So let's see how uncertainty or statements which are not 100% true, can really wreak
havoc with our whole notion of logic. Let's suppose we have predicates or
relationships A, B, and C; and rules such as, you know for all values of X.
If A of X is true, then B of X is true. Similarly, if we have for all values of X;
B of X is true and C of X is true, then normal logic allows us to entail that for
all X, A of X is true, so B of X is true, B of X is true, then C of X is true.
Since this holds for every value of X it holds for all values of X So we have the
inferred rule that A of X implies C of X. And this logical entailment of one rule
from a pair of rules is fundamental. We can simply not reason if we don't have
such freedom to entail new rules from old ones.
But this fundamental principle that we rely on has a problem if the statements
are no longer certain. For example, say that for most X, E of X
implies B of X. An example might be that most firemen are
men. Another statement might be that, for most
X, B of X implies C of X. An example might be, most men have safe
jobs. The, inferred rule for most X, A of X
implies C of X. Does not follow, it is not true that most
firemen have safe jobs. Where is if we'd said that all firemen are
men and all men have safe jobs, we could say that all firemen have safe jobs.
But obviously that's not true. The job of a fireman is not safe and the
reason we have this confusion is that while for most X almost all of A, which is
a set of firemen. We have,
They are men, right, so very few of them are women, because this, this piece over
here. And, for most of these B s, which are the,
those that have, which are the men, most of them have safe jobs.
Which, which essentially include all of these people, including possibly some
firemen, who basically don't go on fire calls but, do only administration The
trouble is that because of this uncertain relationship, you don't have a situation
that most A have safe jobs, only a small number of A have safe jobs.
So this statement, for most A, A of X implies C of X, is simply not true.
So, this basic entailment of A implies C from A implies B and B implies C, a very
fundamental piece of logical inference, simply doesn't hold when we have uncertain
statements. And this creates much more problems than
the fundamental limits of logic. Another problem one can get into with
normal logic if one is not careful is the notion of one event causing another event,
that is causality. That is statements are not necessarily true all the time, but
their truth values change over time because one event causes another event to
become true. For example, if we have a statement like,
if the sprinkler was on, then the grass is wet.
Which we, we might write this as S implies W, sprinkler implies wet.
We might have other statement that if the grass is wet, then it had rained last
night which we might write as if the grass is wet then R, that is it rained last
night. Logical inference would probably blindly
combine these two statements as implies R through logical entailment,
Which states that if the sprinkler is on, then it rained last night which is
blatantly false. So, something has gone wrong because our
statements are no longer about things which are always true, but they're about
things which cause other things. The problem is that causality was treated
differently in each statement, which resulted in an absurdity.
Well it turns out that we've seen causality earlier, without actively having
realized it, when we studied classification.
Let's look at classification again. What were we doing then?
A statement like, if the sprinkler is on then the grass is wet, is also a statement
saying that, The fact the grass is wet is an observable
feature of the event that the sprinkler was on.
Similarly, If it had rained the grass being wet is
again. That W is an observable feature, of the
event, of raining. The trouble is the statement that if W is
observed, then R happened in the past is not a statement about the forward cause
and effect of R having caused wetness, or S having caused wetness, but it's
statement what might've happened if one observed W.
Remember in classification, we were doing something similar, we were observing the
features that, when found in the world and trying to infer their causes.
So, in some sense. This reasoning about R having happened,
having observed W, is like concluding which class of event actually is being
observed. Is it sprinkler or rain?
We are observing the features and concluding what kind of event we are
actually observing. A kind of classification or prediction
using a classifier. Not exactly, but something very similar.
This is an example of abductive reasoning, where one tries to infer the most likely
cause given a set of observations or features.
Abductive reasoning is exactly what we do when we compute the class of an
observation using a classifier. It's also a form of reasoning.
It's not deductive that it is going from sprinkler or rain to wetness, which is
actually the likelihood computation. But is the A Posteriori calculation of
having observed W. What is the most likely cause?
Is it sprinkler or rain? If you view this in the language of
classification, the confusion about having incorrectly concluded that sprinkler
implies, rain goes away. So one needs to, distinguish between
deduction and abduction. Fairly deep, but in the language of
classification, it becomes fairly simple. Let's see how.