A User's Guide to Reason/Metaphysics and Logic

This chapter demonstrates that logic is sufficient to rationally establish a metaphysics, and thus to make metaphysics a science. It provides precise answers to fundamental metaphysical questions: what are being, truth, possibility, necessity, concepts, essence, laws, and reasoning? These concepts are among the most fundamental, because they are necessary for the development of all knowledge. What is Being? Metaphysics is the science of being qua being (Aristotle, Metaphysics Γ), or simply the science of being. It teaches the most general truths, the truths about all beings, therefore the truths about being itself. The being of an individual consists in having properties and relations. A being always has properties and relations. A being that had neither properties nor relations would be nothing; it could not exist. To be is always to have properties and relations, or to be a property or a relation. A concept is a property or a relation. A property, or a quality or a trait, is attributed to a being. A relation exists between several beings. When a relation exists between two beings, it can be considered 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 that are not concepts. Properties and relations are attributed to them, but they cannot be attributed. Individuals in the broad sense are all beings, including concepts. The entire being of an individual (in the strict sense) is to have properties and relations. The being of a concept is to be attributed to beings and to have properties and relations. An individual reveals the universal. “Form is found to be one and identical at the same time in several places. It is as if you were to spread a veil over several human beings and say, ‘The veil remains one in its entirety when it is spread over 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 for many couples of individuals. The same is true 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 assert 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. "It can be considered a surprising oversight that the Logics fail to point out that every judgment expresses such a proposition: the singular is the universal, or, even more specifically: the subject is the predicate (for example, God is absolute spirit). Certainly, the determinations of singularity and universality, of subject and predicate, are also different, but the absolutely universal fact remains that every judgment states them as identical." (Hegel, Encyclopedia of the Philosophical Sciences, The Science of Logic §166, 1830) Russell noted that Hegel does not distinguish between A=B and A is a B. To respect this distinction, it is better to say that an individual reveals the universal rather than is the universal. But this correction does not detract from the relevance of Hegel's observation. Individuals and the binding of concepts Heat and pain can be perceived simultaneously 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 assume 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 is determined by individuals, because the being of an individual is a conjunction of concepts. What is Truth? "It is true that snow is white if and only if snow is white." (Tarski, The concept of truth in formalized languages, 1933, edited in Logic, semantics, metamathematics) One cannot affirm anything without simultaneously affirming its truth. Beings, their properties, and their relations can be named. We speak the truth by naming beings, properties, and relations, and by attributing these properties and relations to the beings that possess them. We speak the truth by saying what beings are. All beings, all properties, and all relations can be named. Everything can be said. Nothing can escape the truth. Truth is as vast as being. This is a principle of panlogism. Since being and truth are coextensive, metaphysics can be defined as the science of truth. It teaches truths about all truths, and therefore, the truth about truth. "We are also right to call philosophy the science of truth." (Aristotle, Metaphysics α, 993b20) "Everything has as much truth as it has being." (ibid., 993b30) An assertion is a statement that can be elementary or compound. An elementary statement attributes a property to one being or a relation to several beings. An elementary statement is true if and only if it attributes a property or a relation to the beings that possess it. "Come now, I will tell you, and you will hear: what are the only paths of inquiry open to the intellect; one, that being is, that non-being is not; the path of certainty, which accompanies truth; the other, that being is not, and that non-being necessarily is; a path where I tell you, you must not be seduced in any way. You cannot know what is not, you cannot grasp it or express it; for thought and being are one and the same." (The poem of Parmenides) Any truth states that it is true in a world; therefore, it states what a world is. By saying that truth is saying about beings what they are, we state what truth is; we therefore state the truth about truth. A falsehood is the negation of a truth. A truth is always equivalent to the negation of its negation, therefore to the negation of a falsehood. "A language will be considered false, both when one says of what is that it is not, and of what is not that it is." (Plato, The Sophist, 241a) Rules for Calculating Truth Compound statements are obtained from elementary statements with logical connectives: not A is the negation of A. A and B is the conjunction of A and B. A or B is the disjunction of A and B. If A then B is the implication of B by A. If ... then is the conditional. For all x, A is