Aquinas's First Cause in formal notation.

[Kuznetzova]

In this thread we will examine Aquinas's argument for the existence of God as the First Cause. We will take the argument and break it down into a formal notation in the interest of making it very transparent. We will follow what Aquinas actually says in notation, and then only later will we backtrack and check for mistakes.


Notes on notation
Entailment arrow versus logical implication.

In normal mathematics, P -> Q stands for the statements that if P is true, then Q must be true.
This is called an implication arrow.

Below I will be using an entailment arrow "-->"
E --> K
Where E and K are phenomenal events. Stating "E", means phenomenal (action) event E took place.
E --> K says that E entails K. In other words, if event E happens, K must happen. This makes no statement of their proximity in space nor their order in time. It only says that if E event takes place, K must also take place. Entailment arrow is essentially an assignment of E to K, where the normal notion of causality is operating between them.

Double entailment
E <--> K
Means that "If event E does not happen, event K cannot and will not happen either". Aquinas is going to use the verbiage that goes, to take away E is to take away K.

!E --> !K

Double entailment results from the logical combination of

( E-->K & !E-->!K )
= E <--> K



Saint Thomas, Summa Theologica, 1225

The second way is from the nature of the efficient cause. In the world of sense we find there is an order of efficient causes. There is no case known (neither is it, indeed, possible) in which a thing is found to be the efficient cause of itself; for so it would be prior to itself, which is impossible. Now in efficient causes it is not possible to go on to infinity, because in all efficient causes following in order, the first is the cause of the intermediate cause, and the intermediate is the cause of the ultimate cause, whether the intermediate cause be several, or only one. Now to take away the cause is to take away the effect. Therefore, if there be no first cause among efficient causes, there will be no ultimate, nor any intermediate cause. But if in efficient causes it is possible to go on to infinity, there will be no first efficient cause, neither will there be an ultimate effect, nor any intermediate efficient causes; all of which is plainly false. Therefore it is necessary to admit a first efficient cause, to which everyone gives the name of God.

In the world of sense we find there is an order of efficient causes.
Ordering in time.
C1 < C2 < C3 < C4 < C5 {1}


(neither is it, indeed, possible) in which a thing is found to be the efficient cause of itself; for so it would be prior to itself, which is impossible.
Ordering is proper and strict.
For all n, Cn = Cn+1 is false. {2}
This excludes using less-than-or-equal in the above.
Equals case is excluded because "prior to itself" is impossible.


in all efficient causes following in order, the first is the cause of the intermediate cause, and the intermediate is the cause of the ultimate cause, whether the intermediate cause be several, or only one.
For all n,
Cn-1 --> Cn {3}
Cn --> Cn+1 {4}
Whether Cn be several or one.
Cn is the intermediate cause.
--> is the entail arrow. Aquinas uses the verbiage "is the cause of".
Aquinas is making a sliding window argument here.
That is, for a given intermediate cause Cn, it was entailed by Cn-1, and it entails Cn+1


Now in efficient causes it is not possible to go on to infinity,
Let B denote a finite, lower bound on n.
B exists. {5}

The way to imagine this is that n stretches across all integers deep into negative numbers stretching towards infinity. n is a sliding subscript on events C. We use "C" because Aquinas names these "causes". One plausible value for B would be B = -34,236,101,782,534,500,000.


Now to take away the cause is to take away the effect.
!Cn-1 --> !Cn {6}
Cn-1 <--> Cn {6*}


But if in efficient causes it is possible to go on to infinity,
If B does not exist, then there does not exist a finite lower bound on n. {7}


there will be no first efficient cause, neither will there be an ultimate effect, nor any intermediate efficient causes;
If B does not exist, then !Cn for all n. {8}
Aquinas is claiming that if there is no lower bound on n, then no events ever took place. In particular he is saying that the lack of the first event not taking place percolates to the rest of them. In this narrow context, the arrow of entailment is percolating in time in the normal forward direction.

Cn-1 --> Cn --> Cn+1 --> Cn+1 --> Cn+3 --> etc

Aquinas asserts !Cn-1, and via application of {6} he gets this thing,

!Cn-1 --> !Cn --> !Cn+1 --> !Cn+1 --> !Cn+3 --> !etc



all of which is plainly false.

