"Why are things as they are and not otherwise" (Johannes Kepler).
"The realm of possible worlds is a philosopher's paradise" (David Kellogg Lewis).
"What really interests me is whether God had any choice in the creation of the world" (Einstein).
Philosophy of Possible Worlds
Why are things the way they are? Why, for example, are there 7 colors, 8 planets and 4 seasons? Why do we have 2 legs and 2 arms? Could things be otherwise?
The idea that "there is only one real world, but many possible alternative worlds" has played an important role in philosophy (especially in philosophical theology) and in science (especially in the development of modal logic, the logic of the necessary and the possible), as well as in both: in philosophy of science.
From the philosophical point of view, there is a controversy about the existence and nature of possible worlds. They may be of the real (physical), imaginative, mental, linguistic or abstract type. In general it is admitted:
That the real world is a particular case among the infinite possible worlds.
That other possible worlds may have different laws.
That, despite the potential diversity of possible worlds, there must be common elements or properties that underlie them. This is the essentialist philosophy of possible worlds.
The Logic of Possible Worlds
From a formal-logical point of view, a possible world must be:
Consistent, that is, free of contradictions. A world is possible if it does not contradict the laws of logic, since the laws of logic are supposed to be universal.
Complete, in the sense that it must cover all its contents. It must also cover all the details, since the slightest variation would give rise to another possible different world.
In the logic of possible worlds there are two types of propositions:
Necessary propositions, which are true in all possible worlds. They are universal truths. They refer to the essence or common core of the possible worlds and do not change from one world to another. Nor do they change with time, they are eternal.
Circumstantial or contingent propositions (which are the opposite of necessary propositions) are sometimes true and sometimes false, depending on the place or world and time. For example, the statement "it is raining". True contingent propositions are true in some possible world.
The Principal Authors
Leibniz, the creator of the concept
The concept of "possible world" has its origin in Leibniz. In his book "Theodicy" (literally, "justification of God") he tries to justify the obvious imperfections of the world, claiming that the world is imperfect because only God is perfect.
According to Leibniz, God possesses a direct vision of all possible worlds. All the details of these worlds are present in his mind, without requiring any intellectual activity on his part. The divine understanding is the region of the infinite possible worlds.
God, out of the infinite possibilities he had, selected this world as having the following characteristics:
It is the most perfect world, the best of all possible worlds, not from the moral point of view, but from the physical and mathematical point of view.
It is the world simplest in hypothesis and richest in phenomena. It is its principle of economy: the "maximin".
It is the most balanced world between variety and homogeneity.
It is the world where nothing happens in leaps and bounds, where nothing happens all at once.
It is the world where the combinations are the best possible. This world is governed by the principle of composability, of the composable. Material things have structures that make some composable and others not. For example, man and horse are not composable to create a centaur (half man, half horse). Abstract objects, on the other hand, are fully composable. The centaur is possible at the abstract level, not real. At the abstract level, there are no limitations, but at the physical level. God created the world following the principle of maximum physical composability.
A possible world is not just any aggregate of possible individuals, but a composable set of possible individuals. What is composable is what can be real. What is not composable is impossible, it cannot be real.
The possible is that which is composable, that is, that which is deprived of contradiction.
The impossible is the contradictory. For example, a square circle is impossible because it is contradictory.
Leibniz was a firm believer in the importance of logic, not only in its proper domain, but as the basis of metaphysics.
The passage from possibility to existence is governed by the "principle of sufficient reason": "nothing is without reason or cause." If there is no sufficient reason for one thing to happen instead of another, nothing happens; the initial situation does not change.
There are necessary and contingent truths.
Necessary truths are those that do not change in any possible world; they are the truths of reason, such as logical and mathematical truths (such as 2+2=4). And the contrary ones are not possible because they imply contradiction.
Contingent truths are those that can change from one possible world to another and in which their contraries do not imply contradiction. Factual truths are contingent, since their opposites may belong to other possible worlds. For example, "Columbus discovered America" is a factual truth, and it is contingent because it could happen or not, within the set of all possibilities.
