Tag Archive | "logic"

An Awesome Logic Primer (Part 2)


 

In the first part of the primer, we discussed the form of a deductive argument, learned the difference between truth and validity, clarified the limits and benefits of logic, learned about logical operators, and were introduced to truth tables, a way of knowing all the truth values that a certain statement can have.

In this part, we are going to move away from the tedium of proving everything with truth tables, going towards handy rules of inference. Think of these as shortcuts when trying to find out if someone’s statements make sense. A list of common fallacies follows, and a short summary of everything in this primer.

Common Properties and Identities (Rules of Inference)

Proving an argument valid or invalid by truth tables becomes very tedious. In the case of more than two statements, for example:

 

P

Q

R

P ∧Q ∧ R

T

T

T

T

T

T

F

F

T

F

T

F

T

F

F

F

F

T

T

F

F

T

F

F

F

F

T

F

F

F

F

F

 

To get all the possible combinations of truth values reflected in the table, we need eight rows. To generalize, we need 2n rows, where n is the number of statements. Of course, we all have lives outside of debating people on the internet, so making these god-fangled tables is not high on anyone’s priority.

Thankfully, logicians who have no life have compiled a list of argument forms which are valid, and these rules of inference(in impressive-sounding Latin!) will now be available to you so you can con your way into someone’s pants by pretending to be a lawyer.

Any argument which can be reduced to these forms must then also be valid.
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Modus Ponens

P→Q
P
∴ Q.

Modus ponendo ponens, translated from Latin to Yoda-speak is, “the way that affirms by affirming”. If P is true, then Q is true. P, therefore Q. It is the subject of the above series of truth tables.

Don’t panic because of that weird-looking triangular dot formation sign there. It’s only a mathematical shorthand for “therefore”. It’s a convenient way to separate premises from the conclusion.
_________________________________________

Modus Tollens

P→Q
~Q
∴~P.

Modus tollendo tollens, again in Yoda-speak is, “the way that denies by denying”. If P, then Q. Not Q, therefore not P. There are two ways to prove that this is valid. One is to use truth tables, and the other is to derive it from modus ponens. If you’re lazy, then just take it on faith. =P

Example: If Alice has friends, she will get invited to the party. She isn’t invited to the party, so Alice must not have any friends.

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Syllogism

P→Q
Q→R
∴P→R.

The syllogism can be spotted everywhere: from your arguments with stubborn kids, to seedy detective novels. It is, after all, the most common form that human reasoning takes.

Example: If he was clobbered to death, the wrench was used to kill him. If the wrench was used to kill him, then the butler killed him. Therefore, if he was clobbered to death, the butler did it.

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Reductio ad Absurdum(aka proof by contradiction, lit. “reduction to the absurd”)

P→Q
P→~Q
∴~P.

The general case of a proof by contradiction. If a statement implies another statement and the opposite of it, then the negation of that statement must be true. This is a staple of debates, and the best way to poke holes in your opponent’s argument.

In real life(Ha! Who are we kidding? On online forums…), P usually takes the form of many premises added together(this means P1∧P2∧P3∧… ), where the number of premises render the absurdity of conclusions obscure.

Example: If there is an invisible pink unicorn, then it must be invisible. If there is an invisible pink unicorn, it must also be pink, and therefore, visible. Therefore, there is no such thing as an invisible pink unicorn.

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This is an incomplete list. For a better one, check out the references at the end of this text.

List of common fallacies
Your face is a common fallacy!

Formal fallacies vs. informal fallacies

A formal fallacy is a mistake in an argument’s form, that is, someone who makes them has obviously skimmed to this part of the primer, missing out on the incredibly detailed and sensual description of lesbian sex between the definitions of the logical conjunction and the logical disjunction.

All formal fallacies stem from invalid arguments, and are actually special cases of non sequitur(Latin for “it does not follow”. Example: rabbits are awesome therefore I am having lunch right now ).

An informal fallacy, however, is usually committed because of false premises or hidden assumptions which are required for the argument to function. That is, it has something to do with an argument’s content.

Since by now you will have the proper tools to perceive formal fallacies, most of the elements of this list will contain informal fallacies. I will also list only a small portion of these fallacies, as easy-to-understand lists are readily available around the net.

