I believe that when we examine virtually any subject, entailment is an important relation which may exist between "things". I've placed this word in scare quotes in order to hopefully suggest that even the apparently inert material objects of our everyday experience may be perceived as events, processes, or relations. Perhaps if I attempt to show why I believe entailment is important then the meaning of the word will also be granted clarity.
Among the reasons why I hold this relation valuable is because if things are related by entailment (i.e. this entails that, or, some thing entails another thing) then truth is preserved across this relation. That is, if we may justifiably believe that fire entails heat, air, and fuel and we also know that fire exists (or is true, or obtains in nature) then we may deduce or conclude the existence of heat, air, and fuel. Or, if we know that the handling of roses entails the risk of thorns and we know someone is handling roses then we may conclude that the risk of thorns exists.
Another reason that I believe entailments are valuable, useful, or relevant relations in a variety of domains is due to their connection with questions which ask, "why?", and the associated answers, "because...". And I suspect upon reading this, some here may be reminded of Aristotle. When we inquire into something by asking "why?" about it and we find answers "because...", we have exposed an entailment. For example, if we ask "why are the planets in orbit around the sun?" and we are able to answer in part "because of gravity" then we may assert that "gravity entails the orbits of planets around the sun." As we consider causes in nature, we are considering natural entailments.
Alright, I wrote the above in order to hopefully address the importance and ubiquity of entailment. In the next section, under the reader's continued patience, I hope to address the expression of the relation. Because it satisfies categorical criteria, we will delve into some mathematical notation. And I'll note at this point that if the reader examines all the statements made in opening post, so far, then they may see that all of them are expressions of entailment. That is, if one cares to look then they will see I have generally attempted to follow the categorical form of...
f : A --> B
f ==> (A --> B )
Under this notation, "f" is the name of a map, "==>" is that which performs or produces the mapping, "A" is the antecedent or domain of the map "-->" is the map, "B" is the consequent or codomain of the map.
Examples of "f" here may include: I believe, I think, I hold, I suspect, I hope, in my opinion, my thinking is, in other words, perhaps, for example, or it may be assumed, or unstated.
Examples of "==>" here may include: that, reason that, under, or of.
Examples of "A" and "B" are comprised of various statements or assertions.
Examples of "-->" here are associated with expressions such as: implies, entails, because, if... then..., as, when, since, upon, gives rise to, requires, necessitates, is due to, thereby, in order to, thus, or it may be unspoken, or a comma.
As we examine an expressed assertion of entailment, we may apply skepticism to it and thereby doubt every aspect of it. That is, under our skeptical examination, we may doubt the validity of: the map's name, the process which produces the map, the antecedent, the map, and the consequent. When I express myself to others according to this form and they, in their turn, express skepticism to me about these assertions then, among other things, it allows me to deduce things about the skeptics themselves. I believe when this form is used, it empowers my reader and me both. I hope that if I employ this general form more frequently, others will also sieze upon it and use it for themselves and thus empower us all.
When writing that poem, another hope was with me that perhaps if others begin to consider entailment then they will one day explain mathematical and natural paradoxes to me. In my studies, it has become ever more plausible to me that paradox is a critical aspect of our world's complexity.
As I suspect that some few members here are swine, and I anticipate nasty, irrelevant or dismissive remarks from them, please allow me first to spit on my own work...
I believe chickens cross the road in order to get to the other side.
Edited by Legion, 19 June 2012 - 12:46 PM.