Western formal logic began in the 5th and 4th centuries BC of the Greeks, who developed syllogism and prepositional systems. The Hellenistic Greeks and Romans did nothing to advance these beginnings, but instead injected a series of rhetoric that haunted the subject until recently. It also began a series of rough textbooks. Later Dark Ages logic began to revive in the 12th century, and by the middle of the 13th century, scholastic logic was well developed. Although it borrowed a lot from Aristotle and a little from Roman allusions to Stoicism, it developed primitive methods in terms of propositional and quantitative logic and the backsliding of the two laws of logic. It borrows very little from rhetoric, but is heavily influenced by grammar. Around the middle of the 15th century, this impulse failed and died out completely within 100 years, replaced by centuries-long incompetent manuals that were often infected with rhetoric and lacked complete ingenuity or serious investigation. Only occasionally, the monotonous desert is interrupted by something interesting, especially by the great genius GW Leibniz. In the mid-19th century, the modern period began with G. Boole and gained new authority through G. Frege's keen analysis. Their work has led to a tremendous increase in new understandings and doctrines of the past, presenting a whole new level of integrity, rigor, and critical control. There are therefore four periods to consider: (1) the Greco-Roman period, (2) the medieval period, (3) the post-Renaissance period, and (4) the modern period.
Aristotle, who claimed to be the founder of logic, said he could not find anything like his philosophical predecessors. This statement seems to make sense. Of course, one can find an atmosphere of intense discussion conducive to this development. Whether founded by Euclid of Megara, a student of Socrates, or platonic academy of the same origin, or in the tradition of 5th-century sophists, the discussion was so strong that it is not surprising that one should begin to reflect on the process of argument, paying attention to repetitive patterns, and summarizing conclusive and non-conclusive approaches in a reflective manner.
Already in Plato one can see the hint of what syllogism would become in Aristotle's hands, and what propositional logic would become in the hands of the Macquarians and Stoics. Roughly speaking, Athens gave birth to the former, while Megara gave birth to the latter. Plato must have been influential, for he developed the concept of universal laws, which had been confirmed in pre-Socrates, but left to Aristotle to implement the first conscious, general, and unambiguous system of formal logic, so that Leibniz could say that he was the first to write mathematics outside of it.
Aristotle's Logic. Aristotle's logical writings, known as Oghenon, were handed down in a systematic order: categories, dealing with terms; explanatory, propositional; prior analysis, general syllogism; posterior analysis, the subject matter, and sophistry refutations, absolute, dialectical, and sophistry syllogism, respectively. Of course, this list does not represent the order of writing, but attempts to determine this through different complex doctrines are somewhat uncertain, since the development of thinkers may not be continuous and homogeneous. Thus, the rebuttal of topics and sophistry, while lacking in the doctrine of syllogism, contains some insights that belong to the higher realm.
Syllogism is a theory of all or part of a kind of interspersed, its laws are presented in the form of schematic diagrams, and the use of letters instead of words in everyday language is a wonderful means of ensuring universality and isolated forms. (It wasn't until Frege took full advantage of the device.) Aristotle first listed his syllogism, classified by patterns called "graphs", in which valid figures alternate with uncertain figures, the latter rejected by counterexamples. His incomplete definition of numbers would have caused great trouble for later writers, while those who focused on words rather than spirits would have been plagued by the incompleteness of a clear list. Aristotle redesigned his system in a number of ways, proposing alternative methods of inference from axioms (thus showing that there is nothing inflexible about a given set of axioms) and making some meta-logical statements. The deduction is either direct, according to the law of conversion, or indirect, by restoring the absurdity of advertising. They proceeded intuitively rather than formally, since Aristotle stated only two or three laws of propositional reasoning, although it is worth noting that he did consciously use propositional variables.
In particular, non-syllogism is distinguished by laws that belong to relational logic, such as " If knowledge is conceived, then the object of knowledge is the object of conception " , a principle proposed by A. De Morgan in the 19th century against the contemporary man who might become Aristotle , which cannot be proved in syllogism. Rebuttals also emanating from subject matter and sophistry emerged laws on identity, which together add up to leibniz's generally recognized "principle of unrecognizable identity." The presence of these things in Aristotle is often overlooked rather than noticed, and their nature is fragmentary. Even the asserted syllogism is not dealt with as thoroughly and universally as the current systematic study of logical thought. Aristotle's modal syllogism is not even fully elaborated and awaits final investigation and evaluation. But his logical beginnings were astonishingly good, and although the merits of certain medieval treatises are unquestionable, before Leibniz there was no existing work (in the absence of a complete Stoic text) comparable.
