Entailment: Its Ubiquity, Importance, And Expression

[Legion]

After writing a poem the other day, Rank Stranger requested that I try to express the thoughts associated with the poem in a more prosaic fashion. What I have written here is an attempt to fulfill his request.

-------------------------

 

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

 

or

 

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.

[bornagainathiest]

Hi Legion!

 

Interesting thoughts.

 

So, here's a handful of questions for you - ones that stem from this thread...

http://www.ex-christ...ed/page__st__60

...where you wrote about our universe's complexity and paradoxical nature.

 

1

.

Can the use of entailments tell us anything useful about nature or do the closed loops of implications (paradoxes) that you say exist in nature, make that impossible?

 

2

.

Does the intrinsic complexity and paradoxicity (!?) of nature prevent us from using reason to understand it? If not, why not? If so, how?

 

3

.

Why won't inferential systems of reason permit closed loops of implication (paradoxes)?

 

4

.

You say that nature is complex and paradoxical. Does that therefore mean that a paradox-friendly system of thought (entailments, perhaps?) is needed to understand it?

 

5

.

The very earliest state of the universe's evolution is it's purest, most pristine and least complex (simplest) condition we know of. Therefore, (presumably) it must also be the least paradoxical natural system we can possibly study.

It must also (presumably) be the system with the least number of possible entailments.

Here are some relevant links.

http://en.wikipedia....ki/Planck_epoch

http://www.aip.org/p...plit/728-1.html

http://www.symmetrym...k-gluon-plasma/

During the Planck epoch there were no such things as the chemical elements, no compounds of them, no atoms, no protons and neutrons (which make up the nucleus of any atom) and nothing had evolved into anything else. Insufficient time had elapsed (10

⁻⁴³

seconds) and the ambient temperature (2 trillion degrees Kelvin) was too high for anything other than quark-gluon 'soup' to exist. Only entailments relevant to these particles (Quarks and Gluons) can possibly apply at this stage in the universe's evolution.

http://en.wikipedia.org/wiki/Quark

http://en.wikipedia.org/wiki/Gluon

This is the simplest our universe ever was.

Now for the question...

 

Would an inferential system of reason be able to meaningfully describe this - a condition of ultimate cosmic simplicity? If not, why not?

 

Thanks,

 

BAA.

[Legion]

Hey there BAA!

 

Let me attempt to answer some of these. I may not be able to do it well, because I'm still learning about entailment, and my comprehension of paradox is very poor. But I'll try.

 

1. I believe entailment provides an excellent view into causality. I think an examination of natural entailment is an examination of causes in nature. Formal (e.g. mathematical) entailment is implication or inference. I don't believe that natural loops of entailment (paradoxes) prevent us from understanding nature, but they may imply that a complete, exhaustive, undertanding of nature is an impossibility. In other words, we can understand ever more facets and depths of nature, but not the whole thing.

 

2. No. I believe we may understand natural paradox by using mathematical paradox to mirror it.

 

3. Some inferential systems will permit loops of implication, but not the formal systems employed by reductionism.

 

4. See #2 above.

 

Thanks for your participation BAA.

[Guest Valk0010]

Well, I am not sure I follow what your saying, because I tend to see things from a different set of what one might call, first principals. So my post will address that base disagreement. Firstly, I think your view of "entailment" relies more heavily then one should on determinism. The two main reasons I doubt determinism and via that causality(for the universe entirely) and your definition of entailment (which as presented is a awfully similar concept) is that, its purely speculative to say things like, I am typing this on my computer right now, because some prior event made me. Its just something you can't know unless one is omniscient. Secondly, we have no way of really understanding the value of action. How could one say murder is wrong, if its cause is some previous morally neutral prior event.

 

Now as to entailment specifically and not issue's surrounding it, I have the following to say. Wouldn't it have a limit as to what is the material world, and would then if one takes a libertarian view to free will or a compatibilist view(the view I see as making the most sense) be in many ways not able to agree with what your saying? I am not sure what your views on are say, the human mind and how it came about. But I suspect, that and the associated things around it, are the one big thing that entailment or causality would reach its limit on.

 

Now as to, in principal, the cause of something helping to determine what something is. I agree, that is sometimes a necessary thing. I am just not sure if your taking something that is limited and trying to peg it unnecessarily and perhaps incorrectly to the universe.

[Galien]

 

 

I believe chickens cross the road in order to get to the other side.

 

Why couldn't you just say that in the first place instead of all the other waffle?