a generalization of A. For all x is the universal quantifier for the variable x. The statement There exists an x ​​such that A is obtained by applying the existential quantifier There exists an x ​​such that to A. Logical connectives are sufficient to state all laws from elementary statements. For example, all human beings are mortal can be stated as for all x, if x is a human being then x is mortal. The truth of compound statements is determined from the truth of the statements that constitute them. For example, If A then B is false if A is true and B is false, but it is true in all other cases. If A then B means that A cannot be true without B also being true. Truth can be calculated in a way similar to arithmetic with numbers: True and True = True True and False = False and True = False and False = False True or True = True or False = False or True = True False or False = False Not True = False Not False = True If True then True = If False then True = If False then False = True If True then False = False The truth of if A then B is simply the falsity of A and not B. It does not depend on any particular causal relationship between the condition A and the consequence B. For example, if it is daytime then 2+2=4 is always true, whether it is daytime or not. Similarly, if 2+2=5 then it's daytime is always true. If A then B means A only if B. It also means B if A. A if and only if B means (if A then B) and (if B then A). If we represent truth by 1 and falsehood by 0, then conjunction is multiplication: 1 x 1 = 1 0 x 0 = 0 x 1 = 1 x 0 = 0 Disjunction resembles addition: 0 + 0 = 0 0 + 1 = 1 + 0 = 1 except that 1 + 1 does not equal 1, whereas True or True = True. Let A(x) be a statement containing the variable x and containing neither for all x nor there exists an x ​​such that. We then say that the occurrences of x in A(x) are free. The occurrences of x in for all x, A(x) are bound by the universal quantifier for all x. Similarly, the occurrences of x in there exists an x ​​such that A(x) are bound by the existential quantifier. An occurrence of a variable is free if and only if it is not bound. When a statement contains one or more free variables, its truth is not determined because the individuals named by the variables are not identified. All free variables must be bound, or replaced by constants, for the truth of a statement to be determined. For all x, A(x) is true if and only if all statements A(a), where a is a constant chosen from the domain of the variable, are true. There exists an x ​​such that A(x) is true if and only if there exists a constant a chosen from the domain of the variable such that A(a) is true. A(a) is the statement obtained by substituting a for all free occurrences of x in A(x). For all x, A(x) resembles an infinite conjunction, the conjunction of all statements A(a) where a is a constant in the domain of the variable. There exists an x ​​such that A(x) resembles an infinite disjunction, the disjunction of all statements A(a) where a is a constant in the domain of the variable. The rules of truth composition show that the truth of statements composed with logical connectives is always determined by the truth of the component statements. When we state a law while leaving its variables x, y, A free, we must understand that they are linked by universal quantifiers, for all x, for all y, for all A, otherwise the truth of the law is not determined. What is it to be possible? To be possible is to be permitted by 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. Logical possibility is absolute possibility. Its fundamental law is coherence, that is, the absence of contradiction. The laws of Nature define natural possibility. Ethical laws define ethical possibility. To be ethically impossible is to be forbidden by ethical laws. Logically possible worlds Mathematics is the science of all logically possible worlds. It can also be called the science of all models, or of all structures, or of all worlds. A concept is fundamental when it is not defined in terms of more fundamental concepts. An atomic fact is the attribution of a fundamental property to an individual or of a fundamental relation to several individuals. A world is a set of atomic facts. An atomic fact can be designated by an atomic statement. An atomic statement is formed 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 it connects. An atomic statement is true of a world if and only if the atomic fact it designates is an element of that world. A set of atomic statements is never contradictory, because atomic statements do not contain negation. Therefore, any set of atomic statements always determines a logically possible world. A logically possible world is absolutely possible. It is an eternal possibility because it does not depend on any condition. From atomic statements, we can construct compound statements with logical connectives and thus formulate all statements about a world. The truth of statements thus composed is completely determined by the truth of the component statements, and therefore ultimately by the truth of the atomic statements. A world is completely determined by the atomic facts that constitute it. A statement is logically possible if and only if it is true of at least one world. A world is also called a model, or 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… 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 the other individuals in the same world. The being of a world is determined by the being of all the individuals in that world, and therefore ultimately by all the atomic facts that constitute it. The properties and relations of an individual in the world constitute