Ci {9}

In "all of which is plainly false", Aquinas asks the reader to simply look around himself at the world. You see events taking place, therefore Ci. (Remember stating Ci is asserting that the event i place). Instantiate one of those phenomenal events with index i.



Therefore it is necessary to admit a first efficient cause, to which everyone gives the name of God

{9} Contradicts {8}. By reductio ad absurdum, B must exist.
Set n=B . CB is the event corresponding to God.

[Kuznetzova]

Now let's comb over Aquinas's logic and see what happens.

In strictly formal terms, Aquinas makes a number of observations that are not entirely necessary for the crux of his argument. The crux of what he is saying is that we can "reverse percolate" entailment into the past given any event in the present moment that takes place. That is to say, because the present moment is happening for real, then we can infer all the previous ones happened for real also. This is an application of {6*} were we seed the bootstrap from the present moment, denoted Ci. In notation:

Ci took place.
Therefore, Ci-n <-- ... <-- Ci-3 <-- Ci-2 <-- Ci-1 <-- Ci


The most dubious and suspect portion of this argument from First Causes is this claim, that the lack of existence of a lower bound on n entails that Cn did not take place. This is a strange argument that can paraphrased:
"If you cannot specify a lower bound to me as a particular, finite B ... then there is no such event that actually happened."

I cannot imagine any modern mathematician taking that seriously. A clever, trained logician may even demonstrate that it is outright false, rather than just a metaphysical opinion on Aquinas's part. An analogous argument would be like saying that the non-existence of a largest prime integer entails that no integers are prime, which is transparently silly. However, Aquinas is saying something slightly more profound. Namely that the primality of small numbers that we can discover must be bootstrapped from some highest prime number. A patent claim of the non-existence of this largest prime would normally percolate to none of them being prime (by that method anyway).

Coming back out of the analogy fog here, Aquinas has set up a chain of causes connected in time, and demanded that this percolation chain must have been "seeded" so-to-speak, from an initial cause, which happened long ago. I do not believe he was able to show this simply by this indexical argument alone.

Now in efficient causes it is not possible to go on to infinity, because in all efficient causes following in order, the first is the cause of the intermediate cause, and the intermediate is the cause of the ultimate cause, whether the intermediate cause be several, or only one.


In our notation, Aquinas is saying
{5} because {3} & {4}
More formally {3 & 4} -> {5} (Sentence "3 and 4" implies 5)

This is where Aquinas took the sliding window argument, and then by some leap of faith, said that this implies that a lower bound, B, exists. As far as I know from my education in math and logic, a sliding window argument is the last thing you would ever use to show that a sequence is bounded. In almost every case, sliding window arguments are used in proofs by induction, where it is understood that the index n, goes to infinity. Induction proofs are seeded by showing n=0 case is true, and then let bootstrapping take care of all the integers (all the way to infinity). This is done to show that a property is true for all integers.

Why would a bootstrapping argument like this entail there is a lower bound? It is very peculiar that Saint Thomas Aquinas thought that the opposite was true. I mean, he did live in the 13th century AD. Formal mathematics and induction had not been completely fleshed out in Europe yet.

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

[Godfree]

In this thread we will examine Aquinas's argument for the existence of God as the First Cause. We will take the argument and break it down into a formal notation in the interest of making it very transparent. We will follow what Aquinas actually says in notation, and then only later will we backtrack and check for mistakes.


Notes on notation
Entailment arrow versus logical implication.

In normal mathematics, P -> Q stands for the statements that if P is true, then Q must be true.
This is called an implication arrow.

Below I will be using an entailment arrow "-->"
E --> K
Where E and K are phenomenal events. Stating "E", means phenomenal (action) event E took place.
E --> K says that E entails K. In other words, if event E happens, K must happen. This makes no statement of their proximity in space nor their order in time. It only says that if E event takes place, K must also take place. Entailment arrow is essentially an assignment of E to K, where the normal notion of causality is operating between them.

Double entailment
E <--> K
Means that "If event E does not happen, event K cannot and will not happen either". Aquinas is going to use the verbiage that goes, to take away E is to take away K.

