Tuesday, December 29, 2015

Camel's in the Desert

Taken by photographer George Steinmetz and featured in a 2005 Turkish edition of National Geographic. At first glance it appears to be an image of dark colored camels crossing the desert but a closer look reveals that these are actually their shadows.

Monday, December 28, 2015

Thursday, December 3, 2015

Wednesday, November 4, 2015

Imaginary Bow and Arrow Leads to Three Day Suspension

Earlier this week a number of local news outlets reported that a first grader at a local Catholic school was suspended for three days for pretending to shoot one or more other students with an imaginary bow and arrow.

According to the boy's parents, Matthew and Martha Miele, the overtly hostile threat of extreme violence happened while their son was playing a game of Power Rangers during recess. A level headed teacher witnessed the brutish intimidation and responsibly brought it to the attention of Principal Joe Crachiolo.  

That afternoon, Martha Miele said she was contacted by Principal Crachiolo regarding the matter.

"I didn't really understand. I had him on the phone for a good amount of time so he could really explain to me what he was trying to tell me. My question to him was 'Is this really necessary? Does this really need to be a three-day suspension under the circumstances that he was playing and he's 6 years old?"

It is more than apparent that the Miele's do not apprehend the seriousness of the situation. Of course all right thinking people understand that imaginary weapons can be just as dangerous as real ones and that childish horseplay is equivalent to threats of violence. Principal Crachiolo appears to understand this, based on a letter he sent to the Miele's which in part stated "I have no tolerance for any real, pretend, or imitated violence. The punishment is an out of school suspension."

So here's to you Principal Crachiolo. Kudos to your tenacity at maintaining your belief that zero tolerance is the best policy even against the flood of evidence that it doesn't make schools safer. Good for you for ignoring the evidence that suspending kids for seemingly frivolous things increases the likelihood that they would have to repeat a grade, which in turn increases the likelihood they will drop out. It's obvious you possess superior judgement skills and the school is fortunate to have you there to keep everyone safe.

WLWT: Child pretends to shoot student with imaginary bow, suspended for 3 days
WCPO: Catholic school suspends 6-year-old for pretending to shoot imaginary bow and arrow at recess

A Generation Later: What We’ve Learned about Zero Tolerance in Schools

Friday, October 9, 2015

Awaji Yumebutai

Awaji Yumebutai 100 step garden

Awaji Yumebutai is an area on the island of Awaji in the Hyōgo Prefecture of Japan which consists of a complex of buildings designed by architect Tadao Ando. The project was built on the remains of a hillside from which the soil had been removed and used for various projects in Osaka. In 1995, during the planning stages to restore the scarred land by turning it into a park, Awaji island was hit with a massive earthquake which claimed the lives of over 6,000 people. The devastation compelled the architect to revise his plans by turning a portion of the area into a memorial which includes the one hundred step garden (Hyakudan-en). In the words of the architect, it is "a symbol to calm the souls of those who lost their lives in the disaster."

Photo by Ken Conley

Photo by Jeffrey Friedl

Photo by Scott Hsu

Shell Garden

Shell Garden. Photo by Jack Chen via Ursula Zitting Pinterist

Shell Garden. Photo from 663highland

Photo from 663highland

Kiseki No Hoshi Greenhouse. Photo by Brodie Karel
Amphitheatre. Photo by Ken Conley

The 100 step garden at night. Photo by wata_masa

The 100 step garden
Wikipedia: Awaji Yumebutai

Wednesday, September 23, 2015

The Stroop Effect

A phenomenon in which individuals take longer to name the color of words printed in a non-matching color, such as the word blue printed in red ink, than when the words are printed in the same color as the word designates, such as the word blue printed in blue ink.

With the following video, try saying the color of the word, not the word that's spelled.

If the printed word and color matched, you would likely be able to say the color much faster. This is because reading words is more automatic than naming colors.

More on the Stroop effect here

Thursday, September 17, 2015

Forest Xylophone

Back in 2011 the Japanese telecom company Docomo created a large forest xylophone to promote the then new Touch Wood SH-08C cell phone. A wooden ball rolled down the steps of the instrument playing Bach's Cantata 147 as it progressed to the bottom.

The xylophone was reintroduced at the 2015 Hokkaido Garden Show where visitors could purchase a ball from a vending machine to play the instruments tune.

Wednesday, September 16, 2015

Denying the Antecedent

Denying the Antecedent is a formal logical fallacy which consists of a conditional premise, a second premise that denies the antecedent of the conditional and a conclusion which denies the consequent of the conditional. The general form of the argument is:

P1. If P, then Q
P2. Not P
C. Therefore, not Q

Since P was never asserted as the only sufficient condition for Q, other factors could account for Q. Therefore, the argument is deductively invalid.

For example:

P1. If Queen Elizabeth is an American citizen, then she is a human being
P2. Queen Elizabeth is not an American citizen
C. Therefore, Queen Elizabeth is not a human being

With this example, both premises are true statements yet the conclusion is false. This of course is due to the fact that being an American citizen is not the only sufficient condition for being a human being. 

Monday, September 14, 2015

Affirming the Consequent

Affirming the consequent is a formal logical fallacy which consists of a conditional premise, a second premise that asserts the consequent of the first conditional premise, and a conclusion which asserts therefore the antecedent of the conditional is true. The general form of the argument is:

P1. If P then Q.
P2. Q
C. Therefore P.

Since P was never asserted as the only sufficient condition for Q, other factors could account for Q. Therefore, in terms of deductive logic, the argument form is invalid.

For example:

P1. If Bill Gates owns Fort Knox, then he is rich.
P2. Bill Gates is rich.
C. Therefore, he owns Fort Knox.

Obviously the consequent that Gates is rich is the result of factors other than owning Fort Knox.

It is important to understand that though affirming the consequent is a fallacy in terms of deductive reasoning, it can be used as a perfectly acceptable form of inference when used inductively or abductively. This of course is due to deductive reasoning's requirement that with a valid argument, if the premises are true then the conclusion must be true. On the other hand, induction and abduction do not have this certainty requirement and instead make inferences based on probability and plausibility.

For example:
P1. If the baby is hungry she will cry
P2. The baby is crying
C. The baby is hungry

This is not deductively valid since there are other reasons the baby may be crying. Perhaps she needs a diaper change or maybe she hit her head. On the other hand, depending on the circumstances, this could be considered a strong abductive argument. If we add more information through additional premises the strength of the argument becomes less ambiguous.

