First principles

Summary :

God provides the means to achieve reason. He gives the truth about the truth and the power to know all, or almost all, truths. He teaches causes, laws, necessity, and possibility. He shows the paths of knowledge and invites us to follow them. He gives the great principles from which we can find all the other good principles and thus found all the sciences. He enlightens us on the nature of the good and on our duties. He teaches the end of reason, the good, and provides the means to attain it.

What is it to be?

Beings have properties and relations.

A concept is a property or a relation. A property, or a quality or a trait, is attributed to a being. A relation is between several beings. When a relation is between two beings, we can consider that it is a property of the couple. A relation between three beings is a property of the triplet, and so on for relations between more beings.

Concepts are beings. They too have properties and relations. Individuals in the strict sense are beings which are not concepts. Properties and relations are attributed to them but cannot be attributed to other beings. Individuals in the broad sense are all beings, including concepts.

The whole being of an individual (in the strict sense) is to have properties and relations. The being of a concept is to be attributed to individuals and to have properties and relations.

An individual reveals the universal

« Form finds itself identical at the same time in several places. It is as if you spread a veil over several human beings and say, « The veil remains one in its totality, when it is stretched on several things. » (Plato, Parmenides, 131b) 

Properties and relations are universals. The same property can be shared by many individuals, it is not the exclusive property of a single individual. A binary relation can be true of many couples of individuals. Likewise for other relations. Properties, relations and conjunctions of properties and relations are universal possibilities. 

The most elementary statements attribute a property to an individual or relate several individuals. They therefore always affirm that an individual reveals the universal, or that several individuals together reveal it. The attribution of the universal to individuals is the fundamental form of thought. Thought is fundamentally the revelation that individuals reveal the universal.

Individuals and the binding of concepts

We can simultaneously perceive heat and pain in two very different ways. In the first case, what is hot is what hurts, heat and pain are bound. In the second case, what is hot is not what hurts, heat and pain are not bound. In the first case, we suppose that there is an individual which binds two properties, to be hot and to hurt. In the second case, we assume that there are two individuals, one which is hot and does not hurt, the other which hurts and is not hot. The binding of concepts (Quine 1992) is determined by individuals, because the being of an individual is a conjunction of concepts.

The truth about the truth

"It is true that snow is white if and only if snow is white." Tarski 1933)

The truth is to say of beings what they are.

By saying of the truth that it is to say of beings what they are, we say what it is, we therefore say the truth about the truth.

Beings, their properties and their relations can be named. We say the truth by naming beings, properties and relations and by attributing these properties and relations to the beings which have them.

All beings, all properties and all relations can be named. Everything can be said. Nothing can escape the power of truth. The true word is as great as being. It is a principle of panlogism.

A proof of the existence of God: it takes divine power for the truth about the truth to be known. Now the truth about the truth can be known, and it is known. So there is divine power.

This proof assumes that only God truly has the power to give the truth about truth. We have this power only by delegation, because he gave it to us.

A world is a set of atomic facts

A concept is fundamental when it is not defined from more fundamental concepts. An atomic, or fundamental, fact is the attribution of a fundamental property to a being of a world or of a fundamental relation to several beings of a world. 

An atomic fact can be designated by an atomic statement. We form an atomic statement by associating the name of a fundamental property with the name of the being to which it is attributed, or the name of a fundamental relation with the names of the beings that it connects.

From atomic statements, we can construct compound statements with logical connectors and thus formulate all the statements about a world. The truth of the statements thus composed is completely determined by that of the component statements, and therefore ultimately by the truth of the atomic statements. A world is completely determined by the atomic facts which constitute it. It can be considered as a set of atomic facts.

If we define a world as a set of atomic facts, an atomic statement is true about a world if and only if the atomic fact it designates is an element of this world.

A set of atomic statements is never contradictory, because atomic statements do not contain negation. Any set of atomic statements therefore always determines a logically possible world, which is also called a model, or sometimes a structure.

For example, the set of the following statements defines the world, or the structure, of natural numbers: 1 follows 0, 2 follows 1, 3 follows 2… It must be understood that this set contains all the atomic truths formed with the names of the natural numbers and the relation of succession. An atomic statement that is not in this set is therefore false.

The being of an individual in a world is determined by all its properties and its relations with other individuals in the same world. The being of a world is determined by the being of all the individuals of this world, therefore ultimately by all the atomic facts which constitute it.

An individual's properties and relations in the world are its entire being in the world. The whole being of a being in the world is its being in the world.

A logically possible world is absolutely possible. It is an eternal possibility because it does not depend on any conditions.

Since an individual's whole being in the world is part of an eternal possibility, it is an eternal possibility. Even a transitory individual, who is born, lives and dies, reveals through his entire existence an eternal possibility.

Remarks :

The principle that a world is a set of atomic facts is borrowed from Ludwig Wittgenstein's Tractatus logico-philosophus. But the definitions here adopted of an atomic fact and an atomic statement are not in the Tractatus. They come from first-order logic.

The principle that an atomic statement is true about a world if and only if the atomic fact it designates is element of that world is borrowed from Alfred Tarski and model theory. In the language of model theory, a world is a model (Keisler 1977). 

David Lewis fears that there could be a circularity in the definition of the concept of logical possibility, because a logically possible world is such that it is impossible that its definition implies a contradiction (Lewis 1986). By defining a logically possible world as a set of atomic statements, this problem of circularity is avoided. The definition of a logically possible world cannot be contradictory because atomic statements never contain negation. If there is no negation, there cannot be any contradiction.

Laws

A set of atomic truths completely determines a universe, a world, a totality, but it does not tell the whole truth about this world. Even if we knew all the atomic truths of the Universe we would not know it very well, because a set of atomic truths is like disintegrated knowledge, which we have reduced to its simplest elements. We want more and better than this knowledge in pieces, without rhyme or reason.

When we know that it is, we not only want to know that it is, we also want to know why it is. But is there a why? Why would there be a why? And why shall we want to know it?

To say why is to say the causes. The scientist is the one who knows the causes. How do we know the causes? Are there really causes to be known?

If X is a cause of Y, the statement that X is always a cause of Y is always a law. To know the causes, we need to know the laws.

The ignorant do not know that the Universe is ordered because they do not know its laws. The scientist discovers the order of the Universe by discovering its laws.

Knowledge of the laws makes all the difference between only superficial knowledge and true knowledge. To truly know a being we want explanations: what is it made of? How did it appear? What are its effects on other beings? We also want to know the final causes when there are any: what are the ends that it must achieve? An explanation is always reasoning based on laws. We therefore need to know the laws to give the explanations which constitute true knowledge.