Peirce, the precursor
Peirce is considered by many to be the forerunner of modern "possible worlds semantics," for he attempted to capture the meaning of modal notions (of necessity and possibility) and provided a foundation for them:
In Leibniz, possible worlds are associated with divine thought. In Peirce, possible worlds constitute an exercise of the freedom of human thought, in particular the capacity to elaborate hypotheses.
For Peirce, a sign is "something that is by someone, for something, in some aspect or disposition." It is at the level of interpreter that the analysis of intelligible worlds appears. Every interpreter introduces a possible world, so that "every world is relative to every formulable utterance". There can be an endless chain of interpreters referring to a sign: each sign is both interpreter of the one that precedes it and interpreted by the one that follows it.
Initially, Peirce interpreted modality as associated with "states of information," a concept that is synonymous with "possible worlds." He distinguishes between essential (or logical) modality and substantial modality. The essential refers to the states of information common to everything, to the deep. The substantial refers to the states of information relative to the real, to the superficial phenomena. Later, Peirce realized that not all types of modality could be explained in terms of states of information.
The foundation of modality is to be found in general laws or principles, which are what determine which states are necessary and which are possible.
Wittgenstein and linguistics
For the first Wittgenstein (the one of the Tractatus), "the world is the totality of facts, not of things". Facts are expressed at the linguistic level, there being a correspondence between the facts of the world and the linguistic forms that express them. With this conception, Wittgenstein gave a "linguistic turn" to philosophy. The traditional conception was that the world is a set or system of objects, each with its corresponding properties or attributes.
With Wittgenstein there is a paradigm shift, a new way of looking at the world:
Through language we conceive, not only how the world is (the facts of the world), but also how it is not and how it could have been.
Language transcends the world. Language is a world of possibilities.
A possible world is a set of possible facts expressed linguistically.
Necessary truths cannot be expressed with language. Only facts, which could be false, can be represented.
The rules of logic are the general principles that dictate the structure of possible facts.
The modal realism of David Kellogg Lewis
David Kellogg Lewis, a student of Quine, was an analytic philosopher, philosopher of language and mind, metaphysician, epistemologist, and philosophical logician. He is best known for his "modal realism," a theory that holds that possible worlds are not just a concept to explain possibility and necessity, but are as real as our own universe. This theory appeared in his 1973 publication "Contrafactuals" and in other articles, but he expresses it more fully in his 1986 work "On the Plurality of Worlds". His main ideas are:
Possible worlds are not concepts, but realities. They exist in the same sense that our universe does. "The inhabitants of other worlds can say that their world is the real one, if for them 'real' is the same as for us."
The world we live in is no more real than any other possible world. All possible worlds have the same ontological level. The real world, from the metaphysical point of view, need not be distinguished among all possible worlds.
There is an infinity of possible worlds, where we can find all possible combinations of events.
The possible worlds are totally incommunicado. There is no spatial, temporal or causal link between them.
A proposition p is possible if, and only if, there exists a possible world in which the proposition p is true.
There are two kinds of modal assertions: 1) De re: the attribution of a possibility or necessity of something to a particular entity; 2) De dicto: the possibility or necessity of this or that happening (for example, that a talking dog exists).
The similarities between universes do not determine identical, but rather "replicas". The replica of something in another hypothetical situation is never identical. The relation that an individual maintains with its replica (in a possible world) is neither symmetrical nor transitive.
Each individual is a replica of itself.
Every individual has at least one replica in some world.
Each individual belongs to only one possible world. One cannot belong to two because one would incur a contradiction. For example, if the Eiffel Tower measures 15 meters and in another world it measures 30, then it measures and does not measure 15 meters, which is contradictory. And it is not valid to say that it measures 15 meters in the world m1, because that would mean that it is part of m1. Measuring 15 meters is an absolute property, not relative to a world.
The theory of modal realism is criticized because it violates the principle of Occam's razor: "not to resort to a multiplicity of entities without necessity".
