Propositional calculus is about the simplest kind of logical calculus in current use. It can be extended in several ways. (Aristotelian "syllogistic" calculus, which is largely supplanted in modern logic, is in some ways simpler – but in other ways more complex – than propositional calculus.) The most immediate way to develop a more complex logical calculus is to introduce rules that are sensitive to more fine-grained details of the sentences being used. First-order logic (a.k.a.
|
|
After a long day of nonsense and topsy-turvy madness, Alice embraces her book on "logical" calculus and smiles back at the wonderful world she had the pleasure of visiting, if only in a dream.
|
|
A psychon was a minimal unit of psychic activity proposed by Warren McCulloch and Walter Pitts in "A Logical Calculus of Ideas Immanent in Nervous Activity" in 1943.McCulloch, W. S. and W. H. Pitts (1943), 'A Logical Calculus of the Ideas Immanent in Nervous Activity', Bulletin of Mathematical Biophysics 7, 115-133. McCulloch was later to reflect that he intended to invent a kind of "least psychic event" with the following properties:McCulloch, W. S. (1961), "What Is a Number, that a Man May Know It, and a Man,that He May Know a Number?" General Semantics Bulletin 26/27, 7–18.
|
|
Vector logicMizraji, E. (1992). Vector logics: the matrix-vector representation of logical calculus. Fuzzy Sets and Systems, 50, 179–185Mizraji, E. (2008) Vector logic: a natural algebraic representation of the fundamental logical gates. Journal of Logic and Computation, 18, 97–121 is an algebraic model of elementary logic based on matrix algebra.
|
|
Via Russell's logical atomism, ordinary language would break into discrete units of meaning. Rational reconstruction, then, would convert ordinary statements into standardized equivalents, all networked and united by a logical syntax. A scientific theory would be stated with its method of verification, whereby a logical calculus or empirical operation could verify its falsity or truth.
|
|
In logic, a many-valued logic (also multi- or multiple-valued logic) is a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values (i.e., "true" and "false") for any proposition. Classical two-valued logic may be extended to n-valued logic for n greater than 2.
|
|
A formal system is used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essentially an "axiomatic system". Fourth-century BCE philologist Pāṇini is credited with the first use of formal system in Sanskrit grammar.
|
|
A formal system (also called a logical calculus, or a logical system) consists of a formal language together with a deductive apparatus (also called a deductive system). The deductive apparatus may consist of a set of transformation rules (also called inference rules) or a set of axioms, or have both. A formal system is used to derive one expression from one or more other expressions.
|
|
In computer science, program synthesis is the task to construct a program that provably satisfies a given high-level formal specification. In contrast to program verification, the program is to be constructed rather than given; however, both fields make use of formal proof techniques, and both comprise approaches of different degrees of automatization. In contrast to automatic programming techniques, specifications in program synthesis are usually non- algorithmic statements in an appropriate logical calculus.
|
|
A formal system (also called a logical calculus, or a logical system) consists of a formal language together with a deductive apparatus (also called a deductive system). The deductive apparatus may consist of a set of transformation rules (also called inference rules) or a set of axioms, or have both. A formal system is used to derive one expression from one or more other expressions. Propositional and predicate calculi are examples of formal systems.
|
|
Because a function may be expressed as a composition, a truth-functional 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 equivalent to . The conditional operator "→" is therefore not necessary for a classical-based logical system if "¬" (not) and "∨" (or) are already in use.
|
|
Thus the connectives "and" and "or" of intuitionistic logic do not satisfy de Morgan's laws as they do in classical logic. Intuitionistic logic substitutes constructability for abstract truth and is associated with a transition from the proof of model theory to abstract truth in modern mathematics. The logical calculus preserves justification, rather than truth, across transformations yielding derived propositions. It has been taken as giving philosophical support to several schools of philosophy, most notably the Anti-realism of Michael Dummett.
|
|
Dick, S. (2005) Towards complex fuzzy logic. IEEE Transactions on Fuzzy Systems, 15,405–414, 2005 Other matrix and vector approaches to logical calculus have been developed in the framework of quantum physics, computer science and optics.Mittelstaedt, P. (1968) Philosophische Probleme der Modernen Physik, Bibliographisches Institut, MannheimStern, A. (1988) Matrix Logic: Theory and Applications. North-Holland, Amsterdam The Indian biophysicist G.N. Ramachandran developed a formalism using algebraic matrices and vectors to represent many operations of classical Jain Logic known as Syad and Saptbhangi.