To be possible is to be permitted by the laws. A system of laws determines a space of possibilities, the space of everything they allow. There are as many forms of possibility as there are systems of laws.

The logical possibility is the absolute possibility. Its fundamental law is coherence, that is to say the absence of contradiction. The laws of Nature define natural possibility. Ethical laws define ethical possibility. To be ethically impossible is to be prohibited by ethical laws.

Necessity and possibility are complementary concepts. What is necessary is what the negation of is not possible. What is possible is what the negation of is not necessary. Laws and all their logical consequences are necessary.

Whatever is necessary is possible but the opposite is not always true. A statement is contingent if and only if it is possible and its negation is possible. A contingent statement is possible without being necessary. Mathematical truths are never contingent, they are always necessarily true and this is an absolute necessity, logical necessity. On the other hand, the truths about events in our world are generally contingent. What happened might not have happened. The laws of Nature do not determine everything that happens, far from it. On their own, they are not enough to deduce the existence of the slightest event.

With laws, we can explain all the possibilities and all the necessities. The search for explanations does not lead to an infinite regression, because we stop at fundamental laws, from which we explain all the others.

We explain a contingency in the same way that we explain a necessity, by looking for conditions of which it is a consequence, but the conditions cannot all be necessary, because a consequence of necessary conditions is also necessary. At least one of the conditions must be contingent. We always explain contingencies on the basis of other contingencies. When we explain contingencies, there is therefore necessarily an infinite regression in the search for causes. The laws explain fatality, the necessary sequence of contingencies, from the beginning until the end of time.

When they are true, the laws are true for all eternity. They logically and chronologically precede everything that actually appears. Everything happens as if the word preceded being, because to be actual a being must be possible, it must be permitted by the laws. “In the beginning the Word. » (John, 1.1)

Logical necessity

Logical laws are logically necessary statements. The principle that a world is a set of atomic facts makes it possible to find all the logical laws, because it makes it possible to define logical possibility and necessity.

A statement is logically possible if and only if it is true in at least one world.

A statement is logically necessary if and only if it is true in all worlds.

"All worlds" above and below means "all the worlds defined with the concepts of the statement".

A statement is logically impossible if and only if it is false in all worlds.

A statement is logically contingent if and only if it is true in at least one world and false in another.

A conjunction of atomic statements is always logically possible and contingent. A contradiction (p and not p) is always logically impossible. The law of excluded middle (p or not p) is logically necessary.

A statement is logically necessary if and only if its negation is logically impossible.

A statement is logically possible if and only if its negation is not logically necessary.

Logical necessity is absolute, unconditional. The truth of logical laws does not depend on any hypothesis.

The conclusion of a reasoning may not depend on any premise, because one can remove the dependence on the premises during the reasoning. For example, if we have drawn the conclusion B from the unique premise A, we can add as a new conclusion (if A then B). This new conclusion does not depend on the premise A. It is therefore without hypothesis. Logical laws are the conclusions of reasoning which do not depend on any hypothesis.

Naturally possible worlds

Naturally possible worlds are the worlds such that the laws of Nature are true in them.

One makes a theory of Nature by postulating fundamental laws of Nature. A naturally possible world is a model of a theory of Nature provided that the postulated laws are true.

If the laws of Nature are formulated with a system of differential equations, the naturally possible worlds are the solutions of the system. The motions of the planets, for example, are naturally possible because they are solutions of the differential equations of Newtonian physics.

Our models of observed reality are never exact models. Their empirical truth, their agreement with observations, always depends on approximations. The agreement between the theoretical models and the observed reality must be good enough for us to be able to say of our models that they are naturally possible worlds, and of our laws that they are laws of Nature, but it is not necessary that the correspondence between the models and the reality be exact.

A statement is naturally necessary if and only if it is true in all naturally possible worlds.

A statement is naturally necessary if and only if it is a logical consequence of the fundamental laws of Nature.

Natural necessity is relative, conditioned by the truth of the fundamental laws of Nature.

Reasoning, what for?

A good reasoning starts from good premises to arrive at a good conclusion by a clear and logical path.

Good premises are good principles or good observations, or conclusions already drawn from good reasoning.

It seems quite obvious that one must reason, but why?

Logical rules always lead from true to true. If the premises are true and the reasoning is logical, the conclusion cannot be false. A logical conclusion cannot give more information than what is already given in the premises. This is why logical reasoning is always tautological. The conclusion affirms what is already affirmed by the premises. But then why reason? It seems that a reasoning can tell us nothing, since a conclusion is only a restatement of what we already know when we know the premises.

Laws are like a concentrate of knowledge, they give an unlimited wealth of information. Applying a law to a particular case is a logical reasoning that reveals what the law teaches in that case. All particular consequences are determined by the laws, but one must reason to discover them. A reasoning reveals explicitly what its premises determine implicitly, it develops what is enveloped, it unfolds what is folded, it reveals what is present but hidden as long as one has not reasoned.

An example: consider the law "if the system S is in the state x at time t then it is in the state f(x) at time t+1, for all times t and all states x" and the observation that the system S is in the state a at time 0. From these two premises we can deduce many consequences: S is in the state f(a) at time 1 , in the state f(f(a))=f^2(a) at time 2, in the state f(f(f(a)))=f^3(a) at time 3... With the principle of infinite induction, we can deduce that S is in the state f^n(a) at time n for all natural numbers n. With a law and an observation of an initial state, we can know the destiny of the system for eternity.

In general a law can be applied to many particular cases, it is general. We can always formulate generalities with a 'for all x'. 'Always' is 'for all time t'. 'Everywhere' is 'for all position x'. 'Never P' is 'for all time t, it is false that P'. 'All minds' is 'for all mind x' or 'for all x, if x is a mind'. The same goes for all truths: 'for all x, if x is a truth', and even all generalities: 'for all x, if x is a generality'.

When beings obey the same laws, we know them all by knowing their laws. It is like knowing a myriad of beings at the same time, as if generalities revealed the whole to us at a single glance. We can thus know very vast totalities: all material beings, all spirits, all worlds, all theories, everything that is naturally or logically possible...

Knowledge of the laws gives reasoning its power. We do science by learning through reasoning what the laws teach.

Nothing new under the sun

The light that comes from distant stars is the same as that of the Sun, or the light we produce on Earth. It always behaves the same way. Everywhere in the Universe, light is always the same and always obeys the same laws. « There is nothing new under the Sun. » (Ecclesiastes) 

Light reveals the properties of matter. A natural substance can always be identified by spectroscopy, the analysis of the light absorbed or emitted. We can know the chemical composition of distant stars by analyzing their light. Light reveals that matter always obeys the same laws everywhere in the Universe.