[Galien]

Well, I am not sure I follow what your saying, because I tend to see things from a different set of what one might call, first principals. So my post will address that base disagreement. Firstly, I think your view of "entailment" relies more heavily then one should on determinism. The two main reasons I doubt determinism and via that causality(for the universe entirely) and your definition of entailment (which as presented is a awfully similar concept) is that, its purely speculative to say things like, I am typing this on my computer right now, because some prior event made me. Its just something you can't know unless one is omniscient. Secondly, we have no way of really understanding the value of action. How could one say murder is wrong, if its cause is some previous morally neutral prior event.

 

Now as to entailment specifically and not issue's surrounding it, I have the following to say. Wouldn't it have a limit as to what is the material world, and would then if one takes a libertarian view to free will or a compatibility view(the view I see as making the most sense) be in many ways not able to agree with what your saying? I am not sure what your views on are say, the human mind and how it came about. But I suspect, that and the associated things around it, are the one big thing that entailment or causality would reach its limit on.

 

Now as to, in principal, the cause of something helping to determine what something is. I agree, that is sometimes a necessary thing. I am just not sure if your taking something that is limited and trying to peg it unnecessarily and perhaps incorrectly to the universe.

 

Does that actually mean anything?

[Guest Valk0010]

Well, I am not sure I follow what your saying, because I tend to see things from a different set of what one might call, first principals. So my post will address that base disagreement. Firstly, I think your view of "entailment" relies more heavily then one should on determinism. The two main reasons I doubt determinism and via that causality(for the universe entirely) and your definition of entailment (which as presented is a awfully similar concept) is that, its purely speculative to say things like, I am typing this on my computer right now, because some prior event made me. Its just something you can't know unless one is omniscient. Secondly, we have no way of really understanding the value of action. How could one say murder is wrong, if its cause is some previous morally neutral prior event.

 

Now as to entailment specifically and not issue's surrounding it, I have the following to say. Wouldn't it have a limit as to what is the material world, and would then if one takes a libertarian view to free will or a compatibility view(the view I see as making the most sense) be in many ways not able to agree with what your saying? I am not sure what your views on are say, the human mind and how it came about. But I suspect, that and the associated things around it, are the one big thing that entailment or causality would reach its limit on.

 

Now as to, in principal, the cause of something helping to determine what something is. I agree, that is sometimes a necessary thing. I am just not sure if your taking something that is limited and trying to peg it unnecessarily and perhaps incorrectly to the universe.

 

Does that actually mean anything?

I may have done a bad job at conveying about 4 different points that deserve there own pages in a two paragraphs. But any question as to what I mean?

[bornagainathiest]

Hey there BAA!

 

Let me attempt to answer some of these. I may not be able to do it well, because I'm still learning about entailment, and my comprehension of paradox is very poor. But I'll try.

 

Attaboy!

 

1. I believe entailment provides an excellent view into causality.

 

May I ask where you derive this belief from? Life experience? Something you've read? Something else?

 

I think an examination of natural entailment is an examination of causes in nature.

 

Ok. But can you justify that statement in anything more than generalized terms? That is, can you use the process of entailment to pinpoint exactly how something is caused in nature? I ask because, if nature is both complex and paradoxical, then won't your description (using entailments) necessarily be both complex and paradoxical? And, being complex and paradoxical, won't your description then be just as difficult to comprehend as the very causes you are trying to understand?

 

(Sorry Legion, but part of me suspects this is why humans need the necessary simplification of formal systems like maths.)

 

Formal (e.g. mathematical) entailment is implication or inference. I don't believe that natural loops of entailment (paradoxes) prevent us from understanding nature, but they may imply that a complete, exhaustive, undertanding of nature is an impossibility. In other words, we can understand ever more facets and depths of nature, but not the whole thing.

 

As you see (from the above) I'm inclined to agree. But if we can only glimpse the reality of nature 'thru a glass, darkly' does the use of entailment actually provide, "an excellent view into causality" as you write or is it just another tool, no better and no worse than, say, an inferential formal system like maths?

 

2. No. I believe we may understand natural paradox by using mathematical paradox to mirror it.

 

Could you describe a worked example of this please?

 

3. Some inferential systems will permit loops of implication, but not the formal systems employed by reductionism.

 

Which inferential systems will permit these loops of implication and why? (Worked examples?)

Which are the formal systems used in reductionism and why won't these permit implicational loops?

 

4. See #2 above.

 

Thanks for your participation BAA.

 

No problem!

 

BAA.

[Legion]

BAA, this is a bit of a delyed response, but I suppose it's better late than never.

 

I think when we ask "why does phenomenon X exist?" and if we are able to answer, for instance, "because of phenomenon Y" then we have revealed an entailment of X by Y, or Y entails X, or Y --> X. This entailment has a bearing on the causes of X. And the many causes can be grouped according to whether they are matrial, efficient, formal, or final causes.

