1.
Logic
–
Logic, originally meaning the word or what is spoken, is generally held to consist of the systematic study of the form of arguments. A valid argument is one where there is a relation of logical support between the assumptions of the argument and its conclusion. Historically, logic has been studied in philosophy and mathematics, and recently logic has been studied in science, linguistics, psychology. The concept of form is central to logic. The validity of an argument is determined by its logical form, traditional Aristotelian syllogistic logic and modern symbolic logic are examples of formal logic. Informal logic is the study of natural language arguments, the study of fallacies is an important branch of informal logic. Since much informal argument is not strictly speaking deductive, on some conceptions of logic, formal logic is the study of inference with purely formal content. An inference possesses a purely formal content if it can be expressed as an application of a wholly abstract rule, that is. The works of Aristotle contain the earliest known study of logic. Modern formal logic follows and expands on Aristotle, in many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal logic captures all of the nuances of natural language, Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference. Symbolic logic is divided into two main branches, propositional logic and predicate logic. Mathematical logic is an extension of logic into other areas, in particular to the study of model theory, proof theory, set theory. Logic is generally considered formal when it analyzes and represents the form of any valid argument type, the form of an argument is displayed by representing its sentences in the formal grammar and symbolism of a logical language to make its content usable in formal inference. Simply put, formalising simply means translating English sentences into the language of logic and this is called showing the logical form of the argument. It is necessary because indicative sentences of ordinary language show a variety of form. Second, certain parts of the sentence must be replaced with schematic letters, thus, for example, the expression all Ps are Qs shows the logical form common to the sentences all men are mortals, all cats are carnivores, all Greeks are philosophers, and so on. The schema can further be condensed into the formula A, where the letter A indicates the judgement all - are -, the importance of form was recognised from ancient times
2.
Greek language
–
Greek is an independent branch of the Indo-European family of languages, native to Greece and other parts of the Eastern Mediterranean. It has the longest documented history of any living language, spanning 34 centuries of written records and its writing system has been the Greek alphabet for the major part of its history, other systems, such as Linear B and the Cypriot syllabary, were used previously. The alphabet arose from the Phoenician script and was in turn the basis of the Latin, Cyrillic, Armenian, Coptic, Gothic and many other writing systems. Together with the Latin texts and traditions of the Roman world, during antiquity, Greek was a widely spoken lingua franca in the Mediterranean world and many places beyond. It would eventually become the official parlance of the Byzantine Empire, the language is spoken by at least 13.2 million people today in Greece, Cyprus, Italy, Albania, Turkey, and the Greek diaspora. Greek roots are used to coin new words for other languages, Greek. Greek has been spoken in the Balkan peninsula since around the 3rd millennium BC, the earliest written evidence is a Linear B clay tablet found in Messenia that dates to between 1450 and 1350 BC, making Greek the worlds oldest recorded living language. Among the Indo-European languages, its date of earliest written attestation is matched only by the now extinct Anatolian languages, the Greek language is conventionally divided into the following periods, Proto-Greek, the unrecorded but assumed last ancestor of all known varieties of Greek. The unity of Proto-Greek would have ended as Hellenic migrants entered the Greek peninsula sometime in the Neolithic era or the Bronze Age, Mycenaean Greek, the language of the Mycenaean civilisation. It is recorded in the Linear B script on tablets dating from the 15th century BC onwards, Ancient Greek, in its various dialects, the language of the Archaic and Classical periods of the ancient Greek civilisation. It was widely known throughout the Roman Empire, after the Roman conquest of Greece, an unofficial bilingualism of Greek and Latin was established in the city of Rome and Koine Greek became a first or second language in the Roman Empire. The origin of Christianity can also be traced through Koine Greek, Medieval Greek, also known as Byzantine Greek, the continuation of Koine Greek in Byzantine Greece, up to the demise of the Byzantine Empire in the 15th century. Much of the written Greek that was used as the language of the Byzantine Empire was an eclectic middle-ground variety based on the tradition of written Koine. Modern Greek, Stemming from Medieval Greek, Modern Greek usages can be traced in the Byzantine period and it is the language used by the modern Greeks, and, apart from Standard Modern Greek, there are several dialects of it. In the modern era, the Greek language entered a state of diglossia, the historical unity and continuing identity between the various stages of the Greek language is often emphasised. Greek speakers today still tend to regard literary works of ancient Greek as part of their own rather than a foreign language and it is also often stated that the historical changes have been relatively slight compared with some other languages. According to one estimation, Homeric Greek is probably closer to demotic than 12-century Middle English is to modern spoken English, Greek is spoken by about 13 million people, mainly in Greece, Albania and Cyprus, but also worldwide by the large Greek diaspora. Greek is the language of Greece, where it is spoken by almost the entire population
3.
Well-formed formula
–
In mathematical logic, a well-formed formula, abbreviated wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can be identified with the set of formulas in the language, a formula is a syntactic object that can be given a semantic meaning by means of an interpretation. Two key uses of formulas are in propositional logic and predicate logic, a key use of formulas is in propositional logic and predicate logics such as first-order logic. In those contexts, a formula is a string of symbols φ for which it makes sense to ask is φ true, once any free variables in φ have been instantiated. In formal logic, proofs can be represented by sequences of formulas with certain properties, although the term formula may be used for written marks, it is more precisely understood as the sequence of symbols being expressed, with the marks being a token instance of formula. Thus the same formula may be more than once. They are given meanings by interpretations, for example, in a propositional formula, each propositional variable may be interpreted as a concrete proposition, so that the overall formula expresses a relationship between these propositions. A formula need not be interpreted, however, to be considered solely as a formula, the formulas of propositional calculus, also called propositional formulas, are expressions such as. Their definition begins with the choice of a set V of propositional variables. The alphabet consists of the letters in V along with the symbols for the propositional connectives and parentheses, the formulas will be certain expressions over this alphabet. The formulas are inductively defined as follows, Each propositional variable is, on its own, If φ is a formula, then ¬φ is a formula. If φ and ψ are formulas, and • is any binary connective, here • could be the usual operators ∨, ∧, →, or ↔. The sequence of symbols p)) is not a formula, because it does not conform to the grammar, a complex formula may be difficult to read, owing to, for example, the proliferation of parentheses. To alleviate this last phenomenon, precedence rules are assumed among the operators, for example, assuming the precedence 1. Then the formula may be abbreviated as p → q ∧ r → s ∨ ¬q ∧ ¬s This is, however, If the precedence was assumed, for example, to be left-right associative, in following order,1. ∨4. →, then the formula above would be rewritten as → The definition of a formula in first-order logic Q S is relative to the signature of the theory at hand. This signature specifies the constant symbols, relation symbols, and function symbols of the theory at hand, the definition of a formula comes in several parts. First, the set of terms is defined recursively, terms, informally, are expressions that represent objects from the domain of discourse
4.