Theophrastos. Aristotle was succeeded by Theophrastus of Eresos as headmaster of the School of Escape. He was primarily known for defining the five syllogisms that came to be known as Baralipton, Celantes, Dabitis, Fapesmo, and Friessomorum. He introduced a non-Aristotle modal syllogism in which conclusions followed the law of assertion of the weakest premise; and he provided an epitaxial proof of the convertibility of the universal negation proposition, perhaps with a spatial model before him. Only fragments remain of his work. They contain references to his "from hypothetical" syllogism, i.e., work with preconditions initiated by his predecessors. Theophrastus may have stimulated Megarian-Stoic's study of propositional logic.
Megarian Stoic School. Materials are only present in fragmented and often hostile reports. Among the Meccarians, O'Brides of Miletus is thought to have discovered a paradox known as the "Liar" or "Epimenides", which Aristotle noticed, and which Theophrastus and Crisipus pondered deeply. It was not until 1937 that a new form was claimed, but it has been discovered that it existed in the Middle Ages. An early version reads: "If you say you're lying and what you're saying in this regard is true, are you lying or telling the truth?" Oblidez reportedly remained hostile to Aristotle, thus depriving later Aristotle of its progressive and complementary influence.
Diodorus Cronus of Iasus (late 4 bc) held views of modalities that are difficult for modern interpreters to interpret. His definition of necessity introduces a temporal variable, namely "neither false nor false". Although it is easy to assume that his definition of implication is that at any time its antecedent is not true and its consequences are false, the text certainly does not prove this. He is the author of the "main argument" on the incompatibility of three modal propositions, which proved unsatisfactory to be reconstructed. Megalastilpo drew new followers, including the influential Zino of Tytion, who founded the Colonnade (°C. 300 BC). Megara's Philo was the first to formulate a truth condition for a material condition, which is true unless its predecessor is true and its subsequent part is false.
The Meccalis seem to have disappeared with the rise of Stoicism, whose logical history was obscured by its second founder, Krishipus of Solly, who died shortly before 200 BC. The most important contribution of the Stoics to logic was the deductive propositional logic system. It is based on five "unprovable emotions" (one should not say "axiom" because they retain the objective meaning of the word used for declarative sentences) and four "subjects" or rules, of which only two are retained. They used ordinal words instead of letters as variables. W. Kneale suggested a convincing reconstruction of the system, which the Stoics claimed to be complete, but it is unclear what their intentions were in doing so.
Later developments. For the rest of ancient logic, it should be mentioned that Cicero— not a logician indeed, but his rhetorical syllogism influenced the logic of the Renaissance and beyond; the manuals of Galen and Apreeus of Madara (2nd century AD); and the Greek critics of Aristotle, especially Alexander of Aphrodites (3rd century) and John Philoponus (6th century). Galen was later credited with inventing the fourth grid of syllogism, but J. Lukasiewicz showed that this was a mistake. Apuleius gives the opposite side, which has become traditional. Alexander shows how to derive the law of transformation from syllogism and the same law through reductio ad absurdum, which offers new possibilities for the axioms of syllogism in the Middle Ages and Leibniz. Philoponus suggests resolving the question of how to define a syllogistic number by referring to the subject (predicate) of the conclusion as the secondary (primary) term and naming the premise there. It was the most economical method, but it was not commonly used until the end of the 17th century. The Trier porphyry (3d century) contributed to his "tree" or genus and species plan, to which he took an epitaxial view, which was included in its genus (see Porphyrin tree).
Boethius was the great communicator of ancient logic to the medieval world. He was a wanderer, but retained some Stoic doctrine, translated much of Oghanon, and wrote works on subject matter or place, as Cicero and Marius Victorinus (4th century) did in the field of rhetoric. His categories of translation and interpretation constituted the early mediavals of Vetus, with other parts of Oganon as a nova. The variables in his paper on hypothetical syllogism are considered propositions, but since the doctrine is basically Theophrastan, and given Boiseus' Aristotle belief, they may be term variables.
