1.
Logic
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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.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
3.
Truth
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Truth is most often used to mean being in accord with fact or reality, or fidelity to an original or standard. Truth may also often be used in modern contexts to refer to an idea of truth to self, the commonly understood opposite of truth is falsehood, which, correspondingly, can also take on a logical, factual, or ethical meaning. The concept of truth is discussed and debated in several contexts, including philosophy, art, Some philosophers view the concept of truth as basic, and unable to be explained in any terms that are more easily understood than the concept of truth itself. Commonly, truth is viewed as the correspondence of language or thought to an independent reality, other philosophers take this common meaning to be secondary and derivative. On this view, the conception of truth as correctness is a derivation from the concepts original essence. Various theories and views of truth continue to be debated among scholars, philosophers, language and words are a means by which humans convey information to one another and the method used to determine what is a truth is termed a criterion of truth. The English word truth is derived from Old English tríewþ, tréowþ, trýwþ, Middle English trewþe, cognate to Old High German triuwida, like troth, it is a -th nominalisation of the adjective true. Old Norse trú, faith, word of honour, religious faith, thus, truth involves both the quality of faithfulness, fidelity, loyalty, sincerity, veracity, and that of agreement with fact or reality, in Anglo-Saxon expressed by sōþ. All Germanic languages besides English have introduced a distinction between truth fidelity and truth factuality. To express factuality, North Germanic opted for nouns derived from sanna to assert, affirm, while continental West Germanic opted for continuations of wâra faith, trust, pact. Romance languages use terms following the Latin veritas, while the Greek aletheia, Russian pravda, each presents perspectives that are widely shared by published scholars. However, the theories are not universally accepted. More recently developed deflationary or minimalist theories of truth have emerged as competitors to the substantive theories. Minimalist reasoning centres around the notion that the application of a term like true to a statement does not assert anything significant about it, for instance, anything about its nature. Minimalist reasoning realises truth as a label utilised in general discourse to express agreement, to stress claims, correspondence theories emphasise that true beliefs and true statements correspond to the actual state of affairs. This type of theory stresses a relationship between thoughts or statements on one hand, and things or objects on the other and it is a traditional model tracing its origins to ancient Greek philosophers such as Socrates, Plato, and Aristotle. This class of theories holds that the truth or the falsity of a representation is determined in principle entirely by how it relates to things, Aquinas also restated the theory as, A judgment is said to be true when it conforms to the external reality. Many modern theorists have stated that this ideal cannot be achieved without analysing additional factors, for example, language plays a role in that all languages have words to represent concepts that are virtually undefined in other languages
4.
Logical conjunction
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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
5.
Logical disjunction
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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
6.
Classical logic
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Classical logic is an intensively studied and widely used class of formal logics. Classical logic was devised as a two-level logical system, with simple semantics for the levels representing true. These judgments find themselves if two pairs of two operators, and each operator is the negation of another, relationships that Aristotle summarised with his square of oppositions. George Booles algebraic reformulation of logic, his system of Boolean logic, with the advent of algebraic logic it became apparent that classical propositional calculus admits other semantics. In Boolean-valued semantics, the values are the elements of an arbitrary Boolean algebra, true corresponds to the maximal element of the algebra. Intermediate elements of the algebra correspond to truth values other than true, the principle of bivalence holds only when the Boolean algebra is taken to be the two-element algebra, which has no intermediate elements. Many-valued logic, including logic, which rejects the law of the excluded middle. Graham Priest, An Introduction to Non-Classical Logic, From If to Is, 2nd Edition, CUP,2008, ISBN 978-0-521-67026-5 Warren Goldfard, Deductive Logic, 1st edition,2003, ISBN 0-87220-660-2
7.
Logical truth
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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
8.
False (logic)
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In logic, false or untrue is the state of possessing negative truth value or a nullary logical connective. In a truth-functional system of propositional logic it is one of two postulated truth values, along with its negation, truth, usual notations of the false are 0, O, and the up tack symbol ⊥. Another approach is used for several formal theories where the false is a propositional constant ⊥, Boolean logic defines the false in both senses mentioned above,0 is a propositional constant, whose value by definition is 0. In a classical propositional calculus, depending on the set of fundamental connectives. Such formulas as p ∧ ¬p and ¬ may be used instead, in both systems the negation of the truth gives false. The negation of false is equivalent to the not only in classical logic and Boolean logic. Because p → p is usually a theorem or axiom, a consequence is that the negation of false is true, the contradiction is a statement which entails the false, i. e. φ ⊢ ⊥. Using the equivalence above, the fact that φ is a contradiction may be derived, for example, contradiction and the false are sometimes not distinguished, especially due to Latin term falsum denoting both. Contradiction means a statement is proven to be false, but the false itself is a proposition which is defined to be opposite to the truth, logical systems may or may not contain the principle of explosion, ⊥ ⊢ φ. A formal theory using ⊥ connective is defined to be consistent if, in the absence of propositional constants, some substitutes such as mentioned above may be used instead to define consistency
9.