!E --> !K

Double entailment results from the logical combination of

( E-->K & !E-->!K )
= E <--> K



Saint Thomas, Summa Theologica, 1225

The second way is from the nature of the efficient cause. In the world of sense we find there is an order of efficient causes. There is no case known (neither is it, indeed, possible) in which a thing is found to be the efficient cause of itself; for so it would be prior to itself, which is impossible. Now in efficient causes it is not possible to go on to infinity, because in all efficient causes following in order, the first is the cause of the intermediate cause, and the intermediate is the cause of the ultimate cause, whether the intermediate cause be several, or only one. Now to take away the cause is to take away the effect. Therefore, if there be no first cause among efficient causes, there will be no ultimate, nor any intermediate cause. But if in efficient causes it is possible to go on to infinity, there will be no first efficient cause, neither will there be an ultimate effect, nor any intermediate efficient causes; all of which is plainly false. Therefore it is necessary to admit a first efficient cause, to which everyone gives the name of God.

In the world of sense we find there is an order of efficient causes.
Ordering in time.
C1 < C2 < C3 < C4 < C5 {1}


(neither is it, indeed, possible) in which a thing is found to be the efficient cause of itself; for so it would be prior to itself, which is impossible.
Ordering is proper and strict.
For all n, Cn = Cn+1 is false. {2}
This excludes using less-than-or-equal in the above.
Equals case is excluded because "prior to itself" is impossible.


in all efficient causes following in order, the first is the cause of the intermediate cause, and the intermediate is the cause of the ultimate cause, whether the intermediate cause be several, or only one.
For all n,
Cn-1 --> Cn {3}
Cn --> Cn+1 {4}
Whether Cn be several or one.
Cn is the intermediate cause.
--> is the entail arrow. Aquinas uses the verbiage "is the cause of".
Aquinas is making a sliding window argument here.
That is, for a given intermediate cause Cn, it was entailed by Cn-1, and it entails Cn+1


Now in efficient causes it is not possible to go on to infinity,
Let B denote a finite, lower bound on n.
B exists. {5}

The way to imagine this is that n stretches across all integers deep into negative numbers stretching towards infinity. n is a sliding subscript on events C. We use "C" because Aquinas names these "causes". One plausible value for B would be B = -34,236,101,782,534,500,000.


Now to take away the cause is to take away the effect.
!Cn-1 --> !Cn {6}
Cn-1 <--> Cn {6*}


But if in efficient causes it is possible to go on to infinity,
If B does not exist, then there does not exist a finite lower bound on n. {7}


there will be no first efficient cause, neither will there be an ultimate effect, nor any intermediate efficient causes;
If B does not exist, then !Cn for all n. {8}
Aquinas is claiming that if there is no lower bound on n, then no events ever took place. In particular he is saying that the lack of the first event not taking place percolates to the rest of them. In this narrow context, the arrow of entailment is percolating in time in the normal forward direction.

Cn-1 --> Cn --> Cn+1 --> Cn+1 --> Cn+3 --> etc

Aquinas asserts !Cn-1, and via application of {6} he gets this thing,

!Cn-1 --> !Cn --> !Cn+1 --> !Cn+1 --> !Cn+3 --> !etc



all of which is plainly false.

Ci {9}

In "all of which is plainly false", Aquinas asks the reader to simply look around himself at the world. You see events taking place, therefore Ci. (Remember stating Ci is asserting that the event i place). Instantiate one of those phenomenal events with index i.



Therefore it is necessary to admit a first efficient cause, to which everyone gives the name of God

{9} Contradicts {8}. By reductio ad absurdum, B must exist.
Set n=B . CB is the event corresponding to God.