Research in the field of logic began to revive at the end of the 11th century, when a great deal of fruitful activity was undertaken, much of which was still to be learned by publishing more texts. Aristotle's complete logic, especially a priori analysis, was only available during the 12th century. Boethius was as influential as Cicero, but grammarians seemed to be more influential than rhetoricians.
Xirth century. Abelard was remembered by his contemporaries as "Aristotle of our time equal or superior among all logicians", pointing out that logic is not the science of using arguments, but the science of discerning the validity of arguments. In his Dialectics, he distinguishes between "precedence" and "outcome" as referring to subjects and predicates in simple propositions and parts of hypothetical propositions. This and other passages show that the advent of propositional logic in the Middle Ages differs from the logic of terminology. Abelard knew that these were different, and that there was an analogy between them—he reported a view that propositional conjunctions and their term analogues had the same meaning, but he refused to accept them. The statement that the hypothesis is called both "effect" and "condition" may raise questions about whether the implication and the reasoning relationship have not yet been clearly distinguished, and that the fact that Albert of Saxony (14th century) distinguishes si and ergo only by their positioning should give rise to one or another view of the theory of "consequences" with caution. Abelard has produced many valid consequences, some even deduced from others, but the Middle Ages never acquired an axiomatic system of propositional logic. One of Abrad's most complex consequences is that "whatever hypothesis the antecedent is concomitant, the outcome is concomitant." This is leibniz's theorem that will be rediscovered and called praeclarum. One should note the metalogic formulation, a style that will remain standard, probably derived from Boethius's De Differentiis topicis, which distinguishes a type of truth from a maxim or metalogical formulation of a kind of truth from examples.
In the 12th century, Adam of Balsham also wrote a highly original work, Ars disserendi, in which one saw the rise of concern for sophistry or logical puzzles, which became a very typical feature of the period. While sophismata can produce a rich system of doctrine under creative hands, the medium is more inclined to perpetuate fragmented treatments than truly systematic ones. Adam made a rare attempt to begin the logic of the problem, in the process concluding that an infinite set could be equal to the proper part of itself.
Thirteenth century. The most famous works of the 13th century are the Introduction to Logic of William of Shyreswood, the Summary of Logic of Peter of Spain (Pope John XXI), and the Commentary of St. Albert the Great and Robert on A priori analysis by Kirwoby. This finally shows that the consequences are already a normal part of the teaching of logic. Peter of Spain became the standard writer throughout the 15th century. Curiously, his summary manual does not contain a chapter on the consequences, but it does have a well-established doctrine of proprietary terminology, as in The Early Shyreswood and similar books. This origin can be vaguely found in the last century, and more can certainly be found. The main property discussed is the hypothesis, i.e. the designation of the subject (and later the predicate) in the proposition. St. Vincent Ferrer's De suppositionibus dialecticis (1372) shows a variety of different logical materials discussed in this regard, including some points of quantitative theory. Once again, considering the necessity of a large number of examples from ordinary speeches favors a fragmented approach.
Post-century. William of Oakham sparked a series of campaigns, in part because his Summa totius logicae was very comprehensive. His influence can be seen even in those who deny his epistemology. Walter Burley, John Buridan, Albert of Saxony, Marsilius of Inghen, Mertonians, Heytesbury and Ralph Strod Ralph Strided) and Richard Ferabrich are some of the most famous writers. In addition to the areas already mentioned, they were very concerned about insolubleity or logical paradoxes, developing many versions of Epimenides, which were known to Adam of Balsham. Many solutions were proposed, including the removal of self-referential propositions from meaningful language (see Antimony).
The 15th century is non-original. Towards its end there is an encyclopedic flea of the Venetian Paul (Paul Veneto), who formed with Mantua and Pergolae of Paul Peter known their contemporaries Sorticolae.
