# kb

id: game-theory-20260505-10
file: data/en.wikipedia.org/wiki/Game_theory-10.md
category: reference
title: Game theory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Game_theory
date_saved: 2026-05-05T03:56:32.715747+00:00
instance: kb-cron
parent: game-theory-20260505-0
summary:
=== Trust game === The Trust Game is an experiment designed to measure trust in economic decisions. It is also called "the investment game" and is designed to investigate trust and demonstrate its importance rather than "rationality" of self-interest. The game was designed by Berg Joyce, John Dickhaut and Kevin McCabe in 1995. In the game, one player (the investor) is given a sum of money and must decide how much of it to give to another player (the trustee). The amount given is then tripled by the experimenter. The trustee then decides how much of the tripled amount to return to the investor. If the trustee is completely self-interested, then they would return nothing. However, experiments have shown that this isn't the expected behavior of the trustee. The outcome instead suggests that people are willing to place trust, by risking some amount of money, in the belief that there will be reciprocity. === Cournot Competition === The Cournot competition model involves players choosing quantity of a homogenous product to produce independently and simultaneously, where marginal cost can be different for each firm and the firm's payoff is profit. The production costs are public information and the firm aims to


id: game-theory-20260505-11
file: data/en.wikipedia.org/wiki/Game_theory-11.md
category: reference
title: Game theory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Game_theory
date_saved: 2026-05-05T03:56:32.715747+00:00
instance: kb-cron
parent: game-theory-20260505-0
summary:
== See also == Applied ethics – Practical application of moral considerations Bandwidth-sharing game – Type of resource allocation game Chainstore paradox – Game theory paradox Collective intentionality – Social concept in philosophy of mind Core (game theory) – Set in game theory Glossary of game theory Intra-household bargaining Kingmaker scenario – Endgame situation in game theory Law and economics – Analysis of law using economic theory Mutual assured destruction – Doctrine of military strategyPages displaying short descriptions of redirect targets Outline of artificial intelligence Parrondo's paradox – Paradox of combining strategies Precautionary principle – Risk management strategy Quantum refereed game Risk management – Identification, evaluation and control of risks Self-confirming equilibrium – Aspect of game theory Tragedy of the commons – Overuse of a shared resource Traveler's dilemma – Non-zero-sum game thought experiment Wilson doctrine (economics) – Argument in economic theory Compositional game theory Lists List of cognitive biases List of emerging technologies List of games in game theory == Notes == == References == == Further reading == Ben-David, S.; Borodin, A.; Karp, R.; Tardos, G.; Wigderson, A. (January 1994). "On the power of randomization in on-line algorithms". Algorithmica. 11 (1): 2–14. doi:10.1007/BF01294260. S2CID 26771869. Downs, Anthony (1957),


id: game-theory-20260505-12
file: data/en.wikipedia.org/wiki/Game_theory-12.md
category: reference
title: Game theory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Game_theory
date_saved: 2026-05-05T03:56:32.715747+00:00
instance: kb-cron
parent: game-theory-20260505-0
summary:
=== Historically important texts === Aumann, R. J.; Shapley, L. S. (1974), Values of Non-Atomic Games, Princeton University Press Cournot, A. Augustin (1838), "Recherches sur les principles mathematiques de la théorie des richesses", Libraire des Sciences Politiques et Sociales Edgeworth, Francis Y. (1881), Mathematical Psychics, London: Kegan Paul Farquharson, Robin (1969), Theory of Voting, Blackwell (Yale U.P. in the U.S.), ISBN 978-0-631-12460-3 Luce, R. Duncan; Raiffa, Howard (1957), Games and decisions: introduction and critical survey, New York: Wiley reprinted edition: R. Duncan Luce; Howard Raiffa (1989), Games and decisions: introduction and critical survey, New York: Dover Publications, ISBN 978-0-486-65943-5 Maynard Smith, John (1982), Evolution and the theory of games, Cambridge University Press, ISBN 978-0-521-28884-2 Nash, John (1950), "Equilibrium points in n-person games", Proceedings of the National Academy of Sciences of the United States of America, 36 (1): 48–49, Bibcode:1950PNAS...36...48N, doi:10.1073/pnas.36.1.48, PMC 1063129, PMID 16588946 Shapley, L.S. (1953), A Value for n-person Games, In: Contributions to the Theory of Games volume II, H. W. Kuhn and A. W. Tucker (eds.) Shapley, L. S. (October 1953). "Stochastic Games". Proceedings of the National Academy of Sciences. 39 (10): 1095–1100. Bibcode:1953PNAS...39.1095S. doi:10.1073/pnas.39.10.1095. PMC 1063912. PMID 16589380. von Neumann, John (1928), "Zur Theorie der Gesellschaftsspiele",