My my , you really are a pleasant surprise ,
while I may have got to my current thinking , on my own and am self taught ,
I can recognize your wisdom and knowledge ,
I often say of intelligent people who seem to be a bit flaky ,
intelligence is no guarantee of wisdom
So the idea that there had to be a first cause , surely this is a ad infinitum ,
that we can say , ok we have your first cause , what caused it ,,????
can't we do that forever ,ie ,
in the beginning there was god , where did god come from ,
I like to put it in laymans terms , we reach the greatest number of people by doing this
To put it simply I would say ,
There was never nothing ,,
otherwise we must answer the question of how do we get something from nothing ,,???
so "there was never nothing ,otherwise nothing is all there would ever be"
conclusion , time must go back forever ,

[Godfree]

Wisdom , many have the feeling that the ancients had the true great wisdom and we must re-discover it ,
I dis-agree , the further back in time we go , in general , the more ignorant we are ,
there have been some great philosophers over time ,
the same or similar intellect today , would draw very different conclusions with what we know now ,
Your probably smarter and know more than most of the "greats "you could quote,
so put it in your words ,
your the one I am interested in .

[Godfree]

Saint Thomas, Summa Theologica, 1225

The second way is from the nature of the efficient cause. In the world of sense we find there is an order of efficient causes. There is no case known (neither is it, indeed, possible) in which a thing is found to be the efficient cause of itself; for so it would be prior to itself, which is impossible. Now in efficient causes it is not possible to go on to infinity, because in all efficient causes following in order, the first is the cause of the intermediate cause, and the intermediate is the cause of the ultimate cause, whether the intermediate cause be several, or only one. Now to take away the cause is to take away the effect. Therefore, if there be no first cause among efficient causes, there will be no ultimate, nor any intermediate cause. But if in efficient causes it is possible to go on to infinity, there will be no first efficient cause, neither will there be an ultimate effect, nor any intermediate efficient causes; all of which is plainly false. Therefore it is necessary to admit a first efficient cause, to which everyone gives the name of God.

Therefore it is easier if we agree to a first efficient cause , lets call that reality shall we ,,??
but is it ,? ,, are any of the "versions "of reality in this world, the real one.?
when you pressure a religious person into what was there before god made the universe ,
they usually reply god , they don't have a problem with the idea that god could be infinite ,,???
so why is it such a hard ask to get them to grasp that the entire universe is infinite ,
while I find the subject very stimulating ,
I am still unsure as to where you stand on exactly what this first efficient cause called god is ,,???
WHAT IS GOD TO YOU ,,,???
are you suggesting we MUST conclude there was a beginning ,,???
I would dis-agree ,
we must conclude you don't get something from nothing ,
your first cause please , describe it , how it causes things , what caused it ,???

[Kuznetzova]

when you pressure a religious person into what was there before god made the universe ,
they usually reply god , they don't have a problem with the idea that god could be infinite ,,???
so why is it such a hard ask to get them to grasp that the entire universe is infinite ,

Right. Also, God and the place God dwells (Heaven) are A O K for existing forever. Earths and universes somehow must have an origin; an origin that the religious folk demand you explain.

while I find the subject very stimulating ,
I am still unsure as to where you stand on exactly what this first efficient cause called god is ,,???
WHAT IS GOD TO YOU ,,,???
are you suggesting we MUST conclude there was a beginning ,,???

I think Aquinas has used an argument that looks passably like mathematical induction. In most normal situations, you would probably use induction to reason about beginnings. Portions in the middle of the argument just don't seem persuasive. Some don't follow from the premise.

I would like to add some additional comments here, since I have your undivided attention.

Prior to the Christian-ization of Europe, the deity in the Bible was very much more unsophisticated than all these ideas about him in modern parlance. Jehovah was more like the mischievous gods of the greeks. It turns out the metaphysical idea that the universe is a giant "machine-like clock" that ticks from cause-to-cause had to wait until much later in history. You have maybe heard that the Founding Fathers of Colonial America were deists. Deism's central foundation was this idea that God was this first cause, that he "set the world into motion". This idea is not found in the Bible whatsoever, because framing it requires enlightenment (post-renaissance) philosophy of later European history. In the grand scheme of things, Aquinas is engaging in a type of scholarly thought that presaged the scientific revolution. In the 13th century, portions of Aquinas's writing, such as Summa Theologica, were some of the most advanced philosophical writings in existence. They seem archaic to us now, because we are post-newton, post-einstein, post-descartes.