its entire being in the world. The entire being of a being in the world is its being in the world. Since the entire being of an individual in the world is part of an eternal possibility, it is an eternal possibility. Even a transient individual, who is born, lives, and dies, reveals an eternal possibility through its entire existence. Notes: The principle that a world is a set, or a totality, of atomic facts is borrowed from Wittgenstein's Tractatus Logico-Philosophus (1921). However, the definitions of an atomic fact and an atomic statement adopted here are not found in the Tractatus. They come from first-order logic. The principle that an atomic statement is true of a world if and only if the atomic fact it designates is an element of that world is borrowed from Wittgenstein's Tractatus, from Tarski, and from model theory. In the language of model theory, a world is a model (Keisler, Fundamentals of model theory, 1977, in Handbook of mathematical logic, edited by Jon Barwise). David Lewis is concerned about circularity in the definition of the concept of logical possibility, because a world is logically possible when its definition cannot possibly imply a contradiction (Lewis, On the plurality of worlds, 1986). By defining a logically possible world from 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 can be no contradiction. Naturally possible worlds The natural sciences are the sciences of naturally possible worlds, including the actual world, because it is naturally possible. A naturally possible world is a world in which the laws of Nature hold true. We construct a theory of Nature by postulating fundamental laws of Nature. A naturally possible world is a model of a theory of Nature, provided that its laws are true of the actual world. If the laws of Nature are formulated with a system of differential equations, the naturally possible worlds are the solutions of the system. The movements of the planets, for example, are naturally possible because they are solutions to the differential equations of Newtonian physics. Our models of observed reality are never exact models. Their empirical truth, that is, their agreement with observations, always depends on approximations. The agreement between theoretical models and observed reality must be good enough for us to say that our models are naturally possible worlds, and that our laws are laws of Nature, but it is not necessary for the correspondence between the models and reality to be exact. A statement is naturally possible if and only if it is true of at least one naturally possible world. What is Necessity? Necessity and possibility are complementary concepts: A statement is necessary if and only if its negation is not possible. A statement is possible if and only if its negation is not necessary. A statement is necessary if and only if it is true in all possible worlds. "All possible worlds" above and below means "all possible worlds defined with the concepts of the statement." Laws are necessary because they are true in all possible worlds, since to be possible is to be permitted by laws. A statement is impossible if and only if it is false in all possible worlds. A statement is contingent if and only if it is true in at least one possible world and false in another. Everything that is necessary is possible, but the converse 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, truths about events in our world are generally contingent. What happened could have not happened. The laws of Nature do not determine everything that happens, far from it. On their own, they are insufficient to deduce the existence of even the smallest event. To believe or desire the impossible is always futile or foolish; it is always to deny necessity, because that which cannot be negated is necessary. Wisdom always requires submitting to necessity, never denying it, and never desiring what it forbids. Logical Necessity Logical laws are logically necessary statements. The principle that a world is a set of atomic facts enables us to find all logical laws because it enables us to define logical possibility and logical necessity. A conjunction of atomic statements is always logically possible and contingent. A contradiction A and not A is always logically impossible. The law of excluded middle A or not A is logically necessary. Logical necessity is absolute, without conditions. The truth of logical laws does not depend on any assumptions. The conclusion of a line of reasoning can be independent of any premise because the dependence on premises can be eliminated during the course of the reasoning. For example, if conclusion B was drawn from the single premise A, one can add the following as a new conclusion: if A then B. This new conclusion does not depend on premise A. It is therefore without hypothesis. Logical laws are the conclusions of reasoning that do not depend on any assumptions. A statement B is a logical consequence of a statement A if and only if, in all logically possible worlds where A is true, B is true. A statement B is a logical consequence of a statement A if and only if if A then B is a logical law. Natural Necessity A statement is naturally necessary if and only if it is true of 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. What is logically necessary is always naturally necessary, but the converse is not true, because the laws of Nature are not logically necessary. What is a concept? "In the beginning was the Word the Word with God God, the Word. He was in the beginning with God. Through him all things were made and without him nothing was made that was made. In him was life the life, the light of mankind and the light