Two-valued logic
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In logic, the semantic principle of bivalence states that every declarative sentence expressing a proposition has exactly one truth value, either true or false. A logic satisfying this principle is called a two-valued logic or bivalent logic, in formal logic, the principle of bivalence becomes a property that a semantics may or may not possess. It is not the same as the law of excluded middle, however, the principle of bivalence is studied in philosophical logic to address the question of which natural-language statements have a well-defined truth value. Reference failures can also be addressed by free logics, the principle of bivalence is related to the law of excluded middle though the latter is a syntactic expression of the language of a logic of the form P ∨ ¬P. The difference between the principle and the law is important because there are logics which validate the law, for example, the three-valued Logic of Paradox validates the law of excluded middle, but not the law of non-contradiction, ¬, and its intended semantics is not bivalent. In classical two-valued logic both the law of excluded middle and the law of non-contradiction hold, many modern logic programming systems replace the law of the excluded middle with the concept of negation as failure. The programmer may wish to add the law of the middle by explicitly asserting it as true, however. The intended semantics of classical logic is bivalent, but this is not true of every semantics for classical logic. In Boolean-valued semantics, the values are the elements of an arbitrary Boolean algebra, true corresponds to the maximal element of the algebra. Intermediate elements of the algebra correspond to truth values other than true, the principle of bivalence holds only when the Boolean algebra is taken to be the two-element algebra, which has no intermediate elements. All finite Boolean algebras are complete, a famous example is the contingent sea battle case found in Aristotles work, De Interpretatione, chapter 9, Imagine P refers to the statement There will be a sea battle tomorrow. The principle of bivalence here asserts, Either it is true that there will be a sea battle tomorrow, aristotle hesitated to embrace bivalence for such future contingents, Chrysippus, the Stoic logician, did embrace bivalence for this and all other propositions. The controversy continues to be of importance in both the philosophy of time and the philosophy of logic. One of the motivations for the study of many-valued logics has been precisely this issue. In the early 20th century, the Polish formal logician Jan Łukasiewicz proposed three truth-values, the true, the false and the as-yet-undetermined and this approach was later developed by Arend Heyting and L. E. J. Brouwer, see Łukasiewicz logic. Issues such as this have also addressed in various temporal logics. Fuzzy logic and some other multi-valued logics have been proposed as alternatives that handle vague concepts better, truth in fuzzy logic, for example, comes in varying degrees. Consider the following statement in the circumstance of sorting apples on a moving belt, upon observation, the apple is an undetermined color between yellow and red, or it is motled both colors
10.
Logical connective
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The most common logical connectives are binary connectives which join two sentences which can be thought of as the functions operands. Also commonly, negation is considered to be a unary connective, logical connectives along with quantifiers are the two main types of logical constants used in formal systems such as propositional logic and predicate logic. Semantics of a logical connective is often, but not always, a logical connective is similar to but not equivalent to a conditional operator. In the grammar of natural languages two sentences may be joined by a grammatical conjunction to form a compound sentence. Some but not all such grammatical conjunctions are truth functions, for example, consider the following sentences, A, Jack went up the hill. B, Jill went up the hill, C, Jack went up the hill and Jill went up the hill. D, Jack went up the hill so Jill went up the hill, the words and and so are grammatical conjunctions joining the sentences and to form the compound sentences and. The and in is a connective, since the truth of is completely determined by and, it would make no sense to affirm. Various English words and word pairs express logical connectives, and some of them are synonymous, examples are, In formal languages, truth functions are represented by unambiguous symbols. These symbols are called logical connectives, logical operators, propositional operators, or, in classical logic, see well-formed formula for the rules which allow new well-formed formulas to be constructed by joining other well-formed formulas using truth-functional connectives. Logical connectives can be used to more than two statements, so one can speak about n-ary logical connective. For example, the meaning of the statements it is raining, comes from Booles interpretation of logic as an elementary algebra. True, the symbol 1 comes from Booles interpretation of logic as an algebra over the two-element Boolean algebra. False, the symbol 0 comes also from Booles interpretation of logic as a ring, some authors used letters for connectives at some time of the history, u. for conjunction and o. Such a logical connective as converse implication ← is actually the same as material conditional with swapped arguments, thus, in some logical calculi certain essentially different compound statements are logically equivalent. A less trivial example of a redundancy is the equivalence between ¬P ∨ Q and P → Q. There are sixteen Boolean functions associating the input truth values P and Q with four-digit binary outputs and these correspond to possible choices of binary logical connectives for classical logic. Different implementations of classical logic can choose different functionally complete subsets of connectives, One approach is to choose a minimal set, and define other connectives by some logical form, as in the example with the material conditional above
11.
Truth function
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In logic, a truth function is a function that accepts truth values as input and produces a truth value as output, i. e. the input and output are all truth values. On the contrary, modal logic is non-truth-functional, a logical connective is truth-functional if the truth-value of a compound sentence is a function of the truth-value of its sub-sentences. A class of connectives is truth-functional if each of its members is, some connectives of a natural language, such as English, are not truth-functional. Connectives of the form x believes that, are typical examples of connectives that are not truth-functional. In both cases, each component sentence is false, but each compound sentence formed by prefixing the phrase Mary believes that differs in truth-value and that is, the truth-value of a sentence of the form Mary believes that. Is not determined solely by the truth-value of its component sentence, the class of classical logic connectives used in the construction of formulas is truth-functional. Their values for various truth-values as argument are usually given by truth tables, truth-functional propositional calculus is a formal system whose formulae may be interpreted as either true or false. In two-valued logic, there are sixteen possible truth functions, also called Boolean functions, truth and falsehood is denoted as 1 and 0 in the following truth tables, respectively, for sake of brevity. Because a function may be expressed as a composition, a logical calculus does not need to have dedicated symbols for all of the above-mentioned functions to be functionally complete. This is expressed in a propositional calculus as logical equivalence of certain compound statements, for example, classical logic has ¬P ∨ Q equivalent to P → Q. The conditional operator → is therefore not necessary for a logical system if ¬ and ∨ are already in use. A minimal set of operators that can express every statement expressible in the propositional calculus is called a minimal functionally complete set, a minimally complete set of operators is achieved by NAND alone and NOR alone. The following are the minimal functionally complete sets of operators whose arities do not exceed 2, some truth functions possess properties which may be expressed in the theorems containing the corresponding connective. Commutativity, The operands of the connective may be swapped without affecting the truth-value of the expression, distributivity, A connective denoted by · distributes over another connective denoted by +, if a · = + for all operands a, b, c. Idempotence, Whenever the operands of the operation are the same, in other words, the operation is both truth-preserving and falsehood-preserving. Absorption, A pair of connectives ∧, ∨ satisfies the law if a ∧ = a for all operands a, b. A set of functions is functionally complete if and only if for each of the following five properties it contains at least one member lacking it, monotonic. Bn ∈ such that a1 ≤ b1, a2 ≤ b2, affine, Each variable always makes a difference in the truth-value of the operation or it never makes a difference
12.