Kripke's essentialism
The philosopher and logician Saul Kripke is one of the leading figures in the field of possible worlds. He defends essentialism, a doctrine that holds that there are certain properties that things have that are immutable and eternal, that they cannot fail to have.
Kripke, inspired by Leibniz, opened a new philosophical field called "semantics of possible worlds", also called "Kripke's semantics". His famous paper "A Completeness Theorem in Modal Logic" [1959] ushered in a new era in modal logic. His semantic model is considered the standard in this type of logic and refers to formal logic constructed in terms of models based on possible worlds. The concept of possible world associates it with logic, with truth values: everything can be a possible world, as long as there is the ability to classify propositions into true and false.
The truth value (true or false) of a proposition is not fixed, but varies according to the world under consideration.
A proposition is true in a world if it correctly describes something in that world and false if it does not.
A proposition is necessarily true if it is true in all possible worlds.
A proposition is possibly true if it is true for at least one possible world.
Possible worlds are always relative to a given world m and accessible only from that world m. The world m2 is accessible from the world m1 if m2 is a possible world relative to m1. But from the point of view of another world, the possible worlds could be different.
Accessibility relations between different possible worlds may or may not fulfill certain properties (such as reflexive, symmetric and transitive).
A frame is a set of worlds where accessibility relations exist between them.
A model is a set of true propositions in all worlds of the frame.
Other aspects of Kripke's semantics are:
Kripke defends a metaphysical thesis: ordinary objects (such as a table) have an essence that belongs to a dimension higher than the material one.
Kripke, intuiting that the problem of necessity was closely connected with that of identity, proved the "necessity theorem of identity" (1963): "If two objects (a and b) are really identical (a=b), then such an identity is valid in all possible worlds (so a=b is necessarily true)." Apparently this is a tautology. But Quine also meant something similar: he defined "p is necessary" as "something that is equal to itself".
For Krike, identity is an essential property of every object and individual. It is also an internal relation that each object maintains with itself. In modal logic, this is expressed as follows: □(x=x) (it is necessary that x be equal to itself).
Therefore, identity statements are necessary. For example,
Necessarily water is H2O.
Necessarily heat is molecular motion.
Necessarily Cicero is Tullius.
Necessarily Hesperus (the evening star) is Phosphorus (the morning star).
In "Naming and Necessity", Kripke distinguishes between the necessary and the a priori:
The notion of necessity is metaphysical. Necessary truths are truths in all possible worlds.
The notion of "a priori" (and also the notion of "a posteriori") belongs to epistemology. The "a priori" is what we know independently of experience.
The following 4 combinations of truths are given: 1) necessary a priori; 2) necessary a posteriori; 3) contingent a priori; 4) contingent a posteriori.
No one disputes that there are a priori necessary and a posteriori contingent truths. Kripke adds the other two:
There are a priori truths that are contingent. For example: water boils at 100 degrees; the standard measure of length is the meter.
There are necessary truths that are a posteriori. For example: the Goldbach conjecture, Fermat's last theorem, the atomic number (79) of gold, the chemical formula (H2O) of water.
We have direct contact with reality and refer to it in three ways:
With proper names. A proper noun (NP) is a term that refers to an object. It has no meaning, only denotes an object, and is not a covert description.
With natural-type terms such as "water," "gold," and "tiger." The world consists of objects that fall into "natural classes". Natural classes are names associated with classes.
With defined descriptions. A defined description (DD) is a way of describing an object or a natural class.
Kripke calls proper names and definite descriptions "designators". And he distinguishes between two types of designators:
Rigid designator. It designates the same object in all possible worlds.
Non-rigid designator. Designates different objects in the different possible worlds.
The proper name (NP) by which we refer to a person is not an essential property of the person, but an accidental property of his identity, so it could be considered a non-rigid designator. But in Kripke's theory, a NP is a rigid designator: it refers to the same individual in the real world, in all possible worlds and in every hypothetical situation. The NP designates the same object in all possible worlds.