Ad Hominem (literally, “to the person”)

Ad hominem attacks neither the form or the premise of the argument, but the person who is making that argument. Note that ad hominem is always a fallacy, but there is a form of attacking the premises which looks very similar to ad hominem- saying, for example, that a person has a vested interest in lying about a premise means that you put a premise in doubt, and not the person.

Example:

A:”What you said about evolution isn’t true because you are a pervert.”

B:”I guess if I say you exist, that’s also not true, then?”

Ad Hominem Tu Quoque (well so is your face)

Could be translated in spirit to “oh yeah? No, you!”, this is asserting that a person’s conclusion is false because it contradicts his actions or a previous statement. Of course, when a person says both Q and ~Q, that probably means that he’s an idiot or a lying shit-faced hypocrite, but unfortunately, it doesn’t say anything about the truth of his current statement.

Example:

A: “Capitalism is evil!!!”

B: “But, you wear branded clothing, and eat at upscale restaurants.”

A: “Well your face is an upscale restaurant.”

_________________________________________
Appeal to Authority (Argumentum ad Verecundiam)

Asserting that something is true because an expert(or someone pretending to be one) said it. This fallacy is akin to a double-edged sword, and it depends on whether the person is a valid authority in that exact subject he is being quoted in.

Example:

A:”Isaac Newton believed in ghosts and occult stuff, so they must be true.”

B:”What.”

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Appeal to Ignorance (Argumentum ad Ignorantiam)

Asserting that since something cannot be proven false, it is therefore true. One of the most rage-inducing fallacies to ever exist.

Example:

A:”You cannot prove that I do not have an invisible dragon in my garage on planet Jupiter. Therefore, it exists.”

B:”My dragon-slaying girlfriend went to Jupiter and made its skin into a trendy purse. So there.”

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Argumentum Verbosium(the argument full of words. I kid not.)

It involves raining down obscure jargon and multiple, often conflicting assertions with the intent to confuse opponents into submission.

Example: Really? Go check the ff page.

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Equivocation

An argument, usually in the form of a syllogism(did you read that part?), where a word with several different meanings is used with a different meaning for each implication. The usual victims of this treatment are abstract words like love, god, nature, etc.

Example:

A:” Being stuck in traffic makes people mad. Mad people get sent to the mental hospital. Being stuck in traffic gets people sent to the mental hospital.”

B:”I don’t even.”

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Appeal to Consequence (Argumentum ad Consequentiam)

Taking a claim to be true because otherwise you will be depressed/the world will end/you will never be loved/etc. A special form of appeal to emotion.

Example:

A: “Being aware that women are still being oppressed in many parts of the world will make me depressed, so I’ll just believe that gender equality has been achieved so I can sleep better at night.”

B:”Good for you.”

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For the impossibly intellectually lazy: a TL;DR
Or for the awesome people who took the time to read, a summary

  • Logic is not a system of all-encompassing laws. It is a formalization of the reasoning that we use every day, kind of like high-level common sense.
  • To simplify things, we construct statements that are either true or false. In the real world, this value depends on many factors, such as semantics and perspective.
  • Even with this simplified system, we can still construct valid arguments and figure out if other people’s arguments are valid or not.
  • The only way for an argument to be invalid is if there’s a possible way that the conclusion is false when you hold all the premises true.
  • A valid argument does not mean the conclusion is true, that depends on the premises. Similarly, an invalid argument does not say anything about the truth of a conclusion.
  • The most important logical symbol is the implication(→), which means “therefore”. It is the usual connector of everyday reasoning.
  • For an argument to be right, it has to be sound- that is, it has to both have true premises and be valid.
  • The most common argument takes the form of a syllogism, which connects statements by implication. (P→Q→R, therefore P→R)
  • Truth tables provide a definitive(but tasking) way of finding out whether the argument is valid. Rules of inference are more efficient, but error-prone. You are prone to error, I mean. Not the rules.
  • Formal fallacies have to do with form, informal fallacies concern everything else.

