Language, Logic and Intuition
We make many decisions during the course of the day. Sometimes these decisions are guided by emotion, sometimes we just rely on a hunch, sometimes we rely on experience, and sometimes we analyze a situation logically and make a decision according to this logical analysis. But very few things in life are easy to analyze in a completely logical way; in most cases, our actual decisions are based on a combination of emotion, experience, and a little bit of logic.
However, when we want a conclusion that isn’t based on any emotion, or hunch, we want a conclusion that is arrived at purely by means of logical argument. This site is devoted to showing how many results that are commonly accepted as being the result of a completely logical argument are in fact flawed because of the failure to acknowledge the significance of the way language is used in the argument - a seemingly innocuous statement can contain subtle errors which render the statement illogical. Unless every aspect of a statement is very carefully analyzed with regard to the use of language by the statement, an ostensibly logical statement may actually contain subtle errors of logic. Even fairly innocuous looking statements can be difficult to analyze, see Natural Language and Reality.
This site explains how intuitive errors may occur; in most cases it is because insufficient attention has been given to the use of language. If you are visiting this site for the first time, I suggest these pages as suitable starting points:
In principle, a logical argument should never rely on an unstated intuitive assumption. It is well known that intuition can lead to erroneous results, and that there are many examples of this having happened. So it should be the case that every logical argument should be carefully examined to ensure that it contains no intuitive assumptions. But there seems to be a blind spot when it comes to the possibility that the way that language is used in an argument might affect the validity of the argument. This possibility is commonly dismissed without any justification for its dismissal. But everything that is referred to by a logical argument must be referred to by symbols that belong to some language. And since that is the case, the fact that those symbols belong to some language is an inherent part of the argument, and is not something that can simply be ignored.
Much of this website deals with the confusion that occurs when levels of language are not clearly delineated. Kurt Gödel set the ball rolling on this in 1931 with his incompleteness theorem which hides its language confusion under an impressive looking facade of complexity. Amazingly, it has long being accepted as correct even though Gödel never actually proved the crucial step in his proof, and although his proof leads to a blatant contradiction, see Gödel’s contradiction. And over the years since that there seems to be an alarming increase in the willingness of certain academics to forgo the need for clear precise logical proofs of any claim, and now there are numerous people who like to call themselves “logicians”, but who are content to simply make a crucial assumption rather than actually make an attempt to prove it, and proceed to base an entire structure of claims based on that assumption. That assumption is that a completely formal language can actually reference itself - that is, that within a completely formal language there can be a sentence that explicitly refers to that entire sentence itself.
Despite their self-appellation as “logicians”, that isn’t logic, and the inane results of these assumptions aren’t logical - they are worthless. For an example of this sort of nonsense, see Halbach and Zhang: Yablo without Gödel.
As site owner I reserve the right to keep my comments sections as I deem appropriate. I do not use that right to unfairly censor valid criticism. My reasons for deleting or editing comments do not include deleting a comment because it disagrees with what is on my website. Reasons for exclusion include:
Frivolous, irrelevant comments.
Comments devoid of logical basis.
Comments with excessive number of different points.
Questions about matters that do not relate to the page they post on. Such posts are not comments.
Comments with a substantial amount of mathematical terms not properly formatted will not be published unless a file (such as doc, tex, pdf) is simultaneously emailed to me, and where the mathematical terms are correctly formatted.
Reasons for deleting comments of certain users:
Bulk posting of comments in a short space of time, often on several different pages, and which are not simply part of an ongoing discussion. Multiple anonymous usernames for one person.
Users, who, when shown their point is wrong, immediately claim that they just wrote it wrong and rewrite it again - still erroneously, or else attack something else on my site - erroneously. After the first few instances, further posts are deleted.
Users who make persistent erroneous attacks in a scatter-gun attempt to try to find some error in what I write on this site. After the first few instances, further posts are deleted.
Difficulties in understanding the site content are usually best addressed by contacting me by e-mail.
Note: a password enables editing of comments, an email enables notification of replies