a journalist” by a leading anti-vaccination campaigner) has now written a complete history of the Wakefield saga. The book goes back some years before the notorious 1998 paper to reveal the involvement of lawyer Barr and vociferous anti-vaccination organisations, through the almost unbelievable litany of lies, corruption and fraud that surrounded Wakefield and the coordinated attempts to use his fraudulent “research” to damage the public’s perception of the safety and efficacy of vaccines, to his elevation to hero status in the anti-vaccination movement and his current incarnation as the director and producer of anti-vaccination films liked the execrable “Vaxxed: From Cover-Up to Catastrophe”. (The use of the word “cover-up” in the title caused irony meters across the world to shatter, given the way that Wakefield et al had covered up his deceit. Also, when the film first came out in 2016 someone commented about the propaganda: “Leni Riefenstahl would have baulked at making something this dishonest”.)

I could summarise the book into something like those old Reader’s Digest condensed novels, but I wouldn’t know what to leave out and this review would be about 300 pages long. The book is an essential read for anyone who has followed Wakefield over the years (and even I, who have followed him very closely, found many new things to wonder and grimace at). It is essential reading for anyone who thinks that scientific and medical research can’t be corrupted by greed and self-interest or to support an agenda. And it is essential reading for anyone who thinks for a nanosecond that the anti-vaccination movement is based on any philosophy that includes honesty, ethics or morality. Strangely, the book also reinforces the claim by anti-vaccinators that all medical research is corrupt and driven by money, although they will make an exception in this case.

You need this book. Buy it! Highly recommended.

– Reviewed by Peter Bowditch

1 Comment

Filed under Book reviews

The Alchemist by Ben Jonson (1610)

Books & Boots

The Alchemist is a plague play. Not only was it written in 1610, when the London theatres were closed (yet again) for (yet another) outbreak of plague, but the plot itself derives from that fact. The master of the house, Lovewit, has (like everyone else who can afford it) fled London and is waiting at his country seat for the plague to abate (his retreat appears to be in Kent; he is said to be waiting in his ‘hop-yards’). In the meantime his housekeeper, Face, has invited a conman, Subtle, and a prostitute, Doll Common, to come and stay in the house in a kind of joint criminal enterprise, persuading a series of gullible victims that Subtle is a renowned alchemist who will supply each of them the Philosopher’s Stone and make their dreams come true… for a price.

The cast

The three crooks

SUBTLE, the alchemist
FACE, the Housekeeper

View original post 6,688 more words

Leave a comment

Filed under Reblogs

Lowitja, the authorised biography of Lowitja O’Donoghue, by Stuart Rintoul

ANZ LitLovers LitBlog

It was Bismarck who said that ‘politics is the art of the possible, the attainable — the art of the next best’.  Well, the two women I most admire in Australian politics are exponents of that art: Penny Wong, who, as I read in Margaret Simons’ recent biography Penny Wong, Passion and Principlesays that you can’t achieve change unless you’re ‘in the room’, even if that means that sometimes you have to settle for less;  and Lowitja O’Donoghue, whose steely determination to represent Indigenous people changed Australia for the better, even though there is still much more to be done.

Stuart Rintoul’s biography traces the story of this remarkable woman’s life, tracked alongside significant events in Australia’s Black History, rendering the biography also a refresher course for those who lived through these events and an education for younger readers who did not.  The book begins in 1979 with…

View original post 1,170 more words

Leave a comment

Filed under Reblogs

A Monarchy Transformed: Britain 1603-1714 by Mark Kishlansky (1996) 7 – the reign of James II

Books & Boots

Because King Charles II died in February 1685 without a son and heir – without, in fact, any legitimate children from his marriage to Catherine of Braganza – the throne passed automatically to his brother, James Duke of York, who ascended the throne as King James II.

James was a professed Roman Catholic and a zealous reformer. He wished to lift the multiple legal restrictions which had been placed on his fellow Catholics and, as a balancing gesture, to lift legal constraints on the Puritans and non-conforming Protestant sects. However, within three short years he managed to alienate almost every party and profession in the country, and especially the powerful Whig politicians.

The crisis came to a head over two big issues. First James made the error of trying seven Anglican bishops for seditious libel. To be precise, in April 1688, encouraged by the Quaker leader William Penn with whom…

View original post 4,010 more words

Leave a comment

Filed under Reblogs

Favorite words

Why Evolution Is True

Many times I’ve posted about “words and phrases I hate,” but now let’s walk on the sunnier side and list the words we like (phrases would be too onerous). I was inspired by the tweet below that Matthew sent me from Jonathan Eisen, evolutionary biologist and brother of Wormageddon instigator Michael Eisen:

These words seem to be chosen because of their sounds, which, I suppose, is the best criterion for having a favorite word. Mine, however are a mixture of sound and meaning. And I don’t have a list, so I’ll just put a few down off the top of my head:

ratiocination (learned from Hitchens)
rodomontade

View original post 37 more words

Leave a comment

Filed under Reblogs

Notes on Nationalism, by George Orwell

ANZ LitLovers LitBlog

My next (NF) book was going to be Michael Ackland’s Henry Handel Richardson, A Life, (because I am still peeved by Brenda Niall’s representation of HHR in Friends and Rivals, Four Great Australian Writers, see why, here) but Orwell’s essay in the Penguin Moderns series was on top of the NF pile… I was sure that his thoughts about nationalism were bound to be pertinent for our age… so HHR will have to wait. (But not for long because these mini-books can be read in a day.)

Orwell writes in his usual acerbic way, starting with his definition of nationalism: the habit of assuming that human beings can be classified like insects and that whole blocks of millions or tens of millions of people can be confidently labelled ‘good’ or ‘bad’.  But more importantly, he says, he means the habit of identifying oneself with a single nation or…

View original post 1,342 more words

Leave a comment

Filed under Reblogs

Increased testing does not explain the increase in US COVID cases

The Logic of Science

The US is experiencing another sharp increase in COVID19 cases. This is a simple fact, but as always seems to be the case in today’s world, this fact is being treated as an opinion. Countless people (including prominent politicians and even the president) are claiming that cases are not actually increasing, and the apparent increase is simply the result of increased testing. This claim is dangerous and untrue, but it also offers a good opportunity to teach some lessons in data analysis. Obviously, an increase in testing will result in an increase in the number of cases that are documented, that much is true, but that doesn’t necessarily mean that the entirety of the increase is from increased testing. So how can we tell whether the true number of cases is increasing? There are multiple ways to examine this, and I’m going to walk through several of them and try…

View original post 2,526 more words

Leave a comment

Filed under Reblogs