shines through the darkness the darkness has not overcome it.” (John 1:1) A concept is a property or a relation. Its defining characteristic is that it is attributed. Concepts 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 would have no meaning. For the truth of a statement to be determined, its meaning must be determined. The same statement can be sometimes true, sometimes false, depending on the various ways of interpreting it. We cannot know anything until we have clearly defined the concepts we use. To produce truth, we must provide the means; we must have clearly defined the concepts. How are concepts defined, or determined? The being of a concept is determined only if all its possible attributions are determined. We know a concept when we know how to attribute it. For the attribution of a concept to be true, there must be laws that determine this attribution. The attribution of observed concepts is determined by the laws of observation. The attribution of the concepts of a theory is determined by the fundamental laws of that theory. If there were no law, there would be no truth to attributions, and therefore no concept. No concept without law. If there were no concept, there would be no being, because to exist, one must have properties and relations. No being without a concept. Thus, we have proven: If there were no law, there would be no being. No being without law. There are always explanations for everything, because explanations are based on laws, and because nothing can exist without law. When they are true, laws are true for all eternity. They logically and chronologically precede everything that currently appears. Everything happens as if the word preceded being, because to be actual, one must be possible, one must be permitted by laws. In the beginning was the Word. (John 1:1) The wise person is the one who knows the causes, but to know the causes is always to know the laws, because the laws are the foundation of being. Knowledge of the laws makes all the difference between merely superficial knowledge and true knowledge. The ignorant person does not know that the Universe is ordered because they do not know its laws. The wise person discovers the order of the Universe by discovering its laws. An observational statement attributes an observed property to an observed individual, or an observed relation between several observed individuals, by naming them. Perceiving is always perceiving simultaneously a concept and one or more beings to which this concept is attributed. An individual in a world is always observed based on its properties and its relations. When we observe a property, we simultaneously observe the individual who has this property. When we observe a relation, we simultaneously observe the individuals who have this relation. But we never observe an individual without observing its properties or relations. A bare 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. 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 world as it currently exists. To observe the world as it currently exists, we do not rely solely on sensory perception; we supplement it with observational 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 itself be an observation. A concept can be attributed not only to beings in the current world, but also to beings in possible worlds. Worlds are possible relative to a system of laws. Possible worlds are those that respect laws. For example, naturally possible worlds are those that respect the laws of Nature. A theory is a system of laws. It is determined by principles: axioms and definitions. Axioms are the fundamental laws that determine fundamental concepts. Definitions determine the concepts defined from 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 serve to observe the current world, but they also provide the means to go much further, because they give us the power to know all possibilities, everything that can be imagined and thought. Theories open the doors to knowledge of all possible worlds. Concepts are determined empirically by providing devices for observing the current world. Concepts are determined purely theoretically by providing 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 in a space and a positive real number z: d(x, y) = z. All instruments for measuring distances determine this concept empirically. Axioms that form the basis of a geometry determine it theoretically. In the empirical sciences, we want our concepts to be determined both empirically and theoretically, because we want theories that explain the current world. A concept is precisely determined when the truth of its attribution is established in all cases. Such precision is rarely achieved and is not necessarily desirable. Conceptual vagueness, partial indeterminacy, can make the use of concepts more flexible and better suited to reality. We simply need to seek the appropriate level of precision, the one that is sufficient to establish the truth of observations or conclusions. What is Essence? The essence of a being is what it essentially is, all its properties and relations that it cannot lose without ceasing to exist. When we ask what X is, we are asking for the essence of X. To say that essence is what it is is to say what essence is, therefore the essence of essence. A property or relation of a being X is essential if and only if X cannot exist without having that property or relation. A property or relation is accidental if and only if it is not essential. The essence of a concept is determined by the laws that govern the attribution of the concept. Adherence to a law is an essential property of a concept, because laws make concepts