Possible Worlds and Logic
When considering the meaning or interpretation of the word “inevitable” or “inevitability,” Japanese dictionaries and other dictionaries explain that it means “it must happen” or “it must be so. However, when asked what is meant by “must be,” it is difficult to come up with a more appropriate and adequate answer.
In philosophy, the concept of “possible worlds” is sometimes used to explain the concepts of “inevitability” and its counterpart “possibility (contingency).
The concept of “possible worlds” is said to have been originally conceived by G. Leibniz, a German philosopher and mathematician active in the 17th and 18th centuries, who used the concept of possible worlds in connection with the mind of God and argued that the world created in reality is “the best of all possible worlds. The world created in reality is “the best of all possible worlds.
In this concept, the real world in which we live is one possible world, but we can also consider, for example, a world in which World War II did not occur as a possible world, or a world (universe) in which the earth will explode (for whatever reason) 10 years from now as a possible world. And, for example, a world in which “inevitably” occurs.
And, for example, the meaning of the statement (philosophically, it is called a proposition) “Inevitably, 2+2=4” is,
In all possible worlds, 2+2=4
and the meaning of the statement “It is possible that 2+2=4” is
There exists a possible world in which 2+2=4.
In this way, we understand the meaning of the sentence “It is possible for 2+2=4” as follows.
Based on this idea of “possible worlds,” possibility and necessity can be analyzed as follows.
- A true proposition is a proposition such that it is true in the real world.
- A possible proposition is a proposition such that it is true in at least one possible world.
- A fortuitous proposition is a proposition such that there exist possible worlds in which it is true and possible worlds in which it is false.
- A necessary proposition is one that is true in all possible worlds.
- Impossible propositions (necessarily false propositions) are propositions that are false in all possible worlds.
The theory of possible worlds was introduced in the 1950s by Saul Kripke and others because the current theory of possible worlds deals with the semantics of possibility and necessity.
If A is a certain sentence (proposition), and “it is necessary” is represented by the symbol □ and “it is possible” by the symbol ◇, “A is necessary” can be represented as “□A” and “A is possible” as “◇A”.
In modal logic, the logic that introduced the aforementioned “necessity” and “possibility (contingency),” such symbolization is used for various considerations. While the aforementioned interpretations of “necessity” and “possibility” are understood as Leibnizian interpretations, modal logic further introduces concepts such as “reachability (accessibility)” between multiple possible worlds and considers the meaning of various “necessity” and “possibility.
In “Making Logic: Part 4, Logic is interesting from here on“, I describe this modal logic, multi-valued logic, and intuitionistic logic as logics that have been extended from classical logic.
The book on “possible worlds” is “Philosophy of Possible Worlds: Considering Existence and Self“.
The following is an excerpt from the introductory section.
Welcome to the “anything goes” worldview – a world of possibilities.
One of the key words that characterize the culture of the 20th and 21st centuries will be “anything goes.
Take, for example, the field of illusionism: in 1917, Marcel Duchamp submitted a regulation toilet bowl as a “work of art” without making any of his own. James Joyce wrote novels in an artificial language that not only transformed English words but also mixed various foreign languages, while Georges Perec and Raymond Queneau wrote novels in a random language that was not only a variant of English but also a mixture of various foreign languages. Georges Perec and Raymond Queneau wrote plays and novels that mechanically transformed letters, words, and grammar. Lucio Fontana mass-produced endless works of art that were nothing more than rips and slices on canvas, while conceptual artists called their daily acts of simply writing dates, breaking things on the street, and leaving apples to slowly rot “works of art.
The idea that any act by a human being could be art, as long as it was given a proper context, quickly became commonplace. In fact, because artists were able to experiment with “anything goes,” the relationship between imagination and action, between perception and institutions, between language and concepts, and many other fundamental themes that shape human culture were vividly reexamined.
On the other hand, with the maturation of civic consciousness and moral systems, a backlash against unrestricted acts of expression has also emerged. In particular, “language control” to impeach discriminatory or political expression is flourishing, which paradoxically also stirs up “freedom of expression.
This movement is also taking place in the culture of religion. As established major religions lose authority and credibility, an eclectic mix of major religions and folk religions, large and small, in various proportions, is being created all over the world, and the number of new religions that have emerged in the last 100 years is estimated to be in the tens or hundreds of thousands. The number of new religions that have sprung up in the last 100 years is said to be in the tens or hundreds of thousands, but the total number of professed believers in each sect is said to be many times the world’s population, and even here, faith is no longer the sole and absolute refuge, but has spread into a convenient form of life that includes morality, metaphysics, rituals, discipline, the occult, psychic powers, and anything else that is possible.
This “anything goes” has achieved remarkable results even in solid genres such as the natural sciences. An example of this is that new phenomena in advanced science are being predicted and confirmed one after another from the pursuit of mathematical freedom and the beauty of equations rather than experiments and observations bound by reality.
