Categorized | Science

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.
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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.”

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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

 
DISCLAIMER: The opinions in this post do not necessarily represent the position of the Filipino Freethinkers.

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