 

In my mind, complex implies paradoxical. So if I use a phrase such as "complex and paradoxical" then I'm being reduntant.

 

To my knowledge, biologist Robert Rosen was the first person to employ mathematical paradox to model natural paradox when he created metabolic, repair models of organisms. I think his first use of these was published in a 1958 paper. But he later simplified their expression as two categorical maps, where metabolism is expressed as f:A-->B and repair is expressed as phi:B-->F.

 

I don't know of all the inferential systems which will support or permit loops of inference, but category theory is among them. Typically a model produced through reduction employs the notion of a state space S and state transitions T such that the model can be expressed as t:S-->S.

[bornagainathiest]

BAA, this is a bit of a delyed response, but I suppose it's better late than never.

 

I think when we ask "why does phenomenon X exist?" and if we are able to answer, for instance, "because of phenomenon Y" then we have revealed an entailment of X by Y, or Y entails X, or Y --> X. This entailment has a bearing on the causes of X. And the many causes can be grouped according to whether they are matrial, efficient, formal, or final causes.

 

In my mind, complex implies paradoxical. So if I use a phrase such as "complex and paradoxical" then I'm being reduntant.

 

To my knowledge, biologist Robert Rosen was the first person to employ mathematical paradox to model natural paradox when he created metabolic, repair models of organisms. I think his first use of these was published in a 1958 paper. But he later simplified their expression as two categorical maps, where metabolism is expressed as f:A-->B and repair is expressed as phi:B-->F.

 

I don't know of all the inferential systems which will support or permit loops of inference, but category theory is among them. Typically a model produced through reduction employs the notion of a state space S and state transitions T such that the model can be expressed as t:S-->S.

 

Thanks for the reply Legion!

 

Now, I hope you won't take this the wrong way, but I'd like to speak plainly and openly to you about certain things you've written recently in this thread and elsewhere in the forum. In short, I'd like to take a critical look at these things. Yes, it's a difficult thing when someone is critical of what you've written and it's often hard to take, however please be assured that my intention is to understand and not to get at you personally.

 

What I'd like to do is to apply the same kind of questioning I'd put to a Christian apologist. Just as I test their claims and assertions to see if they really do bear up under scrutiny, so I'd like to do the same with your words. As I said before, this isn't personal, ok?

 

If you think I'm being unfair or if I'm misrepresenting you in any way, please let me know - I don't want to do you a disservice. I value your presence in this forum and I don't want to rub you up the wrong way or make an enemy of you. That is NOT my intention.

 

I hope you're cool with this.

 

Thanks,

 

BAA.

[Legion]

BAA, because I have spare time, , I wanted to attempt to show how something like the motion of a falling body can be expressed in categorical terms as an endomap and examine causes relative to this expression.

 

If in the category of sets, we let S be the set of all the momenta of an object, and let the set map t be the time of gravity's action, then we may write t:S-->S to express the dynamics of a falling body.

 

At any instant the object has a given momentum, called here m_c, in S, and we may ask, "why does this current momentum exist?" We may answer "because of a previous momentum, m_p, in S", and "because of the time of gravity's action, t". So we may say, t entails that m_p is transformed into m_c, or t(m_p)=m_c. This would be in agreement with the expression t:S-->S. The previous momentum here would be analogous to the material cause of the current momentum, and the time of gravity's action would be analogous to the efficent and formal causes of the current momentum.

 

I hope what I've said here may give us some reference point for discussion if you so desire.

[Legion]

I hope you're cool with this.

 

I guess it depends on other factors. I don't mind skepticism and I don't mind critical thought. But it does rub me the wrong way when others dismiss these considerations without having given them due critical thought.

 

So please be skeptical. Please engage critical thought. And please don't waste my time.

[bornagainathiest]

Thanks, but I'm still struggling to translate what you write into anything 'real'.

 

That's why I've persisted in asking for worked examples. My bad!

I should have asked for worked examples from the real world of atoms, people and planets. You see I just can't make the leap from your letters and notations to the reality I find myself living in.

t:S-->S doesn't mean anything much to me! Sorry 'bout that!

 

You see, for all of it's limitations and formalities, science describes the world I see around me very well. It explains. It reveals processes. It reveals conections. It makes predictions. Sorry again, but I can't see how your entailment loops do any of that.

 

Can you help me out here please?

 

Thanks,

 

BAA.

[bornagainathiest]

Ok Legion,

 

While you're composing a reply, here's the crux of my confusion.

 

If a system of thought can make an accurate prediction about reality, I contend that this is significant, no matter how formal the language it's couched in. So, can your chosen system do this? If so, can you please cite a worked example of it.

