Axiom²

The stuff that I think you should know (part 3) – Reasoning

This third article of the series of articles “The stuff that I think you should know” will explain how to utilize certain properties of logic and other useful related concepts. I will not be explaining logic in depth but with simple examples instead so it’s possible to “intuitively” grasp how it works.

Logic

Logic concerns itself with the form of arguments, if an argument has a certain form that obeys the rules of logic then it’s said to be logically valid.

Arguments

Arguments are formed by premises and conclusion. Premises are phrases that are meant to be analyzed logically in order to reach the conclusion, premises can also be used to increase the amount of support for a certain conclusion.

Deductive reasoning

Example:
Premise 1 – All humans are mammals
Premise 2 – Mary is human
Conclusion – Mary is a mammal

In a valid deductive argument if the premises are true and the language is unambiguous then the conclusion is necessarily true, like in the above example.

If the premise 1 was “All pandas are mammals” then a conclusion would not be logically possible because the premises shared no logical relationship.

If the premise 1 was “All humans are immortal” then the conclusion that “Mary is immortal” would be logically valid, however, because the premise that “All humans are immortal” is false then it should come as no surprise that the conclusion can also be false.

If the conclusion was “Mary is a plant” it would mean that the conclusion is logically invalid for it wasn’t logically derived from the premises (independently of if it’s true or not).

Inductive reasoning

Example:
Only 10 ducks were ever observed and all of them were white
All ducks are white

Inductive reasoning makes a logical “leap” (called an inference) from the premises to the conclusion. In inductive reasoning the premises provide support, the conclusion can vary in terms of likelihood but will never be guaranteed.

If the premise stated 100 ducks instead of 10 ducks the conclusion would be more likely.

If a premise saying that something was found in the ducks DNA that prevents them from being born non-white then the conclusion would be even more likely.

If a single non-white duck is observed, the hypothesis that all ducks are white is false and no amount of supporting evidence can ever sway it towards true again.

Abductive reasoning

Example:
The lawn is wet
If it rains the lawn gets wet
Therefore, it rained

Abductive reasoning, also known as inference to the best explanation, attempts to generate a probable explanation for an observation, informally we may consider this as guessing which is something humans do pretty well. This sort of reasoning is the same that is used in diagnostic medicine where doctors will take an educated guess to find the disease that supposedly caused the observed symptoms. There are typically many possible conclusions in abductive reasoning and it’s up to the person who’s using it to find the simplest and most likely of them. In the example above it would be possible for the conclusion to be wrong even though the premises were correct, there’s more than one cause for a wet lawn, like, sprinklers, garden hoses, plumbing leaks, and so on, so, for this reason, abductive reasoning should be avoided unless necessary (also, because it’s fallacious, more about that later in this article).

Informal discourse

Arguments don’t necessarily have to be stated as the above arguments were, arguments in real life tend to be stated in a much more informal way. Here are some examples of how the first argument of this article could be informally stated:

  • If all humans are mammals and Mary is a human then Mary is a mammal.
  • Mary is a mammal because she’s human and all humans are mammals.
  • How is Mary not a mammal? All humans are mammals and she’s human after all.

Equivalences

Some phrases are logically equivalent to some other phrases like “If it’s a human then it’s a mammal” is equivalent to “If it’s not a mammal then it’s not a human”. There are many types of logical equivalences and sometimes it’s helpful to rephrase in order to make connections or mistakes more apparent. It would be lengthy to go in-depth on this topic but a good way to spot what works and what doesn’t is to think of categories as it was explained in the previous articles.

Example:
“If it’s a human then it’s a mammal” clearly states that the human category belongs to the mammal category, hence,“If it’s not a mammal then it’s not a human” can only be true and equivalent. Other true phrases can be derived from this information even though they aren’t logically equivalent, like “if it’s a mammal it might be a human” and “if it’s a non-human we have no idea if it’s a mammal or not”.

Fallacies

Fallacies are reasoning errors, sometimes intentional sometimes not, it’s not very difficult to identify such errors but an easy trick is questioning ourselves something along the lines of “what does this have to do with what we’re discussing?” or “in what situations is this statement demonstrably false?”. Some of these errors are so common that names were attributed to them, I will list a few:

Burden of proof fallacy – When someone says that their own claim is up to others to prove or disprove.
Ad hominem fallacy – When someone attacks a person (usually with insults or undermining it’s credibility) instead of their argument.
Appeal to ignorance fallacy – When both parties cannot prove that something is or isn’t true one of the parties will assume that his claim is correct by default.
Bandwagon fallacy – When someone says that because a lot of people do and/or believe in something it must, therefore, be correct.
The fallacy fallacy – When someone says that because it’s opponent’s reasoning contains a fallacy then it’s claim must be deemed false.
Cherry picking fallacy – When someone picks data to support their claim while ignoring data that goes against it.
Red herring fallacy – When someone creates a distraction to shift the focus of a discussion.

And many more…

Standard of proof

Should you believe everything people tell you? Of course not, it would be very mentally challenging and very unproductive to do so, but, it would also be extreme and very unproductive to never believe anything, so, where do you draw the line between what you don’t know, what you think you might know and what you think you know, the barely/very/extremely likely or unlikely? In short, when are you certain enough to believe something?

The amount of evidence it takes for you to make these decisions is called a standard of proof, it’s not very helpful to define to yourself what would your standard be for everything you come across but it’s important that it’s reasonably reachable. If there’s a topic that can be subjected to rational inquiry and you don’t have a standard of proof you’ll either believe everything people say about it (be a mindless drone) or you won’t believe in anything about it (be a closed-minded person).

Standards of proof are used all the time in courts depending on the nature of a case, ranging from “more likely than not” to “beyond a reasonable doubt”, however, unlike courts that consider people “innocent until proven guilty” and therefore make the burden of proof rely heavily on the accusation, science and philosophy don’t hold that position for claims, if you claim it, the burden of proof is on you.

Occam’s razor

Occam’s razor is part of a group of tools called philosophical razors which are typically single phrase rules of thumb meant to cut through clutter to get to the bottom of things, it’s also the only razor I’ll mention here even though other useful razors exist. Philosophical razors, just like real razors, should not be used carelessly, they are useful but sometimes they can also cut important things, no conclusion from the usage of a philosophical razor is guaranteed.

Occam’s razor, specifically, is a principle for elegance and simplicity that states that “Among competing hypotheses, the one with the fewest assumptions should be selected”, this means that if two hypothesis have the same power for explaining a given observation the one that assumes less unproven things is considered the most likely candidate by default.

Example:
The car has flat tire.
Option 1 – We ran over something sharp that punctured the tire
Option 2 – Someone drove something sharp into the tire thus puncturing it
Occam’s razor – Both options have the same explanatory power, however, option 1 assumes one thing while option 2 assumes two things, therefore, option 1 is the most likely.

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