A natural substance is pure if it is made of identical molecules or atoms. Natural substances always behave in the same way as soon as they are pure. Pure water always has the properties of pure water. It always obeys the same laws. For it too, there is nothing new under the Sun. More generally, elementary particles, atoms and molecules of the same species are all identical and obey the same laws.

All the points of space are identical. When we know one, we know them all. The same is true for points in space-time. We know infinite spaces in their entirety simply by knowing their laws.

All natural numbers are obtained by adding units that are all identical to each other. We know the constitution of all natural numbers, however large they are, simply by knowing the one. Likewise the elementary constituents of all material beings are identical when they are of the same species. By knowing a small number of elementary species, we know at the same time the constitution of all material systems, even very vast and very complex.

Each number is unique but they all obey the same laws of calculation. When beings are all different, they can also be very similar by obeying the same laws.

All sciences place beings in fundamental categories. Beings of the same category have common properties and obey the same laws. But apart from fundamental physics, beings of the same category are not identical. Beings of the same category can be very similar to each other, but also very different. Each being can have properties that distinguish it from all others.

One always makes theories by imposing laws on beings of the same category. If the beings of the same category do not obey the same laws, one cannot make a theory. That there is nothing new under the Sun, apart from a few individual variations, is a necessary condition for the intelligibility of reality.

The appearance of a unique being is a novelty. So sometimes there is something new under the sun. But that is never completely new. Laws and concepts are not new. An existence is never really new because an individual always reveals an eternal possibility. Everything that is new already existed potentially. So understood, we never invent anything, we only discover possibilities.

What is knowledge?

Knowledge is truth as much as it is accessible to us, as much as God wants to give it to us.

A truth spoken by chance is not knowledge. This is not how God gives the truth. If we had to rely on chance alone to know the truth, we would only be lost souls, we would never really know the truth, since even if we say it by chance, we do not know how to recognize that it is the truth. For a truth to be known, it must appear clearly as a truth, the path which leads to it must show that it is a truth. Such a path makes the truth appear as knowledge, because it shows that it is a truth within our reach, a truth that God wanted to give us. We must therefore have the means to produce truths reliably in order to recognize them as knowledge. We need good methods, good instruments, honesty and good work. The scholar's motto: you will earn the truth by the sweat of your face and you will give birth to it in pain.

For a truth to be knowledge, it must be well produced. We must be good producers of truth. This leads to a definition of knowledge:

A statement is knowledge if and only if it can be produced by a good producer of truth who has worked well in producing it.

The present theory sounds like a truism. Knowledge is what is produced by a knower, when he or she has worked well. And it looks like it puts things upside down. One must not define knowledge from the knower, one must rather define the knower from knowledge. A knower is a mind capable of producing knowledge. But the present theory puts things right. Knowledge is defined from the competence to produce truths in a reliable way, because it is the truth as much as it is accessible to us. Not only the truth, but also the ability to reliably produce it, make knowledge. This is why it is natural to define knowledge from the competence of knowers.

An instrument or device of observation is a means of producing truth. To work well we need good tools. We conclude:

An observation is knowledge if only if it can be produced by a good observation device which has worked well in producing it.

A statement is always equivalent to the observation that it is a true statement. A truth-producing device can therefore always be considered as an observation device.

In Warrant and Proper Function (1993) Plantinga defines knowledge from the proper functioning of cognitive mechanisms. The present definition of knowledge  is borrowed from his theory, but the wording is different. 

A truth-producing device can be a mind, a community of minds (Goldman 1999), or an instrument of observation, but minds are the most fundamental producers of truth. A community produces truths by the cooperation of minds who produce truths. An instrument of observation produces truths when it is well used by a mind. 

Generally a mind is not reliable in all areas where it tries to produce truths. Our skills are differentiated. This is why a truth-producing device must be determined with a domain of competences, not only with a mind, a community or an instrument of observation.

A truth-producing device can be good without being infallible. It is good if and only if it works well most of the time, not necessarily always (Goldman 1986). Even if a good truth-producing device has produced a true statement, this is not necessarily knowledge. Due to unknown circumstances, a succession of errors that compensate each other for example, a good truth producer can produce a true statement even if he or she has not done a good work. In such a case the statement produced is not knowledge, only a truth obtained by chance. The truth of a statement must result from the good work of a truth producer for this statement to be knowledge (Zagzebski 2017).

Is the criterion of low error rate necessary for a producer of truth to be good? Zagzebski (1996 p. 182) offers the following counterexample. A scientist is very creative and makes an important contribution to science one time out of twenty, but she is seriously mistaken the rest of the time. She seems to be a very good producer of knowledge despite her very high error rate. She may even be one of those who advance science the most. But her errors must be corrected in order to arrive at knowledge. When one has nineteen out of twenty chances of being wrong, one does not know. To make a good producer of truth, creativity must be accompanied by a device that reduces its error rate.

A statement is knowledge if and only if it can be known, but a statement can be knowledge without being known. Commonly we assume that knowledge is known by at least one mind, but we can also say of potential knowledge which is not known by anyone, but which could be known, that it is already knowledge waiting to be discovered. Understood in this way, "is knowledge" and "can be known" are synonymous. "can" is to be understood here and below in the sense of natural possibility, not just logical possibility.

When is knowledge known? There is no precise answer, because knowledge can be known in various ways, and because it can be more or less well known. A physicist who uses mathematical formulas without knowing their proofs has knowledge, but it is not as well known as by a mathematician who knows the proofs. The question "when is a statement knowledge?" can, however, receive precise answers. Each of the following five conditions is necessary and sufficient for a statement to be knowledge. It  can be produced by a good producer of truth who has worked well in producing it. It can be produced by an act of intellectual virtue. It is a good principle, a good observation or the conclusion of logical reasoning whose premises are good principles or good observations. It can be recognized as knowledge. It can be well proven.

Intellectual virtues

A virtue is intellectual if it is motivated by the search for truth. A mind becomes a good producer of truth by developing and exercising its intellectual virtues. Honesty, impartiality, consistency, respect for the truth, open-mindedness, courage, perseverance, and many others, enable us to become good producers of truth. A truth producer can be good in the sense that she provides reliable results because her error rate is low, or in the sense that she practices intellectual virtues. There is no need to separate these two meanings. An intellectual virtue must lead to reliable results, otherwise it would not be a virtue. Conversely, one cannot produce reliable results without virtue. The simple use of an observation instrument requires respect for the truth to produce reliable results. We can do almost nothing without the virtues. Even criminals need to practice virtues, at least sometimes, otherwise they could not live. There is no opposition between efficiency or power on the one hand and virtue on the other hand, because nothing is more powerful than virtue. It is the first source of power and the sole source of true power, if we admit that true power is always the power to do good.