what they are. There are no concepts without laws. A theory is a consistent list of axioms and definitions. A theorem of the theory is a logical consequence of its axioms and definitions. A list of statements is consistent if and only if its logical consequences are never contradictions. A contradiction is the conjunction of a statement and its negation. Axioms are the fundamental laws that determine the fundamental concepts of a theory. All other concepts of the theory can be defined from the fundamental concepts. A concept definable from the fundamental concepts is a predicate of the theory. Logic is also called predicate calculus, or concept calculus. The laws that determine a concept defined from the fundamental concepts are logical consequences of its definition and axioms. The essence of a concept is stated by giving its definition, if it can be defined from concepts whose essence is already known, or by giving axioms, the fundamental laws that determine its attribution, if it is a fundamental concept. We express the essence of concepts by creating theories. The axioms and definitions of our theories determine the essence of all our concepts. What is a law? Any statement can be called a law, but laws are generally expected to be universal, or general. A general law can be applied to many specific cases. It can always be formulated with 'for all x'. 'Always' is 'for all times t'. 'Everywhere' is 'for all positions x'. 'Never P' is 'for all times t, it is false that P'. 'All minds' is 'for all mind x' or 'for all x, if x is a mind'. Similarly for all truths: 'for all x, if x is a truth', and even for all generalities: 'for all x, if x is a generality'. When entities obey the same laws, we know them all by knowing their laws. It is to know a myriad of beings simultaneously, as if the laws revealed the totality to us at a single glance. In this way, one can know very vast totalities: all material beings, all minds, all worlds, all theories, everything that is naturally or logically possible... A truly universal law has no exceptions. A single exception is enough to refute it. Yet many rules have exceptions without being bad rules. These are incomplete rules, which do not mention rarely encountered conditions, or probabilistic rules, which give a high probability to their conclusion without guaranteeing its certainty. We believe that Nature obeys laws, but is this truly a justified belief? Isn't it rather wishful thinking? It could be that all the laws of Nature in which we currently believe will be refuted by future observations. And could Nature not be lawless? Matter would not be matter if it did not obey laws. Matter is necessarily detectable; therefore, it must obey laws of detection, which result from the fundamental laws of interaction. Matter that obeyed no laws would be undetectable, and there would be no reason to call it matter. We have absolutely no idea what it could be; it seems inconceivable. It is as if matter and mind were made for each other, because the nature of matter is to obey laws, and the nature of mind is to know laws. Neither matter, nor, a fortiori, life and consciousness, could exist and develop if Nature did not obey laws. We would not be here to discuss it. We should not expect our experiments to definitively prove that Nature obeys laws, which they cannot do, since any law verified today could be refuted tomorrow, but only to help us discover the laws of Nature. We know in advance that Nature obeys laws, but we do not know which ones. Since Nature does not appear to be malicious, but rather generous, it seems that honest work and well-controlled experiments are sufficient to find and prove the laws it obeys. “Many laws of nature can only be known to us through experience; but we cannot learn to know the legitimacy of the connection between phenomena, that is, nature in general, through any experience, since experience itself needs these laws as the foundation of its a priori possibility.” (Kant, Prolegomena to Any Future Metaphysics Which Will Have the Right to Present Itself as a Science, § 37, 1783) All sciences classify beings into categories. Beings of the same category have common properties and obey the same laws. Theories are always formed by imposing laws on beings of the same category. If beings do not obey the same laws, a theory cannot be formed. The truth of laws is a necessary condition for the intelligibility of reality. The light that reaches us from distant stars is the same as that of the Sun, or that which we produce on Earth. It always behaves in 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, that is, the analysis of the light it absorbs or emits. We can know the chemical composition of distant celestial bodies 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 composed of identical molecules or atoms. Natural substances always behave the same way once 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 kind are all identical and obey the same laws. All points in space are identical. When you know one, you know them all. The same is true for points in spacetime. We know infinite spaces in their entirety simply by knowing their laws. All natural numbers are obtained by adding identical units. We know the composition of all natural numbers, however large they may be, simply by knowing one. Similarly, the elementary constituents of all material beings are identical when they are of the same kind. By knowing a small number of elementary species, one simultaneously knows the constitution of all material systems, even very large and very complex ones. What is Reasoning? Logic is the science of reasoning. The best way to