# Circle logic

What I call ‘circle logic’ (not to be confused with circular reasoning) is the situation in which the poles (the two extremes on a linear interval) exhibit the same value of the parameter in question. For example, a patient might not complain because the patient is comfortable and doing well, or because the patient is simply too weak to do so. Of course, this situation can be modeled by any (usually, nonconstant) even function, but it is convenient to choose a well-known function on a finite interval, such as the trigonometric sine function for angle values between -180 degrees and 0 degrees, or a single arch of a cycloid or inverted cycloid.
Another example of circle (or, trig sine) logic: If a fellow invites his girlfriend to dinner at a posh restaurant, it is either to propose marriage, or to break up with her.
Another example:
“Too much or too little wine.
Give him none, he cannot find
truth; give him too much, the
same.”
— Blaise Pascal
Another example: why an item is not found on the store shelves: either because it is so unpopular that it is not worth carrying it, or because it is so popular that it disappears quickly from the shelves by consumers purchasing it.
Another example: You open something slowly fif it is very good, or very bad.

# fif

I use the abbreviation ‘fif’ for ‘if, and only if,’.
Because of the implied necessity of things in foreground, the expression ‘if, and only if,’ is seldom casually encountered, even when its sense is present.
Having an abbreviation for the cumbersome expression ‘if, and only if,’ is not a new idea. What has been traditionally used is ‘iff’, but this has the disadvantage of not being aurally distinct from ‘if’. My abbreviation is aurally distinct from ‘if’.

# Implied necessity of things in foreground

Deliberately placing something in foreground is called posting, and what is so placed is called a post. A key point is that anything in foreground is automatically taken as necessary. This is not difficult to understand. After all, foregrounding something is saying it’s worth the attention of others, and all of us are busy, so why bother us about something that isn’t necessary? (So, the implied necessity of things in foreground can be taken as an theme within Social Psychology.) Therefore a post stating that something is sufficient also means, unless explicitly so stating to the contrary, that it is necessary. That is why in public discourse one seldom sees the expression ‘if, and only if,’: the ‘and only if’ part is universally presumed. Textbooks are very much part of the public discourse, and that is why, even in mathematics, definitions are of the minimal form ‘if’, rather than of the full form ‘if, and only if’, for example, ‘x is a prime if x is an integer > 1 such that…”. Another example is the sign posted for day workers informing them that they can cash their check at a certain convenience store. If you don’t understand the implied necessity and try to cash your check elsewhere, even at the bank, you will find that you wasted your time.

# The inverted cycloid

reference 1 (isochronous pendulum clocks)
reference 2 (Calculus lesson on the the tautochrone and brachistochrone)
Also, the inverted cycloid can give, at least qualitatively, the amount of relevance of something as a function of time, such as that of a cooking pot.