### The Nature of First-Order Logic

What most people call logic is technically called first-order logic. The nomenclature suggests that there must be second-, third-, fourth-, and infinite other higher-order logics. Most people are unaware of these higher-order logics because they violate the three principles of first-order logic—(a) the law of identity, (b) the law of non-contradiction, and (c) the law of excluded middle. As a result, higher-order logics are generally disregarded in all of modern science, which then relies exclusively on first-order logic.

To understand the necessity of many logics, we can take a simple example. Let’s assume that we have to talk about an apple logically. We can look at a specific thing that is an apple and say: “this is an apple”. This statement relies on first-order logic because the three principles of identity, non-contradiction, and excluded middle apply. For instance, if this is an apple, then it cannot be an orange. It cannot be both an apple and an orange. If the universe of fruits was limited to just apples and oranges, and the thing in question was a fruit, then it must be either an apple or an orange; it could not be neither of the two.

First-order logic applies to objects. Each object has one nature, by which it is called. The object is an individual and the nature is a universal. First-order logic maps an individual to a universal. It is assumed that such a mapping can be done for all individuals, such that each individual will have a type.

### The Need for Higher-Order Truths

The situation, however, gets complicated if we ask: How do we determine something’s nature? For example, we can try to define the method by which an apple will be identified. We could say that an apple is red, round, and sweet. Redness is a color, roundness is a shape, and sweetness is a taste. We are using three properties of an apple to decide if it is an apple. Without these properties, we could not know if the thing in question is an apple or an orange. Hence, properties are absolutely essential.

Each object has many properties. We can classify them based on sense perceptions and say that each object has five classes of properties measured by each sense. Within these classes there are many subclasses. For example, color, shape, and size are subclasses of sight. Hardness, roughness, and heaviness are subclasses of touch. Therefore, to be certain that we have an apple, we have to apply all the properties required to determine if something is indeed an apple. Without properties, we cannot know.

The properties of an object are second-order truths about the object; the first-order truth being the nature of the object. However, the first-order truth depends on the second-order truth, since we have to know the properties in order to know the object. For instance, apple is the first-order truth, and shape, color, and taste are second-order truths. The first-order truth depends on the second-order truth.

Similarly, to know that something has a shape, color, and taste, we have to know its precise shape (such as round), its precise color (such as red), its precise taste (such as sweet). Red, round, and sweet are third-order truths about the apple. The second-order truth depends on the third-order truth, just as the first-order truth earlier depended on the second-order truth. If we cannot know that something is red, round, and sweet, then we cannot know that it has color, shape, and taste, and then we cannot know that it is an apple.

### The Distinction of Is and Has

The second-order truths are tied to the first-order truth using the connector has. For example, “the apple has taste, color, shape”. In contrast, the first-order truth uses the connector is, such as in “this is an apple”. However, this transition from first- to second-order truth isn’t used consistently with further higher-order truths. For example, even as the apple has taste, we say that the taste is sweet and the apple is sweet. Similarly, even if the apple was dark red, generally nobody says that red has darkness. We just say that red is dark and the apple is dark red. The has connector is unpopular compared to is.

But if we drop the has connector for higher-order logics, and just stick to the is connector, then we would run into a new problem where “this is apple”, “this is sweet”, and “this is round” would mean that “round is sweet” purely by the use of the identity principle that two things that are equal to one thing are equal to each other. The identity principle doesn’t apply to has, and therefore doesn’t suffer from this issue. But we don’t use has as often and as rigorously as is. Then how are we getting along with the use of is, when clearly that use results in logical absurdities such as equating the taste to the shape?

The answer is that the is connector works in one direction and not in the reverse direction. For example, the apple is red but red is not apple. Or, the red is dark but dark is not red. The is connector is working just like the has connector—in one direction and not in the reverse direction. Therefore, we can continue using the is connector just like we would use the has connector but it is no longer an identity.

### The Nature of Mathematical Logic

If we were to replace all the words with numbers, then identity becomes equality (=). This equality holds for the first-order statements but not for second-, third-, fourth-, or higher-order statements. Therefore, mathematical logic can work with first-order statements of numbers, not with higher-order statements. Since “order” in logic means a hierarchy of concepts, properties, and values, therefore, mathematical logic also cannot work with the hierarchical constructions. It can only work with a one-tier reality.

Since reality has many tiers, therefore, limiting it to one tier would clearly limit mathematics to objects and disallow their properties and values. Mathematics solves this problem by redefining an object as a set of properties. An object is thereby replaced by two curly braces. For example, in classical mechanics, a particle is denoted by a pair of properties as {x, p}, where x is position and p is momentum.