|
|
Neural computation is the hypothetical information processing performed by networks of neurons. Neural computation is affiliated with the philosophical tradition known as Computational theory of mind, also referred to as computationalism, which advances the thesis that neural computation explains cognition. The first persons to propose an account of neural activity as being computational was Warren McCullock and Walter Pitts in their seminal 1943 paper, A Logical Calculus of the Ideas Immanent in Nervous Activity. There are three general branches of computationalism, including classicism, connectionism, and computational neuroscience.
|
|
George Boole In 1854, British mathematician George Boole published a landmark paper detailing an algebraic system of logic that would become known as Boolean algebra. His logical calculus was to become instrumental in the design of digital electronic circuitry. In 1937, Claude Shannon produced his master's thesis at MIT that implemented Boolean algebra and binary arithmetic using electronic relays and switches for the first time in history. Entitled A Symbolic Analysis of Relay and Switching Circuits, Shannon's thesis essentially founded practical digital circuit design.
|
|
Gandon contends that "many of Johnson's insights are today an integral part of philosophy" and that this is so especially of Johnson's doctrine of determinable and determinate. Johnson's work and influence in this latter regard is discussed in the Stanford Encyclopedia of Philosophy entry on Determinables and Determinates by Jessica Wilson.Having also been discussed in that article's precursor Determinates vs. Determinables by David H. Sanford "The Logical Calculus" (1892) reveals the technical capabilities of Johnson's youth, and that he was significantly influenced by the formal logical work of Charles Sanders Peirce.
|
|
This in effect "buries" these quantifiers, which are essential to the inference's validity, within the hyphenated terms. Hence the sentence "Some cat is feared by every mouse" is allotted the same logical form as the sentence "Some cat is hungry". And so the logical form in TL is: :Some As are Bs :All Cs are Ds which is clearly invalid. The first logical calculus capable of dealing with such inferences was Gottlob Frege's Begriffsschrift (1879), the ancestor of modern predicate logic, which dealt with quantifiers by means of variable bindings.
|
|
Many earlier researchers advocated connectionist style models, for example in the 1940s and 1950s, Warren McCulloch and Walter Pitts (MP neuron), Donald Olding Hebb, and Karl Lashley. McCulloch and Pitts showed how neural systems could implement first-order logic: Their classic paper "A Logical Calculus of Ideas Immanent in Nervous Activity" (1943) is important in this development here. They were influenced by the important work of Nicolas Rashevsky in the 1930s. Hebb contributed greatly to speculations about neural functioning, and proposed a learning principle, Hebbian learning, that is still used today.
|
|
A number of axiomatizations of classical propositional logic are due to Łukasiewicz. A particularly elegant axiomatization features a mere three axioms and is still invoked to the present day. He was a pioneer investigator of multi-valued logics; his three-valued propositional calculus, introduced in 1917, was the first explicitly axiomatized non-classical logical calculus. He wrote on the philosophy of science, and his approach to the making of scientific theories was similar to the thinking of Karl Popper. Łukasiewicz invented the Polish notation (named after his nationality) for the logical connectives around 1920.
|
|
A formal system (also called a logical calculus, or a logical system) consists of a formal language together with a deductive apparatus (also called a deductive system). The deductive apparatus may consist of a set of transformation rules (also called inference rules) or a set of axioms, or have both. A formal system is used to derive one expression from one or more other expressions. Formal systems, like other syntactic entities may be defined without any interpretation given to it (as being, for instance, a system of arithmetic).
|
|
Pitts was familiar with the work of Gottfried Leibniz on computing and they considered the question of whether the nervous system could be considered a kind of universal computing device as described by Leibniz. This led to their seminal neural networks paper "A Logical Calculus of Ideas Immanent in Nervous Activity". After five years of unofficial studies, the University of Chicago awarded Pitts an Associate of Arts (his only earned degree) for his work on the paper. In 1943, Lettvin introduced Pitts to Norbert Wiener at the Massachusetts Institute of Technology.