Protips:

  • There are generally two ways to attack an argument, one is questioning validity, and the second is by questioning the premises.
  • To see if an argument is invalid, try to see a scenario where the conclusion is not true but the premises are. This needs wit and imagination.
  • Premises, especially complicated ones, usually can also be dissected and rendered into premises supporting a conclusion. Similarly, arguments which reduce to a tautology can be used as premises.

The Reference List
And suggested readings

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An Awesome Logic Primer (Part 1)


Scope of the Primer
AKA the part with a subtle disclaimer about this not compensating for an actual course in logic

The purpose of this two-part primer is to introduce you, the reader, into the wonderfully complicated and interesting world of logic, the language which every internet intellectual claims to know so much about. Now, taunting others with the words “learn2logic, n00b” and “that’s a fallacy, you moron” will not only be their area of discourse, but also yours!

The basic elements of the part of logic that is discussed here are propositions. In logic, propositions are statements that can take two and only two truth values, true or false. You might say that this doesn’t look like most of the stuff we say in real life, but as this is meant to be an introduction to more complicated stuff which handle natural language [search term for the curious: argumentation theory], it is better to learn the basics and the terminology of logic first.

We also hope to tackle logical quantifiers [first-order logic] in a part 3, as it proves to be quite a handful, and understanding it requires the basic concepts introduced here.

Deductive arguments
The opposite of constructive criticism

As mentioned earlier, a simple proposition is either true or false; it cannot be neither, nor can it be both. A deductive argument is a way of proving that a certain proposition, called the conclusion, follows necessarily from a given set of propositions which are called the premises.

For the sake of argument, we assume the premises to be totally true, forever. That is, given that we accept the premises to be true, then the conclusion must necessarily be true. Another way of viewing this is that if all the premises are true, then there is no way for the conclusion to be false.

To use a common example:

P1: Men are mortal.

P2: Socrates is a man.

C: Therefore, Socrates is mortal.

Propositions P1 and P2 are the premises. Here, the premises are that men are mortal, and Socrates is a man. If we accept these to be true, then Socrates must be mortal.

When it is impossible for the conclusion to be false when the premises are true, then the whole argument is said to be valid. There is a way to test this, which will be explained later on.

Validity vs. Truth

…and Truth vs. Truthiness

A valid argument does not automatically mean that the conclusion is true, as an invalid argument also does not mean that the conclusion is false.

There is a connection between validity and truth though: An argument whose conclusion turns out to be false when the premises are true must always be invalid. That is how we define invalid arguments.

Consider our earlier example, with revised premises:

P1: Men are dinosaurs.
P2: Socrates is a man.
C: Therefore, Socrates is a dinosaur.

Even if we can agree that Socrates is a man (P2), we know that men aren’t dinosaurs (P1). In other words, in a universe where the premises are true, we can conclude that Socrates is a dinosaur, but as we know, this is not that universe.

However, the argument is still a valid one because if we accept P1 and P2 to be true, then C must be true.

An example of an invalid argument with a true conclusion is:

P1: Men are dinosaurs.
P2: Socrates is a dinosaur.
C: Therefore, Socrates is a man.

Why is this invalid? Because it doesn’t say anywhere that all dinosaurs are men. Therefore, it doesn’t mean that just because Socrates is a dino, he’s a man.

Soundness

…which is totally different from sounding; no, don’t Google that.

Soundness is the property to which every argument aspires to. An argument is sound if and only if it is valid, and all its premises are true. Basically, if your argument is sound, then you are right. Go ahead and pat yourself on the back.

The structure of logic:

P1: Oogwars are Tarlarks.
P2: A Mugwarp is an Oogwar.
C: A Mugwarp is a Tarlark.

Notice that the above argument has the exact same form as the one concluding that Socrates is a dinosaur, only that this one consists of terms which could only be gibberish. Since it has the exact same form, it must follow that this argument is valid, given that the premises are true.

Logic isn’t about finding out whether something is true or false, but in finding out whether an argument is valid or invalid. Thus, we are mainly concerned about the form of an argument. Even if I replace the term “Socrates” with “your face”, the argument would still be valid. Arguments by themselves don’t need to even be connected to each other!  We will see this later in the examples.

How do we do this? How do we know that a conclusion logically follows from the premises?