P1. If the baby is hungry she will cry
P2. The baby is crying
P3. The baby eats about every three hours
P4. The baby last ate about three hours ago
P5. The baby does not have a dirty diaper
C. The baby is hungry

Wikipedia: Affirming the Consequent 09/10/15

Reasoning: K. P. Mohanan and Tara Mohanan

Tuesday, September 8, 2015

Ted Cruz's statement in support of Kim Davis

In regards to the arrest of Kentucky Clerk Kim Davis, republican presidential candidate Ted Cruz released an open letter declaring his support for Davis. You can read the full letter here.

Part of the letter included the following:

“For every politician — Democrat and Republican — who is tut-tutting that Davis must resign, they are defending a hypocritical standard. Where is the call for the mayor of San Francisco to resign for creating a sanctuary city — resulting in the murder of American citizens by criminal illegal aliens welcomed by his lawlessness?

"Where is the call for President Obama to resign for ignoring and defying our immigration laws, our welfare reform laws, and even his own Obamacare?

“When the mayor of San Francisco and President Obama resign, then we can talk about Kim Davis.

I believe this could be considered a version of the Tu Quoque fallacy, which essentially is an attempt to dismiss an opponent's argument by pointing out that the person or group making the argument has said or acted in a manner which is inconsistent with it. Generally, the problem with this sort of attack is that just because someone has acted hypocritically doesn't necessarily mean their argument is wrong. 

A classic example of the Tu Quoque is:

Dad: "John, you shouldn't smoke. It is very bad for your health!"
John: "I don't see how you can tell me not to smoke, when you smoke yourself."

Clearly smoking is bad for you and just because the father is being a hypocrite by smoking himself doesn't effect the truthfulness of his argument. Actually, in this particular example, the hypocrisy could serve to strengthen the fathers argument since he has personal experience dealing with addiction and the damaging effects of smoking.

Going back to the Cruz quote, he seems to be implying that if a politician didn't call for the mayor of San Francisco or President Obama to resign, then their inconsistent call for the enforcement of the rule of law should somehow invalidate their argument here. Just as smoking is bad for John's health regardless of whether his father smokes, the law should govern the nation as opposed to the arbitrary decisions of individual government officials. Whether or not a politician has consistently affirmed this principle doesn't change its rightfulness.

Friday, September 4, 2015

Best laid plans of mice and men

The phrase "the best laid plans of mice and men often go awry" or simply "best laid plans" is a simple proverb meaning that even carefully planned projects may sometimes still go wrong.

The saying is an adaptation from a 1785 poem by Robert Burns titled "To a Mouse, on Turning Her Up in Her Nest with the Plough." The line as originally written goes "The best-laid schemes o' mice an' men Gang aft agley." It was also used by John Steinbeck for the title of his 1937 novel "Of Mice and Men."

Friday, August 14, 2015

'Where your tax dollars go', word image critique

The above image has been floating around Facebook, often coming from Bernie Sanders supporters, as an attempt to persuade people that very little of their tax dollars go to welfare and the vast majority goes to corporate subsidies. Before I get too into my back of the napkin level analysis, I want to state that I am critical of corporate subsidies and feel they are a real and growing problem. Having said that I think it's also important to point out what looks to be deceitful propaganda and this word image certainly appears to be just that.

The amount "you pay" in taxes making $50,000 comes to $4597.98 in the above breakdown. Of course what you actually pay is not just based on how much you make but on marital status, number of dependents and a slew of deductions, but regardless, the number is close to what you'd expect your federal withholding would be if you made $50,000 a year. This does not include the amount you'd pay in social security withholding and medicare withholding which will be relevant in the calculation below.

The word image implies that the breakdown it provides is your portion of federal expenditures. If true, it would mean that 87% of all federal spending went to corporations in the form of subsidies. This of course is absolutely absurd.

A quick google search took me to the Center on Budget and Policy Priorities where I found information on federal spending. For 2014, the federal government spent $3.5 trillion dollars. The breakdown of those dollars is $840 billion (24%) social security, $840 billion (24%) medicare/medicaid/CHIP/market subsidies, $630 billion (18%) defense/international security, $385 billion (11%) safety net programs, $245 billion (7%) interest on debt, $560 billion (16%) on everything else including benefits for federal workers, transportation infrastructure, education, science and medical research, etc.

I couldn't find info on how much was actually spent in corporate subsidies in 2014 but did find a 2012 CATO report that estimated it at about $100 billion a year.

So, just for the fun of it, lets see if we can do a rework of the word image using the information above. We are trying to just look at how your federal withholding is allocated which makes this a bit difficult. As social security is mostly paid from social security tax, we will eliminate that from the federal spending breakdown. The medicare/medicaid/CHIP/market subsides are harder to figure as part of it is paid by your medicare withholding and part from federal withholding. It looks like about $200 billion in federal receipts came from medicare withholding in 2014 so well eliminate that amount from the medicare/medicaid/CHIP/market subsidies category in the federal spending breakdown.

After making these adjustments, here are the final results:

If you make $50,000 per year, you pay:

$1177.08 a year for defense
$1195.47 a year for medicare/medicaid/CHIP/market subsidies
$721.88 a year for safety net programs
$459.80 a year for interest on federal debt
$859.82 a year for infrastructure, education, research, etc.
183.68 a year in corporate subsidies

Wednesday, August 12, 2015

'All is Vanity' by Charles Allen Gilbert

Charles Allan Gilbert (September 3, 1873 - April 20, 1929) was an American artist and illustrator. He is most remembered for the widely published illustration above titled 'All is Vanity' (1892). The illustration employs a double image where a woman sitting at a vanity admiring herself in a mirror also appears as a human skull when viewed from a distance or when blurring the eyes by squinting.

Interestingly, Gilbert created the image in 1892 when he was 18 years old but it did not receive recognition until 1902 when he sold the original to LIFE publishing. The image became wildly popular and is the most reproduced optical illusion in history.

Sandlotscience.com: All is Vanity
Wikipedia: Charles Allan Gilbert

Tuesday, August 11, 2015

Poisoning the Well

Poisoning the well is the use of a preemptive abusive or circumstantial ad hominem attack against an opponent with the purpose of discrediting or ridiculing everything they are about to say. It generally has the following form:

1. Unfavorable information (true or false) about person A is presented.
2. Therefore, (explicitly or implicitly) any claims about to be made by person A should be dismissed. 