id: grammar-systems-theory-20260505-0
file: data/en.wikipedia.org/wiki/Grammar_systems_theory-0.md
category: reference
title: Grammar systems theory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Grammar_systems_theory
date_saved: 2026-05-05T03:56:33.967957+00:00
instance: kb-cron
parent: grammar-systems-theory-20260505-0
summary:
Grammar systems theory is a field of theoretical computer science that studies systems of finite collections of formal grammars generating a formal language. Each grammar works on a string, a so-called sequential form that represents an environment. Grammar systems can thus be used as a formalization of decentralized or distributed systems of agents in artificial intelligence. Let A {\displaystyle \mathbb {A} } be a simple reactive agent moving on the table and trying not to fall down from the table with two reactions, t for turning and ƒ for moving forward. The set of possible behaviors of A {\displaystyle \mathbb {A} } can then be described as formal language L A = { ( f m t n f r ) + : 1 ≤ m ≤ k ; 1 ≤ n ≤ ℓ ; 1 ≤ r ≤ k } , {\displaystyle \mathbb {L_{A}} =\{(f^{m}t^{n}f^{r})^{+}:1\leq m\leq k;1\leq n\leq \ell ;1\leq r\leq k\},} where ƒ can be done maximally k times and t can be done maximally ℓ times considering the dimensions of the table. Let G A {\displaystyle \mathbb {G_{A}} } be a formal grammar which generates language L A {\displaystyle \mathbb {L_{A}} } . The behavior of A


id: homeokinetics-20260505-0
file: data/en.wikipedia.org/wiki/Homeokinetics-0.md
category: reference
title: Homeokinetics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Homeokinetics
date_saved: 2026-05-05T03:56:35.253721+00:00
instance: kb-cron
parent: homeokinetics-20260505-0
summary:
Homeokinetics is the study of self-organizing, complex systems. Standard physics studies systems at separate levels, such as atomic physics, nuclear physics, biophysics, social physics, and galactic physics. Homeokinetic physics studies the up-down processes that bind these levels. Tools such as mechanics, quantum field theory, and the laws of thermodynamics provide the key relationships. The subject, described as the physics and thermodynamics associated with the up down movement between levels of systems, originated in the late 1970s work of American physicists Harry Soodak and Arthur Iberall. Complex systems are universes, galaxies, social systems, people, or even those that seem as simple as gases. The basic premise is that the entire universe consists of atomistic-like units bound in interactive ensembles to form systems, level by level, in a nested hierarchy. Homeokinetics treats all complex systems on an equal footing, animate and inanimate, providing them with a common viewpoint. The complexity in studying how they work is reduced by the emergence of common languages in all complex systems. == History == Arthur Iberall, Warren McCulloch and Harry Soodak developed the concept of homeokinetics as a new branch of physics. It began through Iberall's biophysical research for the NASA exobiology program into the dynamics