Truth table
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In particular, truth tables can be used to show whether a propositional expression is true for all legitimate input values, that is, logically valid. A truth table has one column for each variable. Each row of the table contains one possible configuration of the input variables. See the examples below for further clarification, the Com row indicates whether an operator, op, is commutative - P op Q = Q op P. The L id row shows the operators left identities if it has any - values I such that I op Q = Q, the R id row shows the operators right identities if it has any - values I such that P op I = P. The four combinations of values for p, q, are read by row from the table above. The output function for p, q combination, can be read, by row. Key, The following table is oriented by column, rather than by row, There are four columns rather than four rows, to display the four combinations of p, q, as input. P, T T F Fq, T F T F There are 16 rows in this key, the output row for ↚ is thus 2, F F T F and the 16-row key is Logical operators can also be visualized using Venn diagrams. Logical conjunction is an operation on two values, typically the values of two propositions, that produces a value of true if both of its operands are true. The truth table for p AND q is as follows, In ordinary language terms, for all other assignments of logical values to p and to q the conjunction p ∧ q is false. The truth table for p OR q is as follows, Stated in English, if p, then p ∨ q is p, otherwise p ∨ q is q. The truth table associated with the conditional if p then q. Logical equality is an operation on two values, typically the values of two propositions, that produces a value of true if both operands are false or both operands are true. The truth table for p XNOR q is as follows, So p EQ q is true if p and q have the truth value. Exclusive disjunction is an operation on two values, typically the values of two propositions, that produces a value of true if one but not both of its operands is true. The truth table for p XOR q is as follows, For two propositions, XOR can also be written as ∨. The logical NAND is an operation on two values, typically the values of two propositions, that produces a value of false if both of its operands are true
13.
Logical biconditional
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In logic and mathematics, the logical biconditional is the logical connective of two statements asserting p if and only if q, where p is an antecedent and q is a consequent. This is often abbreviated p iff q, the operator is denoted using a doubleheaded arrow, a prefixed E, an equality sign, an equivalence sign, or EQV. It is logically equivalent to ∧, or the XNOR boolean operator and it is also logically equivalent to or, meaning both or neither. The only difference from material conditional is the case when the hypothesis is false, in that case, in the conditional, the result is true, yet in the biconditional the result is false. In the conceptual interpretation, a = b means All a s are b s and all b s are a s, in other words and this does not mean that the concepts have the same meaning. Examples, triangle and trilateral, equiangular trilateral and equilateral triangle, the antecedent is the subject and the consequent is the predicate of a universal affirmative proposition. In the propositional interpretation, a ⇔ b means that a b and b implies a, in other words, that the propositions are equivalent. This does not mean that they have the same meaning, example, The triangle ABC has two equal sides, and The triangle ABC has two equal angles. The antecedent is the premise or the cause and the consequent is the consequence, when an implication is translated by a hypothetical judgment the antecedent is called the hypothesis and the consequent is called the thesis. A common way of demonstrating a biconditional is to use its equivalence to the conjunction of two converse conditionals, demonstrating these separately. When both members of the biconditional are propositions, it can be separated into two conditionals, of one is called a theorem and the other its reciprocal. Thus whenever a theorem and its reciprocal are true we have a biconditional, a simple theorem gives rise to an implication whose antecedent is the hypothesis and whose consequent is the thesis of the theorem. When a theorem and its reciprocal are true we say that its hypothesis is the necessary and sufficient condition of the thesis, that is to say, that it is at the same time both cause and consequence. Logical equality is an operation on two values, typically the values of two propositions, that produces a value of true if and only if both operands are false or both operands are true. The truth table for A ↔ B is as follows, More than two statements combined by ↔ are ambiguous, x 1 ↔ x 2 ↔ x 3 ↔. ↔ x n may be meant as ↔ x n, or may be used to say that all x i are together true or together false, commutativity, yes associativity, yes distributivity, Biconditional doesnt distribute over any binary function, but logical disjunction distributes over biconditional. Idempotency, no monotonicity, no truth-preserving, yes When all inputs are true, falsehood-preserving, no When all inputs are false, the output is not false. Walsh spectrum, Nonlinearity,0 Like all connectives in first-order logic, Biconditional introduction allows you to infer that, if B follows from A, and A follows from B, then A if and only if B
14.
Negation
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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
15.
Bijection
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In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments and images are related or mapped to each other. A function maps elements from its domain to elements in its codomain, given a function f, X → Y The function is injective if every element of the codomain is mapped to by at most one element of the domain. An injective function is an injection, notationally, ∀ x, x ′ ∈ X, f = f ⇒ x = x ′. Or, equivalently, ∀ x, x ′ ∈ X, x ≠ x ′ ⇒ f ≠ f, the function is surjective if every element of the codomain is mapped to by at least one element of the domain. A surjective function is a surjection, notationally, ∀ y ∈ Y, ∃ x ∈ X such that y = f. The function is bijective if every element of the codomain is mapped to by one element of the domain. A bijective function is a bijection, an injective function need not be surjective, and a surjective function need not be injective. The four possible combinations of injective and surjective features are illustrated in the diagrams to the right, a function is injective if every possible element of the codomain is mapped to by at most one argument. Equivalently, a function is injective if it maps distinct arguments to distinct images, an injective function is an injection. The formal definition is the following, the function f, X → Y is injective iff for all x, x ′ ∈ X, we have f = f ⇒ x = x ′. A function f, X → Y is injective if and only if X is empty or f is left-invertible, here f is the image of f. Since every function is surjective when its codomain is restricted to its image, more precisely, every injection f, X → Y can be factored as a bijection followed by an inclusion as follows. Let fR, X → f be f with codomain restricted to its image, a dual factorisation is given for surjections below. The composition of two injections is again an injection, but if g o f is injective, then it can only be concluded that f is injective, a function is surjective if every possible image is mapped to by at least one argument. In other words, every element in the codomain has non-empty preimage, equivalently, a function is surjective if its image is equal to its codomain. A surjective function is a surjection, the formal definition is the following. The function f, X → Y is surjective iff for all y ∈ Y, there is x ∈ X such that f = y. A function f, X → Y is surjective if and only if it is right-invertible, by collapsing all arguments mapping to a given fixed image, every surjection induces a bijection defined on a quotient of its domain
16.