"Don't listen to anything Steve may tell you, he's a socialist."


"Before you listen to my opponent, may I remind you that he has been to prison."

Practically speaking, poisoning the well is a form of ad hominem, and as such, one should follow the guidelines of analyzing an ad hominem to determine if it is being used in a fallacious manner. This essentially means questioning the relevancy of the attack on the claims presented by the person for whom the attack was directed against. 

Thursday, July 23, 2015

Wednesday, July 8, 2015

Abductive Arguments (Inference to the Best Explanation)

An abductive argument (also known as an inference to the best explanation) is an argument in which a hypothesis is inferred from some data on the grounds that it offers the best available explanation of that data.1 Though it may appear as a special type of induction, many philosophers view it as a separate type of inference.

The following example is useful in drawing the distinction between deduction, induction and abduction:

Deductive Reasoning: Suppose a bag contains only red marbles, and you take one out. You may infer by deductive reasoning that the marble is red.

Inductive Reasoning: Suppose you do not know the color of the marbles in the bag, and you take out a handful and they are all red. You may infer by inductive reasoning that all the marbles in the bag are red.

Abductive Reasoning: Suppose you find a red marble in the vicinity of a bag of red marbles. You may infer by abductive reasoning that the marble is from the bag.

Hence we can say that with a deductively valid inference, it is impossible for the premises to be true and the conclusion false. With an inductively strong inference, it is improbable for the premises to be true and the conclusion false. In an abductively weighty inference, it is implausible for the premises to be true and the conclusion false.

Abduction is essentially a kind of guessing by forming the most plausible explanation for a given set of facts or data. It's inference comprises of three steps. First, it begins with the observation of the data, evidence, facts, etc. Second, it forms various explanations that can be given to explain the observations in the first step. Third, it selects the best explanation and draws the conclusion that the selected explanation is acceptable as a hypothesis. Here is the process in standard form:

P1. D exists.
P2. H1 would explain D. 
P3. H1 would offer the best (available) explanation of D. 
C. Therefore, probably, 4. H1

Abductive arguments are commonly used in many areas including law, archaeology, history, science and medical diagnosis. A medical example would include when a doctor examines a patient with certain symptoms and tries to reason from those symptoms to a disease or condition that would explain them. A legal example would be when a police detective gathers evidence then forms a hypothesis as to who committed a crime.

Evaluating Abductive Arguments
The strength of an abductive argument depends of several factors.
1. how decisively H surpasses the alternatives.
2. how good H is by itself, independently of considering the alternatives (we should be cautious about accepting a hypothesis, even if it is clearly the best one we have, if it is not sufficiently plausible in itself)
3. judgments of the reliability of the data
4. how much confidence there is that all plausible explanations have been considered (how thorough was the search for alternative explanations)

Additional factors to consider are:
1. pragmatic considerations, including the costs of being wrong, and the benefits of being right 
2. how strong the need is to come to a conclusion at all, especially considering the possibility of seeking further evidence before deciding.

1. A Practical Study of Argument

2. Abductive, presumptive and plausible arguments

Tuesday, July 7, 2015

Bradford Hill Criteria for Causation (epidemiology)

The Bradford Hill criteria for causation are a group of criteria or guidelines used to help determine if an observed association is potentially causal. They were established in 1965 by the English epidemiologist Sir Austin Bradford Hill.

Research to determine the cause of disease is a principal aim of epidemiology. As most epidemiological studies are observational rather than experimental, a number of possible explanations for an observed association must be considered before a cause-effect relationship can be inferred. In his 1965 paper The environment and disease: association or causation, Hill proposed the following nine guidelines to help assess if a causal relationship exists:

1. Strength: (effect size): A small association does not mean that there is not a causal effect, though the larger the association, the more likely that it is causal.

2. Consistency: (reproducibility): Consistent findings observed by different persons in different places with different samples strengthens the likelihood of an effect.
3. Specificity: Causation is likely if a very specific population at a specific site and disease with no other likely explanation. The more specific an association between a factor and an effect is, the bigger the probability of a causal relationship.

4. Temporality: The effect has to occur after the cause (and if there is an expected delay between the cause and expected effect, then the effect must occur after that delay).

5. Biological gradient: Greater exposure should generally lead to greater incidence of the effect. However, in some cases, the mere presence of the factor can trigger the effect. In other cases, an inverse proportion is observed: greater exposure leads to lower incidence.

6. Plausibility: A plausible mechanism between cause and effect is helpful (but Hill noted that knowledge of the mechanism is limited by current knowledge).

7. Coherence: Coherence between epidemiological and laboratory findings increases the likelihood of an effect. However, Hill noted that "... lack of such [laboratory] evidence cannot nullify the epidemiological effect on associations".

8. Experiment: "Occasionally it is possible to appeal to experimental evidence".

9. Analogy: The effect of similar factors may be considered.

Friday, June 19, 2015

Mill's Methods

Mill's Methods
The nineteenth century philosopher John Stuart Mill devised five methods for reasoning about cause and effect. Though they have serious limitations, they are still useful and widely taught today.

1. The Method of Agreement - Mill wrote "If two or more instances of the phenomenon under investigation have only one circumstance in common, the circumstance in which alone all the instances agree, is the cause (or the effect) of the given phenomenon." In other words, if there is a single circumstance that is present in all positive instances, then we can conclude that this circumstance was the cause of the phenomenon. Note that in textbooks this is often referred to as the direct the method of agreement and only looks at positive instances of the effect in question.

For example, lets say four students dined together at the cafeteria and two of them became ill with food poisoning. The students were questioned about what they ate which resulted in the following list:

Carla            No             Yes          Yes           Yes            Yes
John             Yes            No           No            Yes             Yes
Tom             Yes            Yes          No            No              No
Mary            No             Yes          Yes           No              No

Based on the above information, we can conclude that it was the beans that gave Carla and John food poisoning as this was the only potential cause that was present in both instances.

Though not listed by Mill, some textbooks also refer to what is called the Inverse Method of Agreement (or Negative Method of Agreement). The Inverse Method of Agreement allows one to conclude that a certain circumstance is the cause of the phenomenon under investigation if this circumstance is the only circumstance (of those considered) that is absent in all negative instances.

Using the above example, the inverse method of agreement would lead us to look at the negative instances of Tom and Mary not getting food poisoning. Here we find the beans to be only potential cause which were absent in both cases and can thus conclude them to be the cause.