From the above definition of a deductive argument, we say that a conclusion follows from the premises if, and only if the conclusion is true whenever the premises are true. Therefore, you only need to find out if there is a way that the conclusion is false even when the premises are true.

But this is kind of difficult, isn’t it? Aren’t we using the argument to prove the truth of the conclusion in the first place?

Well, the most common way of proving an argument to be invalid is to use one’s IMAGINATION to think of a situation where the opposite of the conclusion is true, but all the premises are also true. In an above example, can Socrates be immortal even if he’s a man and all men are mortal? There is a formal statement of this, which comes up in the examples from discourse.

Logical Operators

…as opposed to smooth ones.

Logical operators are symbols which we use to define relationships between statements. Naively, we are familiar with them in everyday usage, but their definitions in logic are a tad more formal.

The material implication [If…then, therefore]

Symbol: →, as in P → Q.

True if and only if whenever P is true, Q is true. The basic logical symbol. Note that when P is false, Q can take on any value and the implication is still true.

P: Wheatley is smart.
Q: I’m a potato.

P→Q: If Wheatley is smart, then I’m a potato.

P→Q: Wheatley is smart, therefore I am a potato.

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The material equivalence [If and only if]
Symbol: ↔, as in P  ↔ Q.

A material implication that goes both ways. Also known to all math lovers(snicker) as the equals sign. This means that whenever P is true, Q is true, and whenever P is false, Q is false, and vice versa.

P: I am a creationist.
Q: I fail science forever.

P ↔ Q: Obviously.

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The negation [NOT operator]

Symbol: ~, as in ~P.

It negates the statement. When P is true, ~P is false, when P is false, ~P is true.

P: I eat leaves.

~P: I do not eat leaves.

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The logical conjunction [AND operator]

Symbol: ∧, as in P ∧ Q.
Connects two statements together. True if and only if both statements are true.

P: I love music.
Q: I love books.

P ∧Q: I love music and books.

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The logical disjunction [OR operator]

Symbol: ∨, as in P ∨Q.

Connects two statements together. False if and only if both statements are false. Note here that this is different from the common usage of ‘or’ in language, where you cannot choose both choices. This usage is known as the exclusive disjunction, or XOR.

P: I can talk to you in English.

Q: I can talk to you in Russian.

P ∨Q: I can talk to you in English or in Russian.

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The tautology [T, or 1 in binary]

Symbol: T

A statement that is always true. Note that this and the contradiction aren’t operators, they are pre-defined statements.

The contradiction [F, or 0 in binary]

Symbol: F

A statement that is always false.

Got all that? Take a deep breath for a moment, or step out and get a snack. Shit is about to get real.

Truth Tables

You can’t handle the truth!

A truth table is a chart listing all possible combinations of truth values of a statement or statements. For example, the truth table of the NOT operator:


P ~P
T F
F T

It’s like a shortcut to saying that when P is true, ~P is false, and when P is false, ~P is true.

Now, the truth table of the AND operator:

P

Q

P ∧ Q

T

T

T

T

F

F

F

T

F

F

F

F

Is a clearer way of saying that the new statement made by connecting P and Q with an AND is only true when P and Q are both true.

This is a more interesting truth table, the one for material implication:

P

Q

P→Q

T

T

T

T

F

F

F

T

T

F

F

T

What does this mean? We can restate what the truth table is showing us by saying that the only way that the implication is false is if Q is false while P is true. Sound familiar? When we take P to be all the statements in the premise combined, and Q to be the conclusion, we can see that this is exactly how we determine if a conclusion is valid or invalid!

Time to put the truth table to good use. Let’s find out if this argument is valid or invalid. Consider this example:

P→Q: If I am breathing, then I must be alive.
P: I am breathing.
Q: Therefore, I am alive.

First, the truth table for P→Q:

P

Q

P→Q

T

T

T

T

F

F

F

T

T

F

F

T

Then, the truth table for P ∧ (P→Q):

P

P→Q

P ∧(P→Q)

T

T

T

T

F

F

F

T

F

F

T

F

At last, the truth table for [P ∧ (P→Q)] → Q, to see if the premises, when all together, imply the conclusion:

P ∧ (P→Q)

Q

[P ∧ (P→Q)] → Q

T

T

T

F

F

T

F

T

T

F

F

T

Tada! Look at that beautiful, beautiful last column with T’s on every row. 😀

The above is an example of the rule of inference known as modus ponens. A useful exercise for you would be to construct truth tables for every one of the logical operators above, to get a feel for them.

