[If humanity is defined in the Private language representation of the accumulation of packages, so it is human-made intellectual topology extended ideology. Character is the concentrated expression of intelligence, physical intentional, such as sun and the moon, at a glance; the Western alphabet characters often invisible unintentionally, logic and so on. Smart is the essence of intentionality and formal unity, the pictographic characters from knowing the process is the brief history of the development of natural intelligence of mankind so ....... In addition, Hume asked, it is pushed from the fact that no value to, but this world is a mixed fact and value in the world, I do not know the value of the fact that it can be launched? From language to calculus is the current way of development of artificial intelligence. ]
Abstract: Modern Logic began in Leibniz, split took place in Istanbul and Frege, the formation of the so-called traditional algebraic language and logic of traditional logic, however, whether it is also the traditional algebraic language or traditions from Levin Bu Nici's logic Program (ie: "as a science, mathematics and philosophy" basis) far from. In the Turing machine theory, Turing core elaborated, "automatic machine" and "instruction list language" of these two concepts, both fit nicely Leibniz conception of "rational calculus" and "universal language" can be said that the Turing machine theory closest logic Program of Leibniz, but also out of the traditional algebraic language and traditions, another way to form a new "computing as a logical" tradition (also referred to as "traditional computing" ). "As a logical calculation" it is essentially a "subject to", "the logic of the past" is the responsibility and to humanity as the main object of study is human thinking, all kinds of natural language, "as the logical calculation" sucked computer information processing as a subject of study is the interaction between people and computers as well as handling the computer.
Keywords: logic; Leibniz Program; Turing machine; as a logical calculation
Introduction
If you talk about the relationship between logic and computing, most people would agree with the calculation logic closely related to each other, for example, US computer scientist Mana (Manna) once proposed a "logical calculus that is computer science" point of view, moreover, even Some people believe that "computing is essentially a logic", for example, a famous mathematical logician Mo Shao Kui said: "in fact, they [programming] is the mathematical or logical, or is written in a computer language of mathematical logic, or application of mathematical logic on the computer. " In a sense, the logic of how to calculate the importance can not be overemphasized, these ideas can be defended through the following arguments:
1. From the point of view of the history of computers, computing science originated in logic
Trace the origin of modern computer science, it should be said, it has a close relationship with the logic. As we all know, since Russell and Whitehead co-authored the "Principia Mathematica", the rise of the study of mathematical logic, it is logical to expect even constructed based on the whole mathematics, science and even buildings. In this logic driven doctrine inevitably need to "do it computable" concept formalized. Exploring "can do computable" concept, Church, Godel Turing given almost completely different from each other and because at the same time equivalent definition. Lambda calculus Church the invention, to characterize "computable can do." Godel that "general recursive function" as a definition of "computable can do" in. Turing through the concept of a device description, the definition of "computable do it", which was later known means "Turing machine", which is the theoretical model of modern computers, the computer identifies the birth of modern science.
2. The theoretical basis of the logic of the computer hardware and software system
Boolean logic to become a core integrated circuit design theory, as Herman Goldstein said: "It is through it [Boolean], so that the circuit design into a science and an art from." At the same time, it is precisely because of Boolean logic design ideas to integrate in the switch circuit, will form the famous Moore's Law in the field of integrated circuits, which makes innovation and development of integrated circuit technology can be achieved. First-order logic, on the type of logic programming language and Lambda calculus depth intersect to form a core program design theory. The formal syntax, type system and become the basis for a formal semantics of programming languages. Proof theory model on interactive thought with the computer hardware and software systems of logic, constitute the computer system to verify the correctness of the theory. Human based Hoare logic, discrete logic, Isabelle, Coq theories and tools, can verify the correctness of large hardware and software systems.
This article is not intended to refute the above-mentioned various points of view, I have to admit computing and logic inextricably linked, even though "logical" and "computing" concept itself are not particularly clear. This paper attempts from another perspective calculations and logic binary relationships, not the logic of research on how to calculate, but the elaborate modern "computing" was found, to our understanding of "logic" What are the implications? In other words, we hope to stand "computing" the starting point of discovery, re-archaeological history of the development of logic, and want to do some thinking and inspiration for the development of logic.
First, the logic of Leibniz Program of
the 17th century, logic has changed, Leibniz proposed what logic should do. Leibniz aimed at establishing a universal language for science, the language of science is ideal, appropriate to reflect the nature of the entity in the form of a statement. Leibniz believed that all scientific thought, can turn classified as small and simple, irreducible idea of using them to define all other thoughts, by decomposition and combination of ideas, new discoveries will be possible, as in mathematics the calculation procedure.
Leibniz did not specifically and systematically discussed through logic to explain the logic associated with a major in his "On the universal character" (On the General Characteristic) and "rational calculus basis" (Fundamenta calculi ratiocinatoris) and other papers in.