An act of a good producer of truth who has put the intellectual virtues into practice can be called an act of intellectual virtue. The previous definition of knowledge is therefore equivalent to the following: 

A statement is knowledge if and only if it can be produced by an act of intellectual virtue. 

This definition of knowledge is very close to that presented by Zagzebski in Virtues of the mind (1996).

Knowers produce knowledge because they have a knowledge-producing virtue, an intellectual virtue. With such a truism we have obviously not yet explained anything, we have only named what must be explained, the existence of intellectual virtue, of the capacity to produce truths reliably.

Rules for the direction of the mind

It is in the nature of the mind to give himself ends, which he pursues, and rules, which he applies, not always. A rule can be seen as an end, the end being to obey the rule. It is an end we never fully reach, unless we die, because we can still disobey while we are alive. A system of ends and rules is a program. When its ends are noble, a program is an ideal. To be good producers of truth, we must give ourselves an ideal of reason, a system of ends and rules for the direction of the mind, and realize it. Since reason invites us to use reason, the ideal of reason is a manual for reason.

Why make rules? By making it a rule to take all our steps in the same direction, we go much further than walking like a drunkard who changes direction with each step. The same goes for all good programs. They make us more competent, more powerful, more capable of reaching distant goals.

The programs we give ourselves evolve throughout life. We are continually modifying and supplementing those we have already adopted. We can also give ourselves a research program for programs, giving ourselves rules for finding programs. A manual for reason is precisely a research program for good programs, worthy of reason.

An ideal is not always good. It can be bad because it mixes noble ends with less noble ones, or because it is infeasible and condemns us to frustration. To be feasible, a program must always be adapted to reality, both the outer reality and the inner reality, what we are for ourselves. To give us good programs, we must know both ethical laws and the laws of reality. Ethical laws teach duties and ends worth pursuing. The laws of reality teach how to adapt.

The ideal gives the mind its power. The power of a mind is the power of his or her ideal. In particular, an ideal of knowledge gives a mind the power to produce knowledge, and therefore intellectual virtue.

Good principles are recognized by their fruits

We develop knowledge by making theories and confronting them with observations. A theory is a system of principles. A principle is an axiom or a definition. Theorems of the theory are the conclusions of logical reasoning whose premises are axioms or definitions. Principles are among the main means for developing knowledge. They can be used to program truth-producing devices. We want good principles to make good truth-producing devices.

Without principles our natural means of observation would be the only sources of knowledge. Principles enable us to go much further. Theories make us know all possible worlds and the actual world even before it is actually observed, they increase our ability to observe the avtual world, they teach us the ends which are really worth pursuing, how to acquire knowledge and make good theories. Good principles are always inexhaustible sources of knowledge. They can also be seen as engines or rockets, because they give us power and carry us to the highest heights of knowledge.

The same statement can be a principle of one theory and a theorem of another theory, because a principle can be proved from other principles. The same statement can be an observation and the conclusion of a reasoning because a reasoning can be part of an observation device. We can reason to interpret observations and thus produce new observations.

A statement is knowledge if and only if it is a good observation, a good principle or the conclusion of logical reasoning whose premises are good principles or good observations. This principle is not a definition of knowledge. The concepts of good observation and good principle are defined from the concept of knowledge. A statement is a good observation if and only if it is both an observation and knowledge. A statement is a good principle if and only if it is both a principle and knowledge. 

Since a false statement is never knowledge, good principles and good observations must be true. If a reasoning is logical, the conclusion cannot be false when the premises are true. Logical reasoning where all the premises are true is conclusive proof.

A theory can be considered as an observation device. We observe that statements are true by observing that they are theorems, that they are logically proven from axioms and definitions. “The eyes of the soul, by which it sees and observes things, are nothing but the proofs." (Spinoza, Ethics, Book V, prop. 23, Scolia). Our observation devices are generally fallible, but theories are an exception, because they are infallible if their axioms are true.

« You will recognize them by their fruits. » (Matthew, 7:20)

« We shall see such demonstrations, which do not produce as great a certainty as those of geometry, and which even differ greatly from it, since instead the geometers prove their propositions by certain and incontestable principles, here the principles are verified by the conclusions drawn from them; the nature of these things not suffering that it be done otherwise. It is possible, however, to arrive at a degree of verisimilitude, which is often not much less than complete evidence, when things, which have been proved by these supposed principles, are perfectly connected with the phenomena which experience has pointed out, especially when there are many, and even more so when we form and anticipate new phenomena, which must follow from the hypotheses which are employed, and which we find that in this the effect corresponds to our expectation. If all these proofs of plausibility are found in what I have proposed to treat, as they seem to me to be, it must be a great confirmation of the success of my research, and it is hardly possible that things are not nearly as I represent them. » (Christian Huyghens, Treatise on light, p.2)

A principle is good if and only if it can be used to make good truth-producing devices. We recognize good principles by their fruits, the good knowledge that helps us to adapt to reality, to think well and to live well. Fruits can be considered as proofs that principles are good, but they are not conclusive proofs, because even wrong principles can sometimes bear fruit. Recognizing good principles from their fruits is not infallible, but it can still be a good device for observing knowledge.

We recognize knowledge from the good principles with which it is produced. We recognize good principles from the knowledge that can be produced with them. This approach is circular. To recognize good principles, we have to recognize knowledge, but to recognize knowledge we have to recognize good principles. This circle does not confine us because good principles are not the only criteria for knowledge. We recognize good principles when they help us to become good producers of truths in a consistent set of producers of truth, a community of minds who want knowledge that helps us to adapt to reality, to think well and to live well, to become accomplished minds. This recognition of the accomplishment of the mind is more fundamental than the recognition of good principles, because we recognize good principles by recognizing that they help in our accomplishment.

Since ethics is the knowledge of the good of the mind, it is the knowledge which recognizes the fruits of reason, it is therefore necessary to recognize all the principles of science. Ethics is fundamental to all sciences because it teaches us both how to work well and how to evaluate the fruits of our work.

