I hope you guys enjoyed the April Fools post as much as I enjoyed writing it. The part about finding life in the sub-glacial lakes was true, but I have no knowledge about whether UCLA is planning an expedition, and to my knowledge the 1930s expedition did not occur outside of Lovecraft's novella.
Today I'd like to talk about logic. Symbolic logic is the art of creating a formal system (that is, a system where strings of symbols can have their forms changed according to specific rules) that that matches with specific laws of logic. Aristotle is famed for, among other things, creating an early system of symbolic logic which could parse statements like, "All cats are mammals," and, "All mammals are vertebrates," coming to the conclusion that, "All cats are vertebrates," and, "Some vertebrates are cats," held true while, "All vertebrates are cats," did not. What was so amazing about this was that only rules for organizing symbols were used to come to these conclusions from the premises; of course any human with a left frontal lobe could tell you which conclusions were true and which weren't, but creating a set of rules which tell you this automatically was something worth a bit of bedazzlement.
Any logical problem dealt with a system of premises (things known or accepted as true) and conclusions (things that could be deduced from the premises according to the rules of the formal system), and certain rules were assumed in order to create a system where conclusions could be derived from premises at all. Two of Aristotle's assumptions which have become more contentious in the past few centuries are called the Law of Non-Contradiction and the Law of Excluded Middle.
The Law of Non-Contradiction stated that, in any system of premises and derivable conclusions, there must be no case where both a statement and its negation are both held to be true. For instance, if we had a set of premises that included the statements, "Politicians are in it for the money," and "Politicians want to make the world a better place," we could derive "Politicians are not in it from the money" from the second statement and we'd have a contradiction, which is not allowed per said law. Any logician worth their salt would quickly resolve the issue by adding the word, "Some," to both premises. However, there are some situations where a work-around like this cannot be performed, such as with the premise, "This premise is false." The statement must be false in order for it to be true, so we could derive both, "Premise A is true," and, "Premise A is not true," violating the Law of Non-Contradiction.
The Law of Excluded Middle is a lot like the previous law seen from the other side: it states that in any system there must be no case where both a statement and its negation are both held to be false. In other words, well-formed statements must be true or false, and there is no middle ground (hence the name).
In order to see where this law collapses, we'll have to look at Grelling's paradox. To find this paradox, we create a new word. The first is, 'autological,' and it's define so that a term that is autological if it's definition describes or implies itself. So words like 'pentasyllabic,' 'English,' 'fran ais,' 'sesquipedalian,' and 'last' (being at the end of this list) are all autological, while 'abbreviation,' 'anglais,' 'long,' 'first' (in this case), and 'onomatopoeia' are not. This second list is particularly illustrative of what it means to be the opposite of autological, but more mundane words like 'cat' and 'tarp' are not autological either.
Now, is the word 'autological' itself autological? Since the word implies implying itself, we could safely say that it is, but if we said that it was not, there would be nothing to disprove our assertion, since the word would already have to be assumed to be autological in order to imply itself, itself. That's quite a garble of a sentence, but try it. Assume 'autological' is autological, and any way you can come at the question, your assumption appears true. Assume it's not autological, and your assumption seems just as true. As you can see, this is a case where the Law of Excluded Middle does not apply.
There are various ways logicians have dealt with these two problems, including postulating an 'Indeterminate' truth value aside from 'True' and 'False,' but my favorite way to deal with it is to add two complex truth values to the simple two. The first one I like to call 'Nix,' and it applies to statements like, 'This statement is false,' where whatever you assume in order to evaluate the statement comes up wrong. The second one I call 'Mu,' after the Chinese way of 'unasking' a question - it implies that the question is flawed in that an answer to it must be based on hidden assumptions rather than the quality of the thing itself.
As an example of a 'Mu' statement, consider, 'Cindy is a hard person to get along with.' Now, it may be the case that the Cindy in question here is sensitive to people's expectations, and feels put off when people tense around her. She can be quite friendly when approached with friendliness, but other times she can be a really difficult person. If you approach her with the belief that she's easy-going, she'll mirror your relaxation and you'll find yourself proven right. On the other hand, if you brace yourself for difficulty with her, you're going to have a bad time.
In the next article in this series, we'll talk more about situations where the 'Mu' truth value is useful, about where the names came from, and a bit about Nix's applicability to life. I'd like to end with a quote.