Leibniz first discovered the universal meaning of the symbol: "Human reasoning is always carried out by way of symbols or text in fact, things themselves, or idea of things is always clearly identify the thought was impossible, too. unreasonable, therefore, for economic reasons, need to use the symbol, because every single impression, a geometer, referring to a quadratic curve, he will be forced to recall their definitions and the definitions of these items constitute definition, this is not conducive to new discoveries. If a home in arithmetic calculation process, constantly need to think about the value of all the tokens and passwords he wrote, he will be difficult to complete large-scale computing. Similarly, a judge, in review behavior, abnormal and legal interests of time, can not always thoroughly all of these things have to do a full review, it would be great, it is not necessary. Therefore, we give the contract, given the geometry of the name, in the digital arithmetic symbols given in algebra, a dimension, so that all of the symbols have been found to something, or by Inspection, or by inference, can be fully integrated with the final symbol of these things together, referred to herein symbols, including words, letters, symbols chemistry, astronomy symbols, characters and pictographs, including musical note, shorthand, arithmetic and algebraic symbols and all other symbols of our thinking process will be used. here, the so-called "character" that is written, traceable or engraved text. in addition, the more a symbol to express the concept of what it alleged, the more useful, not only for characterizing, it can also be used for reasoning. "
Based on this, Leibniz insight into: we can assign to all objects in its alphanumeric, which will be able to construct a language or text, it can serve the discovery and artistic judgment, as if in a number of arithmetic, algebra is to the amount role. We will certainly create letters of a human thought, the analysis of the letters of the alphabet and the contrast of the letters of the word, we can find and determine everything.
In Leibniz's insight in, it contains two very important concepts, namely "universal characteristic" and "rational calculus."
The so-called "universal character", not a chemical or symbols of astronomy, nor characters or hieroglyphics of ancient Egypt, but not our everyday language, although our everyday language can be used in reasoning, but it is too ambiguous and can not be used for calculation That is, everyday language can not be the formation and structure of the language of words to detect errors in reasoning. In comparison, the "general character" is the most similar to the arithmetic and algebraic notation in arithmetic and algebraic notation, the reasoning are present in the text of the application, the fallacy of thinking is equivalent to the calculated error. Letters of a universal characteristic of human thought is, by analyzing the links and words by its composition can be found and judge everything.
Another key concept of "rational calculus", which is different from the calculus reasoning, it is a calculation or operation, i.e., according to some predetermined rule, generated by the equation transformation relationship, and a formula is composed of one or more characters. Calculus is closely related to the concept of the "Word Art:" Art Composition and arrangement of the text, in such a way that they characterize thought, that is to say, they have to have so as to have between thinking the relationship between the text. "
In order to achieve this "general character" and "rational calculus," Leibniz trying to establish the concept applies to all alphanumeric. However, due to the interconnectedness of all the wonderful things that clearly alphanumeric various things extremely difficult with the formula set forth. To this end, Leibniz had created an exquisite technique, through which all conclusive relationship can be expressed in numbers and fixed, and can then be further determined in the digital calculations.
Leibniz is further assumed that some number of special characters are already known, wherein some of the unique properties can be generally observed. Thus, catch those numbers and the unique nature of the associated, and then help them prove with amazing dexterity rule directly on all digital logic and can find out if a known argument was formally determined to provide a standard. Anyway, once we took possession of the real alphanumeric thing itself, a virtually certain to prove whether you can get the first judge, without any difficulty, do not fall into the fallacy of danger.
All human thought can be digested several original concept, if the text is assigned to the original concept, the concept of derivative can be formed by forming the character, through which we can extract their conditions and contained therein original concept, and more specifically, their definitions and values, even derived from the definition of the attribute.
It should be said, Leibniz idea of universal character and rational calculus is very "idealistic, utopian," Leibniz himself made just such a vision, although he had a few tried, but did not fully realized, therefore, many of the logician, philosopher Leibniz would this idea called "Leibniz's dream", because, after that, Hilbert, Godel's work It shows that such a perfect language and calculus does not exist, based on this, many scholars assert that Leibniz's dream has been broken.
However, in our view, in fact, ignored by most scholars is, in fact, Leibniz proposed research program is the logic, the logic is seen as a "big science on a real tool" logic studies should study all scientific thought, the idea of science classified as small and simple, irreducible idea, using the universal characteristic defines all other ideas (concepts), word art by decomposition and combination of ideas, new discoveries will possible, as the process calculus mathematics.