Philosopher
–
A philosopher is someone who practices philosophy, which involves rational inquiry into areas that are outside of either theology or science. The term philosopher comes from the Ancient Greek φιλόσοφος meaning lover of wisdom, the coining of the term has been attributed to the Greek thinker Pythagoras. Typically, these brands of philosophy are Hellenistic ones and those who most arduously commit themselves to this lifestyle may be considered philosophers. The separation of philosophy and science from theology began in Greece during the 6th century BC, thales, an astronomer and mathematician, was considered by Aristotle to be the first philosopher of the Greek tradition. While Pythagoras coined the word, the first known elaboration on the topic was conducted by Plato, in his Symposium, he concludes that Love is that which lacks the object it seeks. Therefore, the philosopher is one who seeks wisdom, if he attains wisdom, therefore, the philosopher in antiquity was one who lives in the constant pursuit of wisdom, and living in accordance to that wisdom. Disagreements arose as to what living philosophically entailed and these disagreements gave rise to different Hellenistic schools of philosophy. In consequence, the ancient philosopher thought in a tradition, as the ancient world became schism by philosophical debate, the competition lay in living in manner that would transform his whole way of living in the world. Philosophy is a discipline which can easily carry away the individual in analyzing the universe. The second is the change through the Medieval era. With the rise of Christianity, the way of life was adopted by its theology. Thus, philosophy was divided between a way of life and the conceptual, logical, physical and metaphysical materials to justify that way of life, philosophy was then the servant to theology. The third is the sociological need with the development of the university, the modern university requires professionals to teach. Maintaining itself requires teaching future professionals to replace the current faculty, therefore, the discipline degrades into a technical language reserved for specialists, completely eschewing its original conception as a way of life. In the fourth century, the word began to designate a man or woman who led a monastic life. Gregory of Nyssa, for example, describes how his sister Macrina persuaded their mother to forsake the distractions of life for a life of philosophy. Later during the Middle Ages, persons who engaged with alchemy was called a philosopher - thus, many philosophers still emerged from the Classical tradition, as saw their philosophy as a way of life. Among the most notable are René Descartes, Baruch Spinoza, Nicolas Malebranche, with the rise of the university, the modern conception of philosophy became more prominent
5.
Ludwig Wittgenstein
–
Ludwig Josef Johann Wittgenstein was an Austrian-British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. From 1929 to 1947, Wittgenstein taught at the University of Cambridge, during his lifetime he published just one slim book, the 75-page Tractatus Logico-Philosophicus, one article, one book review and a childrens dictionary. His voluminous manuscripts were edited and published posthumously, Philosophical Investigations appeared as a book in 1953, and has since come to be recognised as one of the most important works of philosophy in the twentieth century. His teacher Bertrand Russell described Wittgenstein as the most perfect example I have ever known of genius as traditionally conceived, passionate, profound, intense, born in Vienna into one of Europes richest families, he inherited a large fortune from his father in 1913. Three of his brothers committed suicide, with Wittgenstein contemplating it too and he described philosophy as the only work that gives me real satisfaction. His philosophy is divided into an early period, exemplified by the Tractatus. The later Wittgenstein rejected many of the assumptions of the Tractatus, ludwigs grandmother Fanny was a first cousin of the famous violinist Joseph Joachim. They had 11 children—among them Wittgensteins father, Karl Otto Clemens Wittgenstein became an industrial tycoon, and by the late 1880s was one of the richest men in Europe, with an effective monopoly on Austrias steel cartel. Thanks to Karl, the Wittgensteins became the second wealthiest family in Austria-Hungary, however, their wealth diminished due to post-1918 hyperinflation and subsequently during the Great Depression, although even as late as 1938 they owned 13 mansions in Vienna alone. Wittgensteins mother was Leopoldine Maria Josefa Kalmus, known among friends as Poldi and her father was a Bohemian Jew and her mother was Austrian-Slovene Catholic—she was Wittgensteins only non-Jewish grandparent. She was an aunt of the Nobel Prize laureate Friedrich Hayek on her maternal side, Wittgenstein was born at 8,30 pm on 26 April 1889 in the so-called Wittgenstein Palace at Alleegasse 16, now the Argentinierstrasse, near the Karlskirche. Karl and Poldi had nine children in all, the children were baptized as Catholics, received formal Catholic instruction, and raised in an exceptionally intense environment. The family was at the center of Viennas cultural life, Bruno Walter described the life at the Wittgensteins palace as an atmosphere of humanity. Karl was a patron of the arts, commissioning works by Auguste Rodin and financing the citys exhibition hall and art gallery. Gustav Klimt painted Wittgensteins sister for her portrait, and Johannes Brahms. For Wittgenstein, who highly valued precision and discipline, contemporary music was never considered acceptable at all, music, he said to his friend Drury in 1930, came to a full stop with Brahms, and even in Brahms I can begin to hear the noise of machinery. He also learnt to play the clarinet in his thirties, a fragment of music, composed by Wittgenstein, was discovered in one of his 1931 notebooks, by Michael Nedo, Director of the Wittgenstein Institute in Cambridge. Three of the five brothers would commit suicide
6.
Propositional logic
–
Logical connectives are found in natural languages. In English for example, some examples are and, or, not”, the following is an example of a very simple inference within the scope of propositional logic, Premise 1, If its raining then its cloudy. Both premises and the conclusion are propositions, the premises are taken for granted and then with the application of modus ponens the conclusion follows. Not only that, but they will also correspond with any other inference of this form, Propositional logic may be studied through a formal system in which formulas of a formal language may be interpreted to represent propositions. A system of rules and axioms allows certain formulas to be derived. These derived formulas are called theorems and may be interpreted to be true propositions, a constructed sequence of such formulas is known as a derivation or proof and the last formula of the sequence is the theorem. The derivation may be interpreted as proof of the represented by the theorem. When a formal system is used to represent formal logic, only statement letters are represented directly, usually in truth-functional propositional logic, formulas are interpreted as having either a truth value of true or a truth value of false. Truth-functional propositional logic and systems isomorphic to it, are considered to be zeroth-order logic, although propositional logic had been hinted by earlier philosophers, it was developed into a formal logic by Chrysippus in the 3rd century BC and expanded by his successor Stoics. The logic was focused on propositions and this advancement was different from the traditional syllogistic logic which was focused on terms. However, later in antiquity, the propositional logic developed by the Stoics was no longer understood, consequently, the system was essentially reinvented by Peter Abelard in the 12th century. Propositional logic was eventually refined using symbolic logic, the 17th/18th-century mathematician Gottfried Leibniz has been credited with being the founder of symbolic logic for his work with the calculus ratiocinator. Although his work was the first of its kind, it was unknown to the larger logical community, consequently, many of the advances achieved by Leibniz were recreated by logicians like George Boole and Augustus De Morgan completely independent of Leibniz. Just as propositional logic can be considered an advancement from the earlier syllogistic logic, one author describes predicate logic as combining the distinctive features of syllogistic logic and propositional logic. Consequently, predicate logic ushered in a new era in history, however, advances in propositional logic were still made after Frege, including Natural Deduction. Natural deduction was invented by Gerhard Gentzen and Jan Łukasiewicz, Truth-Trees were invented by Evert Willem Beth. The invention of truth-tables, however, is of controversial attribution, within works by Frege and Bertrand Russell, are ideas influential to the invention of truth tables. The actual tabular structure, itself, is credited to either Ludwig Wittgenstein or Emil Post
7.