Permutation
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These differ from combinations, which are selections of some members of a set where order is disregarded. For example, written as tuples, there are six permutations of the set, namely and these are all the possible orderings of this three element set. As another example, an anagram of a word, all of whose letters are different, is a permutation of its letters, in this example, the letters are already ordered in the original word and the anagram is a reordering of the letters. The study of permutations of finite sets is a topic in the field of combinatorics, Permutations occur, in more or less prominent ways, in almost every area of mathematics. For similar reasons permutations arise in the study of sorting algorithms in computer science, the number of permutations of n distinct objects is n factorial, usually written as n. which means the product of all positive integers less than or equal to n. In algebra and particularly in group theory, a permutation of a set S is defined as a bijection from S to itself and that is, it is a function from S to S for which every element occurs exactly once as an image value. This is related to the rearrangement of the elements of S in which each element s is replaced by the corresponding f, the collection of such permutations form a group called the symmetric group of S. The key to this structure is the fact that the composition of two permutations results in another rearrangement. Permutations may act on structured objects by rearranging their components, or by certain replacements of symbols, in elementary combinatorics, the k-permutations, or partial permutations, are the ordered arrangements of k distinct elements selected from a set. When k is equal to the size of the set, these are the permutations of the set, fabian Stedman in 1677 described factorials when explaining the number of permutations of bells in change ringing. Starting from two bells, first, two must be admitted to be varied in two ways which he illustrates by showing 12 and 21 and he then explains that with three bells there are three times two figures to be produced out of three which again is illustrated. His explanation involves cast away 3, and 1.2 will remain, cast away 2, and 1.3 will remain, cast away 1, and 2.3 will remain. He then moves on to four bells and repeats the casting away argument showing that there will be four different sets of three, effectively this is an recursive process. He continues with five bells using the casting method and tabulates the resulting 120 combinations. At this point he gives up and remarks, Now the nature of these methods is such, in modern mathematics there are many similar situations in which understanding a problem requires studying certain permutations related to it. There are two equivalent common ways of regarding permutations, sometimes called the active and passive forms, or in older terminology substitutions and permutations, which form is preferable depends on the type of questions being asked in a given discipline. The active way to regard permutations of a set S is to them as the bijections from S to itself. Thus, the permutations are thought of as functions which can be composed with each other, forming groups of permutations
17.
Dual (mathematics)
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Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues theorem is self-dual in this sense under the standard duality in projective geometry, many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and another object of the second type to some family of scalars. From a category theory viewpoint, duality can also be seen as a functor and this functor assigns to each space its dual space, and the pullback construction assigns to each arrow f, V → W its dual f∗, W∗ → V∗. In the words of Michael Atiyah, Duality in mathematics is not a theorem, the following list of examples shows the common features of many dualities, but also indicates that the precise meaning of duality may vary from case to case. A simple, maybe the most simple, duality arises from considering subsets of a fixed set S, to any subset A ⊆ S, the complement Ac consists of all those elements in S which are not contained in A. It is again a subset of S, taking the complement has the following properties, Applying it twice gives back the original set, i. e. c = A. This is referred to by saying that the operation of taking the complement is an involution, an inclusion of sets A ⊆ B is turned into an inclusion in the opposite direction Bc ⊆ Ac. Given two subsets A and B of S, A is contained in Bc if and only if B is contained in Ac. This duality appears in topology as a duality between open and closed subsets of some fixed topological space X, a subset U of X is closed if, because of this, many theorems about closed sets are dual to theorems about open sets. For example, any union of sets is open, so dually. The interior of a set is the largest open set contained in it, because of the duality, the complement of the interior of any set U is equal to the closure of the complement of U. A duality in geometry is provided by the cone construction. Given a set C of points in the plane R2, unlike for the complement of sets mentioned above, it is not in general true that applying the dual cone construction twice gives back the original set C. Instead, C ∗ ∗ is the smallest cone containing C which may be bigger than C. Therefore this duality is weaker than the one above, in that Applying the operation twice gives back a possibly bigger set, the other two properties carry over without change, It is still true that an inclusion C ⊆ D is turned into an inclusion in the opposite direction. Given two subsets C and D of the plane, C is contained in D ∗ if, a very important example of a duality arises in linear algebra by associating to any vector space V its dual vector space V*. Its elements are the k-linear maps φ, V → k, the three properties of the dual cone carry over to this type of duality by replacing subsets of R2 by vector space and inclusions of such subsets by linear maps. That is, Applying the operation of taking the dual vector space twice gives another vector space V**, there is always a map V → V**
18.
De Morgan's laws
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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
19.