The best way to think is to put one truth in front of the other and start again. This is the Cartesian ideal of knowledge. We must start from already known truths and move from truth to truth while respecting the rules of logical reasoning. Logical rules ensure that we always move from truth to truth. But to proceed in this way we must know that our premises, the observations and the principles that we put at the beginning of our reasoning, are true. However, we do not always know from the beginning whether our principles are good. We sometimes don't even know if our observations are good, because good principles are needed to establish their truth. Since we recognize good principles by their fruits, we must wait until the end of the work to recognize that they are really good. In the beginning they are only hypothetical. Recognition of good principles is not instantaneous, it results from a long journey, because we must give the principles time to reveal their truth to us.

Knowledge can always be recognized as knowledge

Do we have to know that we know in order to know? If we have a good truth-producing device, if we believe in the statements it produces and if we do not know that it is a good truth-producing device, we have knowledge without knowing that it is knowledge. We know but we are credulous with respect to our own knowledge. A statement is much better known if we know that it is knowledge. Knowledge can be more or less well known because the device that produces it can be more or less good, but it can also be more or less well known because we know more or less well that it is produced by a good truth-producing device, because our knowledge observation devices can be more or less good.

A statement is knowledge if and only if it can be known. It must be understood that knowledge can be truly well known, that one can have it by knowing that one has it. A statement that could never be recognized as knowledge could not be knowledge, because it could not be truly well known.

A statement is knowledge if and only if it can be recognized as knowledge. 

Several equivalent formulations of this principle can be given. A statement is knowledge if and only if the observation that it is knowledge is knowledge. A statement is knowledge if and only if the statement that it is knowledge is a good observation. A statement is knowledge if and only if there can exist a good device for observing knowledge which works well when it recognizes it as knowledge. 

Good devices for observing knowledge are fundamental to the acquisition of knowledge because they make knowledge really well known when it is known.

To know we must know that we know, we must therefore know that we know that we know, and so on ad infinitum. We could believe that this infinite series of knowledge is a regression which prevents us from always knowing that we know, because statements in infinite number cannot all be known at the same time by a finite mind. But this is to be mistaken about the power of a finite mind. Any generality is equivalent to knowing an infinite number of statements. Knowing an infinite number of statements is not beyond our reach, it is what we do every day as soon as we state a generality.

'X can be known at order 0' equals by definition 'X can be known'. 'X can be known at order n+1' equals by definition 'That X can be known at order n can be known', for any natural number n. With these definitions, we can assert that for any natural number n, X can be known at order n. To know it, we just have to prove it. Can we prove it?

A universal device for observing knowledge is a device for observing all knowledge. It must observe for all truth producers, or truth-producing devices, whether or not they are good producers, and whether they have worked well, or functioned well, in producing what they propose as truths. Can such a device exist?

If it exists, it is like a sun for all minds, which illuminates all the truths accessible to us. It reveals all knowledge. It shows in full light everything that can be known.

A good observation that knowledge is knowledge is also knowledge. A universal system for observing knowledge must therefore be capable of observing itself and recognizing whether the observations it produces are well produced. Is it possible ? Can a good universal device for observing knowledge which observes that it is a good universal device for observing knowledge exist?

Suppose that a good knowledge observation device is capable of observing itself, of recognizing that it indeed produces good knowledge, and of observing that X is knowledge. Then this device can recognize that X can be known at order 0. Furthermore, if it can recognize that X can be known at order n, it can also recognize that it can be known at order n+1, since it is capable of observing itself and therefore of recognizing that it can recognize that X can be known at order n. We can then conclude, with the principle of reasoning by induction, that for any natural number n, this device can recognize that X can be known at order n.

Knowledge observation devices are observation devices for good truth-producing devices. To observe that a truth-producing device is good, one must observe that it does not make too many errors. To observe whether or not it makes errors, we confront it with one or more other truth-producing devices, because the same truth can be produced in many ways. Of course, we need good truth-producing devices to observe whether a truth-producing device is good. But there is no infinite regress, because the consistency of the results provided by independent truth-producing devices is enough to establish that they are good. If we observe that independent truth-producing devices give consistent results most of the time, there is a very small chance that this consistency results from the operation of bad devices, so we can conclude that all devices are good. A truth-producing device is good if and only if it is element of a set of independent devices which give consistent results most of the time. This principle is a criterion for observing good truth-producing devices. 

Independent producers can observe the consistency of results obtained by independent producers. The consistency of the consistency observations can then be observed. The criterion of the consistency of independent producers can thus be applied to itself and it then leads to the observation that it is a good principle, therefore knowledge.

The principle that we recognize good principles by their fruits is a criterion for observing good principles. It bears fruit each time it helps us recognize good principles. When it is thus applied to itself, it leads to the observation that it is itself a good principle.

The great universal principles of the recognition of knowledge provide the means to recognize all knowledge, including itself. By giving such principles, God has given the power to acquire all knowledge.

A community of independent producers of truth, who give themselves the means to recognize good principles and good observation devices, and to observe themselves, is a universal device for observing knowledge. It has the power to recognize all truths, as far as they are accessible to us.

Observing the good fruits and the consistency of results obtained by independent producers does not always lead to certainty, but it is not necessary, because a producer of truth can be good without being infallible.

What is proof?

To know, we have to provide proof. If we don't know how to prove what we're saying, we simply don't know. But what is proof?

Common usage doesn't help us because more or less anything can be called a proof.

An argument is proof of its conclusion. But the premises also are proofs of the conclusion. Even just one of the premises can be called a proof.

An observation is proof. The process of observation proves the truth of what is observed.

The observation of an observation, the observation of a testimony for example, is proof of the existence of both the observation and the fact observed.

A fact can be considered as proof of its causes or its consequences.

A being can be proof. For example, the murder weapon is proof of the crime.

Does a proof have to be infallible to be proof? Is a solid argument that is not enough to establish its conclusion proof? Is bad evidence, a false argument, a fallacy, proof?

In mathematics we always know how to recognize whether reasoning is good proof because it is enough to verify that the premises are axioms or definitions and that the logical rules have been respected. More generally, proof that does not clearly show that it is proof cannot be good proof. A theorem does not bear visible marks that it is a theorem, because reading it is not enough to know whether or not it is a consequence of the axioms and definitions. On the other hand, proof always bears visible marks that it is proof, it must be able to be recognized as proof.

The path that leads to knowledge must show that it is the truth. The process of producing knowledge is therefore always proof that it is knowledge. If a scholar has worked well in producing the truth, his work is good proof of his result. We can therefore conclude:

A statement is knowledge if and only if it can be well proven.

Good work of producing truth must always bear apparent marks that it is good work. If it cannot be recognized as good work, then it is not good work. It’s part of the work to show that it’s a good work. Such work cannot be good without showing it.