Alright, to make a long story short -- deistic creators who act as a "first cause" is a later idea and it did not just fall out of the sky. It turns out there was someone who thought of it first. In this case it was Thomas Aquinas. Our history is complete enough that we can trace this idea to its origin, in Summa Theologica. So what you can do now, is in the presence of a religous nutbar, you can tell them that the first-cause idea has its origin in Aquinas's writing. That would probably scare them away, ( which is sort of the point.)

[Godfree]

when you pressure a religious person into what was there before god made the universe ,
they usually reply god , they don't have a problem with the idea that god could be infinite ,,???
so why is it such a hard ask to get them to grasp that the entire universe is infinite ,

Right. Also, God and the place God dwells (Heaven) are A O K for existing forever. Earths and universes somehow must have an origin; an origin that the religious folk demand you explain.

while I find the subject very stimulating ,
I am still unsure as to where you stand on exactly what this first efficient cause called god is ,,???
WHAT IS GOD TO YOU ,,,???
are you suggesting we MUST conclude there was a beginning ,,???

I think Aquinas has used an argument that looks passably like mathematical induction. In most normal situations, you would probably use induction to reason about beginnings. Portions in the middle of the argument just don't seem persuasive. Some don't follow from the premise.

I would like to add some additional comments here, since I have your undivided attention.

Prior to the Christian-ization of Europe, the deity in the Bible was very much more unsophisticated than all these ideas about him in modern parlance. Jehovah was more like the mischievous gods of the greeks. It turns out the metaphysical idea that the universe is a giant "machine-like clock" that ticks from cause-to-cause had to wait until much later in history. You have maybe heard that the Founding Fathers of Colonial America were deists. Deism's central foundation was this idea that God was this first cause, that he "set the world into motion". This idea is not found in the Bible whatsoever, because framing it requires enlightenment (post-renaissance) philosophy of later European history. In the grand scheme of things, Aquinas is engaging in a type of scholarly thought that presaged the scientific revolution. In the 13th century, portions of Aquinas's writing, such as Summa Theologica, were some of the most advanced philosophical writings in existence. They seem archaic to us now, because we are post-newton, post-einstein, post-descartes.

Alright, to make a long story short -- deistic creators who act as a "first cause" is a later idea and it did not just fall out of the sky. It turns out there was someone who thought of it first. In this case it was Thomas Aquinas. Our history is complete enough that we can trace this idea to its origin, in Summa Theologica. So what you can do now, is in the presence of a religous nutbar, you can tell them that the first-cause idea has its origin in Aquinas's writing. That would probably scare them away, ( which is sort of the point.)

Thomas Aquinas sounds like one of the great thinkers of the past , but as you say , we have moved on ,
Zeitgeist ,, Fear or Love ,,,Final Edition,,Addendum , three fascinating DVD's on religion , capitalism ,
in that doco , they mention Horus , the first jesus , they tried finding a savior for a thousand years ,
before they had tortured enough wanna be's,
So I'm an Atheist , no god , devil , heaven , life after death ghosts ,spirits ,angels ,
and while I'm at it aliens have not visited earth we don't have a Loch Ness monster ,
on and on the bullshit goes , reality is , people can't get enough of this shit ,
Twilight , Vampires , Harry potter ,the world is entrenched in the madness ,
and it will not go away any time soon if the capitalists and hollywood , are religious nutters ,
so my mission , and I have accepted it ,
is to reduce the fantasy bullshit in our society ,
religion , alien encounters , homeopathy ,,that the west is a democracy ,,
we are swimming in so much spin and crap most people wouldn't know where to start ...

[Godfree]

I still don't know what YOU think on these subjects ,
I appreciate the history lesson , and find it all very stimulating ,
but I am interested in what you think , you appear to be trying to justify a claim to god ,
that it is a perfectly rational conclusion to draw from looking at the logic ,
I would say thats no logic at all ,
I have blogged on logic many times on many sites , including Sam Harris , [the end of faith]
had a website called the Reason Project , may be still going,
Logic , the most probable ,
there most probably was no beginning ,
the universe most probably always existed ,
and if you want someone who will believe god exists , Attofishpi , is claiming 100% for god ,
So please put me out of my misery ,
never mind what the greats before us have said ,
what do you say ,,???