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nce of propositions each of which is either an instance of an axiom of a or follows by some rule of a from propositions appearing earlier in the proof thereby disallowing circular reasoning the last proposition is the theorem proved by the proof every nonempty initial segment of a proof is itself a proof whence every proposition in a proof is itself a theorem an axiomatization is sound when every theorem is a tautology and complete when every tautology is a theorem 34 sequent calculus edit main article sequent calculus propositional calculus is commonly organized as a hilbert system whose operations are just those of boolean algebra and whose theorems are boolean tautologies those boolean terms equal to the boolean constant 1 another form is sequent calculus which has two sorts propositions as in ordinary propositional calculus and pairs of lists of propositions called sequents such as a b a c a b c the two halves of a sequent are called the antecedent and the succedent respectively the customary metavariable denoting an antecedent or part thereof is γ and for a succedent δ thus γ a δ would denote a sequent whose succedent is a list δ and whose antecedent is a list γ with an additional proposition a appended after it the antecedent is interpreted as the conjunction of its propositions the succedent as the disjunction of its propositions and the sequent itself as the entailment of the succedent by the antecedent entailment differs from implication in that whereas the latter is a binary operation that returns a value in a boolean algebra the former is a binary relation which either holds or does not hold in this sense entailment is an external form of implication meaning external to the boolean algebra thinking of the reader of the sequent as also being external and interpreting and comparing antecedents and succedents in some boolean algebra the natural interpretation of is as in the partial order of the boolean algebra defined by x y just when x y y this ability to mix external implication and internal implication in the one logic is among the essential differences between sequent calculus and propositional calculus 35 applications edit boolean algebra as the calculus of two values is fundamental to computer circuits computer programming and mathematical logic and is also used in other areas of mathematics such as set theory and statistics 5 computers edit in the early 20th century several electrical engineers who intuitively recognized that boolean algebra was analogous to the behavior of certain types of electrical circuits claude shannon formally proved such behavior was logically equivalent to boolean algebra in his 1937 master s thesis a symbolic analysis of relay and switching circuits today all modern general purpose computers perform their functions using two value boolean logic that is their electrical circuits are a physical manifestation of two value boolean logic they achieve this in various ways as voltages on wires in high speed circuits and capacitive storage devices as orientations of a magnetic domain in ferromagnetic storage devices as holes in punched cards or paper tape and so on some early computers used decimal circuits or mechanisms instead of two valued logic circuits of course it is possible to code more than two symbols in any given medium for example one might use respectively 0 1 2 and 3 volts to code a four symbol alphabet on a wire or holes of different sizes in a punched card in practice the tight constraints of high speed small size and low power combine to make noise a major factor this makes it hard to distinguish between symbols when there are several possible symbols that could occur at a single site rather than attempting to distinguish between four voltages on one wire digital designers have settled on two voltages per wire high and low computers use two value boolean circuits for the above reasons the most common computer architectures use ordered sequences of boolean values called bits of 32 or 64 values e g 01101000110101100101010101001011 when programming in machine code assembly language and certain other programming languages programmers work with the low level digital structure of the data registers these registers operate on voltages where zero volts represents boolean 0 and a reference voltage often 5 v 3 3 v or 1 8 v represents boolean 1 such languages support both numeric operations and logical operations in this context numeric means that the computer treats sequences of bits as binary numbers base two numbers and executes arithmetic operations like add subtract multiply or divide logical refers to the boolean logical operations of disjunction conjunction and negation between two sequences of bits in which each bit in one sequence is simply compared to its counterpart in the other sequence programmers therefore have the option of working in and applying the rules of either numeric algebra or boolean algebra as needed a core differentiating feature between these families of operations is the existence of the carry operation in the first but not the second two valued logic edit other areas where two values is a good choice are the law and mathematics in everyday relaxed conversation nuanced or complex answers such as maybe or only on the weekend are acceptable in more focused situations such as a court of law or theorem based mathematics however it is deemed advantageous to frame questions so as to admit a simple yes or no answer is the defendant guilty or not guilty is the proposition true or false and to disallow any other answer however limiting this might prove in practice for the respondent the principle of the simple yes no question has become a central feature of both judicial and mathematical logic making two valued logic deserving of organization and study in its own right a central concept of set theory is membership an organization may permit multiple degrees of membership such as novice associate and full with sets however an element is either in or out the candidates for membership in a set work just like the wires in a digital computer each candidate is either a member or a nonmember just as each wire is either high or low algebra being a fundamental tool in any area amenable to mathematical treatment these considerations combine to make the algebra of two values of fundamental importance to computer hardware mathematical logic and set theory two valued logic can be extended to multi valued logic notably by replacing the boolean domain 0 1 with the unit interval 0 1 in which case rather than only taking values 0 or 1 any value between and including 0 and 1 can be assumed algebraically negation not is replaced with 1 x conjunction and is replaced with multiplication xy and disjunction or is defined via de morgan s law 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 boolean operations edit the original application for boolean operations was mathematical logic where it combines the truth values true or false of individual formulas natural language edit natural languages such as english have words for several boolean operations in particular conjunction and disjunction or negation not and implication implies but not is synonymous with and not when used to combine situational assertions such as the block is on the table and cats drink milk which naïvely are either true or false