id: image-analysis-20260505-0
file: data/en.wikipedia.org/wiki/Image_analysis-0.md
category: reference
title: Image analysis
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Image_analysis
date_saved: 2026-05-05T03:56:36.441246+00:00
instance: kb-cron
parent: image-analysis-20260505-0
summary:
Image analysis or imagery analysis is the extraction of meaningful information from images; mainly from digital images by means of digital image processing techniques. Image analysis tasks can be as simple as reading bar coded tags or as sophisticated as identifying a person from their face. Computers are indispensable for the analysis of large amounts of data, for tasks that require complex computation, or for the extraction of quantitative information. On the other hand, the human visual cortex is an excellent image analysis apparatus, especially for extracting higher-level information, and for many applications — including medicine, security, and remote sensing — human analysts still cannot be replaced by computers. For this reason, many important image analysis tools such as edge detectors and neural networks are inspired by human visual perception models. == Digital == Digital Image Analysis or Computer Image Analysis is when a computer or electrical device automatically studies an image to obtain useful information from it. Note that the device is often a computer but may also be an electrical circuit, a digital camera or a mobile phone. It involves the fields of computer or machine vision, and medical imaging, and makes heavy use of pattern recognition, digital


id: information-theory-20260505-0
file: data/en.wikipedia.org/wiki/Information_theory-0.md
category: reference
title: Information theory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Information_theory
date_saved: 2026-05-05T03:56:37.735412+00:00
instance: kb-cron
parent: information-theory-20260505-0
summary:
Information theory is the mathematical study of the quantification, storage, and communication of a particular type of mathematically defined information. The field was established and formalized by Claude Shannon in the 1940s, though early contributions were made in the 1920s through the works of Harry Nyquist and Ralph Hartley. Information theory was initially formed in the context of telecommunication but soon found a wide range of other applications. It is now at the intersection of mathematics, statistics and computer science, and has applications in diverse fields ranging from electrical engineering and physics to neurobiology. As a simple example of the concept, if one flips a fair coin and does not yet know the outcome (heads or tails), then they lack a certain amount of information. After looking at the coin, they gain information about the outcome. For a fair coin, the probability of either heads or tails is 1/2 and the amount of information is expressed as − log 2 ⁡ ( 1 / 2 ) {\displaystyle -\log _{2}(1/2)} = 1 bit of information. A key concept in information theory is information entropy. In Shannon's formulation entropy is equal to the lack of information about an event. In the above


id: information-theory-20260505-1
file: data/en.wikipedia.org/wiki/Information_theory-1.md
category: reference
title: Information theory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Information_theory
date_saved: 2026-05-05T03:56:37.735412+00:00
instance: kb-cron
parent: information-theory-20260505-0
summary:
The landmark event establishing the discipline of information theory and bringing it to immediate worldwide attention was the publication of Claude Shannon's classic paper "A Mathematical Theory of Communication" in the Bell System Technical Journal in July and October 1948. Historian James Gleick rated the paper as the most important development of 1948, noting that the paper was "even more profound and more fundamental" than the transistor. He came to be known as the "father of information theory". Shannon outlined some of his initial ideas of information theory as early as 1939 in a letter to Vannevar Bush. Prior to this paper, limited information-theoretic ideas had been developed at Bell Labs, all implicitly assuming events of equal probability. Harry Nyquist's 1924 paper, Certain Factors Affecting Telegraph Speed, contains a theoretical section quantifying "intelligence" and the "line speed" at which it can be transmitted by a communication system, giving the relation W = K log m (recalling the Boltzmann constant), where W is the speed of transmission of intelligence, m is the number of different voltage levels to choose from at each time step, and K is a constant. Ralph Hartley's 1928 paper, Transmission of Information, uses the word information as


id: information-theory-20260505-2
file: data/en.wikipedia.org/wiki/Information_theory-2.md
category: reference
title: Information theory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Information_theory
date_saved: 2026-05-05T03:56:37.735412+00:00
instance: kb-cron
parent: information-theory-20260505-0
summary:
Intuitively, the entropy H ( X ) {\displaystyle H(X)} of a discrete random variable X is a measure of the amount of uncertainty associated with the value of X {\displaystyle X} when only its distribution is known. A high entropy indicates the outcomes are more evenly distributed, making the result harder to predict. For example, if one transmits 1000 bits (0s and 1s), and the value of each of these bits is known to the receiver (has a specific value with certainty) ahead of transmission, no information is transmitted. If, however, each bit is independently and equally likely to be 0 or 1, 1000 shannons of information (more often called bits) have been transmitted. ==== Properties ==== A key property of entropy is that it is maximized when all the messages in the message space are equiprobable. For a source with n possible symbols, where p i = 1 n {\textstyle p_{i}={\frac {1}{n}}} for all i {\displaystyle i} , the entropy is given by: H ( X ) = log 2 ⁡ ( n ) {\displaystyle H(X)=\log _{2}(n)} This maximum value represents the most unpredictable state. For a source that emits a sequence of N {\displaystyle N} symbols that are


