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
In the world of sense we find there is an order of efficient causes.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.
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.