An observation device cannot be good without showing that it is good. If an observation device cannot be observed to be good, then it is not good.

A good observation that a statement is knowledge is always also good proof that it is knowledge. The observation that truth-producing work is good work is proof of the conclusion of the work. We provide proof not only by good work but also by observing that it is good work.

Good proof does not always provide certainty, because it can be good without us being certain that it is good. For example, an observation can be good, because it is produced by a good observation device which worked well in producing it, but the knowledge thus produced is not known with certainty, because the observation device is good without being infallible.

Providing proof is always inviting a mind to realize that it is capable of knowing for oneself. The proof, if it is truly good, is a universal path, a path which shows all minds how their own resources give them the means to reach the truth. Providing proof is a form of respect for all minds, it is a way of honoring their ability to know the truth.

Knowing without being certain that we know

Can we know without knowing that we know?

If we have a good truth-producing device, if we believe in the truths it produces, and if we don't know that it is a good truth-producing device, is it to know or not to know?

We have sense organs and a brain with which we produce true observations. But we have to be knowledgeable to know that sensory perception produces truth. If we believe that our perceptions produce truth without knowing it, do we have any knowledge?

If we believe that well-produced truths are truths without knowing that they have been well-produced, we are no different from a gullible person who believes any nonsense spouted by a liar, a crook or a charlatan, so we we don't know. To know, we have to know that we know.

We can know more or less well than we know, because our devices for observing knowledge can be more or less good.

If I believe that I know because I have seen and seen well, then I know and I know that I know, because my capacity to recognize that I have seen well, that good conditions of observation have been met, is a good device for observing knowledge, even if it is not infallible.

We cannot know without knowing that we know, but we may not know it very well.

If I have a good truth-producing device D, if it has functioned well in producing the truth, if I also have a good truth-producing device D' about truth producers, capable of recognizing that D is a good producer of truth, and capable of recognizing itself as a good truth-producing device, then I know while knowing that I know, but if D' is not infallible, then I know without being certain that I know. We can therefore know without being certain that we know.

God has given us truth, but not always certainty.

Certainty gives a feeling of power, as if we don't need anyone but ourselves to know, as if we don't need God to be powerful.

Certainty leads to intolerance. If I have a good knowledge observation system that provides certain results then I can exclude everyone that does not respect my rules, everything that does not pass my knowledge observation test. Any candidate for knowledge that I do not recognize as a good candidate is automatically excluded: go away, you have no place in the field of knowledge.

A community of independent producers of truth, who give themselves the means to recognize good principles and good observation systems, is a universal system for observing knowledge, but it does not lead to certainty and intolerance. . On the contrary, it imposes the law of hospitality: we do not know in advance all the good principles and all the good methods of observation, and we cannot know them. Science is necessarily unpredictable. We cannot know in advance what it will become and how we will recognize it. Our principles of observation of knowledge are universal, but in general, they do not give a quick and certain answer. We must give time to principles, hypotheses, theories and even observation systems so that they can bear fruit. Faced with a new hypothesis, we have a duty to welcome it. A candidate for knowledge is not excluded, but invited to show what he is worth. Certainty and intolerance lead to ignorance because they prevent us from correctly observing knowledge. Without a spirit of hospitality, willing to let all forms of truth come, and to give them the means to reveal to us all that they can reveal, we are incapable of recognizing knowledge, and therefore incapable of knowing.

How do we know the concepts?

Concepts (properties and relations) are the meanings of words and expressions that name concepts. For example the property of being a tree is the meaning of the word tree. Without concepts our words could not have meaning.

For the truth of a statement to be determined, its meaning must be determined. The same statement can sometimes be true, sometimes false, depending on the various ways of interpreting it. We cannot know anything until we have clearly determined the concepts we are using. To produce the truth we must give ourselves the means, we must have clearly determined the concepts. 

How are concepts defined, or determined?

We know a concept when we know how to attribute it.

A fact is always determined by the attribution of a property to an individual, or of a relation between several individuals, therefore by the attribution of a concept.

An observation statement attributes an observed property to an observed individual, or an observed relation between several observed individuals, by naming them. An observation statement is always the observation of a fact.

To attribute concepts is to observe facts.

Sensory perception devices are producers of observations about the world in which we live. They are therefore fundamental producers of truth, because without them we could not know the actual world.

To perceive is always to perceive at the same time a concept and one or more beings to whom we attribute this concept. We always perceive a being by attributing to it a perceived property, or a perceived relation with other beings. The perception of beings is always the perception of facts.

An individual in a world is always observed from its properties and relations. When we observe a property, we at the same time observe the individual which has this property. When we observe a relation, we also observe the individuals who have this relation. But we never observe an individual without observing their properties or their relations. A naked individual, without properties or relations, cannot be observed, it cannot exist for an observer. The observation of concepts is more fundamental than the observation of individuals, because an individual is never observed without concepts being attributed to it.

To observe the actual world, we are not satisfied with sensory perception, we supplement it with observation instruments and theories. A theory can be used to produce new observations from observations already made, because the conclusion of reasoning based on observations can also be an observation.

A concept can be attributed not only to beings in the actual world, but also to beings in possible worlds. This is why concepts are determined with laws. Worlds are possible relative to a system of laws. Possible worlds are those that respect the laws. For example, naturally possible worlds are worlds that respect the laws of Nature. A theory is a system of laws. It is determined with principles: axioms and definitions. Axioms are the fundamental laws that determine fundamental concepts. Definitions determine the concepts defined from the fundamental concepts. Theorems are the logical consequences of principles. Concepts are attributed to possible beings if and only if this attribution is a theorem.

Theories are used to observe and explain the actual world, but they also enable us to go much further, because they give the power to know all the possibilities, everything that can be imagined and thought. Theories open the doors of perception of all possible worlds.

The universe of all possible universes is like a space in which each point is itself a space.

We determine concepts empirically by providing observation systems for the actual world. We determine concepts in a purely theoretical way by giving principles that enable us to reason with them. For example, the concept of distance can be identified with the ternary relation between two points x and y of a space and a positive real number z: d(x, y) = z. All distance measuring instruments empirically determine this concept. Axioms which found a geometry determine it theoretically.

In empirical sciences, we want our concepts to be determined both empirically and theoretically, because we want theories that explain the actual world.

A concept is precisely determined when the truth of its attribution is determined in all cases. Such precision is rarely achieved and is not necessarily desirable. Conceptual vagueness, partial indetermination, can make the use of concepts more flexible and better adapted to reality. We must only seek the appropriate precision, that which is sufficient to establish the truth of the observations or conclusions.