Intuitionistic logic
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In particular, systems of intuitionistic logic do not include the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic. Formalized intuitionistic logic was developed by Arend Heyting to provide a formal basis for Brouwers programme of intuitionism. From a proof-theoretic perspective, Heyting’s calculus is a restriction of classical logic in which the law of excluded middle and double negation elimination have been removed. Excluded middle and double negation elimination can still be proved for some propositions on a case by case basis, however, several systems of semantics for intuitionistic logic have been studied. One semantics mirrors classical Boolean-valued semantics but uses Heyting algebras in place of Boolean algebras and these, however, are technical means for studying Heyting’s deductive system rather than formalizations of Brouwer’s original informal semantic intuitions. Yet such semantics persistently induce logics properly stronger than Heyting’s logic, some authors have argued that this might be an indication of inadequacy of Heyting’s calculus itself, deeming the latter incomplete as a constructive logic. In the semantics of classical logic, propositional formulae are assigned values from the two-element set. This is referred to as the law of excluded middle, because it excludes the possibility of any truth value besides true or false. In contrast, propositional formulae in intuitionistic logic are not assigned a truth value and are only considered true when we have direct evidence, hence proof. Operations in intuitionistic logic therefore preserve justification, with respect to evidence and provability, Intuitionistic logic is one of the set of approaches of constructivism in mathematics. The use of constructivist logics in general has been a topic among mathematicians. A very common objection to their use is the lack of two central rules of classical logic, the law of excluded middle and double negation elimination. To prohibit existence statements and the principle of excluded middle is tantamount to relinquishing the science of mathematics altogether, despite the serious challenges presented by the inability to utilize the valuable rules of excluded middle and double negation elimination, intuitionistic logic has practical use. One reason for this is that its restrictions produce proofs that have the existence property, one reason that this particular aspect of intuitionistic logic is so valuable is that it enables practitioners to utilize a wide range of computerized tools, known as proof assistants. These tools assist their users in the verification of large-scale proofs, one example of a proof which was impossible to formally verify before the advent of these tools is the famous proof of the four color theorem. That proof was controversial for some time, but it was verified using Coq. The syntax of formulas of intuitionistic logic is similar to propositional logic or first-order logic, however, intuitionistic connectives are not definable in terms of each other in the same way as in classical logic, hence their choice matters. In intuitionistic propositional logic it is customary to use →, ∧, ∨, ⊥ as the basic connectives, in intuitionistic first-order logic both quantifiers ∃, ∀ are needed
20.
Fuzzy logic
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Fuzzy logic is a form of many-valued logic in which the truth values of variables may be any real number between 0 and 1. By contrast, in Boolean logic, the values of variables may only be the integer values 0 or 1. Fuzzy logic has been employed to handle the concept of partial truth, furthermore, when linguistic variables are used, these degrees may be managed by specific functions. The term fuzzy logic was introduced with the 1965 proposal of set theory by Lotfi Zadeh. Fuzzy logic had however been studied since the 1920s, as infinite-valued logic—notably by Łukasiewicz, Fuzzy logic has been applied to many fields, from control theory to artificial intelligence. Classical logic only permits conclusions which are true or false. However, there are also propositions with variable answers, such as one might find when asking a group of people to identify a color. In such instances, the truth appears as the result of reasoning from inexact or partial knowledge in which the answers are mapped on a spectrum. Humans and animals often operate using fuzzy evaluations in many everyday situations, instead the person instinctively applies quick fuzzy estimates, based upon previous experience, to determine what output values of force, direction and vertical angle to use to make the toss. Take, for example, the concepts of empty and full, the meaning of each of them can be represented by a certain fuzzy set. The concept of emptiness would be subjective and thus would depend on the observer or designer, a basic application might characterize various sub-ranges of a continuous variable. For instance, a measurement for anti-lock brakes might have several separate membership functions defining particular temperature ranges needed to control the brakes properly. Each function maps the temperature value to a truth value in the 0 to 1 range. These truth values can then be used to determine how the brakes should be controlled, in this image, the meanings of the expressions cold, warm, and hot are represented by functions mapping a temperature scale. A point on that scale has three truth values—one for each of the three functions, the vertical line in the image represents a particular temperature that the three arrows gauge. Since the red arrow points to zero, this temperature may be interpreted as not hot, the orange arrow may describe it as slightly warm and the blue arrow fairly cold. While variables in mathematics usually take numerical values, in fuzzy logic applications non-numeric values are used to facilitate the expression of rules. A linguistic variable such as age may accept values such as young, because natural languages do not always contain enough value terms to express a fuzzy value scale, it is common practice to modify linguistic values with adjectives or adverbs
21.
Unit interval
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In mathematics, the unit interval is the closed interval, that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. In addition to its role in analysis, the unit interval is used to study homotopy theory in the field of topology. In the literature, the unit interval is sometimes applied to the other shapes that an interval from 0 to 1 could take. However, the notation I is most commonly reserved for the closed interval, the unit interval is a complete metric space, homeomorphic to the extended real number line. As a topological space it is compact, contractible, path connected, the Hilbert cube is obtained by taking a topological product of countably many copies of the unit interval. In mathematical analysis, the interval is a one-dimensional analytical manifold whose boundary consists of the two points 0 and 1. Its standard orientation goes from 0 to 1, the unit interval is a totally ordered set and a complete lattice. The size or cardinality of a set is the number of elements it contains, the unit interval is a subset of the real numbers R. However, it has the same size as the whole set, the cardinality of the continuum. Moreover, it has the number of points as a square of area 1, as a cube of volume 1. The number of elements in all the sets is uncountable. The interval, with two, demarcated by the positive and negative units, occurs frequently, such as in the range of the trigonometric functions sine and cosine. This interval may be used for the domain of inverse functions, for instance, when θ is restricted to then sin is in this interval and arcsine is defined there. Sometimes, the unit interval is used to refer to objects that play a role in various branches of mathematics analogous to the role that plays in homotopy theory. For example, in the theory of quivers, the interval is the graph whose vertex set is. One can then define a notion of homotopy between quiver homomorphisms analogous to the notion of homotopy between continuous maps. In logic, the interval can be interpreted as a generalization of the Boolean domain, in which case rather than only taking values 0 or 1. Algebraically, negation is replaced with 1 − x, conjunction is replaced with multiplication, interpreting these values as logical truth values yields a multi-valued logic, which forms the basis for fuzzy logic and probabilistic logic. In these interpretations, a value is interpreted as the degree of truth – to what extent a proposition is true, or the probability that the proposition is true
22.