Once the study of this logic is realized, it would be like Leibniz said: "Once people establish the concept of alphanumeric largest part of that time humans will have a new tool to improve its intelligence capabilities , far better than the optical tool to strengthen the human eye, and the reason it is superior to the vision, the vision microscope and a telescope and replace the same. it's great use of, interference with the compass as a sailor, it brings all those engaged in the constellation than at sea human and experimental investigation of the use of much larger, from which in turn will produce what, depending on the hand of fate, but it will certainly bring important and excellent results. "
Second, as a logical algebra and logic of language as
1967, Jean Van Haye Knott reviewed by logic from Frege to Godel's literature, the proposed development division of logic is as logical as algebra and two logical paths language. Among them, Frege's logic is leading as the language of logic, and the logic of Boolean algebra as the development of logic.
In Van Haye Knott view, Frege's logic is different from a key concept of Boolean logic is the logic of universality. The first is the full name of this universal quantifier theory, propositional calculus which is missing, this is the full name makes Frege logic enables the entire scientific knowledge symbol rewritten. Meanwhile, in Frege's system of universal logic it is self-expression, the world body can be divided into objects and functions items. Frege's world is a world that is this world, it is fixed, at this point, with De Morgan Boolean world is free to change.
In 1997, the Xindi Ka "Peirce's position in the history of the theory of logic article echoed Van Haye once Knott's" language and logic as a logical calculus of "distinction, further stated: Frege with the early and mid Russell, early Wittgenstein, the Vienna Circle, Quinn, Heidegger equivalent belongs to a tradition, in their eyes, only large and a world that is the real world, the real world is the only explanation of our language, the majority may not exist the world, and thus deny the legitimacy of modal logic fundamentally deny the truth decidability; and Peirce with the history of Istanbul, Schroeder, Levin Heim, Tarski, Godel, such as the late Carl Knapp the other is a relatively traditional, they support, including possible worlds theory (modal logic), model theory, logical semantics and meta-theory logic, including a series of theories.
Whether it is the traditional algebraic logic, or the logic of the traditional language, they will be attributed to its logical origin Leibniz, they have each think they are the Leibniz inherited orthodoxy. Interestingly enough, these two traditional inheritance of Leibniz's, are more concerned about the concept of Leibniz's "universal character", but relatively ignored or belittled Leibniz the concept of "rational calculus" of both traditions have pointed out that each other just to achieve "rational calculus", and their only truly "universal character." In fact, the traditional algebraic logic and the logic of the traditional language have failed to achieve Leibniz had depicted "real tool on Big Science," the logic program. Perhaps Russell and Whitehead did try very close in their vast "Principia Mathematica" great book, but eventually Godel's incompleteness theorems mercilessly crushed. It seems that Leibniz's logic Program has entered a dead end, as some logicians said, Leibniz dream is gone.
Third, computing Turing's turn
in the study of logic, it seems to ignore Turing's work (although in computer science research, Turing was posthumously calculate the father of science). Turing by careful observation basic arithmetic, give a general definition of calculating automatic machine, roughly as follows:
Automaton is a "black box", specific details are not given, but it has the following properties. It has a limited number of states, which can be expressed in this way: show the total number of states, say n, followed enumerate them, 1,2, ..., n. Core operating characteristics include those described automaton how it changes state, i.e., from a state i, into another state j. This state needs to change some kind of interaction with the outside world. For the purposes of the machine, so that the whole external world consists of a long paper tape. This paper makes a bandwidth, for example one inch, and make it into a long one inch squares. On each square of paper tape, can do or not do a sign, say a point, and assume that a point can be erased or written,. Little squares labeled 1 is referred to, is referred unmarked squares 0. (We could use more signs, but Turing showed that it does not matter, does not bring any substantial help to the universality of the discussion.) In describing the relative position of the automatic tape machines, tape always assumed a square can be read directly automaton, and the robot can be moved forward or backward tape, say, move one square. Specifically, in a state so that the automatic machine i (= 1,2, ..., n), and a read e (= 0,1) on the tape. Then, it will switch to state j (= 1,2, ..., n), moving tape squares p (p = 0, 1,! "1; +1 grid represents a forward movement" represents a rearward moving a grid), and read the new squares labeled f (= 0,1; 0 denotes an erasure, where 1 indicates a marker point). To j, p, f is expressed as i, e function, which describes such a complete definition of the automaton.