Divine generosity

« It is not possible that the divinity be envious. » (Aristotle, Metaphysics, Book A, 983a) 

« Only what is finally perfectly determined is at the same time exoteric, conceivable, capable of being learned and of being the property of all. The intelligible form of science is the path towards it which is open and offered to all and made the same for all, and achieving reasonable knowledge through understanding is the just demand of the consciousness which comes to join science. » (Hegel, Phenomenology of Spirit, Preface, p.XV)

As soon as good principles are recognized, they are adopted by all those who understand that they make more competent, stronger, more lucid. If they are really good, really useful, they naturally impose themselves on all those to whom they serve. By inventing or developing good schemas and good theories with good principles, one can make oneself useful for all minds. Fruits of reason are universal. Good observations, good principles and good reasoning are good for everyone. If one mind can reap the fruits of good principles, then all minds can reap the same fruits. When we seek good principles, we seek a good for all minds, we put into practice the great principle of ethics, that the good of a mind is to live for the good of all minds.

The great principles reveal the power of reason. They give all minds the means to acquire all knowledge, to understand all minds and to reveal all the benefits of reason. By learning what the great principles teach us, we also learn that we can think for the good of all minds. To be good for all minds is not an unattainable ideal. This is the reality of rational thought.

The good of a mind, the best for a mind, is to live for the good of all minds, for the best, for all minds.

This ethical principle does not impose a complete renouncement of one's own interest. Doing one's own good is good. But a mind that does not use its power for the good of others is a weak mind who does not develop its potential, a mind that is not accomplisheded. When we really know how to serve, we do our own good at the same time as that of others, and we reveal that we are really competent, really strong.

That the good of a mind is to live for the good of all minds can be understood as a communist or socialist principle: the good is a good community. There is no good without a good society. But it can also be adopted by a conservative right, who can affirm, for example, that the best for all is the protection of private interests.

That the good of a mind is to live for the good of all minds, has an immediate consequence: to want the good of a mind it is necessary to want him to want the good, since his good is to live for the good , and therefore to want it. For example, parents want the good of their children by wanting them to become honest citizens. But how do we do it? And are we capable of doing it? Isn't that asking for the impossible?

We want the power to know the good and to do it, and we want to give this power, since giving it is good. But is it possible? Must we not be God to have the power to know the good, to do it and to give this power?

Keeping the best for oneself, not giving it, is not the best but the worst: depriving others of the best. The best is not to keep the best for oneself but to give it. The power to give the best is the best. Therefore the power to give the power to give the best is also the best.

Everything happens just like if reason was a generous divinity, who gives wisdom to those who really want to know it. The first truth about reason is precisely that it is generous. It is not jealous, it does not deprive us of the best. It would not be the best if it deprived any of us of the best.

God's greatest goodness is to teach us how to be good. And in teaching goodness, he at the same time teaches us how to teach goodness, because the power to teach goodness is part of goodness.

We have the power to give power by delegation. The first source of power is always God. It is only God who truly has the power to give power. We have the power to give power only if God wants it.

That the good of a mind is to live for the good of all minds leads to a criterion for recognizing all knowledge, because a science cannot be a science without being a good for all minds. By knowing that rational knowledge must be a good for all minds, we have the fundamental knowledge which provides the means to recognize all knowledge.

The principle that the good of a mind is to live for the good of all minds is itself a good for all minds, therefore a manifestation of divine generosity.

Reason is a good for all minds. If we defend, if we teach or if we illustrate reason, we do good for all minds.

The universality of criticism is a consequence of the principle that reason is a good for all minds. When one claims to offer knowledge, all objections must be welcomed. A mind is always entitled to share its dissatisfaction, because he is himself a criterion for recognizing good knowledge, because good knowledge must help him to think well and to live well in order to be truly good. A claim to science which does not offer hospitality to all criticism denies science and reason. No science without the freedom of criticism. This is one of the most fundamental rules of the development of science.

For knowledge to be shared, it must draw only on common resources, accessible to all. One might think that it is a very restrictive limit, that by depriving oneself of private resources, one also deprives oneself of the best of knowledge, but the exact opposite is true. Our intelligence is the most powerful just when it is limited to common resources. It is by helping each other that we discover best the power of our intelligence, that we develop the best knowledge and that we make reason live.

Is reason divine?

Why say that reason is given to us by God? Why not say that we give it to ourselves?

We do science, we bring all knowledge into existence when we develop it, teach it and discuss it. Without our work as producers of truth, reason would not exist. It is our invention and our work.

“God made human beings in his image and they repaid him.” Are the wisdom and generosity that we attribute to God projections of only human wisdom and generosity?

Reason is necessary. It cannot not be what it is. We don't decide what it is. It does not depend on our arbitrary decisions or our good pleasure.

We don't invent reason, we discover it. It is what it is from all eternity. When we do science, we discover an eternal possibility.

When we discover reason, we discover at the same time that we are capable of discovering it, but we do not decide how to discover it. It is reason that shows us how to discover reason. It is therefore false to say that we make reason; the truth is that it makes us.

If one believes that the power to do good is the only true force, and that reason alone gives this power, then reason is the first source of all force. Without reason, we are without force. With reason we find the force and the force to give ourselves the force, but we do not choose what this force is.

Since the first source of all force, reason, directs us to the good, it is natural to identify it with divine power.

Our force is only to receive all the force that God gives us, to let this power act in us. God gives the force but we cannot demand it, only hope for it, let it come, and sometimes exercise it. We do not have the force to give ourselves the force unless God gives us that force, nor do we have the force to choose what it is.

Complement: undecidable theories

A knowledge observation system cannot be universal, infallible and answer all the questions that require a yes or no answer. If such a device existed, we could ask it "are you going to answer no to this question?" It can only give a false answer.

A theory of knowledge can be universal and infallible. First-order logic is such a theory. But a universal and infallible theory of knowledge cannot answer all the questions that arise about itself. This is a consequence of a theorem of Tarski: a true theory cannot contain a universal truth predicate for itself.

The proof of Tarski's theorem is by reduction to absurdity: suppose that T is a true theory such that all the statements of T are in the domain of variation of the variables of T and that is true is a universal truth predicate that can be defined in T. For a statement of T with a free variable A(x), we can ask whether or not it is true of itself. For example, x is a statement is true of itself, since x is a statement is a statement. x is not a statement is not true of itself, since x is not a statement is a statement. If is true is a universal truth predicate that can be defined in T, a statement A(x) equivalent to x is not true of itself can be defined in T. Is it true of itself? Is A(A(x)) true? It follows from the definition of A(x) that it is true of itself if and only if it is not true of itself. This leads to a contradiction: A(A(x)) and not A(A(x)) is a theorem of T but is false. The initial hypothesis of the existence of a true theory T in which we can define a universal truth predicate for itself is therefore false.