Sets use the connector has to say that a particle has the property of position and momentum. Or, the property x has the value of 10 while the property p has the value of 15. Now we get a solution where the is connector is used only for equality while the has connector is used for set memberships. By this process, logic is restricted to first-order entities whereas set theory is used for all higher-order entities.

In mathematical logic, I cannot say that “the particle is 10” like I could say “the apple is red” in ordinary language. Instead, I have to say: “The particle has the property of position, the property has a value of 10”. A strict hierarchy from the object to property to value is maintained by this construction.

### The Problem of Quantum Mechanics

This construction, however, collapses in quantum mechanics when a quantum ensemble “collapses” into one state. That collapse is precisely equivalent to saying “the apple is red” for one time, place, situation, and person while saying “the apple is sweet” for another time, place, situation, and person. This collapse in which the whole (apple) becomes one part (red, round, or sweet) is possible in ordinary language but not in mathematical language. Thus, quantum mechanics cannot describe ordinary experience.

Quantum mechanics now resorts to assigning probabilities to different statements such as P (apple is red) = 0.1, P (apple is round) = 0.3, P (apple is sweet) = 0.2, etc. The whole reduces to one part for a given time, place, situation, and person, but this reduction is not universal truth. It is a spatial-temporal-situational-personal truth. By using probabilities, we make it appear like it is a universal truth.

However, despite this appearance, the set theoretic model of science is broken because all members in the set of apple := {red, round, sweet} are not observable at once. We cannot say that apple has some members is an objective truth if it is not true for all times, places, situations, and persons. With the collapse of objectivity and universal truth, everything beginning with logic collapses. But this problem is not understood because objective-universal truth is replaced by probabilities.

### Reality as Non-Objective Potentials

To solve this problem, we have to revise the notion of reality as potentials. An object is now a collection of potentials, one of which is first-order object, some of which are second-order properties, many of which are third-order values of properties, and so on. An observation of these potentials requires the selection of some potentials by a choice, to produce facts. Sentences can pertain to the reality, the choice, or the facts. We have to know that these three kinds of statements are not identical.

For example, when we say that an apple is red, round, and sweet, we are talking about three potentials which can be observed one after another by some consciousness that uses a choice to experience them. At the point of observation, we can say that the apple is red, because three potentials—apple, color, red—have been activated at once. Strictly speaking, there is a hierarchy between them, so we should be saying that the apple has color which has redness. But there is no harm in saying that apple is red if we don’t invert the proposition to claim that red is apple. Either type of connector works if we are careful.

However, there is another serious problem because there is nothing in logic that says that apple is an object, color is a property, and red is the value of a property. For logic, color could be an object, red could be its property, and dark red could be the value. Similarly, an apple could be a property of the object fruit. Thus, we cannot fix what we mean by first-, second-, or third-order logic unless we say that there is indeed a first object relative to which all other entities are second, third, fourth, and so on. When we cannot fix the first-order entity then we cannot even use first-order logic universally because for someone else the same thing is a second-, third-, or fourth-order entity.

This problem also requires us to treat reality as potentials. For instance, the apple being an object is just one of the potentials of the apple. The apple being a property of fruit is yet another potential. Likewise, redness can potentially be an object, a property, a value, and so on. Everything is potentially infinite things, because it can potentially be treated as first-, second-, third-, or successive order truth for some time, place, situation, and person. It doesn’t become a universal truth. Only that which is the original object remains the first-order truth universally.

### Three Kinds of Incompatible Claims

We can illustrate this viewpoint by the idea of a face and a mask. The face is the object and the mask is a property. As successive masks cover a face, each mask can potentially be called a face for the next-level mask. For someone looking at the face covered by numerous masks, the third-order mask is required to know the second-order mask, which is required to know the first-order mask, which is required to know the face.

Thus, all properties, values, and other higher-order attributes described by second-, third-, and higher-order logics are like masks while the object described by first-order logic is like a face. When we see the face covered with the mask, we say that the face is the mask but what we really mean is that the face has a mask on. The mask on the face can change. At that point, the face would be known by the new mask, although what we really mean is that the face is now wearing a different mask.