2. The Method of Difference - "If an instance in which the phenomenon under investigation occurs and an instance in which it does not occur, have every circumstance in common save one, that one occurring only in the former, the circumstance in which alone the two instances differ, is the effect, or the cause, or an indispensable part of the cause, of the phenomenon." In other words, if there is a positive and a negative instance where the presence or absence of all possible causes are the same except one cause which is present in the positive instance and absent in the negative instance, then it can be concluded to be the cause of the phenomenon. Note that the method of difference looks at both  positive and negative instances of the effect in question.

Using the food poisoning example above there are two relevant instances where the method of difference can be applied:

Carla            No             Yes          Yes           Yes            Yes
Mary            No             Yes          Yes           No              No

Since the only potential cause in which they differ is present in the positive instance and absent in the negative instance, we can conclude it was the beans that caused the food poisoning.

3. The Joint Method of Agreement & Difference - "if two or more instances in which the phenomenon occurs have only one circumstance in common, while two or more instances in which it does not occur have nothing in common save the absence of that circumstance; the circumstance in which alone the two sets of instances differ, is the effect, or cause, or a necessary part of the cause, of the phenomenon."  There seems to be a fair amount of controversy over this method among those scholars that examine such things. The biggest criticisms seem to be that The joint method/indirect method is not really a combination of the method of agreement and method of difference. Also, the definition above as provided by Mill is restrictive in that it does not allow full achievement of the intended purpose of the joint method. A more usable amended joint method of agreement & difference is provided by Skorupski:

"If two or more instances in which the phenomenon occurs have a circumstance in common, while in two or more instances in which the phenomenon does not occur that circumstance is absent, and if there is no other circumstance or combination of circumstances which is present in all the instances in which the phenomenon occurs, and absent in all the instances in which it does not occur, then the given circumstance is the effect, or the cause, or an indispensable part of the cause, of the phenomenon."

This can be summarized as the circumstance which alone is present in all the positive instances and absent in all the negative instances.

Here is a modified version of the food poisoning example which demonstrates the amended joint method:

Carla            No             Yes          Yes           Yes            Yes
Ann              Yes            Yes          No            Yes            Yes
Doug            Yes            No           No            No              No
Byron           No             Yes          No            No              No

With this example, the method of agreement does not give a unique answer since there are two positive circumstances (fries and beans) present in both positive instances. The method of difference also does not provide an answer since there is not a positive and negative instance where all causes are the same except a single cause which is positive in one instance and negative in the other. However, using the amended joint method we find that the beans are the cause as they are the only circumstance which is present in all positive instances and absent in all negative instances.

4. The Method of Residue - "Subduct from any phenomenon such part as is known by previous inductions to be the effect of certain antecedents, and the residue of the phenomenon is the effect of the remaining antecedents."

5. The Method of Concomitant Variation - "Whatever phenomenon varies in any manner whenever another phenomenon varies in some particular manner, is either a cause or an effect of that phenomenon, or is connected with it through some fact of causation." 

Friday, June 12, 2015

Causal Inductive Arguments

A causal inductive argument is an inductive argument in which the conclusion claims that one event(s) causes another.

Causality is the relationship between an event (cause) and a second event (effect), where the second event is understood to be the consequence of the first. Intuitively,

Types of Causes
The term "cause" can be used in several different ways:

1. Necessary Cause - A necessary cause (or condition) is one that is required to be present for the effect to occur. This relationship can be written as, C is the cause of E in the sense that C is a necessary condition of E. That is to say, without C, E will not occur. This relationship implies that the presence of E necessarily implies the presence of C. The presence C, however does not imply that E will occur.

For example, if a professor says that one can pass his class only by completing all the assignments, then completing the assignments is a necessary cause of the effect of passing the class. It should be noted that completing the assignments won't guarantee passing as there are other things (causes) that must happen such as having scores that average out to a passing grade.

2. Sufficient Cause - A sufficient cause is one that by itself is enough for the effect to occur. This relationship can be written as, C is the cause of E in the sense that C is a sufficient condition of E. That is to say, given C, E will occur. However, another cause may alternatively cause E. Thus the presence of E does not imply the presence of C.

For example, boiling a potato is a sufficient condition for cooking a potato, but it is not a necessary condition since there are many ways of cooking potatoes, such as baking or frying them.

3. Necessary & Sufficient Cause - A necessary and sufficient cause leads to an effect that always occurs when the condition is met and never occurs unless the condition is met. This relationship can be written as, C is the cause of E in the sense that C is a necessary and sufficient condition of E. That is to say, without C, E will not occur, and with C, E will occur.

For example, being a male sibling is necessary and sufficient for being a brother.

4. Contributing Cause - Commonly, when we speak of one event causing another we are referring to it being a contributing cause. This relationship can be written as, C is causally relevant to E. It is a condition that makes E more likely to occur than it would be were C not there.

A Contributing cause is neither necessary nor sufficient in and of itself to bring about a certain effect.

For example, being physically inactive is a general contributing causal factor to being overweight. It is not a necessary condition as some overweight people are physically active. Nor is it a sufficient condition as some physically inactive people are not overweight. Nevertheless, it is causally relevant being one of a number of contributing factors.

Distinguishing Between Correlation and Causation
A correlation is an association of two variables. When judging an association between variable, three possibilities exist:

1) Positive correlation - if a higher proportion of Qs than non-Qs are H, then there is a positive correlation between being Q and being H. In other words, Q and H increase and decrease in synchrony (parallel).
2) Negative correlation - if a smaller proportion of Qs than non-Qs are H, then there is a negative correlation between being Q and being H. In other words, Q tends to increase when H decreases and vice versa.
3) No correlation - if about the same proportion of Qs as non-Qs are H, then there is no correlation between being Q and being H.

The phrase 'correlation does not imply causation' is commonly used in science and statistics to emphasize that a correlation does not necessarily imply that one event causes the other. The reason for this is that a positive correlation generally allows for the existence of four possibilities:

1. Q is a cause of H. 
2. H is a cause of Q. 
3. The positive correlation of Q and H is a coincidence. 
4. Some other factor, X, is a cause of both Q and H

To automatically infer that a positive correlation between Q and H means that Q causes H is to disregard the other three possibilities. This is why correlation alone is generally thought to be insufficient grounds to establish cause.