|
|
Arithmetic operations are to be performed one binary digit at a time. He estimates addition of two binary digits as taking one microsecond and that therefore a 30-bit multiplication should take about 302 microseconds or about one millisecond, much faster than any computing device available at the time. Von Neumann's design is built up using what he call "E elements," which are based on the biological neuron as model,Von Neumann credits this model to Warren McCulloch and Walter Pitts, A logical calculus of the ideas immanent in nervous activity, Bull. Math. Biophysics, Vol.
|
|
Numerous papers spearheaded the coalescing of the field. In 1935 Russian physiologist P. K. Anokhin published a book in which the concept of feedback ("back afferentation") was studied. The study and mathematical modelling of regulatory processes became a continuing research effort and two key articles were published in 1943: "Behavior, Purpose and Teleology" by Arturo Rosenblueth, Norbert Wiener, and Julian Bigelow; and the paper "A Logical Calculus of the Ideas Immanent in Nervous Activity" by Warren McCulloch and Walter Pitts. In 1936, Ștefan Odobleja published "Phonoscopy and the clinical semiotics".
|
|
Mathematics in physics would reduce to symbolic logic via logicism, while rational reconstruction would convert ordinary language into standardized equivalents, all networked and united by a logical syntax. A scientific theory would be stated with its method of verification, whereby a logical calculus or empirical operation could verify its falsity or truth. In the late 1930s, logical positivists fled Germany and Austria for Britain and America. By then, many had replaced Mach's phenomenalism with Otto Neurath's physicalism, and Rudolf Carnap had sought to replace verification with simply confirmation.
|
|
Because Leibniz was a mathematical novice when he first wrote about the characteristic, at first he did not conceive it as an algebra but rather as a universal language or script. Only in 1676 did he conceive of a kind of "algebra of thought", modeled on and including conventional algebra and its notation. The resulting characteristic included a logical calculus, some combinatorics, algebra, his analysis situs (geometry of situation), a universal concept language, and more. What Leibniz actually intended by his characteristica universalis and calculus ratiocinator, and the extent to which modern formal logic does justice to calculus, may never be established.
|
|
He is remembered for his work with Joannes Gregorius Dusser de Barenne from Yale and later with Walter Pitts from the University of Chicago. Here he provided the foundation for certain brain theories in a number of classic papers, including "A Logical Calculus of the Ideas Immanent in Nervous Activity" (1943) and "How We Know Universals: The Perception of Auditory and Visual Forms" (1947), both published in the Bulletin of Mathematical Biophysics. The former is "widely credited with being a seminal contribution to neural network theory, the theory of automata, the theory of computation, and cybernetics".
|
|
Alice is sitting by the riverbank all alone reading her book entitled The Principles of Logical Calculus when she decides she simply could not stand to read any further. She's bored and she hates being stuck in that "horrible place", as she refers to her beautiful outdoor garden. Suddenly out of nowhere, a talking white rabbit wearing a waistcoat and a pocket watch appears, and Alice sees that he is in a big hurry. Running with curiosity, Alice follows the white rabbit down his rabbit hole, and the girl soon finds herself falling deep into the hole.
|
|
Platon Poretsky Platon Sergeevich Poretsky (; October 3, 1846 in Elisavetgrad – August 9, 1907 in Chernihiv Governorate) was a noted Russian astronomer, mathematician, and logician. Graduated from Kharkov University, he worked in Astrakhan and Pulkovo. Later, as an astronomer at Kazan University, following the advice of his older colleague Professor of Mathematics A.V. Vasiliev at Kazan University (father of Nicolai A. Vasiliev) to learn the works of George Boole, Poretsky developed "logical calculus" and through specific "logical equations" applied it to the theory of probability. Thus, he extended and augmented the works of logicians and mathematicians George Boole, William Stanley Jevons and Ernst Schröder.
|
|
All of the works of Epimenides are now lost, and known only through quotations by other authors. The quotation from the Cretica of Epimenides is given by R.N. Longenecker, "Acts of the Apostles", in volume 9 of The Expositor's Bible Commentary, Frank E. Gaebelein, editor (Grand Rapids, Michigan: Zondervan Corporation, 1976–1984), page 476. Longenecker in turn cites M.D. Gibson, Horae Semiticae X (Cambridge: Cambridge University Press, 1913), page 40, "in Syriac". Longenecker states the following in a footnote: An oblique reference to Epimenides in the context of logic appears in "The Logical Calculus" by W. E. Johnson, Mind (New Series), volume 1, number 2 (April, 1892), pages 235–250.