The above description is Turing's computer, later known as "Turing Machine." Based on Turing machines, Turing gives a more intuitive "can do computable function" definition: "If the value of a function can be obtained by performing a machine, then the function is called computable can do it . " The Church - Turing thesis, with Turing computable defined recursively defined and Godel original Church definable equivalent. Thus, characterizing one Turing machine has become calculable, the theoretical model for all computers. Despite the concern, however few people noticed Turing machine in addition to the characterization of a computable, Turing in my view, it is a kind of symbolic logic. A speech in London Mathematical Society Turing once held in 1947, explained some of his views on symbolic logic and mathematical philosophy: "I expect a digital computer will eventually be able to inspire our considerable interest in symbolic logic and philosophy of mathematics AC language between humans and these machines, namely: instruction list language, formed a kind of symbolic logic machine with fairly accurate way to explain what we tell them everything, without reservation, that there is no sense of humor at all mankind must accurately convey their meaning to these machines, otherwise there will be trouble in fact, human beings can communicate with any precise language with these machines, namely: in essence, we are able to carry out any symbolic logic and machine exchange, as long as they explain the symbolic logic of the machine assembly instruction sheet. this means that the logic system has a wider scope than ever before. As for the philosophy of mathematics, because the machine itself will do more and more mathematics, human interest will continue to shift the focus to philosophical questions. "
Turing machine has two main components: "automatic machine" and "instruction list language." Wherein the instruction list language is a language described in the state transition table, i.e., a state automaton description of the conversion, and read the language of movement. Turing that instruction list language is a language between humans and machines, forming a symbolic logic. Instruction List language here that we later developed various types of programming languages.
It should be said, Turing machine "automatic machine" and "instruction list language" is the interpretation of Leibniz's "universal character" and "rational calculus" of. Turing is Bifuleige, Boolean and Russell practice more successfully Leibniz dream. In Turing's scheme:
"Programming language" is a "universal characteristic" an implementation (≈ universal programming language text);
"Automata" "rational calculus" in one implementation (≈ rational calculus Automaton).
In programming languages, all equivalent with Turing machine, or can be calculated in any computable function programming language called Turing complete, most programming languages are Turing complete. Tautology, for any (Turing complete) programming language is concerned, it can solve any computable problem. Turing complete programming language Bifuleige in the expressive power of predicate logic stronger, be able to describe a variety of disciplines including issues of physics, mathematics, chemistry, astronomy and biology to achieve rewrite the entire scientific notation.
The essential difference between logic and the logic of Frege and Boolean Turing is that in addition to achieving "universal character" in a programming language other than Turing logic and one for "rational calculus" automated calculation mechanism, namely "automatic machine . " In other words, no matter how complex the program, we are able to achieve in "automatic machine" automatic calculation mechanism.
Turing to Leibniz's "universal character" and "rational calculus" Effective integration of a logic to bring a "calculated to", it can be said, in addition to algebra and logic as a logical language, Turing a logical opened up a "calculation logic as" a new path. This examination of the logic is essentially a "subject to", "the logic of the past" is the responsibility and to humanity as the main object of study is to characterize human thinking and human thinking all sorts of natural language, "as calculated logic "sucked computer as the main body of information processing, research is the interaction between people and computers as well as handling the computer.
It should be said, every time to re-examine the logic, will promote a new philosophical reflection. In the history of philosophy, because Frege opened the traditional language of logic, philosophy initiated in the so-called "linguistic turn", we will Frege regarded as the founder of the philosophy of language, it is because the "linguistic turn" makes language philosophers will talk about Frege. In recent years, philosophical circles at home and abroad, the rise of thinking "calculated to" or "information to" a surge of philosophy, these reflections are derived from the Survey of the Turing machine computing and related technologies. I think, standing logic perspective, one should seriously look at Turing's "logical calculations", the development of logic in a timely manner to promote the path computed.
Fourth, the conclusion
from Leibniz wrote "On the universal characteristic" So far, it has been 300 years. In this 300 years among modern mathematical logic and Frege along Boolean algebra with the two traditional language development, however, difficult to find, whether it is the algebraic tradition, language or tradition, and Leibniz had conceived of logic Program are far from.
In our calculation of the Turing machine, study calculus has been very much respect, however, it seems that most people ignore symbolic logic characteristics Turing machine, according to Turing himself in 1947 in his speech mentioned, in fact, Fig. the instruction list language Lingji formed a symbolic logic, we can be referred to as a Turing logic. Turing logic bypass the logical framework Frege, also bypass Istanbul - Peirce - logic tradition Schroeder to Leibniz as a starting point, since they re-established a new "calculated as logic "tradition. It should be said, a major motivation Leibniz engaged in research is to study the logic of human thought symbolic system, which led to his commitment to ponder questions such as the nature of the logical proposition, the concept of truth, but also led him in an attempt to Wikipedia the book is the idea of all forms of knowledge. Today, even though computing technology developed rapidly, even breaking the data storage and network technology limitations, constructed like Cyc, DBpedia and other such large knowledge base, knowledge base, however these are described entities, relationships, attributes, etc. the use of Boolean logic or Frege's still along the old path, resulting in yet really, even partially achieved the envisaged Leibniz system. Return to Leibniz, along the "traditional computing" Turing, re-examine the physical, body, and other metaphysical significance of the problem, maybe we are able to review the "Leibniz's Dream" pave a new path.
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