Though a correlation alone is not enough evidence to establish causation, the absence of a correlation does establish the absence of a causal relationship.  This is true since correlation is a necessary aspect of causation even though it is not sufficient for it. The general form of this argument is:

P1. If Q is a cause of H, Q must be positively correlated with H. 
P2. It is not the case that Q is positively correlated with H. 
C. Therefore, It is not the case that Q is a cause of H.

Cogent Causal Arguments
We've established that correlation is a necessary condition of arguing for causality yet alone is not sufficient evidence. To establish a cogent causal argument, premised on a positive correlation, it is necessary to provide evidence which seeks to exclude the other possibilities which correlation allows for. There are various methods from diverse fields of science and philosophy available to help investigate causal claims, some of which are listed below.

1. Mill's Methods
2. Bradford Hill criteria for causation

A Practical Study of Argument

Logical Reasoning

Khan Academy: Fundamentals: Necessary and Sufficient Conditions

A Preferred Treatment of Mill's Methods

Mill (Arguments of the Philosophers)

The Logic of Causal Conclusions: How we know that fire burns, fertilizer helps plants grow, and vaccines prevent disease

Causality and Causation: The Inadequacy of the Received View
A Short History of ‘Causation’










Thursday, June 4, 2015

Appeal to Authority

An appeal to authority is an argument that something is true because someone of authority says it is true. The basic form of the argument is:

P1. Person X has asserted claim P
P2. Person X is an authority on subject K
C. Therefore P is acceptable

In practice, there are many instances where it is reasonable to accept inductive arguments where an authority is used to to support a claim. This is something I believe most would find intuitively true given that we rely on the advice and counsel of experts all the time.

The difference between a legitimate appeal to authority versus one which is fallacious is generally dependent on whether the authority being cited is an expert on the matter under consideration, whether there is general agreement among experts in the area of knowledge under consideration and whether the area of knowledge under consideration is credible.

Govier provides the following form of an acceptable appeal to authority:

1. Expert X has asserted claim P
2. X is a reliable and credible person in this context 
3. P falls within area of specialization K
4. K is a genuine area of knowledge
5. X is an expert, or authority, in K. 
6. The experts in K agree about P 
7. P is acceptable

Given that the above guidelines provide for acceptable appeals to authority, then a violation of one or more of these conditions would lead to what is commonly referred to as a fallacious appeal to authority. Some ways an appeal to authority can go wrong or be weakened include:

1. The authority cited is not really an expert or is not an expert in the area pertaining to the issue at hand.
2. The authority is an "expert" in an area which is not a genuine area of knowledge (An "expert" in homeopathy promoting a treatment does not carry weight as homeopathy is not a genuine area of knowledge).
3. The authority's opinion is unrepresentative of what the majority of experts believe to be true about the subject.
4. There is widespread disagreement among experts on the subject.

Introduction to Logic and Critical Thinking 
A Practical Study of Argument
Fallacy Files: Appeal to Authority

To review later:

Thursday, May 28, 2015

Statistical Syllogism

A statistical syllogism is an inductive argument in which a statistical generalization is applied to a particular case. For example:

Most surgeons carry malpractice insurance.
Dr. Jones is a surgeon.
Therefore, Dr. Jones likely carries malpractice insurance.

This sort of argument can be written in the general form:

P1. Most A's are B
P2. x is an A
C. Therefore, probably x is a B

When the proportions are known the form can be written as:

P1. Z percent of A's are B
P2. x is an A
C. Therefore, it is probable to the .Z degree that x is B

In the general forms presented above, A is called the reference class,  B the attribute class and x is the individual object.

We often use informal versions of the statistical syllogism in everyday reasoning. For instance, if you read in the New York times that the President is visiting China and you believe it to be true, on what basis do you justify this belief? Most people understand that you can't believe everything you read in a newspaper but recognize that certain kinds of reports published in certain newspapers tend to be true. This is one of those kind of reports so it is likely true.

Strength/Weakness of a Statistical Syllogism
There are two primary standards which determine the strength of a statistical syllogism. First is the strength condition which is, the closer to 100% the reference class is to the attribute class the greater the confidence in the truth of the conclusion. Conversely, the closer to 0% the weaker the argument.

Second is the available evidence condition (also called the rule of total evidence) which requires using all available evidence in constructing or assessing such arguments. With statistical syllogisms this essentially means questioning if there is additional relevant information available concerning the individual object (x) that has not been included in the premises? Another way of saying this is that the individual object must be included in the reference class most specifically relevant to the conclusion. Failure to use all available evidence is commonly referred to as the Fallacy of Incomeplete Evidence.

For example:

P1. Sixty percent of students at the University believe in God.
P2. Fred is a student at the University.
C.  It is sixty percent probable Fred believes in God.

But if we also know that Fred is a history major and that only forty percent of history majors believe in God then it would not be appropriate to use the reference class in the example since it excludes this relevant information.

Due care must be taken when judging individuals using statistical syllogisms as their misuse can contribute to stereotyping and prejudice.

A Practical Study of Argument

Critical Thinking: An Introduction to Basic Skills

Critical Reasoning and Philosophy: A Concise Guide to Reading, Evaluating and Writing Philosophical Works

Wednesday, May 20, 2015

Analogical Arguments

An analogical argument is the use of a comparison between two or more things which have some similarity and from this basis inferring that they share some other property. The central topic which we want to draw a conclusion about is often referred to as the primary subject and the thing(s) to which the primary subject is compared to is called the analogue. The things the analogues and primary subject have in common are referred to as shared attributes. The attribute which the analogues possess that is being inferred to the primary subject are called the target attribute.

As described by Govier "An argument based on analogy begins by using one case (usually agreed on and relatively easy to understand) to illuminate or clarify another (usually less clear). It then seeks to justify a conclusion about the second case on the basis of considerations about the first. The grounds for drawing the conclusion are the relevant similarities between the cases, which show a commonality of structure."

The general form of an analogical argument is:

P1. A has properties p, q, r
P2. B has properties p, q, r
P3. A has property s
C. Therefore B probably has property s


Analogue = A
Primary Subject = B
Shared Attributes = p, q, r
Target Attribute = s

For example:

P1. John's brother and parents smoked two packs of cigarettes a day and ate fatty foods.
P2. John smoked two packs of cigarettes a day and ate fatty foods.
P3. John's brother and parents all died prematurely of heart attacks.
C. Therefore, John will probably die prematurely of a heart attack.