Contradiction
–
In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other. Illustrating a general tendency in applied logic, Aristotles law of noncontradiction states that One cannot say of something that it is, by extension, outside of classical logic, one can speak of contradictions between actions when one presumes that their motives contradict each other. By creation of a paradox, Platos Euthydemus dialogue demonstrates the need for the notion of contradiction, in the ensuing dialogue Dionysodorus denies the existence of contradiction, all the while that Socrates is contradicting him. I in my astonishment said, What do you mean Dionysodorus, the dictum is that there is no such thing as a falsehood, a man must either say what is true or say nothing. Indeed, Dionysodorus agrees that there is no such thing as false opinion, there is no such thing as ignorance and demands of Socrates to Refute me. Socrates responds But how can I refute you, if, as you say, note, The symbol ⊥ represents an arbitrary contradiction, with the dual tee symbol ⊤ used to denote an arbitrary tautology. Contradiction is sometimes symbolized by Opq, and tautology by Vpq, the turnstile symbol, ⊢ is often read as yields or proves. In classical logic, particularly in propositional and first-order logic, a proposition φ is a contradiction if, since for contradictory φ it is true that ⊢ φ → ψ for all ψ, one may prove any proposition from a set of axioms which contains contradictions. This is called the principle of explosion or ex falso quodlibet, in a complete logic, a formula is contradictory if and only if it is unsatisfiable. Therefore, a proof that ¬ φ ⊢ ⊥ also proves that φ is true, the use of this fact constitutes the technique of the proof by contradiction, which mathematicians use extensively. This applies only in a logic using the excluded middle A ∨ ¬ A as an axiom, in mathematics, the symbol used to represent a contradiction within a proof varies. A consistency proof requires an axiomatic system a demonstration that it is not the case both the formula p and its negation ~p can be derived in the system. Posts solution to the problem is described in the demonstration An Example of a Successful Absolute Proof of Consistency offered by Ernest Nagel and they too observe a problem with respect to the notion of contradiction with its usual truth values of truth and falsity. They observe that, The property of being a tautology has been defined in notions of truth, yet these notions obviously involve a reference to something outside the formula calculus. Therefore, the mentioned in the text in effect offers an interpretation of the calculus. This being so, the authors have not done what they promised, namely, proofs of consistency which are based on models, and which argue from the truth of axioms to their consistency, merely shift the problem. Given some primitive formulas such as PMs primitives S1 V S2, so what will be the definition of tautologous
8.
Logical truth
–
Logical truth is one of the most fundamental concepts in logic, and there are different theories on its nature. A logical truth is a statement which is true, and remains true under all reinterpretations of its components other than its logical constants and it is a type of analytic statement. All of philosophical logic can be thought of as providing accounts of the nature of logical truth, Logical truths are truths which are considered to be necessarily true. This is to say that they are considered to be such that they could not be untrue and it must be true in every sense of intuition, practices, and bodies of beliefs. However, it is not universally agreed that there are any statements which are necessarily true, a logical truth is considered by some philosophers to be a statement which is true in all possible worlds. This is contrasted with facts which are true in this world, as it has historically unfolded, later, with the rise of formal logic a logical truth was considered to be a statement which is true under all possible interpretations. Empiricists commonly respond to this objection by arguing that logical truths, are analytic, Logical truths, being analytic statements, do not contain any information about any matters of fact. Other than logical truths, there is also a class of analytic statements. The characteristic of such a statement is that it can be turned into a logical truth by substituting synonyms for synonyms salva veritate, can be turned into No unmarried man is married. By substituting unmarried man for its synonym bachelor, in his essay, Two Dogmas of Empiricism, the philosopher W. V. O. Quine called into question the distinction between analytic and synthetic statements, in his conclusion, Quine rejects that logical truths are necessary truths. Instead he posits that the truth-value of any statement can be changed, including logical truths, considering different interpretations of the same statement leads to the notion of truth value. The simplest approach to truth values means that the statement may be true in one case, in one sense of the term tautology, it is any type of formula or proposition which turns out to be true under any possible interpretation of its terms. This is synonymous to logical truth, however, the term tautology is also commonly used to refer to what could more specifically be called truth-functional tautologies. Not all logical truths are tautologies of such a kind, Logical constants, including logical connectives and quantifiers, can all be reduced conceptually to logical truth. For instance, two statements or more are logically incompatible if, and only if their conjunction is logically false, one statement logically implies another when it is logically incompatible with the negation of the other. A statement is true if, and only if its opposite is logically false. The opposite statements must contradict one another, in this way all logical connectives can be expressed in terms of preserving logical truth
9.
First-order logic
–
First-order logic – also known as first-order predicate calculus and predicate logic – is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. This distinguishes it from propositional logic, which does not use quantifiers, Sometimes theory is understood in a more formal sense, which is just a set of sentences in first-order logic. In first-order theories, predicates are associated with sets. In interpreted higher-order theories, predicates may be interpreted as sets of sets, There are many deductive systems for first-order logic which are both sound and complete. Although the logical relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem, first-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, no first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axioms systems that do fully describe these two structures can be obtained in stronger logics such as second-order logic, for a history of first-order logic and how it came to dominate formal logic, see José Ferreirós. While propositional logic deals with simple declarative propositions, first-order logic additionally covers predicates, a predicate takes an entity or entities in the domain of discourse as input and outputs either True or False. Consider the two sentences Socrates is a philosopher and Plato is a philosopher, in propositional logic, these sentences are viewed as being unrelated and might be denoted, for example, by variables such as p and q. The predicate is a philosopher occurs in both sentences, which have a structure of a is a philosopher. The variable a is instantiated as Socrates in the first sentence and is instantiated as Plato in the second sentence, while first-order logic allows for the use of predicates, such as is a philosopher in this example, propositional logic does not. Relationships between predicates can be stated using logical connectives, consider, for example, the first-order formula if a is a philosopher, then a is a scholar. This formula is a statement with a is a philosopher as its hypothesis. The truth of this depends on which object is denoted by a. Quantifiers can be applied to variables in a formula, the variable a in the previous formula can be universally quantified, for instance, with the first-order sentence For every a, if a is a philosopher, then a is a scholar. The universal quantifier for every in this sentence expresses the idea that the if a is a philosopher. The negation of the sentence For every a, if a is a philosopher, then a is a scholar is logically equivalent to the sentence There exists a such that a is a philosopher and a is not a scholar
10.
