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@@ -8,7 +8,7 @@ The term "critical thinking" gets thrown around a lot in schools, but children a
 
 The point of teaching the formal, symbolic logic starting at a young age is not so kids, teens and young adults become good at truth tables. The point is they'll internalize logic like any other concept. The pattern recognition part of their brain will automatically recognize valid arguments when they see them. It will also recognize invalid forms of argument and logical fallacies without consciously doing any heavy lifting. That's where the most value is in teaching logic.
 
-When I studied philosophy in community college, I remember there was an art student. He had a great personality and was a very likable person. Whenever he got called on to answer a question though, he was never able to produce the right answer. It was clear to me that he never learned how to think logically. I wondered what it must be like to be a young adult never having learned that. There are also plenty of functioning older adults out there that never learned how to think logically. To be clear, studying formal logic isn't a prerequisite for logical thought. What I find to be the case with nearly everyone without training in formal logic is that they have an intuitive sense of how to reason, but there's important pieces of the puzzle they're missing. That's what I'm going to focus on in this post, the things that those without experience in formal logic get confused about. In my posts, I try not to assume prior knowledge, so I'm going to explain a bit about logic before I explain some of those missing pieces. If you're already familiar with logic, click here.
+When I studied philosophy in community college, I remember there was an art student. He had a great personality and was a very likable person. Whenever he got called on to answer a question though, he was never able to produce the right answer. It was clear to me that he never learned how to think logically. I wondered what it must be like to be a young adult never having learned that. There are also plenty of functioning older adults out there that never learned how to think logically. To be clear, studying formal logic isn't a prerequisite for logical thought. What I find to be the case with nearly everyone without training in formal logic is that they have an intuitive sense of how to reason, but there are important pieces of the puzzle they're missing. That's what I'm going to focus on in this post, the things that those without experience in formal logic get confused about. In my posts, I try not to assume prior knowledge, so I'm going to explain a bit about logic before I explain some of those missing pieces. If you're already familiar with logic, click here.
 
 # Logic
 Logic is the study of [rules of inference](https://en.wikipedia.org/wiki/Rule_of_inference). Rules of inference allow you to draw conclusions based on premises. In other words, starting with a statement A, you can conclude statement B. For example, the earth is round is a true statement. Therefore the earth is round or up is down is also a true statement. In fact, I could replace the statement up is down with any proposition Z and the earth is round or Z would still be true. I used the rule of inference "addition" to draw my conclusion, so I'm guaranteed that it's true no matter what Z is. I can apply another rule of inference to get humans have 3 legs therefore either the earth is round or up is down. That is also a true statement. It sounds strange because the normal way of understanding "therefore" is as a causal relationship. In this context, it's a strictly logical implication, not causal. Despite how strange it sounds, humans have 3 legs therefore either the earth is round or up is down logically follows from the earth is round.
@@ -19,10 +19,10 @@ To test your skills in logic, I suggest trying out some logic puzzles such as [K
 Now that I've talked about what logic is, I want to talk about some of the important aspects of logic people commonly get confused about.
 
 ## There Are Only 2 Ways an Argument Can Be Disproved
-The first way to disprove an argument is by showing that one of the premises is false. The other way is showing that [the structure of the argument is invalid](https://rationalwiki.org/wiki/Logical_validity). People are used to thinking of arguments in terms of "arguments for" and "arguments against". That's why it's easy to get confused here. It's the attitude "There's some good arguments for a proposition and some good arguments against it and it's my job to weigh the pros and cons". But, in logic, an argument is either sound or unsound. The property of [soundness](https://rationalwiki.org/wiki/Soundness) means that the premises are true and it has valid form. If the conclusion of an argument derives from valid rules of inference based on the premises, then the only way to disprove the argument is to show one of the premises is false. If all the premises are true and the form is valid, then the argument is sound and the conclusion is true. There's no "arguments for" and "arguments against", or "maybe it's wrong some other way". There's no two ways about it. No if, ands or buts. If an argument is sound, the conclusion necessarily follows.
+The first way to disprove an argument is by showing that one of the premises is false. The other way is showing that [the structure of the argument is invalid](https://rationalwiki.org/wiki/Logical_validity). People are used to thinking of arguments in terms of "arguments for" and "arguments against". That's why it's easy to get confused here. It's the attitude "There are some good arguments for a proposition and some good arguments against it and it's my job to weigh the pros and cons". But, in logic, an argument is either sound or unsound. The property of [soundness](https://rationalwiki.org/wiki/Soundness) means that the premises are true and it has valid form. If the conclusion of an argument derives from valid rules of inference based on the premises, then the only way to disprove the argument is to show one of the premises is false. If all the premises are true and the form is valid, then the argument is sound and the conclusion is true. There's no "arguments for" and "arguments against", or "maybe it's wrong some other way". There's no two ways about it. No if, ands or buts. If an argument is sound, the conclusion necessarily follows.
 