id: information-theory-20260505-3
file: data/en.wikipedia.org/wiki/Information_theory-3.md
category: reference
title: Information theory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Information_theory
date_saved: 2026-05-05T03:56:37.735412+00:00
instance: kb-cron
parent: information-theory-20260505-0
summary:
=== Mutual information (transinformation) === Mutual information measures the amount of information that can be obtained about one random variable by observing another. It is important in communication where it can be used to maximize the amount of information shared between sent and received signals. The mutual information of X relative to Y is given by: I ( X ; Y ) = E X , Y [ S I ( x , y ) ] = ∑ x , y p ( x , y ) log ⁡ p ( x , y ) p ( x ) p ( y ) {\displaystyle I(X;Y)=\mathbb {E} _{X,Y}[SI(x,y)]=\sum _{x,y}p(x,y)\log {\frac {p(x,y)}{p(x)\,p(y)}}} where SI (Specific mutual Information) is the pointwise mutual information. A basic property of the mutual information is that: I ( X ; Y ) = H ( X ) − H ( X | Y ) . {\displaystyle I(X;Y)=H(X)-H(X|Y).\,} That is, knowing Y {\textstyle Y} , we can save an average of I(X; Y) bits in encoding X {\displaystyle X} compared to not knowing Y {\displaystyle Y} . Mutual information is symmetric: I ( X ; Y ) = I ( Y ; X ) = H ( X


id: information-theory-20260505-4
file: data/en.wikipedia.org/wiki/Information_theory-4.md
category: reference
title: Information theory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Information_theory
date_saved: 2026-05-05T03:56:37.735412+00:00
instance: kb-cron
parent: information-theory-20260505-0
summary:
I ( X n → Y n ) ≜ ∑ i = 1 n I ( X i ; Y i | Y i − 1 ) {\displaystyle I(X^{n}\to Y^{n})\ \triangleq \ \sum _{i=1}^{n}I(X^{i};Y_{i}|Y^{i-1})} , where I ( X i ; Y i | Y i − 1 ) {\displaystyle I(X^{i};Y_{i}|Y^{i-1})} is the conditional mutual information I ( X 1 , X 2 , . . . , X i ; Y i | Y 1 , Y 2 , . . . , Y i − 1 ) {\displaystyle I(X_{1},X_{2},...,X_{i};Y_{i}|Y_{1},Y_{2},...,Y_{i-1})} . In contrast to mutual information, directed information is not symmetric. The I ( X n → Y n ) {\displaystyle I(X^{n}\to Y^{n})} measures the information bits that are transmitted causally from X n {\displaystyle X^{n}} to Y n {\displaystyle Y^{n}} . The Directed information has many applications in problems where causality plays an important role such as capacity of channel with feedback, capacity of discrete memoryless networks with feedback, gambling with causal side information, compression with causal side information, real-time control communication settings, and in statistical physics. === Other quantities === Other important information theoretic quantities include the Rényi entropy and the Tsallis entropy (generalizations of the concept