Subset
–
In mathematics, especially in set theory, a set A is a subset of a set B, or equivalently B is a superset of A, if A is contained inside B, that is, all elements of A are also elements of B. The relationship of one set being a subset of another is called inclusion or sometimes containment, the subset relation defines a partial order on sets. The algebra of subsets forms a Boolean algebra in which the relation is called inclusion. For any set S, the inclusion relation ⊆ is an order on the set P of all subsets of S defined by A ≤ B ⟺ A ⊆ B. We may also partially order P by reverse set inclusion by defining A ≤ B ⟺ B ⊆ A, when quantified, A ⊆ B is represented as, ∀x. So for example, for authors, it is true of every set A that A ⊂ A. Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and superset, respectively and this usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if x ≤ y then x may or may not equal y, but if x < y, then x definitely does not equal y, and is less than y. Similarly, using the convention that ⊂ is proper subset, if A ⊆ B, then A may or may not equal B, the set A = is a proper subset of B =, thus both expressions A ⊆ B and A ⊊ B are true. The set D = is a subset of E =, thus D ⊆ E is true, any set is a subset of itself, but not a proper subset. The empty set, denoted by ∅, is also a subset of any given set X and it is also always a proper subset of any set except itself. These are two examples in both the subset and the whole set are infinite, and the subset has the same cardinality as the whole. The set of numbers is a proper subset of the set of real numbers. In this example, both sets are infinite but the set has a larger cardinality than the former set. Another example in an Euler diagram, Inclusion is the partial order in the sense that every partially ordered set is isomorphic to some collection of sets ordered by inclusion. The ordinal numbers are a simple example—if each ordinal n is identified with the set of all ordinals less than or equal to n, then a ≤ b if and only if ⊆. For the power set P of a set S, the partial order is the Cartesian product of k = |S| copies of the partial order on for which 0 <1. This can be illustrated by enumerating S = and associating with each subset T ⊆ S the k-tuple from k of which the ith coordinate is 1 if and only if si is a member of T
11.
Immanuel Kant
–
Immanuel Kant was a German philosopher who is considered a central figure in modern philosophy. Kant took himself to have effected a Copernican revolution in philosophy and his beliefs continue to have a major influence on contemporary philosophy, especially the fields of metaphysics, epistemology, ethics, political theory, and aesthetics. Politically, Kant was one of the earliest exponents of the idea that peace could be secured through universal democracy. He believed that this will be the outcome of universal history. Kant wanted to put an end to an era of futile and speculative theories of human experience, Kant argued that our experiences are structured by necessary features of our minds. In his view, the shapes and structures experience so that, on an abstract level. Among other things, Kant believed that the concepts of space and time are integral to all human experience, as are our concepts of cause, Kant published other important works on ethics, religion, law, aesthetics, astronomy, and history. These included the Critique of Practical Reason, the Metaphysics of Morals, which dealt with ethics, and the Critique of Judgment, Immanuel Kant was born in 1724 in Königsberg, Prussia. His mother, Anna Regina Reuter, was born in Königsberg to a father from Nuremberg. His father, Johann Georg Kant, was a German harness maker from Memel, Immanuel Kant believed that his paternal grandfather Hans Kant was of Scottish origin. Kant was the fourth of nine children, baptized Emanuel, he changed his name to Immanuel after learning Hebrew. Young Kant was a solid, albeit unspectacular, student and he was brought up in a Pietist household that stressed religious devotion, humility, and a literal interpretation of the Bible. His education was strict, punitive and disciplinary, and focused on Latin and religious instruction over mathematics, despite his religious upbringing and maintaining a belief in God, Kant was skeptical of religion in later life, various commentators have labelled him agnostic. Common myths about Kants personal mannerisms are listed, explained, and refuted in Goldthwaits introduction to his translation of Observations on the Feeling of the Beautiful and Sublime. It is often held that Kant lived a strict and disciplined life. He never married, but seemed to have a social life — he was a popular teacher. He had a circle of friends whom he met, among them Joseph Green. A common myth is that Kant never traveled more than 16 kilometres from Königsberg his whole life, in fact, between 1750 and 1754 he worked as a tutor in Judtschen and in Groß-Arnsdorf
12.
Analytic truth
–
Logical truth is one of the most fundamental concepts in logic, and there are different theories on its nature. A logical truth is a statement which is true, and remains true under all reinterpretations of its components other than its logical constants and it is a type of analytic statement. All of philosophical logic can be thought of as providing accounts of the nature of logical truth, Logical truths are truths which are considered to be necessarily true. This is to say that they are considered to be such that they could not be untrue and it must be true in every sense of intuition, practices, and bodies of beliefs. However, it is not universally agreed that there are any statements which are necessarily true, a logical truth is considered by some philosophers to be a statement which is true in all possible worlds. This is contrasted with facts which are true in this world, as it has historically unfolded, later, with the rise of formal logic a logical truth was considered to be a statement which is true under all possible interpretations. Empiricists commonly respond to this objection by arguing that logical truths, are analytic, Logical truths, being analytic statements, do not contain any information about any matters of fact. Other than logical truths, there is also a class of analytic statements. The characteristic of such a statement is that it can be turned into a logical truth by substituting synonyms for synonyms salva veritate, can be turned into No unmarried man is married. By substituting unmarried man for its synonym bachelor, in his essay, Two Dogmas of Empiricism, the philosopher W. V. O. Quine called into question the distinction between analytic and synthetic statements, in his conclusion, Quine rejects that logical truths are necessary truths. Instead he posits that the truth-value of any statement can be changed, including logical truths, considering different interpretations of the same statement leads to the notion of truth value. The simplest approach to truth values means that the statement may be true in one case, in one sense of the term tautology, it is any type of formula or proposition which turns out to be true under any possible interpretation of its terms. This is synonymous to logical truth, however, the term tautology is also commonly used to refer to what could more specifically be called truth-functional tautologies. Not all logical truths are tautologies of such a kind, Logical constants, including logical connectives and quantifiers, can all be reduced conceptually to logical truth. For instance, two statements or more are logically incompatible if, and only if their conjunction is logically false, one statement logically implies another when it is logically incompatible with the negation of the other. A statement is true if, and only if its opposite is logically false. The opposite statements must contradict one another, in this way all logical connectives can be expressed in terms of preserving logical truth
13.