 ## How Logical Fallacies Work
-A logical fallacy is an error in reasoning. It can be [formal](https://en.wikipedia.org/wiki/Formal_fallacy) or [informal](https://en.wikipedia.org/wiki/Informal_fallacy). Formal fallacies have to do with the structure of an argument. If an argument has bad structure, it is invalid. Informal fallacies have to do with the content of an argument. In my experience, it's more rare for people to commit formal fallacies. This is because there are so many more ways to commit informal fallacies than there are ways to commit formal fallacies. There are only a few ways to structure an argument improperly, but there are virtually endless ways to get the content wrong since the content can be anything at all. Take a look at [yourlogicalfallacyis.com](https://yourlogicalfallacyis.com). It's good to become familiar with informal fallacies by name and be able to call them out in real time. To challenge yourself, try doing that during a live presidential debate. There's so many logical fallacies in those it's impossible to keep up, at least for me.
+A logical fallacy is an error in reasoning. It can be [formal](https://en.wikipedia.org/wiki/Formal_fallacy) or [informal](https://en.wikipedia.org/wiki/Informal_fallacy). Formal fallacies have to do with the structure of an argument. If an argument has bad structure, it is invalid. Informal fallacies have to do with the content of an argument. In my experience, it's more rare for people to commit formal fallacies. This is because there are so many more ways to commit informal fallacies than there are ways to commit formal fallacies. There are only a few ways to structure an argument improperly, but there are virtually endless ways to get the content wrong since the content can be anything at all. Take a look at [yourlogicalfallacyis.com](https://yourlogicalfallacyis.com). It's good to become familiar with informal fallacies by name and be able to call them out in real time. To challenge yourself, try doing that during a live presidential debate. There are so many logical fallacies in those it's impossible to keep up, at least for me.
 
 The thing people get confused about when they're unfamiliar with logical fallacies is they think fallacies are a minor problem for an argument, similar to the "arguments for" and "arguments against" I talked about earlier. They see the fallacy as the "argument against" part. That's completely the wrong way to think about logical fallacies. The presence of a single logical fallacy in an argument means that argument is toast. A logical fallacy is not a "counterpoint" to an argument. It fully invalidates the argument. An entirely new argument is needed to prove the conclusion.
 
@@ -78,6 +78,6 @@ I've said that for an argument to be valid, the premises must be true. But how d
 In the early 1920's, famous German mathematician [David Hilbert](https://en.wikipedia.org/wiki/David_Hilbert) put forward a proposal calling for the axiomatization of mathematics. He wanted to make all mathematical truths reducible to an agreed upon set of axioms such that all true statements could be proved, but no false statements could be proved. In 1931, one of the most significant logicians in history, [Kurt Gödel](https://en.wikipedia.org/wiki/Kurt_G%C3%B6del), showed that no set of axioms is capable of proving all truths about the arithmetic of natural numbers. See [Gödel's Incompleteness Theorems](https://stopa.io/post/269). Gödel used mathematical logic to show that there are some places mathematical logic cannot go. Boiled down, he proved that logic cannot prove everything. This is also true in computing. See [The Halting Problem](https://en.wikipedia.org/wiki/Halting_problem). The essence of the trick seems to be, no matter which logic you're talking about, to find a way to encode [the liar paradox](https://rationalwiki.org/wiki/Liar_paradox) in the system. A prerequisite for that is somehow getting the logical system to talk about itself. Gödel found a very fascinating theorem and I would recommend for anyone interested to look more in depth at it.
 
 # Conclusion
-That's all I've got for this post. I think I've packed in a lot of information and good examples to research. Even if you never learn logic, I believe by reading this post you get a sense of what logic is all about and how to at least recognize some common informal fallacies and misunderstandings. I tried to include plenty of useful external links. This post is barely scratching the surface though. For some readers, just scratching the surface is good enough. But for all I know, the next Gödel might be reading this. In 2011, [a 25-year old math problem about superpermutations was solved by an anonymous 4chan user](https://yewtu.be/embed/OZzIvl1tbPo?local=true). If that doesn't show that cleverness can come from anywhere, I don't what does.
+That's all I've got for this post. I think I've packed in a lot of information and good examples to research. Even if you never learn logic, I believe by reading this post you get a sense of what logic is all about and how to at least recognize some common informal fallacies and misunderstandings. I tried to include plenty of useful external links. This post is barely scratching the surface though. For some readers, just scratching the surface is good enough. But for all I know, the next Gödel might be reading this. In 2011, [a 25-year-old math problem about superpermutations was solved by an anonymous 4chan user](https://yewtu.be/embed/OZzIvl1tbPo?local=true). If that doesn't show that cleverness can come from anywhere, I don't what does.
 
 I hope you enjoyed the post. If there's anything that you think I should have covered in this post or that I should talk in the future, [let me know about it](mailto:nick@nicholasjohnson.ch).