id: information-theory-20260505-5
file: data/en.wikipedia.org/wiki/Information_theory-5.md
category: reference
title: Information theory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Information_theory
date_saved: 2026-05-05T03:56:37.735412+00:00
instance: kb-cron
parent: information-theory-20260505-0
summary:
→ Message W Encoder f n → E n c o d e d s e q u e n c e X n Channel p ( y | x ) → R e c e i v e d s e q u e n c e Y n Decoder g n → E s t i m a t e d m e s s a g e W ^ {\displaystyle {\xrightarrow[{\text{Message}}]{W}}{\begin{array}{|c| }\hline {\text{Encoder}}\\f_{n}\\\hline \end{array}}{\xrightarrow[{\mathrm {Encoded \atop sequence} }]{X^{n}}}{\begin{array}{|c| }\hline {\text{Channel}}\\p(y|x)\\\hline \end{array}}{\xrightarrow[{\mathrm {Received \atop sequence} }]{Y^{n}}}{\begin{array}{|c| }\hline {\text{Decoder}}\\g_{n}\\\hline \end{array}}{\xrightarrow[{\mathrm {Estimated \atop message} }]{\hat {W}}}} Here X {\displaystyle X} represents the space of messages transmitted, and Y {\textstyle Y} the space of messages received during a unit time over our channel. Let p(y|x) be the conditional probability distribution function of Y {\textstyle Y} given X {\displaystyle X} . We will consider p(y|x) to be an inherent fixed property of our communications channel (representing the nature of the noise of our channel). Then the joint distribution of X {\displaystyle X} and Y {\textstyle Y} is completely determined by our channel and by our choice of f(x), the marginal distribution of messages we choose to send over the channel. Under these


id: information-theory-20260505-6
file: data/en.wikipedia.org/wiki/Information_theory-6.md
category: reference
title: Information theory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Information_theory
date_saved: 2026-05-05T03:56:37.735412+00:00
instance: kb-cron
parent: information-theory-20260505-0
summary:
Information theoretic concepts apply to cryptography and cryptanalysis. Turing's information unit, the ban, was used in the Ultra project, breaking the German Enigma machine code and hastening the end of World War II in Europe. Shannon himself defined an important concept now called the unicity distance. Based on the redundancy of the plaintext, it attempts to give a minimum amount of ciphertext necessary to ensure unique decipherability. Information theory leads us to believe it is much more difficult to keep secrets than it might first appear. A brute force attack can break systems based on asymmetric key algorithms or on most commonly used methods of symmetric key algorithms (sometimes called secret key algorithms), such as block ciphers. The security of all such methods comes from the assumption that no known attack can break them in a practical amount of time. Information theoretic security refers to methods such as the one-time pad that are not vulnerable to such brute force attacks. In such cases, the positive conditional mutual information between the plaintext and ciphertext (conditioned on the key) can ensure proper transmission, while the unconditional mutual information between the plaintext and ciphertext remains zero, resulting in absolutely secure communications. In other


id: logic-20260505-0
file: data/en.wikipedia.org/wiki/Logic-0.md
category: reference
title: Logic
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Logic
date_saved: 2026-05-05T03:56:39.118518+00:00
instance: kb-cron
parent: logic-20260505-0
summary:
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to work." Premises and conclusions express propositions or claims that can be true or false. An important feature of propositions is their internal structure. For example, complex propositions are made up of simpler propositions linked by logical vocabulary like ∧ {\displaystyle \land } (and) or → {\displaystyle \to


id: logic-20260505-1
file: data/en.wikipedia.org/wiki/Logic-1.md
category: reference
title: Logic
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Logic
date_saved: 2026-05-05T03:56:39.118518+00:00
instance: kb-cron
parent: logic-20260505-0
summary:
Formal logic uses formal languages to express, analyze, and clarify arguments. They normally have a very limited vocabulary and exact syntactic rules. These rules specify how their symbols can be combined to construct sentences, so-called well-formed formulas. This simplicity and exactness of formal logic make it capable of formulating precise rules of inference. They determine whether a given argument is valid. Because of the reliance on formal language, natural language arguments cannot be studied directly. Instead, they need to be translated into formal language before their validity can be assessed. The term "logic" can also be used in a slightly different sense as a countable noun. In this sense, a logic is a logical formal system. Distinct logics differ from each other concerning the rules of inference they accept as valid and the formal languages used to express them. Starting in the late 19th century, many new formal systems have been proposed. There are disagreements about what makes a formal system a logic. For example, it has been suggested that only logically complete systems, like first-order logic, qualify as logics. For such reasons, some theorists deny that higher-order logics are logics in the strict sense. === Informal logic === When


