Mathematics,  Philosophy,  Research

How Meanings Change the Use of Logic

Anyone who has even a preliminary encounter with logic believes that if A = B, then B = A. This is commonly expressed as the belief that if A is B, then B is A. This widespread use of logic is easily shown to be false when we employ two concepts, one of which is more general than the other. For example, “all men are mortal” doesn’t imply that “all mortals are men”, because “mortal” is more general than “men”. This post explores the implications of this fact for the understanding of both mind and matter.

The Problem of Logical Inversions

Logical inversions are commonly used in all of mathematics. For instance, if X = Y + Z, then, Y + Z = X. You need only two numbers to find the third, provided their relationship given by the above equation is true. The essential premise underlying this use is that X, Y, and Z are all entities of the same type. For instance, if X is $10 and Y is $3 then, by the above definition, Z must be $7. Let’s call this the quantitative use of logic, because it involves the use of quantities and all terms in this equation are of the same type.

However, sometimes, this quantitative use involves different types of entities. For instance, if Y is 5 oranges, and Z is 8 apples, then X must be 13 fruits. You cannot simply add apples and oranges, unless you recognize that they are instances of a type called “fruit”. Additions are permitted only on the same type of entities, not on different types.

Therefore, if you are adding different types of entities, you must first find a more abstract entity of which the entities being added are instances or sub-types. For example, to add apples and oranges, you must view them as sub-classes of fruits. Even to perform ordinary mathematical operations, therefore, we need to keep a conceptual hierarchy in mind, because without that conceptual hierarchy we could just add two entities and produce a new number, although that number would not represent a new type of object. For instance, you cannot add planets and galaxies, or tables and shirts, unless you identify them as instances of a general class called “things” or “objects”.

And when you have constructed this conceptual hierarchy, the logical inversions cannot be applied. For instance, if “mortals” is a more general class than “men”, then we can say that “all men are mortals”, but we cannot say that “all mortals are men”.

This is very counterintuitive for many people, because they have come to believe in the notion that if A = B, then B = A. The fact is that logical inversions work when they deal in the same type of entity, but they fail when applied to different types of entities. The logic that employs different kinds of entities can be called qualitative logic, as opposed to the quantitative logic that deals only in the same kinds of entities. Logical inversion works for quantitative logic but fails for qualitative logic.

Quantitative Logic in Machines

Quantitative logic is used in modern science and in all computers. This logic cannot deal with concepts since, as we’ve seen, logical inversions are allowed in quantitative reasoning, but forbidden in qualitative reasoning. Thinking is characterized by the use of concepts, and an implicit understanding of which concept is more abstract than another. We also use logic, but we don’t always employ the inverted reasoning when using concepts.

Can machines think? They certainly can, provided these machines can also avoid the inverted reasoning. In other words, thinking machines will use qualitative logic rather than the quantitative logic. The qualitative logic includes the quantitative logic, because if we employ an understanding of types, then we also know when two entities are of the same type, and can therefore be inverted as in quantitative logic. However, quantitative logic cannot deal with qualitative reasoning, because of the inversion problem.

Does this mean that humans have some special capability to reason that machines cannot? The short answer is “no”. It is possible to build machines, provided they can use qualitative logic. But to build such a machine, we must have a method to detect types, and thereby understand the cases in which reasoning can be inverted, and the cases in which it cannot be inverted. Machines can reason like humans if they use qualitative logic, and computers can emulate this logic provided we change science from quantities to qualities. This will not only create new technology, but also explain human thinking.

The Genesis of Mathematical Paradoxes

The above inversion problem lies at the root of numerous mathematical, logical, computing theory, and set theory paradoxes. At the heart of each paradox lies the problem of using the same term to denote two different kinds of entities—one more general than the other. If you are interested in a detailed discussion of these paradoxes, and how they emerge, you can refer to Gödel’s Mistake which describes the role of meaning in mathematics, and illustrates how paradoxes arise when this meaning is discarded.