From the 13th century onwards, syllogism was considered a special or even a small one. The so-called Aristotle ideas that developed in the next period—valid arguments are always syllogism—were rather foreign to the Middle Ages. This topic is of course dealt with in detail in Aristotle's commentary and needs to be dealt with in detail in the summarizing commentary, but in more general papers the syllogism is only a consequence. The common way to define terms is to generalize the Boysius method, the first premise is the big premise of the definition, where the extreme term is the big premise. This is very different from Philipponus' approach, with signs that someone can correctly calculate its consequences, but without a uniform and systematic formulation. Various mnemonics were experimented with in the 13th century, and the familiar "Barbara, Serraren, etc." also appeared in people's sights. It happened in Hillswood.
Around 1440, the first recorded sound of the logically new era or non-era was heard. L. Valla, a prominent humanist scholar, subsequently rejected the third dimension of syllogism on the grounds that women, children, and illogicians generally do not argue that way. Perhaps this was the first time that everyday language was called the standard of logical doctrine. Obviously, all meaning of syllogism as a deductive system has been lost; in fact, Walla said that Aristotle's main deductive method, transformation, is only "a remedy for pathological syllogism." In contrast to Abrad, R. Agricola's dialectical invention was that he firmly thrust the contradictory "topic" tradition onto the path of rhetoric. P. melanchthon, written in 1521, elaborated Cicero's syllogism before Aristotle. Older doctrines were quickly abandoned or ridiculed. G. savonarola has repeatedly told the 16th century in countless reprints that any controversy from connecting to its part is a dignus explosion.
Ram doctrine controversy. In the middle of the 16th century, vernacular logic began to emerge, such as T. Wilson's The Rule of Reason (1551) and Peter Ramus's Dialectique (1555). The last writer's view of logical reform has caused widespread and enduring debate. His simplified syllogism and novel terminology gave rise to lengthy reviews of rare and new technical scholastic philosophies. Aristotle found that apart from the sins of Ramism and the fourth grid of syllogism, there was little to discuss, and few people recognized that this was a problem that needed to be solved by definition. Sextus Empiricus appeared in Latin in 1569, but did not rediscover Stoic logic.
Occasionally, there are intermittent situations in the cloud. J. Hospinianus (1515 – 75) thoroughly studied syllogism on the basis of combinations, and G. Cardano explained his Dialectics with geometric arguments. J. Junge (Logica Hamburgensis, 1638) showed a deductive interest in syllogism and some appreciation for the logic of Aristotle relations. In 1662, A. geulincx pleaded for the restoration of medieval doctrine. In the same year, A. Arnauld and P. Nicole's Port Royal Logic was published. The authors were anti-rhetorical and anti-Ramis, worshipped geometry, and did much to strengthen the theory of syllogism. At the same time, they opened the way to introduce epistemological and psychological discussions into logic books. H. Aldridge, in his Artis logicae compendium (1691), correctly lists 24 syllogistic emotions in four numbers and methodically proves that all other emotions are invalid.
Leibniz and after. At the same time, GW leibniz has begun to develop entirely new ideas. A polymath known for his monism in philosophy invented calculus in mathematics, and before he was 20 years old, he was troubled by the idea that logic could develop mathematically. Others before him had discerned kinship (e.g., Roger Bacon) but did not use mathematical symbols. Leibniz experimented with various versions of the logical calculus, which he hoped to use in conjunction with a rationally constructed universal language. He also envisioned an encyclopedia that would be perfected as science progressed and would unify the entire human body of knowledge at any stage. In forming these projects, Leibniz found that J. Wilkins, G. Dalgarno, and other contemporaries of the encyclopedia, T. Zwinger and JH Alsted, and Raymond Lull of calculus. But his own ideals transcend any of them, especially in analyzing ideas into the simplest parts; language will reflect this, the encyclopedia exists, and calculus does the opposite, in order to efficiently discover new combinations. Leibniz's efforts in his logical calculus were frustrated by the difficulty of empty terms (which the medieval also noticed) and doubts about the relationship between extension and connotation. He foresaw L. Euler's circular diagram and JH Lambert (1728) his efforts in logical calculus were frustrated by the difficulty of empty terms (which the medieval also noticed) and doubts about the relationship between extension and connotation. He foresaw L. Euler's circular diagram and JH Lambert (1728) his efforts in logical calculus were frustrated by the difficulty of empty terms (which the medieval also noticed) and doubts about the relationship between extension and connotation. He foresaw L. Euler's circular diagram and JH Lambert (1728 – 77).