Gottlob Frege
–
Friedrich Ludwig Gottlob Frege was a German philosopher, logician, and mathematician. Considered a major figure in mathematics, he is responsible for the development of modern logic and he is also understood by many to be the father of analytic philosophy, where he concentrated on the philosophy of language and mathematics. Though largely ignored during his lifetime, Giuseppe Peano and Bertrand Russell introduced his work to generations of logicians. Frege was born in 1848 in Wismar, Mecklenburg-Schwerin and his father Carl Alexander Frege was the co-founder and headmaster of a girls high school until his death. In childhood, Frege encountered philosophies that would guide his future scientific career, Frege studied at a gymnasium in Wismar and graduated in 1869. His teacher Gustav Adolf Leo Sachse, who was a poet, played the most important role in determining Freges future scientific career, Frege matriculated at the University of Jena in the spring of 1869 as a citizen of the North German Confederation. In the four semesters of his studies he attended approximately twenty courses of lectures and his most important teacher was Ernst Karl Abbe. Abbe was more than a teacher to Frege, he was a trusted friend, after Freges graduation, they came into closer correspondence. His other notable university teachers were Christian Philipp Karl Snell, Hermann Karl Julius Traugott Schaeffer, Frege married Margarete Katharina Sophia Anna Lieseberg on 14 March 1887. Though his education and early work focused primarily on geometry. His Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Halle a/S, the Begriffsschrift broke new ground, including a rigorous treatment of the ideas of functions and variables. Previous logic had dealt with the constants and, or. Freges conceptual notation however can represent such inferences, one of Freges stated purposes was to isolate genuinely logical principles of inference, so that in the proper representation of mathematical proof, one would at no point appeal to intuition. If there was an element, it was to be isolated and represented separately as an axiom, from there on. Already in the 1879 Begriffsschrift important preliminary theorems, for example a generalized form of law of trichotomy, were derived within what Frege understood to be pure logic and this idea was formulated in non-symbolic terms in his The Foundations of Arithmetic. Later, in his Basic Laws of Arithmetic, Frege attempted to derive, by use of his symbolism, most of these axioms were carried over from his Begriffsschrift, though not without some significant changes. The one truly new principle was one he called the Basic Law V, the crucial case of the law may be formulated in modern notation as follows. Let denote the extension of the predicate Fx, i. e. the set of all Fs, then Basic Law V says that the predicates Fx and Gx have the same extension iff ∀x
14.
Tractatus Logico-Philosophicus
–
The Tractatus Logico-Philosophicus is the only book-length philosophical work published by the Austrian philosopher Ludwig Wittgenstein in his lifetime. G. E. Moore originally suggested the works Latin title as homage to the Tractatus Theologico-Politicus by Baruch Spinoza. Wittgenstein wrote the notes for the Tractatus while he was a soldier during World War I and completed it when a prisoner of war at Como and it was first published in German in 1921 as Logisch-Philosophische Abhandlung. The Tractatus was influential chiefly amongst the logical positivists of the Vienna Circle, such as Rudolf Carnap, Bertrand Russells article The Philosophy of Logical Atomism is presented as a working out of ideas that he had learned from Wittgenstein. The Tractatus employs a notoriously austere and succinct literary style, the work contains almost no arguments as such, but rather consists of declarative statements that are meant to be self-evident. The statements are hierarchically numbered, with seven basic propositions at the primary level, Wittgensteins later works, notably the posthumously published Philosophical Investigations, criticised many of the ideas in the Tractatus. There are seven main propositions in the text and these are, The world is everything that is the case. What is the case is the existence of states of affairs, a logical picture of facts is a thought. A thought is a proposition with a sense, a proposition is a truth-function of elementary propositions. The general form of a proposition is the form of a truth function. This is the form of a proposition. Whereof one cannot speak, thereof one must be silent, the first chapter is very brief,1 The world is all that is the case. 1.1 The world is the totality of facts, not of things,1.11 The world is determined by the facts, and by their being all the facts. 1.12 For the totality of facts determines what is the case,1.13 The facts in logical space are the world. 1.2 The world divides into facts,1.21 Each item can be the case or not the case while everything else remains the same. This along with the beginning of two can be taken to be the relevant parts of Wittgensteins metaphysical view that he use to support his picture theory of language. These sections concern Wittgensteins view that the sensible, changing world we perceive does not consist of substance, Proposition two begins with a discussion of objects, form and substance. 2 What is the case—a fact—is the existence of atomic facts,2.01 An atomic fact is a combination of objects
15.
Bertrand Russell
–
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, writer, social critic, political activist and Nobel laureate. At various points in his life he considered himself a liberal, a socialist, and a pacifist and he was born in Monmouthshire into one of the most prominent aristocratic families in the United Kingdom. In the early 20th century, Russell led the British revolt against idealism and he is considered one of the founders of analytic philosophy along with his predecessor Gottlob Frege, colleague G. E. Moore, and protégé Ludwig Wittgenstein. He is widely held to be one of the 20th centurys premier logicians, with A. N. Whitehead he wrote Principia Mathematica, an attempt to create a logical basis for mathematics. His philosophical essay On Denoting has been considered a paradigm of philosophy, Russell mostly was a prominent anti-war activist, he championed anti-imperialism. Occasionally, he advocated preventive nuclear war, before the opportunity provided by the monopoly is gone. He went to prison for his pacifism during World War I, in 1950 Russell was awarded the Nobel Prize in Literature in recognition of his varied and significant writings in which he champions humanitarian ideals and freedom of thought. Bertrand Russell was born on 18 May 1872 at Ravenscroft, Trellech, Monmouthshire and his parents, Viscount and Viscountess Amberley, were radical for their times. Lord Amberley consented to his wifes affair with their childrens tutor, both were early advocates of birth control at a time when this was considered scandalous. Lord Amberley was an atheist and his atheism was evident when he asked the philosopher John Stuart Mill to act as Russells secular godfather, Mill died the year after Russells birth, but his writings had a great effect on Russells life. His paternal grandfather, the Earl Russell, had asked twice by Queen Victoria to form a government. The Russells had been prominent in England for several centuries before this, coming to power, Lady Amberley was the daughter of Lord and Lady Stanley of Alderley. Russell often feared the ridicule of his grandmother, one of the campaigners for education of women. Russell had two siblings, brother Frank, and sister Rachel, in June 1874 Russells mother died of diphtheria, followed shortly by Rachels death. In January 1876, his father died of bronchitis following a period of depression. Frank and Bertrand were placed in the care of their staunchly Victorian paternal grandparents and his grandfather, former Prime Minister Earl Russell, died in 1878, and was remembered by Russell as a kindly old man in a wheelchair. His grandmother, the Countess Russell, was the dominant family figure for the rest of Russells childhood, the countess was from a Scottish Presbyterian family, and successfully petitioned the Court of Chancery to set aside a provision in Amberleys will requiring the children to be raised as agnostics. Her favourite Bible verse, Thou shalt not follow a multitude to do evil, the atmosphere at Pembroke Lodge was one of frequent prayer, emotional repression, and formality, Frank reacted to this with open rebellion, but the young Bertrand learned to hide his feelings
16.