Here I will illustrate this problem through the easily accessible paradox called the Barber’s Paradox described by Bertrand Russell. The paradox is the following innocuous statement: “A Barber Shaves All Those Who Don’t Shave Themselves”. To create a paradox, we must use the term “barber” in two different ways—(1) to denote a class of people who shave others, and (2) as an individual who belongs to this class. The class of people is obviously more general than the individual who belongs to this class, quite like “mortal” is more general than the “men” who belong to this class. In the case of “all men are mortal”, we have two distinct words to denote two types of entities—one more general than the other—and therefore the reason for the paradox becomes evident upon inversion. However, when the same word alternately denotes a class and an individual, this paradox is harder to decode. Yet, the problem in the two cases remains the same.

The Barber’s Paradox relies on this confusion. It asks: Does the barber shave himself? Supposing that the barber does not shave himself, then by the definition that a barber must shave all those who don’t shave themselves, he must shave himself. If, however, he shaves himself, then he should not have shaved himself because he is a barber. Clearly, the problem here is that Mr. Barber is not always a barber. There are two distinct entities—a class and an individual—the former is more general and the latter is particular. The logic that says Mr. Barber is a barber cannot be inverted to claim that a barber is Mr. Barber. In other words, A is B, but B is not A. Or, A = B, but B ≠ A.

The problem of concepts is therefore the following: if we use inversions while employing concepts, then we will end up in logical paradoxes. If we don’t include the hierarchy of concepts in reasoning, we won’t know when to apply inversions (because the entities are of the same type) and when not to use them (because the entities are of different types). Logical reasoning would be like human thinking if the conceptual hierarchy was involved, because then we would know that we can walk up the hierarchy and claim that “all men are mortal” but we can’t walk down to claim “all mortals are men”.

Thinking Machines Need a New Science

To make a machine that thinks, the machine must carry the conceptual hierarchy. It must now be able to detect which entity is more abstract, and which entity is an instance of that abstract idea. If this machine is given two symbols—one of which represents an idea more abstract than the other—the machine must detect which one is more abstract than the other. In other words, the machine must detect the relationship between symbols, simply by measuring the positions of the symbols. This is impossible in current science because a machine cannot detect the meaning of the symbol simply from its physical address in a computer, and therefore current science can never create a thinking machine.

By implication, we also cannot understand human thinking, which avoids logical paradoxes because it forbids logical inversions on all concepts, since it holds a conceptual hierarchy. Many scientists, and most non-scientists, don’t understand this problem which arises from the logical inversion of concepts. They don’t realize that thinking is not a question of finding a material configuration but involves a problem in logic, which can be avoided only when material objects have meanings, and these meanings can be derived from the measurement of physical states. We cannot, therefore, keep meanings in a separate “world”, while this world is material. Rather, we must collapse the difference between matter and meaning, and all objects should be seen as symbols of meaning. The physical state of the symbol must itself denote meaning. In other words, we would no longer be concerned with the physical state, but only with meanings.

What is Semantic Science?

A semantic science requires a fundamental revision to the notion of space and time. The revision is that all locations in space and all events in time are not the same, because the objects at these locations are themselves meanings—which can be more or less abstract. Space and time in this view become an inverted tree structure rather than a box structure.

Reasoning in this space is walking down the inverted tree (i.e. from root to leaf) due to which we establish the relationship between the more and the less abstract concepts, although the more abstract concept precedes the less abstract concept. Perception involves walking up the inverted tree, where we try to infer the more abstract concepts from the less abstract concepts. But, ultimately, these inferences can be flawed if we begin in some presuppositions which aren’t the true abstractions. The problem in modern science is that it has begun from a fundamental presupposition, namely that all the world is uniform, and all objects are of the same type. That supposition results in the use of quantitative logic, and conceptual variety must be forbidden by definition.

The implications of semantics for mathematics, physics, logic, computing theory, and the nature of mind are many, but they all stem from a simple problem of logical inversion, which needs neither sophisticated theories nor complicated philosophy. The problem is logic itself, when this logic is used in conjunction with concepts.