For example, "The inventor of bifocals" might not be thought of as a rigid designator, since in our world it designates Benjamin Franklin, but in another world it could be a different person. But according to Kripke, "Benjamin Franklin" is a rigid designator because Benjamin Franklin could have been a different kind of person, but he could not have been someone different.
The object denoted by a rigid designator need not exist in every possible world, that is, it need not be a necessary object. We can imagine a possible world in which the same designator would be used differently. For example, a world in which a person with the name "Benjamin Franklin" existed, but it would be different from the Benjamin Franklin of our world. "The inventor of bifocals" would be the inventor of bifocals in that possible world, who could be someone else or no one. Benjamin Franklin could have had another name such as "Richard Franklin."
Kripke's so-called "modal argument" is based on the following central idea. Normally, a proper name NP appearing in a proposition can be replaced by a DD definite description, without affecting the meaning of that proposition. For example, the name "Aristotle" can be replaced by "the author of Nicomachean Ethics", "the most famous teacher of Alexander the Great", and so on. But in Kripke's theory, the substitution in a proposition of a name for a description modifies the modal profile of the proposition and, therefore, its meaning.
For example, the propositions
Necessarily, if Aristotle existed, Aristotle was Aristotle.
Necessarily, if Aristotle existed, Aristotle was Alexander the Great's most famous teacher.
Proposition 1 is trivially true because there is no possible world in which Aristotle was not Aristotle. Proposition 2 is clearly false because in some world it is possible that Aristotle would not have been Alexander the Great's teacher.
Imaginary Mathematics
Imaginary mathematics can also be framed within the philosophy of possible worlds.
In formal axiomatic systems it is possible to define any consistent set of axioms and rules of derivation, whether or not there are interpretations (models) that can be applied to the real world. In this sense a formal axiomatic system can have the character of "imaginary". Recall that non-Euclidean geometries were initially called "imaginary", since it was believed that they did not correspond to reality.
Imaginary mathematics is more generic and abstract than traditional mathematics, in the same way that non-Euclidean geometry without the restriction of the fifth postulate (that of parallels) is more general than Euclidean geometry, since the latter is a particular case of the former. The historical delay in recognizing the existence of alternative geometries to Euclidean geometry came from the identification between geometric space and physical space. But mathematics transcends the physical world.
Imaginary mathematics is thus definitively detached from the real world. Nevertheless, of course, there is mathematics that can be applied to the real world. But the mathematical universe is wider than the real one. For the Platonists the mathematical world, as part of the world of ideas, is also "real", although belonging to a higher dimension.
It is at this level of abstraction/generalization that Bertrand Russell's ironic statement, "Mathematics is the science in which we do not know what we are talking about, nor whether what we say is true", can have a place.
Cantor was one of the energetic advocates of the freedom of mathematics to create or develop abstract entities where all possibilities are open. Cantor referred to "free mathematics" rather than "pure" mathematics.
The Conceptual Challenge of Possible Worlds
The concept of "possible world" has been subject to multiple discussions and interpretations, because it is an ambiguous, even mysterious and metaphysical concept. Quine was especially critical, since he said that possible worlds had an obscure conceptual foundation.
Kripke's theory (the semantics of possible worlds) has sharpened the controversy. Although it is considered an advance in the formal semantics of modality, several problems are detected:
In the concepts of necessity and possibility, there are signs of circularity. For example, the definition of "possible" as "true in a possible world".
Formal developments do not clearly respond to the modal concepts of necessity and possibility.
The concept of "possible world" is associated with logic, with truth values. It is not really "semantic" in the strict sense.
The concept of "possible world" cannot be explained because it cannot be reduced to simpler elements.
The fact is that so far no clear and conclusive answer has been given as to the true nature of possible worlds, for numerous questions arise, such as the following:
What exactly is a possible world?
Are possible worlds real, abstract, mental, linguistic, or imaginative? Do they belong to a Platonic realm?
What kinds of entities "inhabit" a possible world?
Can an entity inhabit several possible worlds at the same time?