Stephen Cole Kleene
–
Stephen Cole Kleene /ˈkleɪniː/ KLAY-nee was an American mathematician. Kleenes work grounds the study of functions are computable. A number of concepts are named after him, Kleene hierarchy, Kleene algebra, the Kleene star, Kleenes recursion theorem. He also invented regular expressions, and made significant contributions to the foundations of mathematical intuitionism, although his last name is commonly pronounced /ˈkliːniː/ KLEE-nee or /ˈkliːn/ kleen, Kleene himself pronounced it /ˈkleɪniː/ KLAY-nee. His son, Ken Kleene, wrote, As far as I am aware this pronunciation is incorrect in all known languages, I believe that this novel pronunciation was invented by my father. Kleene was awarded the BA degree from Amherst College in 1930 and he was awarded the Ph. D. in mathematics from Princeton University in 1934. His thesis, entitled A Theory of Positive Integers in Formal Logic, was supervised by Alonzo Church, in the 1930s, he did important work on Churchs lambda calculus. In 1935, he joined the department at the University of Wisconsin–Madison. After two years as an instructor, he was appointed assistant professor in 1937, while a visiting scholar at the Institute for Advanced Study in Princeton, 1939–40, he laid the foundation for recursion theory, an area that would be his lifelong research interest. In 1941, he returned to Amherst College, where he spent one year as a professor of mathematics. During World War II, Kleene was a lieutenant commander in the United States Navy. He was an instructor of navigation at the U. S. Naval Reserves Midshipmens School in New York, in 1946, Kleene returned to Wisconsin, becoming a full professor in 1948 and the Cyrus C. MacDuffee professor of mathematics in 1964 and he was chair of the Department of Mathematics and Computer Science, 1962–63, and Dean of the College of Letters and Science from 1969 to 1974. The latter appointment he took on despite the considerable student unrest of the day and he retired from the University of Wisconsin in 1979. In 1999 the mathematics library at the University of Wisconsin was renamed in his honor, Kleenes teaching at Wisconsin resulted in three texts in mathematical logic, Kleene and Kleene and Vesley, often cited and still in print. Kleene wrote alternative proofs to the Gödels incompleteness theorems that enhanced their status and made them easier to teach. Kleene and Vesley is the classic American introduction to intuitionist logic, Kleene served as president of the Association for Symbolic Logic, 1956–58, and of the International Union of History and Philosophy of Science,1961. In 1990, he was awarded the National Medal of Science, the importance of Kleenes work led to the saying that Kleeneness is next to Gödelness
17.
Tautology (logic)
–
In logic, a tautology is a formula that is true in every possible interpretation. Philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921, a formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions, a formula that is neither a tautology nor a contradiction is said to be logically contingent. Such a formula can be either true or false based on the values assigned to its propositional variables. The double turnstile notation ⊨ S is used to indicate that S is a tautology, Tautology is sometimes symbolized by Vpq, and contradiction by Opq. Tautologies are a key concept in logic, where a tautology is defined as a propositional formula that is true under any possible Boolean valuation of its propositional variables. A key property of tautologies in propositional logic is that a method exists for testing whether a given formula is always satisfied. The definition of tautology can be extended to sentences in predicate logic, in propositional logic, there is no distinction between a tautology and a logically valid formula. The set of formulas is a proper subset of the set of logically valid sentences of predicate logic. In 1800, Immanuel Kant wrote in his book Logic, The identity of concepts in analytical judgments can be explicit or non-explicit. In the former case analytic propositions are tautological, here analytic proposition refers to an analytic truth, a statement in natural language that is true solely because of the terms involved. In 1884, Gottlob Frege proposed in his Grundlagen that a truth is analytic if it can be derived using logic. But he maintained a distinction between analytic truths and tautologies, in 1921, in his Tractatus Logico-Philosophicus, Ludwig Wittgenstein proposed that statements that can be deduced by logical deduction are tautological as well as being analytic truths. Henri Poincaré had made remarks in Science and Hypothesis in 1905. It has got to be something that has some quality, which I do not know how to define. Here logical proposition refers to a proposition that is using the laws of logic. During the 1930s, the formalization of the semantics of propositional logic in terms of truth assignments was developed, the term tautology began to be applied to those propositional formulas that are true regardless of the truth or falsity of their propositional variables. Some early books on logic used the term for any proposition that is universally valid, propositional logic begins with propositional variables, atomic units that represent concrete propositions
18.
Logical disjunction
–
In logic and mathematics, or is the truth-functional operator of disjunction, also known as alternation, the or of a set of operands is true if and only if one or more of its operands is true. The logical connective that represents this operator is written as ∨ or +. A or B is true if A is true, or if B is true, or if both A and B are true. In logic, or by means the inclusive or, distinguished from an exclusive or. An operand of a disjunction is called a disjunct, related concepts in other fields are, In natural language, the coordinating conjunction or. In programming languages, the short-circuit or control structure, or is usually expressed with an infix operator, in mathematics and logic, ∨, in electronics, +, and in most programming languages, |, ||, or or. In Jan Łukasiewiczs prefix notation for logic, the operator is A, logical disjunction is an operation on two logical values, typically the values of two propositions, that has a value of false if and only if both of its operands are false. More generally, a disjunction is a formula that can have one or more literals separated only by ors. A single literal is often considered to be a degenerate disjunction, the disjunctive identity is false, which is to say that the or of an expression with false has the same value as the original expression. In keeping with the concept of truth, when disjunction is defined as an operator or function of arbitrary arity. Falsehood-preserving, The interpretation under which all variables are assigned a value of false produces a truth value of false as a result of disjunction. The mathematical symbol for logical disjunction varies in the literature, in addition to the word or, and the formula Apq, the symbol ∨, deriving from the Latin word vel is commonly used for disjunction. For example, A ∨ B is read as A or B, such a disjunction is false if both A and B are false. In all other cases it is true, all of the following are disjunctions, A ∨ B ¬ A ∨ B A ∨ ¬ B ∨ ¬ C ∨ D ∨ ¬ E. The corresponding operation in set theory is the set-theoretic union, operators corresponding to logical disjunction exist in most programming languages. Disjunction is often used for bitwise operations, for example, x = x | 0b00000001 will force the final bit to 1 while leaving other bits unchanged. Logical disjunction is usually short-circuited, that is, if the first operand evaluates to true then the second operand is not evaluated, the logical disjunction operator thus usually constitutes a sequence point. In a parallel language, it is possible to both sides, they are evaluated in parallel, and if one terminates with value true
19.