think is to put one truth in front of another and start again. A line of reasoning is a sequence of statements. The statements are the premises, the conclusion, possibly provisional hypotheses, and intermediate steps. The premises are axioms, definitions, hypotheses, or conditions. For a line of reasoning to be logically sound, it is necessary and sufficient that all the intermediate steps and the conclusion be obvious logical consequences of the preceding steps, the premises, and the provisional hypotheses on which they depend. The conclusion must depend only on the premises, not on the provisional hypotheses. A logical consequence of a list of premises can never be false if the premises are true. Definition: A is a logical consequence of premises P if and only if there is no logically possible world about which premises P are true and A is false. The premises P are any finite list of premises, including the null list: no premise. A logical law, or a tautology, is a logical consequence of the null list. This means it is a law that does not depend on any assumptions, an unconditional, categorical, absolute law. A logical law can never be false. If a line of reasoning is logically correct, all the intermediate steps leading to the conclusion are true, provided that the conditions on which they depend are true. Logic always leads from true to true, never from true to false. The twelve fundamental rules of deduction provide all the obvious logical consequences necessary for any logically sound argument: The rule of detachment: B is a logical consequence of the two premises A and If A then B. If B is a logical consequence of premises P and A, then If A, then B is a logical consequence of premises P. Proof by contradiction: If B and the negation of B are both logical consequences of premises P and A, then the negation of A is a logical consequence of premises P. A is a logical consequence of the negation of the negation of A. A and B is a logical consequence of the two premises A and B. A and B are both logical consequences of the single premise A and B. A or B and B or A are both logical consequences of A. C is a logical consequence of the three premises A or B, if A then C, and if B then C. Applying a general law to a specific case: A(a) is a logical consequence of For all x, A(x). a is a constant or a variable. A(a) is obtained by substituting a for all free occurrences of x in A(x). Generalization from an example without condition: If A(x) is a logical consequence of premises P in which the variable x has no free occurrences, then For all x, A(x) is a logical consequence of premises P. Proof of existence by example: There exists an x ​​such that A(x) is a logical consequence of A(a). a is a constant or a variable. x is a variable that has no free occurrences in A(a). A(x) is obtained by substituting x for one or more occurrences of a in A(a), not necessarily all of them. A(a) is a logical consequence of There exists an x ​​such that A(x). a is a new constant. It should not be mentioned in the previous steps of the reasoning. A(a) is obtained by substituting a for all free occurrences of x in A(x). Be careful to distinguish between If A then B and its converse If B then A. A is not a logical consequence of B and If A then B. Believing otherwise is a very common logical fallacy. The difference between a statement and its converse is the difference between if and only if. If A then B is equivalent to A only if B not to A if B. The generalization rule from an example without condition allows us to generalize from a hypothetical example: let C(x) be conditions on a hypothetical being x. If A(x) is a logical consequence of the premises P and C(x) then If C(x) then A(x) is a logical consequence of the premises P. If x has no free occurrences in the premises P, then for all x, if C(x), then A(x) is a logical consequence of the premises P. It can be proven that the fundamental rules of deduction are always true, based on the definition of the concept of logical consequence and the rules for calculating truth for logical operators. Logical rules always lead from one truth to another. If the premises are true and the reasoning is logical, the conclusion cannot be false. A logical conclusion cannot therefore provide 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, what is the point of reasoning? It seems that reasoning can teach us nothing, since a conclusion is merely a restatement of what we already know when we know the premises. Laws are like a concentrated source of knowledge; they provide an unlimited wealth of information. Applying a law to a specific case is a logical argument that reveals what the law teaches in that case. All the specific consequences are determined by the laws, but reasoning is necessary to discover them. Reasoning explicitly reveals what its premises implicitly determine; it unfolds what is concealed, it unveils what is present but hidden until reasoning is applied. An example: consider the law "if the system S is in state x at time t, then it is in state f(x) at time t+1, for all times t and all states x," and the observation that the system S is in state a at time 0. From these two premises, we can deduce numerous consequences: S is in state f(a) at time 1, in state f(f(a))=f²(a) at time 2, in state f(f(f(a)))=f³(a) at time 3, and so on. Using the principle of mathematical induction, we can deduce that S is in state fn(a) at time n for all natural numbers n. With a law and an observation of an initial state, we can know the fate of the system for eternity. Knowledge of laws gives reasoning its power. We do science by learning through reasoning what the laws teach.