id: logic-20260505-2
file: data/en.wikipedia.org/wiki/Logic-2.md
category: reference
title: Logic
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Logic
date_saved: 2026-05-05T03:56:39.118518+00:00
instance: kb-cron
parent: logic-20260505-0
summary:
Premises and conclusions are the basic parts of inferences or arguments and therefore play a central role in logic. In the case of a valid inference or a correct argument, the conclusion follows from the premises, or in other words, the premises support the conclusion. For instance, the premises "Mars is red" and "Mars is a planet" support the conclusion "Mars is a red planet". For most types of logic, it is accepted that premises and conclusions have to be truth-bearers. This means that they have a truth value: they are either true or false. Contemporary philosophy generally sees them either as propositions or as sentences. Propositions are the denotations of sentences and are usually seen as abstract objects. For example, the English sentence "the tree is green" is different from the German sentence "der Baum ist grün" but both express the same proposition. Propositional theories of premises and conclusions are often criticized because they rely on abstract objects. For instance, philosophical naturalists usually reject the existence of abstract objects. Other arguments concern the challenges involved in specifying the identity criteria of propositions. These objections are avoided by seeing premises and conclusions not as propositions but as sentences, i.e. as


id: logic-20260505-3
file: data/en.wikipedia.org/wiki/Logic-3.md
category: reference
title: Logic
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Logic
date_saved: 2026-05-05T03:56:39.118518+00:00
instance: kb-cron
parent: logic-20260505-0
summary:
Logic is commonly defined in terms of arguments or inferences as the study of their correctness. An argument is a set of premises together with a conclusion. An inference is the process of reasoning from these premises to the conclusion. But these terms are often used interchangeably in logic. Arguments are correct or incorrect depending on whether their premises support their conclusion. Premises and conclusions, on the other hand, are true or false depending on whether they are in accord with reality. In formal logic, a sound argument is an argument that is both correct and has only true premises. Sometimes a distinction is made between simple and complex arguments. A complex argument is made up of a chain of simple arguments. This means that the conclusion of one argument acts as a premise of later arguments. For a complex argument to be successful, each link of the chain has to be successful. Arguments and inferences are either correct or incorrect. If they are correct then their premises support their conclusion. In the incorrect case, this support is missing. It can take different forms corresponding to the different types of reasoning. The strongest form of support corresponds to deductive reasoning.


id: logic-20260505-4
file: data/en.wikipedia.org/wiki/Logic-4.md
category: reference
title: Logic
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Logic
date_saved: 2026-05-05T03:56:39.118518+00:00
instance: kb-cron
parent: logic-20260505-0
summary:
==== Ampliative ==== Ampliative arguments are arguments whose conclusions contain additional information not found in their premises. In this regard, they are more interesting since they contain information on the depth level and the thinker may learn something genuinely new. But this feature comes with a certain cost: the premises support the conclusion in the sense that they make its truth more likely but they do not ensure its truth. This means that the conclusion of an ampliative argument may be false even though all its premises are true. This characteristic is closely related to non-monotonicity and defeasibility: it may be necessary to retract an earlier conclusion upon receiving new information or in light of new inferences drawn. Ampliative reasoning plays a central role in many arguments found in everyday discourse and the sciences. Ampliative arguments are not automatically incorrect. Instead, they just follow different standards of correctness. The support they provide for their conclusion usually comes in degrees. This means that strong ampliative arguments make their conclusion very likely while weak ones are less certain. As a consequence, the line between correct and incorrect arguments is blurry in some cases, such as when the premises offer weak but non-negligible


id: logic-20260505-5
file: data/en.wikipedia.org/wiki/Logic-5.md
category: reference
title: Logic
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Logic
date_saved: 2026-05-05T03:56:39.118518+00:00
instance: kb-cron
parent: logic-20260505-0
summary:
=== Definitory and strategic rules === The main focus of most logicians is to study the criteria according to which an argument is correct or incorrect. A fallacy is committed if these criteria are violated. In the case of formal logic, they are known as rules of inference. They are definitory rules, which determine whether an inference is correct or which inferences are allowed. Definitory rules contrast with strategic rules. Strategic rules specify which inferential moves are necessary to reach a given conclusion based on a set of premises. This distinction does not just apply to logic but also to games. In chess, for example, the definitory rules dictate that bishops may only move diagonally. The strategic rules, on the other hand, describe how the allowed moves may be used to win a game, for instance, by controlling the center and by defending one's king. It has been argued that logicians should give more emphasis to strategic rules since they are highly relevant for effective reasoning. === Formal systems === A formal system of logic consists of a formal language together with a set of axioms and a proof system used to draw inferences from these axioms. In logic, axioms