Here is another example in non-standard language:

Tom goes to Las Vegas for his first time. He goes into huge casino with lots of slot machines, gambling tables, bars and an all you can eat buffet. He goes into a second huge casino that also has lots of slot machines, gambling tables and bars. He becomes hungry and remembers the first casino had an all you can eat buffet and concludes that this casino probably as one as well.

Evaluating Analogical Arguments
The strength or weakness of an analogical argument depends upon a number of considerations:

Similarity - Verify that the properties proposed as being shared among the comparison group (shared attributes) do indeed exist. As analogical arguments are rarely actually presented in the form above, it may even be necessary to first list just how it is assumed the comparison groups are similar. Here is a simple example. "John is like Mike. Mike is smart. Therefore John must be smart". In this example none of the assumed similarities between John and Mike have been presented. Before the argument can carry any weight these similarities must be listed and verified.

Relevance - The more relevant the shared attributes are to the target attribute, the stronger the argument. Here is an example of an analogical argument which lacks relevance. "Book A and Book B both have a hardbound cover, pages, words on the pages and numbers at the bottom of the pages. Book A is a boring story. Therefore we can assume that Book B has a boring story." Though I have given a number of similar properties between Book A and Book B, none of these properties are relevant and thus do nothing to increase the probability that Book B is boring.

Number - The more shared attributes the primary subject and analogues share in common with each other, the stronger the argument. This is based on the notion that the more two things are alike, the more likely they also share the property stated in the conclusion. As stated above, relevance plays a key role in determining how much weight these similarities are given.

Disanalogy - Relevant disanalogies or dissimilarities must also be considered when determining the strength or weakness of an analogy. For example if I say, "I have known three people who have had surgery at this hospital with the same surgeon and they have all turned out successfully. Therefore Jane's surgery will also be a success." But what if the three success stories all had minor surgery and Jane is scheduled for a high risk procedure? This of course would be a very relevant disanalogy.

Critical Thinking Web: Analogical Arguments

Monday, May 18, 2015

Hasty Generalization

The hasty generalization is an informal fallacy in which an inductive generalization is made from a sample that is inadequate to support the generalization in the conclusion. As discussed in the post on inductive generalization, this may be because the sample is too small or biased.

Hasty generalizations often result from anecdotal arguments, which are short stories typically taken from the personal experience of the arguer. Generally, these anecdotal arguments describe only one or a few episodes which are then used to generalize about the population.

For example:

"Acupuncture works. My friend Tom tried it and he said it cured his back pain.".


"Smoking isn't harmful. My dad smoked a pack a day and lived until 95."

The Nizkor Project: Hasty Generalization
Fallacy Files: Hasty Generalization

Thursday, May 14, 2015

Inductive Generalization

An inductive generalization is an argument that moves from particular premises to a generalized claim. As defined by Trudy Govier "In inductive generalizations, the premises describe a number of observed objects or events as having some particular feature, and the conclusion asserts, on the basis of these observations, that all or most objects or events of the same type will have that feature."

P1 - Pavlovian conditioning caused dog Fido to salivate when a bell rings.
P2 - Pavlovian conditioning caused dog Rover to salivate when a bell rings. 
P3 - Pavlovian conditioning caused dog Spot to salivate when a bell rings.
P4 - (etc.)
C - Therefore, Pavlovian conditioning causes all dogs to salivate when a bell rings.

It seems intuitive that the strength of the example above largely relies upon how many particular instances Pavlovian conditioning resulted in a dog salivating. A thousand instances of a salivating dog would be a stronger argument than only ten instances. This leads us to the concept of sample.

"In inductive generalizations, features that have been observed for some cases are projected to others. Following established practice in statistics and in science, we call the observed cases the sample and the cases we are trying to generalize about the population." Statistical sampling methodologies are beyond the scope of this post but the basic idea is that the strength of an inductive generalization largely depends on sample size and how representative it is.  

In general, increased sample size is associated with a decrease in sampling error as it is more likely to represent the population (though there are diminishing returns). For more on why this is so, see the law of large numbers and central limit theorem.

A representative sample is one in which the selected segment closely parallels the whole population in terms of the characteristics that are under examination (for example, if one third of the population has relevant characteristic X, then one third of the sample should have characteristic X). We try to make samples representative by choosing them in such a way that the variety in the sample will reflect variety in the population.

Sampling methods include Random Sampling, Stratified Sampling, Systematic Sampling, Convenience Sampling, Quota Sampling and Purposive Sampling.

Monday, May 11, 2015

Induction: Inductively Strong & Inductively Cogent

Induction is the process of reasoning from one or more premises to reach a conclusion which is likely, though not certainly true. Hence, ainductive argument is one in which if the premises were true, then the conclusion is likely to be true.

"In the most general sense, Inductive reasoning is that in which we extrapolate from experience to what we have not experienced. The assumption behind inductive reasoning is that known cases can provide information about unknown cases."1 Govier goes on to describe inductive arguments as having the following characteristics:

1. The premises and the conclusion are all empirical propositions.
2. The conclusion is not deductively entailed by the premises.
3. The reasoning used to infer the conclusion from the premises is based on the underlying assumption that the regularities described in the premises will persist.
4. The inference is either that unexamined cases will resemble examined ones or that evidence makes an explanatory hypothesis probable.

Inductively Strong Arguments - An inductively strong (forceful) argument is one in which, if the premises were considered to be true, the conclusion is probably true. In other words, if we assume the premises are true the likelihood that the conclusion of an inductively strong argument is true is greater than 50%. If the probability that the conclusion is true is 50% or less, than the argument is inductively week.

Inductively Cogent Arguments - An inductively cogent argument is one which is inductively strong and all of its premises are actually true. 

To fit into our informal logic model, instead of requiring that the premises be true we would require that they be acceptable. This, of course, is due to the difficulty often encountered in establishing with certainty whether or not something is true. 

Types of inductive arguments include inductive generalization, statistical syllogism and analogical arguments.

1 A Practical Study of Argument

Critical Reasoning and Philosophy, A Concise Guide to Reading, Evaluating, and Writing Philosophical Works 

Probability and Induction

Thursday, May 7, 2015


An outline is a method of presenting the main and subordinate ideas of a document by organizing them hierarchically.

Alphanumeric outline
An alphanumeric outline uses Roman numerals, capitalized letters, Arabic numerals and lowercase letters as prefix headings. 