Does space exist in possible worlds? If there is space −and logically there must be a space− what is the nature of that space?
Does time exist in possible worlds; are possible worlds dynamic?
Can possible worlds be accessed? If so, what kind of access is it: mental, linguistic, or imaginative?
What can we know about possible worlds? Can we know everything about them?
Do possible worlds only describe counterfactual situations, i.e., contrary to what happened in the real world?
How are possible worlds related to each other? Can they communicate or interact with each other?
Are there higher-order possible worlds, i.e., can one possible world contain other possible worlds? If so, is there a supreme possible world that contains them all?
Are there virtual possible worlds, that is, possible worlds made up of parts of possible worlds?
According to the theory of absolute time −or no-time−, everything that is possible to happen already exists in an absolute temporal dimension, where there is no distinction between past, present and future. The temporal relations (simultaneity, precedence, etc.) between objects are relations like any other relations. When something happens or we live an experience, the only thing we are doing is accessing something that already exists previously. Absolute time is the deep realm, the world where all possibilities reside.
One can also associate the possible with the imaginable. According to the theory of imaginary worlds, everything we can imagine already exists in a higher dimension. This theory goes beyond the conception of absolute time.
MENTAL and Possible Worlds
As opposed to the semantics of possible worlds as a fuzzy concept, MENTAL offers an extremely simple conceptual and formal model of possible worlds that answers all of the above questions.
Simplicity.
It is simpler to consider all possible worlds than a particular world. For example, the set of infinite natural numbers 1, 2, 3,... is very simple, although a particular number of (say) 5000 million digits is extraordinarily complex. The complex is given in the particular. The simple is given in the general or universal.
The necessary and the possible.
MENTAL represents the true essence of possible worlds, for the possibilities reside in the degrees of freedom, which are the primary archetypes.
In MENTAL, what is necessary is the absolute or essential, which are the primitives, which constitute the foundation of all possible worlds. By means of these primitives we can create concrete expressions, but the primitives themselves are not expressible by language.
The possible is associated with the possible expressions or manifestations of the primitives. Therefore, the necessary is associated with the profound, and the possible with the superficial. Necessity and possibility are opposite concepts, which are united in MENTAL. Possibility is linked to composability (the combinatorics of instances of primitives), which is of an abstract type. At the abstract conceptual level there are no combinatorial restrictions, as Leibniz claimed.
Absolute (inexpressible) necessity resides in the primary archetypes. Relative (expressible) necessity is realized by generic expressions (parameterized or not) that are always being evaluated and do not change unless expressly modified. For example,
⟨( n>5 → n=5 )⟩ // n cannot be greater than 5
⟨( f(x) = (2*x + 1) )⟩ // function definition
Possible world = abstract space.
A possible world is a perfectly conceptually defined and rigorously formalized abstract world. In MENTAL, a possible world is an abstract space, the space in which expressions reside and where relationships exist between them. The abstract space is constructed by means of the relations between expressions. In this respect, there is some agreement with Peirce when he states that "states of information" are synonymous with "possible worlds".
Level of reality.
A possible world is real, but its existence is linguistic, mental, and imaginative. All emanate from the Platonic realm of semantic primitives. In MENTAL, possible worlds are not philosophical speculations, but are constructible worlds, expressible with language, though necessarily abstract in character. The theory of possible worlds is purely theoretical and speculative. With MENTAL theory and practice are united
According to Kripke's metaphysical thesis, ordinary objects have an essence that belongs to a higher dimension than the material one. Indeed, everything (the physical and the mental) share the same essence, which are the same primary archetypes.
The source of the possible worlds.
In the MENTAL model there is a single source from which all expressions and all possible worlds emanate: the primitives, which constitute the essence of all possible worlds, including the real one.
Ω symbolizes the possible, all possible expressions, all possible worlds. Everything is already in Ω. Every manifested expression has its foundation and support in Ω;
The infinite possibilities (expressions) that can be formed with primitives already exist in the field of all possibilities. MENTAL is modal realism, but not in the sense given by David Kellogg Lewis, but in the sense that every expression already exists, but at the potential level. When we refer to an expression, we are accessing something that already existed previously, although the existence is not material, but mental and abstract.