id: logic-20260505-6
file: data/en.wikipedia.org/wiki/Logic-6.md
category: reference
title: Logic
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Logic
date_saved: 2026-05-05T03:56:39.118518+00:00
instance: kb-cron
parent: logic-20260505-0
summary:
In Aristotelian logic, the subject can be universal, particular, indefinite, or singular. For example, the term "all humans" is a universal subject in the proposition "all humans are mortal". A similar proposition could be formed by replacing it with the particular term "some humans", the indefinite term "a human", or the singular term "Socrates". Aristotelian logic only includes predicates for simple properties of entities. But it lacks predicates corresponding to relations between entities. The predicate can be linked to the subject in two ways: either by affirming it or by denying it. For example, the proposition "Socrates is not a cat" involves the denial of the predicate "cat" to the subject "Socrates". Using combinations of subjects and predicates, a great variety of propositions and syllogisms can be formed. Syllogisms are characterized by the fact that the premises are linked to each other and to the conclusion by sharing one term in each case. Thus, these three propositions contain three terms, referred to as major term, minor term, and middle term. The central aspect of Aristotelian logic involves classifying all possible syllogisms into valid and invalid arguments according to how the propositions are formed. For example, the syllogism "all men are


id: logic-20260505-7
file: data/en.wikipedia.org/wiki/Logic-7.md
category: reference
title: Logic
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Logic
date_saved: 2026-05-05T03:56:39.118518+00:00
instance: kb-cron
parent: logic-20260505-0
summary:
Modal logic is an extension of classical logic. In its original form, sometimes called "alethic modal logic", it introduces two new symbols: ◊ {\displaystyle \Diamond } expresses that something is possible while ◻ {\displaystyle \Box } expresses that something is necessary. For example, if the formula B ( s ) {\displaystyle B(s)} stands for the sentence "Socrates is a banker" then the formula ◊ B ( s ) {\displaystyle \Diamond B(s)} articulates the sentence "It is possible that Socrates is a banker". To include these symbols in the logical formalism, modal logic introduces new rules of inference that govern what role they play in inferences. One rule of inference states that, if something is necessary, then it is also possible. This means that ◊ A {\displaystyle \Diamond A} follows from ◻ A {\displaystyle \Box A} . Another principle states that if a proposition is necessary then its negation is impossible and vice versa. This means that ◻ A {\displaystyle \Box A} is equivalent to ¬ ◊ ¬ A {\displaystyle \lnot \Diamond \lnot A} . Other forms of modal logic introduce similar symbols but associate different meanings with them to apply modal logic to other fields. For example, deontic logic concerns


id: logic-20260505-8
file: data/en.wikipedia.org/wiki/Logic-8.md
category: reference
title: Logic
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Logic
date_saved: 2026-05-05T03:56:39.118518+00:00
instance: kb-cron
parent: logic-20260505-0
summary:
Informal logic is usually carried out in a less systematic way. It often focuses on more specific issues, like investigating a particular type of fallacy or studying a certain aspect of argumentation. Nonetheless, some frameworks of informal logic have also been presented that try to provide a systematic characterization of the correctness of arguments. The pragmatic or dialogical approach to informal logic sees arguments as speech acts and not merely as a set of premises together with a conclusion. As speech acts, they occur in a certain context, like a dialogue, which affects the standards of right and wrong arguments. A prominent version by Douglas N. Walton understands a dialogue as a game between two players. The initial position of each player is characterized by the propositions to which they are committed and the conclusion they intend to prove. Dialogues are games of persuasion: each player has the goal of convincing the opponent of their own conclusion. This is achieved by making arguments: arguments are the moves of the game. They affect to which propositions the players are committed. A winning move is a successful argument that takes the opponent's commitments as premises and shows how one's own conclusion follows