These include the theory of relativity, which dismantled the common sense of “absolute space-time” by assuming that simultaneity and the progress of time can change depending on the choice of coordinate system, and quantum mechanics, which shook up the concept of “only one reality” by assuming that there is not only one actual path taken by elementary particles but many paths and states that can coexist. In quantum physics in particular, many positions are divided over the interpretation of the theory, and the “anything goes” debate is very active. (For more information on quantum physics, see “Quantum Physics, Artificial Intelligence, and Natural Language Processing.)
If mathematics supports the progress of the natural sciences, then philosophy and logic are the foundation that supports the progress of the humanities. In the fields of philosophy and logic, too, the spirit of “anything is possible,” unrestricted by the nature of reality, is achieving remarkable results, and the central weapon in this spirit is the concept of “possible worlds.
Logic is not limited to the numbers and quantities handled by mathematics, but it explores the possibilities of all kinds of beings and concepts – ways in which various beings coexist, ways in which they are connected, and relationships among concepts that are not bound by the contingency of reality, as long as they all make sense. The device called “possible worlds” is a materialization of such abstract concepts of countless possibilities, each of which is an independent entity.
In the same way that the aforementioned artist’s “no-rule” activities brought to light a variety of values and revealed hidden truths by actually performing them, rather than just thinking about them in his head, so too, the myriad “possible worlds” are treated as independent entities, arranged, increased or decreased, combined, divided, rearranged, etc., and thus become the materialization of philosophy and logic. In the same way that we have seen the hidden truth, many problems in philosophy and logic are solved by treating each of the myriad “possible worlds” as independent entities, arranging, increasing, decreasing, synthesizing, dividing, and rearranging them.
In “Philosophy of Possible Worlds: Considering ‘Existence’ and ‘Self,'” we will discuss why “possible worlds” were necessary for this “possible world,” how the possible worlds used in philosophy relate to each other, what exactly “possible worlds” are, and what the most extreme of the possible world ideas, “possibilityism,” is. Finally, he gives an interpretation from these perspectives to the age-old ultimate questions of “why life exists, why I exist, and why the universe looks the way it does,” and concludes with a discussion of “probability,” in which “possibility” is subdivided and quantified. He ends with the story of the
Chapter 1: What Can Possible Worlds Do?
Chapter 2: Network of Possible Worlds
Chapter 3 What is a possible world?
Chapter 4: Is there really a possible world?
Chapter 5 Natural Science and Possible Worlds
Chapter 6 Outside of Possible Worlds
Chapter 7 Application - Solving Difficult Problems in Possible Worlds
Appendix Possible Worlds Book Guide
Possible Worlds, Probability Theory and Artificial Intelligence Technology
Thus, the idea of Possible Worlds (PW) is a concept used primarily in philosophy and logic to refer to a possible world that is different from the real world. This is to say that it is a world in which different elements and events may unfold in different ways, unconstrained by physical constraints or laws.
Probability Theory (Probability Theory), on the other hand, is a field of mathematics that deals with uncertainty and randomness and provides a framework for predicting the probability of events occurring and their outcomes. From this perspective, probability theory not only evaluates the probability of events in the real world, but also considers the probability of events in the possible world.
Artificial Intelligence (AI) technology is a generic term for technologies that enable computer systems to perform intelligent tasks, which can be divided into subfields such as machine learning, deep learning, natural language processing, and computer vision. AI is the use of probability theory and statistical methods The main purpose of AI is to analyze events and data in the real world and make predictions and decisions by utilizing probability theory and statistical methods.
In this artificial intelligence technology, possible worlds and probability theory play an important role: AI systems are required to analyze real-world data and generate possible outcomes and predictions, while probability theory is used for statistical modeling of data and evaluating the reliability of predictions. The concept of possible worlds could also be applied when AI systems make decisions based on multiple scenarios and possible actions.
This is specifically the case when considering a self-driving car, where the AI system analyzes sensor data and surrounding conditions to plan possible collision avoidance strategies, and this planning is based on probabilistic models, considering different possible worlds (e.g., different movements of other vehicles or pedestrian behavior), and the AI system uses probabilistic The system can be interpreted as making a theoretical prediction or inference and selecting the best possible world.
Approaches that consider such multiple possibilities are described in “Answer Set Programming” (ASP), The theory of fusion of probability and logic described in , the approach of probabilistic generative models described in “Fusion of Probability and Logic (1) Bayesian Networks, KBMC, PRM, and SRL” and “Fusion of Probability and Logic (2) PLL (Probabilistic Logic Learning)“. , and generative models using deep learning described in “Overview of Automatic Sentence Generation Using Huggingface“, etc. can be implemented using a variety of tools.
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