Necessarily true
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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
23.
Heyting algebra
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In mathematics, a Heyting algebra is a bounded lattice equipped with a binary operation a → b of implication such that c ∧ a ≤ b is equivalent to c ≤ a → b. From a logical standpoint, A → B is by definition the weakest proposition for which modus ponens. Equivalently a Heyting algebra is a lattice whose monoid operation a⋅b is a ∧ b. Like Boolean algebras, Heyting algebras form a variety axiomatizable with finitely many equations, Heyting algebras were introduced by Arend Heyting to formalize intuitionistic logic. As lattices, Heyting algebras are distributive, the open sets of a topological space form such a lattice, and therefore a Heyting algebra. In the finite case every nonempty distributive lattice, in particular every nonempty finite chain, is complete and completely distributive. It follows from the definition that 1 ≤0 → a, although the negation operation ¬a is not part of the definition, it is definable as a →0. The definition implies that a ∧ ¬a =0, making the content of ¬a the proposition that to assume a would lead to a contradiction. It can further be shown that a ≤ ¬¬a, although the converse, ¬¬a ≤ a, is not true in general, that is, double negation does not hold in general in a Heyting algebra. Heyting algebras generalize Boolean algebras in the sense that a Heyting algebra satisfying a ∨ ¬a =1, complete Heyting algebras are a central object of study in pointless topology. The internal logic of a topos is based on the Heyting algebra of subobjects of the terminal object 1 ordered by inclusion. Every Heyting algebra whose set of non-greatest elements has a greatest element is subdirectly irreducible and it follows that even among the finite Heyting algebras there exist infinitely many that are subdirectly irreducible, no two of which have the same equational theory. Hence no finite set of finite Heyting algebras can supply all the counterexamples to non-laws of Heyting algebra, nevertheless, it is decidable whether an equation holds of all Heyting algebras. Heyting algebras are often called pseudo-Boolean algebras, or even Brouwer lattices, although the latter term may denote the dual definition. A Heyting algebra H is a lattice such that for all a and b in H there is a greatest element x of H such that a ∧ x ≤ b. This element is the relative pseudo-complement of a with respect to b and we write 1 and 0 for the largest and the smallest element of H, respectively. In any Heyting algebra, one defines the pseudo-complement ¬a of any element a by setting ¬a =, by definition, a ∧ ¬ a =0, and ¬a is the largest element having this property. However, it is not in general true that a ∨ ¬ a =1, thus ¬ is only a pseudo-complement, not a true complement, a complete Heyting algebra is a Heyting algebra that is a complete lattice
24.
Boolean algebra (structure)
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In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets and it is also a special case of a De Morgan algebra and a Kleene algebra. The term Boolean algebra honors George Boole, a self-educated English mathematician, booles formulation differs from that described above in some important respects. For example, conjunction and disjunction in Boole were not a pair of operations. Boolean algebra emerged in the 1860s, in papers written by William Jevons, the first systematic presentation of Boolean algebra and distributive lattices is owed to the 1890 Vorlesungen of Ernst Schröder. The first extensive treatment of Boolean algebra in English is A. N. Whiteheads 1898 Universal Algebra, Boolean algebra as an axiomatic algebraic structure in the modern axiomatic sense begins with a 1904 paper by Edward V. Huntington. Boolean algebra came of age as serious mathematics with the work of Marshall Stone in the 1930s, and with Garrett Birkhoffs 1940 Lattice Theory. In the 1960s, Paul Cohen, Dana Scott, and others found deep new results in mathematical logic and axiomatic set theory using offshoots of Boolean algebra, namely forcing, a Boolean algebra with only one element is called a trivial Boolean algebra or a degenerate Boolean algebra. It follows from the last three pairs of axioms above, or from the axiom, that a = b ∧ a if. The relation ≤ defined by a ≤ b if these equivalent conditions hold, is an order with least element 0. The meet a ∧ b and the join a ∨ b of two elements coincide with their infimum and supremum, respectively, with respect to ≤, the first four pairs of axioms constitute a definition of a bounded lattice. It follows from the first five pairs of axioms that any complement is unique, the set of axioms is self-dual in the sense that if one exchanges ∨ with ∧ and 0 with 1 in an axiom, the result is again an axiom. Therefore, by applying this operation to a Boolean algebra, one obtains another Boolean algebra with the same elements, furthermore, every possible input-output behavior can be modeled by a suitable Boolean expression. The smallest element 0 is the empty set and the largest element 1 is the set S itself, starting with the propositional calculus with κ sentence symbols, form the Lindenbaum algebra. This construction yields a Boolean algebra and it is in fact the free Boolean algebra on κ generators. A truth assignment in propositional calculus is then a Boolean algebra homomorphism from this algebra to the two-element Boolean algebra, interval algebras are useful in the study of Lindenbaum-Tarski algebras, every countable Boolean algebra is isomorphic to an interval algebra. For any natural n, the set of all positive divisors of n, defining a≤b if a divides b
25.