|
|
His best known logico-mathematical book is An Outline of Mathematical Logic: Fundamental Results and Notions Explained with All Details published in Polish in 1961 and in English in 1969. His book Fonctions Récursives became the standard handbook at the French universities. His other book Zarys arytmetyki teoretycznej (An Outline of Theoretical Arithmetic) became the basis for the Mizar system by the University of Białystok's team of a notable computer scientist Andrzej Trybulec. In Poland, Grzegorczyk was the first who popularized logical calculus by the book Logika popularna (Popular logic), also translated into Czech in 1957 and Russian in 1965, and the problems of decidability theory by the book Zagadnienia rozstrzygalności (Decidability problems).
|
|
Later, Zinoviev developed a general theory of succession (the theory of inference), which was significantly different from classical and intuitionistic mathematical logic. According to Wessel, its originality was the introduction of the two-place predicate "from... logically follows..." into the formula for logical following, in fact, metatermine. The theory of logical calculus and the remaining sections of logic (the theory of quantifiers and predication, the logic of classes, normative and epistemic logic) were built on the basis of the theory. The work "Complex Logic" (1970) presented a systematic consideration of the formal apparatus for analyzing concepts, statements and evidence; a strict quantifier theory was formulated in the monograph that corresponded to intuitive assumptions; the properties of quantifiers were investigated.
|
|
Walter Harry Pitts, Jr. (23 April 1923 – 14 May 1969) was a logician who worked in the field of computational neuroscience.Smalheiser, Neil R. "Walter Pitts" , Perspectives in Biology and Medicine, Volume 43, Number 2, Winter 2000, pp. 217–226, The Johns Hopkins University Press He proposed landmark theoretical formulations of neural activity and generative processes that influenced diverse fields such as cognitive sciences and psychology, philosophy, neurosciences, computer science, artificial neural networks, cybernetics and artificial intelligence, together with what has come to be known as the generative sciences. He is best remembered for having written along with Warren McCulloch, a seminal paper in scientific history, titled "A Logical Calculus of Ideas Immanent in Nervous Activity" (1943).
|
|
A physical symbol system (also called a formal system) takes physical patterns (symbols), combining them into structures (expressions) and manipulating them (using processes) to produce new expressions. The physical symbol system hypothesis (PSSH) is a position in the philosophy of artificial intelligence formulated by Allen Newell and Herbert A. Simon. They wrote: This claim implies both that human thinking is a kind of symbol manipulation (because a symbol system is necessary for intelligence) and that machines can be intelligent (because a symbol system is sufficient for intelligence). The idea has philosophical roots in Hobbes (who claimed reasoning was "nothing more than reckoning"), Leibniz (who attempted to create a logical calculus of all human ideas), Hume (who thought perception could be reduced to "atomic impressions") and even Kant (who analyzed all experience as controlled by formal rules).
|
|
Vernon's objective is to provide a framework to design cognitive system by elaborating a comprehensive and structured overview of the entire research area concerned with cognitive systems; referring to it as a pre-paradigmatic discipline. He differentiates two global approaches: the cognitivist and the emergent. As the first one has its origins in cybernetics (1943-53) and is based on logical calculus immanent in nervous activity, the second one comes from the study of self-organized systems (1958), focuses on embodiment and can be refined in three subcategorizes: Connectionist, Dynamical, and Enactive. While he is engaged in measuring up those four foregoing paradigms, he is also advocating the Enactive Systems Model to offer the framework by which successively richer orders of cognitive capability can be achieved, and by which the system itself will becomes part of an existing world of meaning (ontogeny) or shapes a new one (phylogeny).
|
|
The title page of the original 1879 edition Begriffsschrift (German for, roughly, "concept-script") is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book. Begriffsschrift is usually translated as concept writing or concept notation; the full title of the book identifies it as "a formula language, modeled on that of arithmetic, of pure thought." Frege's motivation for developing his formal approach to logic resembled Leibniz's motivation for his calculus ratiocinator (despite that, in the foreword Frege clearly denies that he achieved this aim, and also that his main aim would be constructing an ideal language like Leibniz's, which Frege declares to be a quite hard and idealistic—though not impossible—task). Frege went on to employ his logical calculus in his research on the foundations of mathematics, carried out over the next quarter century.
|
|