Well, that’s it for part 1! I recommend taking a short break before going on to part 2, and of course, trying out some exercises.

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Three Primary Truths


three pansiesI stumbled upon this article and found this:

There are three truths which must be accepted at the beginning of any investigation into the problem of knowledge and truth:

  • The First Fact: The fact of our existence. “I exist.”
  • The First Principle: The principle of contradiction. “A thing cannot be and not-be at the same time in the same respect.”
  • The First Condition: The essential capability of the mind to know truth. “My intellect can reason and discover truth.”

These primary truths cannot be “proved” by a positive demonstration because they are presupposed and involved in every demonstration. They are so evident that any attempt to doubt or deny them would already mean they have been affirmed and accepted. They are, therefore, fully grounded in reason and no reasonable person can dispute them consistently.

To some it might sound like circular reasoning, but these truths are indeed self-evident in any logical discussion. Let’s see:

  • The First Fact: The fact of our existence. “I exist.”

The fact that we are able to read this proves the first fact. Now what more proof do we need in order to know that we are reading this indeed?

  • The First Principle: The principle of contradiction. “A thing cannot be and not-be at the same time in the same respect.”

Of course. Otherwise, something can be both true and false at the same time in the same respect. If that’s the case, then there is no longer any point in logical discussions because anything can be right and wrong at the same time, so everybody is just wasting time. And this goes hand in hand with:

  • The First Condition: The essential capability of the mind to know truth. “My intellect can reason and discover truth.”

And that’s why we are here at Filipino Freethinkers, to discover truth through Reason. (We appreciate theists who check their dogmas upon entering the site, because doctrine and ‘authority’ have no influence here. But those who don’t believe in reason are wasting their time here and it would be better spent praying for divine enlightenment.)

On the other hand, religion only subscribes to the First Fact (our existence), but not to the First Principle (contradiction) and the First Condition (intellect’s ability to discover truth). Religious dogmas have so many contradictions that are conveniently answered by “our minds are too finite to grasp God’s infinite wisdom”.

As freethinkers, we know that belief is no longer a matter of choice, but of conclusion; no matter how the religious (including our parents) try to proselytize, as long as what they preach is unscientific, illogical, or irrational, they cannot force us to believe. Well the most they could do is to make us (falsely) claim belief. We do not choose to be atheists, agnostics, or deists; we just become, most likely as a result of freethinking.

Now the question is, do we choose to become freethinkers? Is it a matter of choice when we base our beliefs on science, logic, and reason instead of authority, tradition, or dogma? Or is it a matter of conclusion (same with becoming atheists, agnostics, or deists)? I think it’s the latter, when we realize that science, logic, and reason are more reliable in terms of finding the truth than authority, tradition, or dogma, but I would like to hear other points of view. How do people become freethinkers in the first place?

And with this we invite everyone to write. We’ve been coming up with fewer articles lately but that’s probably because of the film fest, after which there will be a lot to write about. 🙂

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Towards healthier skepticism: Correlation does not imply causation


This post will attempt to repeat, clarify, and elucidate the need for the remembrance and understanding of the phrase “correlation does not imply causation”. Scientific studies will be given, and the words in the phrase, which vary in meaning depending on usage, will be defined accordingly.

Scientific studies

Please take a moment to go through the following actual, summarized scientific research results:

1) In a previous scientific research using quantitative assessment, numerous epidemiological studies showed that women who were taking combined hormone replacement therapy (HRT) also had a lower-than-average incidence of coronary heart disease (CHD), leading doctors to propose that HRT was protective against CHD.

2) From a study at the University of Pennsylvania Medical Center, young children who sleep with the light on are much more likely to develop myopia in later life.

We will get back to them in a moment. Now we focus on correlation or co-relation, and why scientists, statisticians and skeptics, at the very least, should always try to maintain and promote the phrase “Correlation does not imply causation”.

Read the full story

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