Topos
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In mathematics, a topos is a category that behaves like the category of sheaves of sets on a topological space. Topoi behave much like the category of sets and possess a notion of localization, the Grothendieck topoi find applications in algebraic geometry, the more general elementary topoi are used in logic. Since the introduction of sheaves into mathematics in the 1940s a major theme has been to study a space by studying sheaves on a space and this idea was expounded by Alexander Grothendieck by introducing the notion of a topos. An important example of this idea to date is the étale topos of a scheme. A theorem of Giraud states that the following are equivalent, There is a small category D, C is the category of sheaves on a Grothendieck site. A category with these properties is called a topos, here Presh denotes the category of contravariant functors from D to the category of sets, such a contravariant functor is frequently called a presheaf. Girauds axioms for a category C are, C has a set of generators. Furthermore, colimits commute with fiber products, in other words, the fiber product of X and Y over their sum is the initial object in C. All equivalence relations in C are effective, the last axiom needs the most explanation. If X is an object of C, an equivalence relation R on X is a map R→X×X in C such that for any object Y in C, since C has colimits we may form the coequalizer of the two maps R→X, call this X/R. The equivalence relation is effective if the canonical map R → X × X / R X is an isomorphism, girauds theorem already gives sheaves on sites as a complete list of examples. Note, however, that nonequivalent sites often give rise to equivalent topoi, as indicated in the introduction, sheaves on ordinary topological spaces motivate many of the basic definitions and results of topos theory. The category of sets is an important special case, it plays the role of a point in topos theory, indeed, a set may be thought of as a sheaf on a point. More exotic examples, and the raison dêtre of topos theory, to a scheme and even a stack one may associate an étale topos, an fppf topos, a Nisnevich topos. Topos theory is, in sense, a generalization of classical point-set topology. One should therefore expect to see old and new instances of pathological behavior, for instance, there is an example due to Pierre Deligne of a nontrivial topos that has no points. If X and Y are topoi, a morphism u. Note that u∗ automatically preserves colimits by virtue of having a right adjoint, by Freyds adjoint functor theorem, to give a geometric morphism X → Y is to give a functor u∗, Y → X that preserves finite limits and all small colimits
26.
Subobject classifier
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In category theory, a subobject classifier is a special object Ω of a category, intuitively, the subobjects of an object X correspond to the morphisms from X to Ω. As the name suggests, what a subobject classifier does is to identify/classify subobjects of an object according to which elements belong to the subobject in question. Because of this role, the classifier is also referred to as the truth value object. This is why the subobject classifier is used in the categorical description of logic. Every function from X to Ω arises in this fashion from precisely one subset A, to be clearer, consider a subset A of S, where S is a set. In facts, χA is true precisely on the elements of A, in this way, the collection of all subsets of S, denoted by P, and the collection of all maps from S to Ω =, denoted by ΩS, are isomorphic. To categorize this notion, recall that, in theory, a subobject is actually a pair consisting of an object. Accordingly, true refers to the object 1 and the arrow, true, the subset A of S can now be defined as the pullback of true and the characteristic function χA, shown on the following diagram, Defined that way, χ is morphism SubC → HomC. By definition, Ω is a subobject classifier if this morphism is an isomorphism, for the general definition, we start with a category C that has a terminal object, which we denote by 1. Every elementary topos, defined as a category with finite limits and power objects, necessarily has a subobject classifier. For the topos of sheaves of sets on a topological space X, it can be described in these terms, For any open set U of X, Ω is the set of all open subsets of U. Roughly speaking an assertion inside this topos is variably true or false, a quasitopos has an object that is almost a subobject classifier, it only classifies strong subobjects. For a small category C, the subobject classifer in the topos of presheaves S e t C o p is given as follows, for any c ∈ C, Ω is the set of sieves on c. Artin, Michael, Alexander Grothendieck, Jean-Louis Verdier, toposes and Local Set Theories, an Introduction. Topoi, The Categorial Analysis of Logic, north-Holland, Reprinted by Dover Publications, Inc. Sketches of an Elephant, A Topos Theory Compendium, sheaves in Geometry and Logic, a First Introduction to Topos Theory. Pedicchio, Maria Cristina, Tholen, Walter, eds, special topics in order, topology, algebra, and sheaf theory. Encyclopedia of Mathematics and Its Applications
27.
Agnosticism
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Agnosticism is the philosophical view that the existence of God or the supernatural are unknown and unknowable. Agnosticism is a doctrine or set of rather than a religion. English biologist Thomas Henry Huxley coined the word agnostic in 1869, the Nasadiya Sukta in the Rigveda is agnostic about the origin of the universe. Agnosticism is of the essence of science, whether ancient or modern and it simply means that a man shall not say he knows or believes that which he has no scientific grounds for professing to know or believe. Consequently, agnosticism puts aside not only the part of popular theology. On the whole, the bosh of heterodoxy is more offensive to me than that of orthodoxy, because heterodoxy professes to be guided by reason and science, and orthodoxy does not. Agnosticism, in fact, is not a creed, but a method, positively the principle may be expressed, In matters of the intellect, follow your reason as far as it will take you, without regard to any other consideration. And negatively, In matters of the intellect do not pretend that conclusions are certain which are not demonstrated or demonstrable, being a scientist, above all else, Huxley presented agnosticism as a form of demarcation. A hypothesis with no supporting objective, testable evidence is not an objective, as such, there would be no way to test said hypotheses, leaving the results inconclusive. His agnosticism was not compatible with forming a belief as to the truth, or falsehood, karl Popper would also describe himself as an agnostic. Others have redefined this concept, making it compatible with forming a belief, george H. Smith rejects agnosticism as a third alternative to theism and atheism and promotes terms such as agnostic atheism and agnostic theism. Agnostic was used by Thomas Henry Huxley in a speech at a meeting of the Metaphysical Society in 1869 to describe his philosophy, early Christian church leaders used the Greek word gnosis to describe spiritual knowledge. Agnosticism is not to be confused with religious views opposing the ancient religious movement of Gnosticism in particular, Huxley used the term in a broader, Huxley identified agnosticism not as a creed but rather as a method of skeptical, evidence-based inquiry. In recent years, scientific literature dealing with neuroscience and psychology has used the word to mean not knowable, in technical and marketing literature, agnostic can also mean independence from some parameters—for example, platform agnostic or hardware agnostic. Scottish Enlightenment philosopher David Hume contended that meaningful statements about the universe are always qualified by some degree of doubt and he asserted that the fallibility of human beings means that they cannot obtain absolute certainty except in trivial cases where a statement is true by definition. A strong agnostic would say, I cannot know whether a deity exists or not, a weak agnostic would say, I dont know whether any deities exist or not, but maybe one day, if there is evidence, we can find something out. Therefore, their existence has little to no impact on human affairs. Agnostic thought, in the form of skepticism, emerged as a philosophical position in ancient Greece
28.