Relationship between possible worlds.
All possible expressions and all possible worlds share the same primitives, although manifested in different ways. All possible worlds are deeply united by the primitives.
Abstract time.
Expressions of a possible world can be static or dynamic. But, in either case, it is abstract time, linked to the expressions of sequence (linear time) and set (parallel time).
General principles and laws.
Peirce certainly intuited that the foundation of modality is to be found in general laws or principles, although he did not go so far as to state what those principles were. In MENTAL, the principles are the universal semantic primitives, and the laws are the various general axioms relating those primitives.
According to Wittgenstein, the rules of logic are the general principles that determine the structure of facts. But the general principles are the primary archetypes. Logic is only one of the dimensions of reality and language.
Possible expressions.
MENTAL is not oriented to propositions, but to expressions, those allowed by language. The truth or falsity of an expression is not considered, since these concepts are of a superficial type. Instead, the concept of "existence" and "non-existence" is used. Since every non-null expression exists, this allows reasoning about all kinds of expressions and not only those of a logical type. And the truth of an expression is its meaning, its semantics.
The true reality.
The true reality is what is essential, what is common, what is shared by all possible worlds. If a mathematical object is necessary in all possible worlds, then it is real. Since the primitives of MENTAL are necessary in all possible worlds, then the primitives of MENTAL constitute true reality.
Imaginary mathematics.
In MENTAL it goes beyond imaginary mathematics. With the language one can define or specify imaginary mathematics (plural) based, not only on arbitrary axioms, but on any kind of imaginary expressions (numbers, sets, sequences, rules, functions, programs, etc.), as well as more or less imaginary domains (non-diophantine arithmetic, imaginary algebra, imaginary logic, etc.). The possibility of MENTAL to specify all kinds of expressions and combine them without restrictions leads to a richer and more flexible mathematics.
Essentialism.
MENTAL is clearly aligned with essentialist philosophy, since it is based on essential concepts (the primitives), which must necessarily be valid in all possible worlds.
In the same way that MENTAL clarifies and reveals the true essence or nature of different subjects (computation, imaginary numbers, paradoxes, etc.), it also reveals the true essence of possible worlds.
Language and possible worlds.
According to Wittgenstein, possibilities reside in language. Through language we conceive the real world and possible worlds. With MENTAL we can express all possibilities.
Consciousness.
Mind (internal world) and nature (external world) share the same primary archetypes. This is precisely why the world is intelligible because a link is established between the internal world and the external world, a link that is associated with consciousness. On these archetypes the concepts are formed at the mental level. Therefore:
We relate to things at a deep level, at the level of the primary archetypes. The primary archetypes are inscrutable, unknowable at the rational level, but we can intuit them.
We can only know the manifestations of the primary archetypes. We can know them precisely because they are grounded in the deep level of the primary archetypes.
The things we know we can express at the linguistic level, for in language the primary archetypes are also present, providing the connecting scheme between the deep and the surface world.
The deep world is the necessary world. The surface worlds are the possible worlds.
The "Magna Carta" of possible worlds.
With MENTAL, freedom of creation and development is clearly defined. Freedom is conditioned only by the primary archetypes, which represent the degrees of freedom or mental dimensions. It also has the property of being "pure" in the sense that we are always using pure resources of consciousness. MENTAL transcends the physical and the psychic to represent the foundation, the "Magna Carta", of the possible worlds.
Bibliography
Barrow, John D. Imposibilidad. Los límites de la ciencia y la ciencia de los límites. Gedisa, 1999.
Bruner, Jerome Seymour. Realidad mental y mundos posibles: Los actos de la imaginación que dan sentido a la experiencia. Gedisa, 2004.
DeWitt, Bryce S.; Graham, Neill (eds.) The Many-Worlds Interpretation of Quantum Mechanics. Princeton University Press, 1973.