The Chicago Manual of Style (CMS) uses the following outline format:

I. (Roman numeral) 
     A. (Capital letter) 
          1. (Number) 
               a) (Lowercase letter followed by closing parenthesis) 
                    (1) (Number enclosed in parenthesis) 
                         (a) (Lowercase letter enclosed in parenthesis) 
                              i) (Roman numeral with lowercase letters followed by a closing parenthesis)

The Modern Language Association (MLA) uses essentially the same method except the first lowercase letter is followed by a period instead of a closing parenthesis.

Decimal outline
The decimal outline uses only numbers as prefix headings making it easier to see how every item relates within the hierarchy. It uses the following outline format:

Here is a sample decimal outline:

1.0 Choose Desired College
     1.1 Visit and evaluate college campuses 
     1.2 Visit and evaluate college websites 
          1.2.1 Look for interesting classes 
          1.2.2 Note important statistics

Wednesday, May 6, 2015

Favorite podcast episodes

Here are some of my favorite podcast episodes. I've been listening to podcast for years and continue to, so I'll be adding newly and rediscovered favorites.

The Living Room (Diane’s new neighbors across the way never shut their curtains, and that was the beginning of an intimate, but very one-sided relationship.)

Radiolab: Season 7, episode 1
Animal Minds
-Animal Blessings
-Spindle Cells
-Sharing is Caring? Or is it a Sin?

Radiolab: Season 7, episode 2
Lucy (Emotionally Stirring, Thought Provoking)

Radiolab: Season 3, episode 4
Part 3: Clive

Radiolab: Season 12, episode 2

Radiolab: Staph Retreat

Radiolab: Season 4, episode 2
Whole episode good but I found the section titled Deception to be fascinating

Rationally Speaking
Episode 134 - "Science drives moral progess" Michael Shermer

Reply All
#43 The Law That Sticks
#47 Quite Already

Episode 1
-Dark Thoughts (Thought Provoking, Psychological) 
-Locked-In Man (Emotionally Stirring, Thought Provoking, Psychological)

When Willpower Isn't Enough
How to Be Less Terrible at Predicting the Future

Econtalk: Phil Rosenzweig on Leadership, Decisions, and Behavioral Economics

Tuesday, April 21, 2015

Kentucky: Neither straights or gays can marry the same sex so ban is not discriminatory

It's been reported that the administration of Kentucky Governor Steve Beshear has filed a brief with the US Supreme Court defending it's ban on gay marriage. In it they argue
"Kentucky’s marriage laws treat homosexuals and heterosexuals the same and are facially neutral. Men and women, whether heterosexual or homosexual, are free to marry persons of the opposite sex under Kentucky law, and men and women, whether heterosexual or homosexual, cannot marry persons of the same sex under Kentucky law,"
I'll be a bit more generous than some of the news outlets that reported on this by noting that the brief is  42 pages so there may or may not be other more rational arguments presented (I didn't take the time to read it). Regardless, it is hard to believe that the argument above could possibly be presented as a serious justification for upholding it's ban on gay marriage.

Imagine a law which states that it is now illegal to practice Christianity and then arguing that it doesn't discriminate against those who are Christian because it also applies to Muslims, Buddhist, and atheist. This sort of specious legal reasoning really drives me crazy. I can't imagine that the Supreme Court will give it any credence.

HuffPo: Kentucky: Our Same-Sex Marriage Ban Isn't Anti-Gay Because It Applies To Straight People, Too

Thursday, March 19, 2015

Dragon Ball Z: Light of Hope

A very impressive fan film adaptation of "The History of Trunks" TV-Special that is part of the Dragon Ball Z cartoon series. You can read more about it here. Looking forward to the next installment.

Thursday, March 5, 2015

Did a Texas city fire It's police department and hire a private security firm?

Occasionally I like to take the time to look into stories that I find a little bit fishy. I do it as a sort of exercise which usually lends credence to the old saying "Don't believe everything you read."

I got into this one after reading a post a friend made on Facebook which linked to a story on The Free Thought Project titled Texas Town Experiences 61% Drop in Crime After Firing Their Police Department. The article essentially says that in 2012, the city of Sharpstown "fired their cops" and "hired S.E.A.L. Security Solutions, a private firm, to patrol their streets." A representative of the security firm is quoted saying "Since we've been in there, an independent crime study that they've had done [indicates] we've reduced the crime by 61%” in just 20 months.

After looking into it a bit I found similar stories had been released within the last couple of days such as Texas City Gets Rid of Police Dept., Hires ‘SEAL Security’ — Guess What Reportedly Happened to Crime from the Blaze, Texas Town Fires Entire Police Department, Crime Drops by 61%, from Infowars and Texas town sees crime drop by almost two thirds after firing police, hiring private security from Rare. As the titles suggest, these stories all either directly state or indirectly imply that the city of Sharpstown fired their police department and hired a private security firm to take its place. But is this actually what happened? In a word, no.

According to Wikipedia, Sharpstown is a master-planned community in Greater Sharpstown, Southwest Houston, Texas which is served by two Houston Police Department patrol divisions. For many years, the Sharpstown Civic Association had contracted with Harris County Constables for additional patrols but due to budgetary issues ended the relationship in 2012. It is this arrangement for additional patrols that the above articles are referring to when they state that the city fired their police department.

I'm not sure if this is a matter of shoddy journalism or a deliberate act to fabricate a story which would be more appealing to their readers? As it took very little effort for me to discover the truth, I'm inclined to think it's the latter. The sad thing is, there was a perfectly good story here which didn't need to be based on the false claim that the town fired their police department. It could have been titled something like, City of Sharpstown turns to private security to help patrol its streets. The article could then talk about how the constables used to be contracted to patrol the area but that the security firm does the job at half the cost and with better results. See, no need for the lies and false implications.


Popular Constables On Patrol Program (COPS) Discontinued

Friday, February 6, 2015

Propositional Logic / Sentential Logic

I. Introduction
Propositional logic (also called propositional calculus, sentential calculus, sentential logic, etc.) is a branch of logic that studies ways of joining simple (atomic) propositions to form more complicated propositions using logical connectives, and the logical relationships of these propositions.

To highlight the difference between term and propositional logic, think of the sentence "All dogs are mammals". Using term logic we would say the fundamental units of the sentence are the categories of dogs and mammals. In contrast, with propositional logic the fundamental unit of the sentence is the entire statement or proposition "All dogs are mammals". It's important not to equate propositions with sentences as a sentence can have more than one statement. For instance, the sentence “All dogs are mammals, and all cats are mammals too” clearly contains the two propositions.