### Refuting Some Misconceptions

The face-mask reality is called Brahman and Māyā in Vedic texts. It has been misinterpreted as truth and illusion, although there is indeed a sense in which the mask is an illusion and the face is the reality. The misunderstanding is that the mask is separate from the face, when the mask is actually a part of the face and has been produced or manifested from the face. The mask is one of the many personas of the face so it is reality. Since the masks are seen one by one and appear separate from each other, therefore, the illusion is the separation of the masks. All masks are aspects of the face.

This misunderstanding is clearly understood when we treat the face as a concept. For example, if mammal is the face then dog is one of the masks but there are other masks like cows and horses. The masks called dogs, cows, and horses are parts of mammal and yet faces of the mammal. The dogs, cows, and horses are not illusions although they are masks covering the mammal in the sense that we cannot see the complete mammal by seeing either dogs, cows, or horses. But while seeing a dog, horse, or cow we also cannot say that the mammal is invisible. The masks of dogs, cows, and horses reveal the mammal and yet not fully. Hence, the mask is neither fully an illusion nor fully the reality. The mask is just a later-order truth compared to the face. It cannot be equated to the original truth nor can it be called an illusion.

Seeing the mask means seeing diversity and seeing the face means seeing the unity. When both diversity and unity are seen, then the diversity ceases to be an illusion. The diversity appears as an illusion only if we think that the diversities are separate from each other and the face. But in the face-mask paradigm, each individual observer is a mask of a face and what he sees are also masks of a face. Those who can see the masks as an aspect of the face, are in knowledge. Others are in illusion. This means seeing oneself as simultaneously a part of the face and yet a covering of the complete face.

The three seemingly incompatible claims can therefore also be restated as: (a) there is a face, (b) there are many masks, and (c) the masks are at once a part of the face and the covering of the face. The inside of the mask is congruent with the face and the outside of the mask appears uniquely different. By this understanding Brahman becomes the whole truth and Māyā becomes an aspect of the whole truth.

### Revisions to the Logical Paradigm

Aristotelian logic has no distinction between face and mask. What we see is the reality. Since we see many different things, therefore, there is diversity without a unity. In this logic of many separate things, there can be nothing common between two things—i.e., there can be no face behind the mask. Two things are either identical or different. They cannot be simultaneously identical and different. Each different thing must also be an independently existing reality rather than a potential to be seen.

Factually, this conception of reality applies only to the face and not to any of the masks. The face is the universal truth and the masks are the temporal, spatial, situational, and personal truths. The masks are neither false nor the universal truth. Therefore, universalist thinking is false for all the masks.

The face-mask paradigm requires us to see many tiers of truth. The original face is the Absolute Truth and the successive masks are Relative Truths, parts of the Absolute Truth, and yet, Individual Truths. We have to treat the same thing as a Relative Truth, a Partial Truth, and an Individual Truth, and contrast it to the Absolute Truth, a Complete Truth, which is also an Individual Truth. One Individual Truth is Absolute, Complete, and Original. Other Individual Truths are Relative, Partial, and Derived. A Relative, Partial, and Derived truth cannot be a thing-in-itself, defined by itself, for itself, and by itself. It is defined relative to the Absolute Truth. All thing-in-itself conceptions of truth become fundamentally flawed and false.

### Vedic Theology Reduces to Logic

When we start talking about face and mask, the mask being a part of the face, and yet distinct from the face, the distinction between Absolute and Relative truth, the distinction between Whole and Partial truth, then the whole of Vedic philosophy is transformed into an exercise of an alternative logic. By logic, we mean the process by which the Absolute Truth becomes Relative Truths, or how the face produces many masks, enjoys wearing many masks, and reveals or hides different aspects of the face through those masks. This is not the logic of many independent truths. It is the logic of one Absolute Truth expanding into many Relative Truths that are individual truths but not independent truths.

Infinite individual truths merge into one Absolute truth—if they are just individual truths rather than independent truths. Infinite independent truths just remain infinite independent truths and no universal truth can ever be found except by saying that all independent truths are the same truth. Effectively, universalism operates by destroying diversity to establish a universal truth. However, the Vedic system embraces diversity, and establishes unity within it. Unity in diversity is not universal one truth.

Those who want to understand unity in diversity have to discard the logical thinking rooted in universal one truth. Their logic is a hindrance to understanding the Absolute Truth. Their ideas of reality are contrary to conceptions of face and mask. Their notions of objectivity are alien to the idea of potentiality and choice. Everything in Western thinking is a hindrance to understanding Vedic philosophy. If that thinking is discarded, then the whole of Vedic philosophy is nothing more than an exercise in logic. But it is not Aristotelian logic. Even the universalization of logic is a Western theme, not a Vedic one.