Logical conjunction
–
In logic and mathematics, and is the truth-functional operator of logical conjunction, the and of a set of operands is true if and only if all of its operands are true. The logical connective that represents this operator is written as ∧ or ⋅. A and B is true only if A is true and B is true, an operand of a conjunction is a conjunct. Related concepts in other fields are, In natural language, the coordinating conjunction, in programming languages, the short-circuit and control structure. And is usually denoted by an operator, in mathematics and logic, ∧ or ×, in electronics, ⋅. In Jan Łukasiewiczs prefix notation for logic, the operator is K, logical conjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both of its operands are true. The conjunctive identity is 1, which is to say that AND-ing an expression with 1 will never change the value of the expression. In keeping with the concept of truth, when conjunction is defined as an operator or function of arbitrary arity. The truth table of A ∧ B, As a rule of inference, conjunction introduction is a classically valid, the argument form has two premises, A and B. Intuitively, it permits the inference of their conjunction, therefore, A and B. or in logical operator notation, A, B ⊢ A ∧ B Here is an example of an argument that fits the form conjunction introduction, Bob likes apples. Therefore, Bob likes apples and oranges, Conjunction elimination is another classically valid, simple argument form. Intuitively, it permits the inference from any conjunction of either element of that conjunction, therefore, A. or alternately, A and B. In logical operator notation, A ∧ B ⊢ A. falsehood-preserving, yes When all inputs are false, walsh spectrum, Nonlinearity,1 If using binary values for true and false, then logical conjunction works exactly like normal arithmetic multiplication. Many languages also provide short-circuit control structures corresponding to logical conjunction. Logical conjunction is used for bitwise operations, where 0 corresponds to false and 1 to true,0 AND0 =0,0 AND1 =0,1 AND0 =0,1 AND1 =1. The operation can also be applied to two binary words viewed as bitstrings of length, by taking the bitwise AND of each pair of bits at corresponding positions. For example,11000110 AND10100011 =10000010 and this can be used to select part of a bitstring using a bit mask. For example,10011101 AND00001000 =00001000 extracts the fifth bit of an 8-bit bitstring
20.
Negation
–
Negation is thus a unary logical connective. It may be applied as an operation on propositions, truth values, in classical logic, negation is normally identified with the truth function that takes truth to falsity and vice versa. In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition p is the proposition whose proofs are the refutations of p. Classical negation is an operation on one logical value, typically the value of a proposition, that produces a value of true when its operand is false and a value of false when its operand is true. So, if statement A is true, then ¬A would therefore be false, the truth table of ¬p is as follows, Classical negation can be defined in terms of other logical operations. For example, ¬p can be defined as p → F, conversely, one can define F as p & ¬p for any proposition p, where & is logical conjunction. The idea here is that any contradiction is false, while these ideas work in both classical and intuitionistic logic, they do not work in paraconsistent logic, where contradictions are not necessarily false. But in classical logic, we get an identity, p → q can be defined as ¬p ∨ q. Algebraically, classical negation corresponds to complementation in a Boolean algebra and these algebras provide a semantics for classical and intuitionistic logic respectively. The negation of a proposition p is notated in different ways in various contexts of discussion and fields of application. Among these variants are the following, In set theory \ is also used to indicate not member of, U \ A is the set of all members of U that are not members of A. No matter how it is notated or symbolized, the negation ¬p / −p can be read as it is not the case p, not that p. Within a system of logic, double negation, that is. In intuitionistic logic, a proposition implies its double negation but not conversely and this marks one important difference between classical and intuitionistic negation. Algebraically, classical negation is called an involution of period two and this result is known as Glivenkos theorem. De Morgans laws provide a way of distributing negation over disjunction and conjunction, ¬ ≡, in Boolean algebra, a linear function is one such that, If there exists a0, a1. An ∈ such that f = a0 ⊕ ⊕, another way to express this is that each variable always makes a difference in the truth-value of the operation or it never makes a difference. Negation is a logical operator
21.
Law of the excluded middle
–
In logic, the law of excluded middle is the third of the three classic laws of thought. It states that for any proposition, either that proposition is true, the law is also known as the law of the excluded third, in Latin principium tertii exclusi. Another Latin designation for this law is tertium non datur, no third is given, the principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as, ∗2 ⋅11. The principle should not be confused with the principle of bivalence. The principle of excluded middle, along with its complement, the law of contradiction, are correlates of the law of identity, some systems of logic have different but analogous laws. For some finite n-valued logics, there is a law called the law of excluded n+1th. If negation is cyclic and ∨ is a max operator, then the law can be expressed in the language by. It is easy to check that the sentence must receive at least one of the n truth values, Other systems reject the law entirely. For example, if P is the proposition, Socrates is mortal, then the law of excluded middle holds that the logical disjunction, Either Socrates is mortal, or it is not the case that Socrates is mortal. is true by virtue of its form alone. That is, the position, that Socrates is neither mortal nor not-mortal, is excluded by logic. An example of an argument that depends on the law of excluded middle follows and we seek to prove that there exist two irrational numbers a and b such that a b is rational. It is known that 2 is irrational, clearly this number is either rational or irrational. If it is rational, the proof is complete, and a =2 and b =2, but if 22 is irrational, then let a =22 and b =2. Then a b =2 =2 =22 =2, in the above argument, the assertion this number is either rational or irrational invokes the law of excluded middle. An intuitionist, for example, would not accept this argument without further support for that statement and this might come in the form of a proof that the number in question is in fact irrational, or a finite algorithm that could determine whether the number is rational. By non-constructive Davis means that a proof that actually are mathematic entities satisfying certain conditions would have to provide a method to exhibit explicitly the entities in question. For example, to prove there exists an n such that P, under both the classical and the intuitionistic logic, by reductio ad absurdum this gives not for all n, not P. The classical logic allows this result to be transformed into there exists an n such that P, indeed, David Hilbert and Luitzen E. J. Brouwer both give examples of the law of excluded middle extended to the infinite
22.
Contraposition
–
In logic, contraposition is an inference that says that a conditional statement is logically equivalent to its contrapositive. The contrapositive of the statement has its antecedent and consequent inverted and flipped, for instance, the proposition All bats are mammals can be restated as the conditional If something is a bat, then it is a mammal. Now, the law says that statement is identical to the contrapositive If something is not a mammal, then it is not a bat. The contrapositive can be compared with three other relationships between conditional statements, Inversion, ¬ P → ¬ Q If something is not a bat, then it is not a mammal. Unlike the contrapositive, the truth value is not at all dependent on whether or not the original proposition was true. The inverse here is not true. Conversion, Q → P If something is a mammal, then it is a bat, the converse is actually the contrapositive of the inverse and so always has the same truth value as the inverse, which is not necessarily the same as that of the original proposition. Negation, ¬ There exists a bat that is not a mammal, If the negation is true, the original proposition is false. Here, of course, the negation is false, note that if P → Q is true and we are given that Q is false, ¬ Q, it can logically be concluded that P must be false, ¬ P. This is often called the law of contrapositive, or the modus tollens rule of inference, according to this diagram, if something is in A, it must be in B as well. So we can all of A is in B as, A → B It is also clear that anything that is not within B cannot be within A. This statement, ¬ B → ¬ A is the contrapositive, therefore, we can say that →. Practically speaking, this may make much easier when trying to prove something. Alternatively, we can try to prove ¬ B → ¬ A by checking all girls without brown hair to see if they are all outside the US. This means that if we find at least one girl without brown hair within the US, we will have disproved ¬ B → ¬ A, to conclude, for any statement where A implies B, then not B always implies not A. Proving or disproving either one of these statements automatically proves or disproves the other. A proposition Q is implicated by a proposition P when the relationship holds, This states that, if P, then Q, or, if Socrates is a man. In a conditional such as this, P is the antecedent, one statement is the contrapositive of the other only when its antecedent is the negated consequent of the other, and vice versa
23.