id: logic-20260505-9
file: data/en.wikipedia.org/wiki/Logic-9.md
category: reference
title: Logic
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Logic
date_saved: 2026-05-05T03:56:39.118518+00:00
instance: kb-cron
parent: logic-20260505-0
summary:
The term "mathematical logic" is sometimes used as a synonym of "formal logic". But in a more restricted sense, it refers to the study of logic within mathematics. Major subareas include model theory, proof theory, set theory, and computability theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic. However, it can also include attempts to use logic to analyze mathematical reasoning or to establish logic-based foundations of mathematics. The latter was a major concern in early 20th-century mathematical logic, which pursued the program of logicism pioneered by philosopher-logicians such as Gottlob Frege, Alfred North Whitehead, and Bertrand Russell. Mathematical theories were supposed to be logical tautologies, and their program was to show this by means of a reduction of mathematics to logic. Many attempts to realize this program failed, from the crippling of Frege's project in his Grundgesetze by Russell's paradox, to the defeat of Hilbert's program by Gödel's incompleteness theorems. Set theory originated in the study of the infinite by Georg Cantor, and it has been the source of many of the most challenging and important issues in mathematical logic. They include Cantor's theorem, the status of the Axiom of Choice, the question


id: logic-20260505-10
file: data/en.wikipedia.org/wiki/Logic-10.md
category: reference
title: Logic
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Logic
date_saved: 2026-05-05T03:56:39.118518+00:00
instance: kb-cron
parent: logic-20260505-0
summary:
Logic was developed independently in several cultures during antiquity. One major early contributor was Aristotle, who developed term logic in his Organon and Prior Analytics. He was responsible for the introduction of the hypothetical syllogism and temporal modal logic. Further innovations include inductive logic as well as the discussion of new logical concepts such as terms, predicables, syllogisms, and propositions. Aristotelian logic was highly regarded in classical and medieval times, both in Europe and the Middle East. It remained in wide use in the West until the early 19th century. It has now been superseded by later work, though many of its key insights are still present in modern systems of logic. Ibn Sina (Avicenna) was the founder of Avicennian logic, which replaced Aristotelian logic as the dominant system of logic in the Islamic world. It influenced Western medieval writers such as Albertus Magnus and William of Ockham. Ibn Sina wrote on the hypothetical syllogism and on the propositional calculus. He developed an original "temporally modalized" syllogistic theory, involving temporal logic and modal logic. He also made use of inductive logic, such as his methods of agreement, difference, and concomitant variation, which are critical to the scientific method. Fakhr al-Din


id: mathematical-linguistics-20260505-0
file: data/en.wikipedia.org/wiki/Mathematical_linguistics-0.md
category: reference
title: Mathematical linguistics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Mathematical_linguistics
date_saved: 2026-05-05T03:56:40.346427+00:00
instance: kb-cron
parent: mathematical-linguistics-20260505-0
summary:
Mathematical linguistics is the application of mathematics to model phenomena and solve problems in general linguistics and theoretical linguistics. Mathematical linguistics has a significant amount of overlap with computational linguistics. == Discrete mathematics == Discrete mathematics is used in language modeling, including formal grammars, language representation, and historical linguistic trends. === Set theory === Semantic classes, word classes, natural classes, and the allophonic variations of each phoneme in a language are all examples of applied set theory. Set theory and concatenation theory are used extensively in phonetics and phonology. === Combinatorics === In phonotactics, combinatorics is useful for determining which sequences of phonemes are permissible in a given language, and for calculating the total number of possible syllables or words, based on a given set of phonological constraints. Combinatorics on words can reveal patterns within words, morphemes, and sentences. === Finite-state transducers === Context-sensitive rewriting rules of the form a → b / c _ d, used in linguistics to model phonological rules and sound change, are computationally equivalent to finite-state transducers, provided that application is nonrecursive, i.e. the rule is not allowed to rewrite the same substring twice. Weighted FSTs found applications in natural language processing, including machine translation,