When we use first-order logic to reason about the truth of first-order logic, we use a truth predicate that is not universal. The domain of statements to which it can be attributed is a limited domain, it is not the domain of all the statements of the theory which we use to reason.

A theory of knowledge can be universal by being a theory of all knowledge, but never by being all knowledge about all knowledge. Omniscience is not within our reach.

A true theory cannot prove that it is true because it cannot state it, because it does not contain a truth predicate for all its statements. But the truth of a true theory T can be proven in a theory T+, more powerful than T, which contains a truth predicate for T: x is a true statement defined with the concepts of T. If T+ is true, x is a true statement defined with the concepts of T+ cannot be a predicate of T+ but only of a more powerful theory T++, and so on.

A model M of a theory T is a set of atomic facts defined with the fundamental concepts of T for which all the axioms of T are true. All the theorems of T are also true for the model M. A true theory can never prove the existence of a model of itself. If it did so it would have a truth predicate for itself: x is true for the model and according to Tarski's theorem it would be false. The completeness theorem of first-order logic states that a coherent theory always has a model. If a theory can prove its own coherence and if it can prove the existence of a model for any coherent theory then it is false. Gödel proved that we can do without the second condition: a true theory can never prove its own coherence. This is Gödel's second incompleteness theorem.

If a theory T is powerful enough to be a theory of natural numbers, we can always define in T a universal predicate of provability. x can be proven in T can be defined in T by numbering the formulas of T or naming them in some way, because the set of all theorems of a theory is always recursively enumerable. Now a universal truth predicate in T cannot be defined in T if T is true. Therefore the universal predicate of provability in T is not a universal predicate of truth in T. We thus obtain from Tarski's theorem Gödel's first incompeteness theorem (the two theorems were discovered independently in 1931): a true theory powerful enough to be a theory of natural numbers always makes it possible to state truths that it cannot prove.  But it would be wrong to conclude that there are truths that absolutely cannot be proven, because a truth that a theory T cannot prove can be proven in a theory T+ more powerful than T. T+ in turn makes it possible to state truths that it cannot prove.

The proof of Gödel's first incompleteness theorem is similar to that of Tarski. From the predicate x can be proven in T we can define in T a predicate G(x) equivalent to that x is true of itself cannot be proven in T. Can we prove in T G(G(x)) ? By definition of G(x), G(G(x)) is true if and only if it cannot be proven in T. If it could be proven in T then it would be false. If T is true, all statements that can be proven in T are true, so G(G(x)) cannot be proven in T. Therefore it is true. We thus prove that G(G(x)) is true and cannot be proven in T under the hypothesis that T is true.

Gödel's proof of G(G(x)) can be stated in a theory T+ which contains the predicate: x is a true statement defined with the concepts of T. T+ can prove the truth of T, but not of T+. The proof of G(G(x)) cannot be stated in the theory T, because T cannot prove the truth of T.

Gödel's and Tarski's theorems prove that omniscience is not within our reach, because no theory is sufficient to prove all truths. But if we conclude that there are truths that no theory can prove, we commit a logical error. for any theory T, there is a truth V such that V cannot be proven in T is not equivalent to there is a truth V such that for any theory T, V cannot be proven in T, because for all x there is an y such that is not equivalent to there is an y such that for all x. No one can know all truths does not imply that there are truths that no one can know.

Proving that the set of theorems of a theory is recursively enumerable is the laborious part of Gödel's proof. It's laborious but it's obvious. The recursive enumerability of the set of theorems is a consequence of the principle that a proof can always be recognized as a proof:

A set E is recursively enumerable if and only if there can be an infallible observation device which always answers yes when asked whether an element of E is an element of E.

A set E is decidable if and only if there can be an infallible observation device which always answers yes when asked whether an element of E is an element of E and always no when asked whether a being which is not element of E is element of E.

A set is decidable if and only if it and its complement are recursively enumerable.

If a recursively enumerable set is a set of natural numbers, we consider its complement in the set of all natural numbers to know if it is decidable. If it is a set of statements, we consider its complement in the set of all the statements formulated with the same letters.

The set of all proofs in a theory is always decidable, because a proof can always be recognized as proof.

A theorem of a theory is the conclusion of a proof in that theory. The set of all theorems of a theory is always recursively enumerable, because the set of all proofs in a theory is decidable. It is enough to make an observation device which observes all the sequences of statements of the theory to decide whether they are proofs, starting with the shortest sequences of statements. Such an observation device always ends up finding a proof if there is at least one, but if there is no proof, it never answers, it continues eternally looking for a proof that does not exist, it is not capable of answering that there is no proof.

If a true theory is powerful enough to be a theory of natural numbers, then the set of all its theorems is undecidable. The theory is undecidable. This can be proven from the proof of Gödel's first incompleteness theorem. If a theory T is decidable and if it is powerful enough to be a theory of natural numbers then all true statements which assert that a statement cannot be proven in T can be proven in T, because the set of statements which cannot be proven in T is recursively enumerable. But then the statement G(G(x)) could be proven in T when it is false. So T cannot be true. A theory that is decidable and powerful enough to be a theory of natural numbers is therefore always false.

We only have to reason about all the names formed with a single letter I, II, III, ...  in order to reason about all the natural numbers. A true theory is therefore always undecidable if it provides the means to reason about the most elementary truths.

We must distinguish two concepts of undecidability: the undecidability of a theory and the undecidability of a statement in a theory. A statement is undecidable in a theory if and only if neither it nor its negation can be proven in that theory. A theory is undecidable if and only if the set of its theorems is not decidable. 

If all the statements that can be formulated with the concepts of a theory are decidable in the theory then the theory is decidable, because we can always prove that a statement is not a theorem by proving that its negation is a theorem. On the other hand, if a theory is decidable, it is not always true that all the statements that can be formulated with the concepts of the theory are always decidable in this theory. It is enough that we can always decide whether or not a statement is a theorem for a theory to be decidable; it is not necessary that a statement or its negation always be a theorem.

In Gödel's proof, we prove that G(G(x)) is true and undecidable in the theory T but this is not enough to prove that T is undecidable. To do this, it is necessary to prove that the set of non-theorems (the statements which cannot be proven in T) is not recursively enumerable.

The theory of recursive enumerability and undecidability is presented in many books. Smullyan's doctoral thesis, Theory of formal systems (1961), is an excellent presentation.