Universally Valid Templates: one more time for the Deep Learners who appreciate proofs
Introduction
In the previous post I argued that most of the knowledge that an intelligent agent needs to perform most higher-level cognitive tasks (language understanding, planning, reasoning, etc.) is not learned, but is acquired. Unlike skills (say playing guitars), that are individually learned, and learned differently due to our unique individual observations and experiences, universally valid knowledge is not learned incrementally, but is acquired, and it is not ‘approximately’ acquired nor acquired with some probability. Universally Valid Cognitive Templates (UVCTs) that are agnostic to our individual experiences (and are also agnostic to culture, time, place, etc.) are acquired in one shot — either we come to know of them or we don’t. In short, that knowledge — that a 4-year old uses to function in the world, to reason and to understand language, is knowledge that we have no choice but following — you can think of this knowledge as knowledge that emanates from the universal laws — the “laws of nature”.
In the previous post (and briefly in “AI Cannot Ignore Symbolic Logic, and Here’s Why”), I mentioned a few of these Universally valid Cognitive Templates (UVCTs) such as the logic of events (e.g., a sub-event e2 of a parent event e1 must have the same time and location of e1), the logic of the “contained-in” relationship, the universality of transitivity, the logic of “parthood”, etc. Here I will discuss one Universally Valid Cognitive Template that is usually a problematic logical operation — namely, entailment (or, implication) — that we all questioned when we first learned it. Many “learned” people think we invented this rule/template. But far from it, it is a template that simply respects the laws of nature. I will show here that in fact the logic of this operation is consistent with physical reality and thus it was not invented but we simply discovered it! In the next post I will also discuss in some details another culture- and language-agnostic phenomenon that is universal and that is never learned, but it simply obeys metaphysics — or, the logic of ontological categories of the world we live in. If some data-driven, machine learning extremists still want to ignore mathematical proofs, then I suggest they get into politics, advertising/marketing, or comedy — dancing could also be a career option, since they are good at hand waving. In any case, they definitely should not be involved in science nor even in engineering.
Entailment (Implication)
In figure 1a below is a truth table that explains the semantics (functional behavior) of the entailment logical operation (if P then Q, or P implies Q, or P entails Q).
The truth table (that extensionally lists all possible data variations) represents the the “intension” that sums up this logic: P implies Q is false (F) only when P is true (T) and Q is F. In a sense, the semantics of the “implies” operation says that from truth you cannot infer falsehood, but anything else goes. It does sound strange, and it did baffle me when I had my first exposure to prepositional logic. How could a statement like
(1) if there are pink flying pigs, then the moon is made of cheese.
be true. Well, without getting into the technical side of things, but yes (1) is true because it can never be disproven (it can never be proven false), and the reason it can never be proven false is because you will never see pink flying pigs to see if (conditional) conclusion holds—and, since you will never be able to disprove (1), (1) is true. (Incidentally, I use this trick on people who tell me since I cannot disprove the existence of the Abrahamic God, then He must exist —I usually reply “then there are “unicorn pink flying elephants”, unless you can prove they do not exist!)
Humor aside, what I want to prove now is that the truth table of implication operation shown above in figure 1a was not invented by logicians, but was simply discovered (they stumbled upon it, like all of mathematical truths that simply explain the laws of nature). In fact, the truth table of implication corresponds not just to abstract reality but even to physical reality.
Why the Meek (us, meager Humans) Did Not Invent Entailment, We Simply Discovered It
First, let us talk briefly about the notion of a set. We can make a set (a grouping) of anything we want. But the notion of a set itself, is not our invention. Sets exist in nature — there just are sets of things. Now any set we want to think of is a set that is defined by some predicate (criteria for membership) — so I can think (conceive) of the set German cars, and in this case an object “is a member” of this set if it is a car and the car is manufactured by a German car company. Moreover, my VW would then be a member of this set, because it is a a car and it is is manufactured by a German car company. So, sets are natural objects that model/explain a natural phenomenon.
More to our point, however, we can always convert between a logical symbol (or a predicate) and a set (e.g., the set of apples is the set of all objects that the “predicate” apple is true of). In other words, a set is just the extension of some predicate/criteria of membership. Now, then, what is the set equivalent of P implies Q? Well, P implies Q is true in set-theoretic terms if the set corresponding to P is a subset of the set corresponding to Q. More formally,
Examples abound. Think of “if laptop then electronic device” which is equivalent to saying “the set of all laptops is a subset of all electronic devices” or think of “if apple then fruit” which is equivalent to saying the set of all apples is a subset of the set of all fruits”, etc. Now, what does it mean to say A is a subset of B, pictorially? It means we have some space (that contains all A’s) that is a subspace (a space contained) in the space that contains all the B’s (see figure 1b).
With that picture in mind, let us now prove that the logical operation of implication was not invented, but discovered. Let us also prove that this universal template, like all Universally Valid Cognitive Templates are not things we experience, or observe, or incrementally learn, they are just natural “laws of the universe” we live in.
Below in figure 2 we show why P implies Q is false in one situation only. Basically, the laws of nature (or “reality”) explains the truth table in figure 2a.
Note that it can happen that P is true (we are in P), and that Q is true (we are in Q) — this is the black circle, it can also happen that P is false (we are not in P) but Q is true (we are in Q) — this is the green circle, and it can also happen that P is false (we are not in P) and also Q is false (we are also not in Q) — this is the yellow circle (note that U stands for the universe of discourse). The only situation that cannot happen is the one where P is true (we are in P), but Q is false (we are not in Q). The last combination cannot even physically happen. In other words, the logic of implication/entailment confirms with the logic of our physical reality. None of this was “invented” it was simply discovered.
Final Remarks
I have over three posts tried to convince extremists on the empirical side (data-driven, statistical and machine learning folks) that we (humans) give ourselves to much credit to the point that we came to believe that individual experiences/observations are what make up our cognitive apparatus. But nothing else is farther from the truth. Most of what we “know” is imprinted in us, is not individual, is not experienced, and is not “approximate” nor probabilistic. The logic of the universe is not something we can violate.
Stop this childish charade of learning from data — which is nothing more than discovering patterns in data and computational statistics , and start working on the actual hard problems, if you really want to utter the phrase “artificial intelligence”.
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https://medium.com/ontologik
Recall that the set-theoretic equivalence of logical implication is the subset relation. We can always convert between