History of logic
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The history of logic deals with the study of the development of the science of valid inference. Formal logics developed in ancient times in China, India, Greek methods, particularly Aristotelian logic as found in the Organon, found wide application and acceptance in Western science and mathematics for millennia. The Stoics, especially Chrysippus, began the development of predicate logic, christian and Islamic philosophers such as Boethius and William of Ockham further developed Aristotles logic in the Middle Ages, reaching a high point in the mid-fourteenth century. The period between the fourteenth century and the beginning of the century saw largely decline and neglect. Empirical methods ruled the day, as evidenced by Sir Francis Bacons Novum Organon of 1620, valid reasoning has been employed in all periods of human history. However, logic studies the principles of reasoning, inference. It is probable that the idea of demonstrating a conclusion first arose in connection with geometry, the ancient Egyptians discovered geometry, including the formula for the volume of a truncated pyramid. Ancient Babylon was also skilled in mathematics, while the ancient Egyptians empirically discovered some truths of geometry, the great achievement of the ancient Greeks was to replace empirical methods by demonstrative proof. Both Thales and Pythagoras of the Pre-Socratic philosophers seem aware of geometrys methods, fragments of early proofs are preserved in the works of Plato and Aristotle, and the idea of a deductive system was probably known in the Pythagorean school and the Platonic Academy. The proofs of Euclid of Alexandria are a paradigm of Greek geometry, the three basic principles of geometry are as follows, Certain propositions must be accepted as true without demonstration, such a proposition is known as an axiom of geometry. Every proposition that is not an axiom of geometry must be demonstrated as following from the axioms of geometry, the proof must be formal, that is, the derivation of the proposition must be independent of the particular subject matter in question. Further evidence that early Greek thinkers were concerned with the principles of reasoning is found in the fragment called dissoi logoi and this is part of a protracted debate about truth and falsity. Thales was said to have had a sacrifice in celebration of discovering Thales Theorem just as Pythagoras had the Pythagorean Theorem, Indian and Babylonian mathematicians knew his theorem for special cases before he proved it. It is believed that Thales learned that an angle inscribed in a semicircle is a right angle during his travels to Babylon, before 520 BC, on one of his visits to Egypt or Greece, Pythagoras might have met the c.54 years older Thales. The systematic study of proof seems to have begun with the school of Pythagoras in the sixth century BC. Indeed, the Pythagoreans, believing all was number, are the first philosophers to emphasize rather than matter. He is known for his obscure sayings and this logos holds always but humans always prove unable to understand it, both before hearing it and when they have first heard it. But other people fail to notice what they do when awake, in contrast to Heraclitus, Parmenides held that all is one and nothing changes
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Paradox
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A paradox is a statement that, despite apparently sound reasoning from true premises, leads to a self-contradictory or a logically unacceptable conclusion. A paradox involves contradictory yet interrelated elements that exist simultaneously and persist over time, some logical paradoxes are known to be invalid arguments but are still valuable in promoting critical thinking. Some paradoxes have revealed errors in definitions assumed to be rigorous, others, such as Currys paradox, are not yet resolved. Examples outside logic include the Ship of Theseus from philosophy, paradoxes can also take the form of images or other media. Escher featured perspective-based paradoxes in many of his drawings, with walls that are regarded as floors from other points of view, and staircases that appear to climb endlessly. In common usage, the word often refers to statements that may be both true and false i. e. ironic or unexpected, such as the paradox that standing is more tiring than walking. Common themes in paradoxes include self-reference, infinite regress, circular definitions, patrick Hughes outlines three laws of the paradox, Self-reference An example is This statement is false, a form of the liar paradox. The statement is referring to itself, another example of self-reference is the question of whether the barber shaves himself in the barber paradox. One more example would be Is the answer to this question No, contradiction This statement is false, the statement cannot be false and true at the same time. Another example of contradiction is if a man talking to a genie wishes that wishes couldnt come true, vicious circularity, or infinite regress This statement is false, if the statement is true, then the statement is false, thereby making the statement true. Another example of vicious circularity is the group of statements. Other paradoxes involve false statements or half-truths and the biased assumptions. This form is common in howlers, for example, consider a situation in which a father and his son are driving down the road. The car crashes into a tree and the father is killed, the boy is rushed to the nearest hospital where he is prepared for emergency surgery. On entering the suite, the surgeon says, I cant operate on this boy. The apparent paradox is caused by a hasty generalization, for if the surgeon is the boys father, the paradox is resolved if it is revealed that the surgeon is a woman — the boys mother. Paradoxes which are not based on a hidden error generally occur at the fringes of context or language, paradoxes that arise from apparently intelligible uses of language are often of interest to logicians and philosophers. Russells paradox, which shows that the notion of the set of all sets that do not contain themselves leads to a contradiction, was instrumental in the development of modern logic
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Edward N. Zalta
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Edward N. Zalta is a Senior research scholar at the Center for the Study of Language and Information. He received his Ph. D. in philosophy from the University of Massachusetts Amherst in 1980, Zalta has taught courses at Stanford University, Rice University, the University of Salzburg, and the University of Auckland. Zalta is also the Principal Editor of the Stanford Encyclopedia of Philosophy, Zaltas most notable philosophical position is descended from the position of Alexius Meinong and Ernst Mally, who suggested that there are many non-existent objects. On Zaltas account, some objects exemplify properties, while others merely encode them, while the objects that exemplify properties are discovered through traditional empirical means, a simple set of axioms allows us to know about objects that encode properties. For every set of properties, there is one object that encodes exactly that set of properties. This allows for a formalized ontology