After Leibniz, for example by Lambert, his contemporaries GJ Holland and GF Castillon, many attempts were made to construct satisfactory symbolic calculus. Sir William Hamilton claimed to prioritize quantification predicates, but this had already been done by Leibniz and the man just mentioned. A. de morgan made a real breakthrough, with CS Peirce calling it "the unquestioned father of kinship logic."
In 1847, the same year that de Morgan's formal logic appeared, George Boole published Mathematical Analysis of Logic, followed by Studies in the Laws of Thought in 1854. Since then, with the refinement of calculus, there has been a steady clarification of the idea of interdependence. The thorough systematization and study of logical concepts has become unprecedentedly possible.
Boolean algebra. Boolean algebras, where 1 – x represents the object class in the discursive universe, 1, does not belong to the x class, where the equation x (1 – x ) = 0 represents the principle of non-contradiction, is rich enough for all traditional class reasoning patterns, although some (e.g., dependent) need to declare that the class in question is not empty. The system can also be interpreted in the true value function domain or the probability domain. WS Jevons (1835 – 82) shows that inclusive alternation provides some advantages over the exclusivity used by Booleans; it gives the law x + x = x, removal coefficient. In 1869, he built a logic machine in a new way; The logical diagram proposed by Venn in 1881 also reflects the new approach. CS peirce was a very ingenious and creative thinker who extended boolean algebra with the now-customary inclusion symbols (Similar notations used by Lambert and JD Gergonne), which he also interpreted as prepositional meanings. In 1885, he devised a truth table test of the necessary truth values of a formula and "extended" boolean systems into relational logic by introducing essentially new concepts; here he also developed De Morgan's work with the help of OH Mitchell. Peirce also shows how to define all truth-valued functional connectors (neither... by means of joint exclusions). Nor did it... ), rediscovered by HM Sheffer more than 30 years later. The Vorlesungen U BER modular algebra De Logic E. Schrö der incorporated boolean systems into various improvements made in intervals and further developed Ideas about The Relationship Pierce. Since this represented the pinnacle of Boolean thought, the resulting system is now known as the Boole-Schröder algebra.
Frege and after. At the same time, in 1879, the G. Frege's Begriffsschrift, probably the most insightful and original work of logic ever published. On the one hand, Frege is explicitly concerned with eliminating the influence of all rhetoric and even traditional grammar, and on the other hand, providing an accurate analysis of reasoning in a more thorough way than through a system such as Boolean. As Aristotle's syllogism did, the Boolean-Schroeder system used an unexpressed intuitive logic. This basic logic was successfully formalized by Frege, using only the rules of modus ponens and substitution variables to derive valid propositional formulas from axioms (later considered too extravagant). Frege's connecting words are made up of vertical and horizontal lines; although his expressions can be read mechanically in terms of negation and conjunction, the space they occupy has forbidden their universal use. There are also more compact symbols, such as S. Lesniewski's "wheel", which are closer to the intended meaning on the diagram and easier to use for calculations. Frege applied his propositional system to propositional functions and analyzed them, giving rules for the use of quantifiers and discussing the different properties of whether variables are governed by quantifiers. In these systems, logic has finally matured.
Frege's goal was to analyze and codify mathematical reasoning in a deductive manner. G. Peano actually applied the new method to mathematics and introduced improvements in symbolism. B. Russell and AN Whitehead joined the ideas of Frege and Peano in the Creation of Principles of Mathematics (1910 – 13), the most comprehensive exposition of logical and mathematical ideas ever written. In 1917, J. Lukasiewicz published his first ideas on multivalued logic (inspired by Aristotle and published in 1920, when E. Post's independent investigations in the same field also appeared). S. Jaskowski and G. Gentzen, and K. G's natural reasoning system ö del's proof of the completeness of predicate logic, which appeared in 1930 by ģ ödel's epimonic adaptation of Epimenides in 1931 showed that systems of mathematical principles were undecidable, and continued to be adapted to show that many other systems were also present, notably A. Tarski. A. Church in 1936 showed that the predicate calculus had this property.