the meanings of these logical connectives often have the meaning of their logical counterparts however with descriptions of behavior such as jim walked through the door one starts to notice differences such as failure of commutativity for example the conjunction of jim opened the door with jim walked through the door in that order is not equivalent to their conjunction in the other order since and usually means and then in such cases questions can be similar the order is the sky blue and why is the sky blue makes more sense than the reverse order conjunctive commands about behavior are like behavioral assertions as in get dressed and go to school disjunctive commands such love me or leave me or fish or cut bait tend to be asymmetric via the implication that one alternative is less preferable conjoined nouns such as tea and milk generally describe aggregation as with set union while tea or milk is a choice however context can reverse these senses as in your choices are coffee and tea which usually means the same as your choices are coffee or tea alternatives double negation as in i don t not like milk rarely means literally i do like milk but rather conveys some sort of hedging as though to imply that there is a third possibility not not p can be loosely interpreted as surely p and although p necessarily implies not not p the converse is suspect in english much as with intuitionistic logic in view of the highly idiosyncratic usage of conjunctions in natural languages boolean algebra cannot be considered a reliable framework for interpreting them digital logic edit boolean operations are used in digital logic to combine the bits carried on individual wires thereby interpreting them over 0 1 when a vector of n identical binary gates are used to combine two bit vectors each of n bits the individual bit operations can be understood collectively as a single operation on values from a boolean algebra with 2 n elements naive set theory edit naive set theory interprets boolean operations as acting on subsets of a given set x as we saw earlier this behavior exactly parallels the coordinate wise combinations of bit vectors with the union of two sets corresponding to the disjunction of two bit vectors and so on video cards edit the 256 element free boolean algebra on three generators is deployed in computer displays based on raster graphics which use bit blit to manipulate whole regions consisting of pixels relying on boolean operations to specify how the source region should be combined with the destination typically with the help of a third region called the mask modern video cards offer all 2 2 3 256 ternary operations for this purpose with the choice of operation being a one byte 8 bit parameter the constants src 0xaa or 0b10101010 dst 0xcc or 0b11001100 and msk 0xf0 or 0b11110000 allow boolean operations such as src dst msk meaning xor the source and destination and then and the result with the mask to be written directly as a constant denoting a byte calculated at compile time 0x80 in the src dst msk example 0x88 if just src dst etc at run time the video card interprets the byte as the raster operation indicated by the original expression in a uniform way that requires remarkably little hardware and which takes time completely independent of the complexity of the expression modeling and cad edit solid modeling systems for computer aided design offer a variety of methods for building objects from other objects combination by boolean operations being one of them in this method the space in which objects exist is understood as a set s of voxels the three dimensional analogue of pixels in two dimensional graphics and shapes are defined as subsets of s allowing objects to be combined as sets via union intersection etc one obvious use is in building a complex shape from simple shapes simply as the union of the latter another use is in sculpting understood as removal of material any grinding milling routing or drilling operation that can be performed with physical machinery on physical materials can be simulated on the computer with the boolean operation x y or x y which in set theory is set difference remove the elements of y from those of x thus given two shapes one to be machined and the other the material to be removed the result of machining the former to remove the latter is described simply as their set difference boolean searches edit search engine queries also employ boolean logic for this application each web page on the internet may be considered to be an element of a set the following examples use a syntax supported by google nb 1 doublequotes are used to combine whitespace separated words into a single search term nb 2 whitespace is used to specify logical and as it is the default operator for joining search terms search term 1 search term 2 the or keyword is used for logical or search term 1 or search term 2 a prefixed minus sign is used for logical not search term 1 search term 2 parentheses can help broaden search when terms 2 and 3 are similar terms synonyms search term 1 and search term 2 or search term 3 see also edit mathematics portal boolean algebras canonically defined boolean differential calculus booleo cantor algebra heyting algebra list of boolean algebra topics logic design principia mathematica three valued logic łukasiewicz logic vector logic notes edit not all search engines support the same query syntax additionally some organizations such as google provide specialized search engines that support alternate or extended syntax see syntax cheatsheet the now defunct google code search used to support regular expressions but no longer exists doublequote delimited search terms are called exact phrase searches in the google documentation references edit boole george 2011 07 28 the mathematical analysis of logic being an essay towards a calculus of deductive reasoning boole george 2003 1854 an investigation of the laws of thought prometheus books isbn 978 1 59102 089 9 the name boolean algebra or boolean algebras for the calculus originated by boole extended by schröder and perfected by whitehead seems to have been first suggested by sheffer in 1913 edward vermilye huntington new sets of independent postulates for the algebra of logic with special reference to whitehead and russell s principia mathematica in transactions of the american mathematical society 35 1933 274 304 footnote page 278 peirce charles s 1931 collected papers vol 3 harvard university press p 13 isbn 978 0 674 13801 8 cite book isbn date incompatibility help 1 2 3 4 5 6 7 givant steven r halmos paul richard 2009 introduction to boolean algebras undergraduate texts in mathematics springer pp 21 22 isbn 978 0 387 40293 2 nelson eric s 2011 the yijing and philosophy from leibniz to derrida journal of chinese philosophy 38 3 377 396 doi 10 1111 j 1540 6253 2011 01661 x lenzen wolfgang leibniz logic in fieser james dowden bradley eds internet encyclopedia of philosophy issn 2161 0002 oclc 37741658 1 2 3 dunn j michael hardegree gary m 2001 algebraic methods in philosophical logic oxford university press p 2 isbn 978 0 19 853192 0 bimbó katalin dunn j michael 2008 generalized galois logics relational semantics of nonclassical logical calculi csli publications isbn 978 1 57586 573 7 stone m h 1936 the theory of representation for boolean algebras transactions of the...
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