II. Propositions
A proposition can be defined as a declarative sentence, or part of a sentence, that is capable of having a truth-value of either true or false, but not both. For the purpose of this post, the term "proposition" and "statement" are used interchangeably.

For example, "Paris is the capital of France." is a proposition with a truth value of "true" while the sentence, "Everyone born on Monday has purple hair" has a truth value of "false."

Some sentences are not propositions such as commands like "Close the door" or questions like "Is it hot outside?"

The smallest indivisible units in propositional logic are statements referred to as simple or atomic propositions. These are statements which are either true or false and cannot be broken down into other simpler statements. For example, "The dog ran" is an atomic proposition. Atomic propositions can be used to form complex propositions by using connective words (see connectives below). For example, "the dog ran and the cat hid" is a complicated proposition which combines two atomic propositions using the connective word "and".

III. The Language of Propositional Logic
Classical truth-function propositional logic utilizes a simple symbolic language to represent propositions expressed in natural language such as English. Simple (atomic) statements are represented by capital letters 'A', 'B', 'C', etc. The logical signs '∧', '∨', '→', '↔', and '~' are used in place of the truth-functional operators, "and", "or", "if... then...", "if and only if", and "not", respectively. Parentheses are used to group propositions similar to how they are used in algebra and arithmetic.

A. Connectives (logical operators)
Connectives (logical operators) are words or phrases used either to modify a statement or join simple statements together to form a complex statement. Though there are many connectives, the five basic ones are:


Here is a chart of which shows the logical operator (connectives formal name), the symbol and its natural language usage.

As there are no absolute standards in regards to symbols
 used in Propositional Logicdifferent authors use different symbols.
This link provides a list of alternate symbols you may come across.

1. Negation
Not-P.     For example, “It is not green.”

The negation of statement P, simply written "~P" in language PL, is regarded as true if P is false, and false if P is true. Unlike the other operators, negation is applied to a single statement. The corresponding chart can therefore be drawn more simply as follows:

Though the word "not" is generally thought of as the English equivalent of "~", it can be expressed in many ways such as:

It is not the case that...
It would be false to say that...
...failed to...

2. Conjunction 
p and q.     For example, “It is wet and it is cold.”

Conjunction is a truth-functional connective similar to "and" in English and written "∧" in PL. When dealing with a conjunction, you must consider both p and q. That is, a conjunction is true if and only if both conjuncts are true.

Though "and" is generally thought as the English equivalent to "∧" there are many ways it can be expressed. Some include:

...as well as...
...despite the fact that...

Though these may not seem similar to the word "and" it is important to remember that propositional logic treats as a conjunction any sentence whose truthfulness depends on both conjuncts being true and false if any or both conjuncts are false. For instance, the sentence "John loves Mary even though she barely tolerates him" is true only if both propositions "John loves Mary" and "She barely tolerates him" are true. If either or both are false, the whole proposition is false.

3. Disjunction
p or q (or both).     For example, “It is wet or it is cold.”

Disjunction is a truth-functional connective similar to "or" in English. The disjunction of two statements p and q, written in PL as "(p ∨ q)", is true if either p is true or q is true, or both p and q are true, and is false only if both p and q are false.

Though we say that "or" is the rough English equivalent to PL "∨", it should be noted that "∨" is used in the inclusive sense. More often than not when the word "or" is used to join together two English statements, we only regard the whole as true if one side or the other is true, but not both, as with the statement "Either we can buy the toy robot, or we can buy the toy truck; you must choose!" This is called the exclusive sense of "or". However, in PL, the sign "v" is used inclusively such as with the statement "Her grades are so good that she's either very bright or studies hard" which does not exclude the possibility of both.

Though the word "or" is generally used (imperfectly) as the English equivalent to "∨", there are other ways it can be expressed such as:


4. Conditional (material implication)
If p, then q.     For example, “If it is green, then it is heavy.”

The conditional is a connective similar to "if_then_" statements in English and generally represented as "→" in PL. The first simple statement in a conditional is referred to as antecedent and the second simple statement is known as the consequent. For example, with the statement "If you flip the light switch, the lights will go out" the antecedent is "You flip the light switch" and the consequent is "the lights will go out".

A conditional statement asserts that if the antecedent p is true, the consequent q will be true as well. In other words, a conditional statement is only false if the antecedent p is true and the consequent q is false.

5. Biconditional (material equivalence)
p if and only if q.    For example "A triangle is equilateral if and only if it is a triangle with three equal sides".

The biconditional is a connective similar to "if and only if" statements in English and generally represented as "↔" in PL. A biconditional is regarded as true if the antecedent and consequent are either both true or both false, and is regarded as false if either have different truth-values.

B. Scope, Parentheses and the Main Connective
Whenever more than one connective is used in a statement, there is a chance of ambiguity. Consider the statement S∨C∧T where S is "I will show you stamps", C is "I will make you some coffee" and T is "I will give you $1000". Hence, in English it would be written, "I will show you my stamps or make you coffee and give you $1000".

As it is presented, the statement is ambiguous and could be interpreted in two different ways:

1) "I will show you my stamps or make you coffee but in any event, I will give you $1000".

2) "Either I will show you my stamps or make you coffee and give you $1000".

The problem here is that we don't know what the scope is of either of the two connectives used in the statement. Does the disjunction in S∨C∧T connect S to C or does it connect S to C∧T? Does the conjunction in S∨C∧T connect C to T or does it connect S∨C to T?

To deal with this problem, parentheses are used to establish scope. So going back to our two interpretations of S∨C∧T, we would write:

1) "I will show you my stamps or make you coffee but in any event, I will give you $1000" as ((S∨C)∧T).

2) "Either I will show you my stamps or make you coffee and give you $1000" as (S∨(C∧T)).

*Note that in actual practice you will often find the outermost parentheses are omitted. 

In propositional logic, the main connective is the connective with the greatest scope in a statement. For example, with ((S∨C)∧T) we see that the scope of ∨ is limited to S and C while the scope of ∧ encompasses the entire statement. Therefore, in this example ∧ is the main operator.

Internet Encyclopedia of Philosophy: Propositional Logic
Introduction to Logic (second edition): Harry Gensler
Critical Thinking: An Appeal to Reason by Peg Tittle: Supplemental Chapter: Propositional Logic
Wikipedia: Negation
Wikipedia: Logical conjunction
Wikipedia: Logical disjunction
Logic Self-Taught: A Workbook