De Morgan's laws
–
In propositional logic and boolean algebra, De Morgans laws are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician, the rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation. Applications of the rules include simplification of logical expressions in computer programs, De Morgans laws are an example of a more general concept of mathematical duality. The negation of conjunction rule may be written in sequent notation, the negation of disjunction rule may be written as, ¬ ⊢. De Morgans laws are shown in the compact form above, with negation of the output on the left. A clearer form for substitution can be stated as, ≡ ¬, ≡ ¬ and this emphasizes the need to invert both the inputs and the output, as well as change the operator, when doing a substitution. In set notation, De Morgans laws can be remembered using the mnemonic break the line, De Morgan’s laws commonly apply to text searching using Boolean operators AND, OR, and NOT. Consider a set of documents containing the words “cars” and “trucks”, Document 3, Contains both “cars” and “trucks”. Document 4, Contains neither “cars” nor “trucks”, to evaluate Search A, clearly the search “” will hit on Documents 1,2, and 3. So the negation of that search will hit everything else, which is Document 4, evaluating Search B, the search “” will hit on documents that do not contain “cars”, which is Documents 2 and 4. Similarly the search “” will hit on Documents 1 and 4, applying the AND operator to these two searches will hit on the documents that are common to these two searches, which is Document 4. A similar evaluation can be applied to show that the two searches will return the same set of documents, Search C, NOT, Search D. The laws are named after Augustus De Morgan, who introduced a version of the laws to classical propositional logic. De Morgans formulation was influenced by algebraization of logic undertaken by George Boole, nevertheless, a similar observation was made by Aristotle, and was known to Greek and Medieval logicians. For example, in the 14th century, William of Ockham wrote down the words that would result by reading the laws out, jean Buridan, in his Summulae de Dialectica, also describes rules of conversion that follow the lines of De Morgans laws. Still, De Morgan is given credit for stating the laws in the terms of formal logic. De Morgans laws can be proved easily, and may seem trivial. Nonetheless, these laws are helpful in making inferences in proofs
24.
Syllogism
–
A syllogism is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two or more propositions that are asserted or assumed to be true. In its earliest form, defined by Aristotle, from the combination of a statement and a specific statement. For example, knowing that all men are mortal and that Socrates is a man, Syllogistic arguments are usually represented in a three-line form, All men are mortal. In antiquity, two theories of the syllogism existed, Aristotelian syllogistic and Stoic syllogistic. Aristotle defines the syllogism as. a discourse in which certain things having been supposed, despite this very general definition, in Aristotles work Prior Analytics, he limits himself to categorical syllogisms that consist of three categorical propositions. From the Middle Ages onwards, categorical syllogism and syllogism were usually used interchangeably and this article is concerned only with this traditional use. The use of syllogisms as a tool for understanding can be dated back to the logical reasoning discussions of Aristotle, the onset of a New Logic, or logica nova, arose alongside the reappearance of Prior Analytics, the work in which Aristotle develops his theory of the syllogism. Prior Analytics, upon re-discovery, was regarded by logicians as a closed and complete body of doctrine, leaving very little for thinkers of the day to debate. Aristotles theories on the syllogism for assertoric sentences was considered especially remarkable, Aristotles Prior Analytics did not, however, incorporate such a comprehensive theory on the modal syllogism—a syllogism that has at least one modalized premise. Aristotles terminology in this aspect of his theory was deemed vague and in many cases unclear and his original assertions on this specific component of the theory were left up to a considerable amount of conversation, resulting in a wide array of solutions put forth by commentators of the day. The system for modal syllogisms laid forth by Aristotle would ultimately be deemed unfit for practical use, boethius contributed an effort to make the ancient Aristotelian logic more accessible. While his Latin translation of Prior Analytics went primarily unused before the twelfth century and his perspective on syllogisms can be found in other works as well, such as Logica Ingredientibus. With the help of Abelards distinction between de dicto modal sentences and de re modal sentences, medieval logicians began to shape a coherent concept of Aristotles modal syllogism model. For two hundred years after Buridans discussions, little was said about syllogistic logic, the Aristotelian syllogism dominated Western philosophical thought for many centuries. In the 17th century, Sir Francis Bacon rejected the idea of syllogism as being the best way to draw conclusions in nature. Instead, Bacon proposed a more inductive approach to the observation of nature, in the 19th century, modifications to syllogism were incorporated to deal with disjunctive and conditional statements. Kant famously claimed, in Logic, that logic was the one completed science, though there were alternative systems of logic such as Avicennian logic or Indian logic elsewhere, Kants opinion stood unchallenged in the West until 1879 when Frege published his Begriffsschrift. This introduced a calculus, a method of representing categorical statements by the use of quantifiers, in the last 20 years, Bolzanos work has resurfaced and become subject of both translation and contemporary study
25.
Truth value
–
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth. In classical logic, with its intended semantics, the values are true and untrue or false. This set of two values is called the Boolean domain. Corresponding semantics of logical connectives are truth functions, whose values are expressed in the form of truth tables, logical biconditional becomes the equality binary relation, and negation becomes a bijection which permutes true and false. Conjunction and disjunction are dual with respect to negation, which is expressed by De Morgans laws, assigning values for propositional variables is referred to as valuation. In intuitionistic logic, and more generally, constructive mathematics, statements are assigned a value only if they can be given a constructive proof. It starts with a set of axioms, and a statement is true if you can build a proof of the statement from those axioms, a statement is false if you can deduce a contradiction from it. This leaves open the possibility of statements that have not yet assigned a truth value. Unproven statements in Intuitionistic logic are not given a truth value. Indeed, you can prove that they have no truth value. There are various ways of interpreting Intuitionistic logic, including the Brouwer–Heyting–Kolmogorov interpretation, see also, Intuitionistic Logic - Semantics. Multi-valued logics allow for more than two values, possibly containing some internal structure. For example, on the interval such structure is a total order. Not all logical systems are truth-valuational in the sense that logical connectives may be interpreted as truth functions, but even non-truth-valuational logics can associate values with logical formulae, as is done in algebraic semantics. The algebraic semantics of intuitionistic logic is given in terms of Heyting algebras, Intuitionistic type theory uses types in the place of truth values. Topos theory uses truth values in a sense, the truth values of a topos are the global elements of the subobject classifier. Having truth values in this sense does not make a logic truth valuational