 

Please note that I'm not trying to put one system above another. I've already stated that they are probably complementary. The use of entailments no doubt reveals much about causality, in ways that science cannot. However, can it only do this retroactively or does it have predictive power too?

 

I seek to learn what I can, not to waste your time.

 

Thanks,

 

BAA.

[Legion]

Post #11 may be a good place to start BAA. If you direct questions at it, I will try to answer.

[bornagainathiest]

Oh shuggles!

 

Gotta run Legion.

 

Catch you later. Promise.

 

BAA.

[Legion]

The use of entailments no doubt reveals much about causality, in ways that science cannot.

 

Um, I think in so far as causality is synonymous with natural entailment, science IS the study of natural entailment.

[Legion]

Catch you later. Promise.

 

Alrighty.

[RankStranger]

I have a couple of questions along the same lines as what BAA has asked:

 

1. What is gained by making these statements in this formal language rather than just writing them in a sentence?

 

2. I'm of the opinion that just about any action you can name has multiple 'causes'- including known causes, known unknown causes, and unknown unknown causes (I'm channeling Rumsfeld here, but I think he made a valid point). Hell, there might be infinite causes of any given action. And if that's the case, how does entailment lead us anywhere except by implication, intuition, and the like? I just don't see a useful potential here.

[bornagainathiest]

Post #11 may be a good place to start BAA. If you direct questions at it, I will try to answer.

 

Back again!

 

#11?

 

Ummm... with all due respect Legion, I'm with the Rank Stranger when it comes to the content of #11.

 

What's the advantage of my learning a new language from you if exploring causality can be done in plain old English?

Also, since you suggest that science is the study of natural entailment, why not just stick with science?

As I mentioned earlier, science delivers the goods and it's results are (pretty much) accessible to anyone.

Can you see what we're both getting at?

 

Now for the (potentially) unpalatable part of my message. Sorry if this upsets you, but if it's testing you want Legion, testing you'll get.

 

When a Christian apologist makes a claim about the content of the Bible, I expect them to make good on their claim in a way that I can understand. If they tell me that I have to look at scripture in their 'special' way to understand the truth of their claim, then I'm immediately suspicious about their initial claim. I suppose you'd be too, right?

 

Recently Legion, you've made two claims...

 

" I believe entailment provides an excellent view into causality."

 

"I think an examination of natural entailment is an examination of causes in nature."

 

...but (as far as I can see) you've failed to present anything that others (without your 'special' way of understanding) can comprehend as proof of these claims. As such they remain squarely in the realm of your belief-system, inaccessible by anyone else. Thus, we are forced to take it on 'faith' from you that entailment does what you say it does. Please correct me here if you think this is an unfair comparison with said Christian apologist, above.

 

Do you see the dilemma?

We can't understand how entailment works using your 'special' way of notation and you don't seem to be able to step outside of that 'special' way and talk to us in language we'll understand.

 

That's why I persist in asking you for real world examples of this method - because those are things I can get my head around. If you can't produce these examples, then we're back at square 1. I can't see the usefulness of entailment-ology and you can't give me anything other than examples of entailment-ology in the language of entailment, which is entailment-ese.

 

Therefore, something that's a closed book to me remains something that produces no tangible benefits that I can apprehend. Is there a way that you can open the book for me that doesn't require me to take language lessons?

 

Thanks,

 

BAA.

[Legion]

Alright, if you guys can see no value in it, then I guess that's that.

 

I've found others with whom to study this. So it's cool.

[RankStranger]

Alright, if you guys can see no value in it, then I guess that's that.

 

I've found others with whom to study this. So it's cool.

 

I'm willing to be convinced. I just don't see it.

[Guest Valk0010]

Alright, if you guys can see no value in it, then I guess that's that.

 

I've found others with whom to study this. So it's cool.

 

I'm willing to be convinced. I just don't see it.

Neither do I, and I was hoping my reasons for that would be addressed.

[RankStranger]

Alright, if you guys can see no value in it, then I guess that's that.

 

I've found others with whom to study this. So it's cool.

 

I'm willing to be convinced. I just don't see it.

Neither do I, and I was hoping my reasons for that would be addressed.

 

And I didn't say that there was no value in it- I just said that I don't see it. And I asked questions to clarify.

[Guest Valk0010]

Alright, if you guys can see no value in it, then I guess that's that.

 

I've found others with whom to study this. So it's cool.

 

I'm willing to be convinced. I just don't see it.

Neither do I, and I was hoping my reasons for that would be addressed.

 

And I didn't say that there was no value in it- I just said that I don't see it. And I asked questions to clarify.

We are saying similar things.