Provabilitylogic is a modal logic , in which the box or necessity operator is interpreted as it is provable that . The point is to capture the notion of a proof predicate of a reasonably rich formal theory , such as Peano arithmetic . There are a number of provability logics, some of which are covered in the literature mentioned in the References section. The basic system is generally referred to as GL for Kurt G del G del Martin Hugo L b L b or L or K4W. It can be obtained by adding the modal version of L b s theorem to the logic K or K4 . It was pioneered by Robert M. Solovay in 1976. Since then until his passing in 1996 the prime inspirer of the field was George Boolos . Significant contributions to the field have been made by Sergei Artemov, Lev Beklemishev, Giorgi Japaridze , Dick de Jongh , Franco Montagna, Vladimir Shavrukov, Albert Visser and others. Interpretability logic s present natural extensions of provabilitylogic. See also Interpretability logic Kripke semantics References George Boolos , The Logic of Provability . Cambridge University Press, 1993. http www.csc.villanova.edu japaridz Giorgi Japaridze and Dick de Jongh, http www.csc.villanova.edu japaridz Text prov.pdf The logic of provability . In Handbook of Proof Theory , S. Buss, ed. Elsevier, 1998, pp. 475 546 ... www.phil.uu.nl preprints preprints PREPRINTS preprint234.pdf Provabilitylogic . In http dx.doi.org 10.1007 1 4020 3521 7 3 Handbook of Philosophical Logic , D. Gabbay and F. Guenthner, eds., vol. 13, 2nd ed., pp. 189 360. Springer, 2005. Per Lindstr m , Provabilitylogic a short introduction . Theoria 62 1996 , pp. 19 61. Craig Smory ski, Self reference and modal logic . Springer, Berlin, 1985. Robert M. Solovay , Provability Interpretations of Modal Logic , Israel Journal of Mathematics, Vol. 25 1976 287 304. http plato.stanford.edu entries logicprovabilityProvabilitylogic , from the Stanford Encyclopedia of Philosophy . Category Modal logic Category Proof theory logic stub es L gica ... more details
In mathematical logic , the Hilbert Bernays provability conditions , named after David Hilbert and Paul Bernays , are a set of requirements for formalized provability predicates in formal theories of arithmetic Smith 2007 224 . These conditions are used in many proofs of Kurt G del s G del s second incompleteness theorem second incompleteness theorem . They are also closely related to axioms of provabilitylogic . The conditions Let T be a formal theory of arithmetic with a formalized provability predicate Prov n , which is expressed as a formula of T with one free number variable. For each formula &phi in the theory, let &phi be the G del number of &phi . The Hilbert Bernays provability conditions are If T proves a sentence &phi then then T proves Prov &phi . For every sentence , T proves Prov &rarr Prov Prov T proves that Prov &phi &rarr &psi and Prov &phi imply Prov &psi References Smith, Peter 2007 . An introduction to G del s incompleteness theorems . Cambridge University Press. ISBN 978 0 521 67453 9 mathlogic stub Category Mathematical logic ... more details
Other uses Philosophy sidebar Logic from the Greek wiktionary logik ref possessed of reason ... . Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy , mathematics , semantics , and computer science . Logic examines general forms which argument s may take, which forms are valid, and which are fallacies . In philosophy, the study of logic ... of valid inference s within some formal language . ref name stanford logic onthology Logic is also ... ref Logic was studied in several ancient civilizations, including India , ref For example, Nyaya ... at 2200 years. ref and Ancient Greece Greece . Logic was established as a discipline by Aristotle , who gave it a fundamental place in philosophy. The study of logic was part of the classical Trivium education trivium . Logic is often divided into two parts, inductive reasoning and deductive reasoning . Nature The concept of Argument form logical form is central to logic, it being held that the validity ... Aristotelian syllogistic logic and modern symbolic logic are examples of formal logics. Informal logic ... important branch of informal logic. The dialogues of Plato ref cite book author Plato authorlink ... isbn 0 14 015040 4 ref are good examples of informal logic. Mathematical formalism Formal logic is the study ... study of logic. Modern formal logic follows and expands on Aristotle. ref cite book author Aristotle ... Library year 2001 isbn 0 375 75799 6 chapter Posterior Analytics ref In many definitions of logic ... of informal logic vacuous, because no formal logic captures all of the nuance of natural language. Symbolic logic is the study of symbolic abstractions that capture the formal features of logical ... A. G. last Hamilton title Logic for Mathematicians publisher Cambridge University Press year 1980 isbn 0 521 29291 3 ref Symbolic logic is often divided into two branches propositional logic and predicate logic . Mathematical logic is an extension of symbolic logic into other areas, in particular to the study ... more details
Interpretability logics comprise a family of modal logic s that extend provabilitylogic to describe interpretability and or various related metamathematical properties and relations such as weak interpretability , sub 1 sub conservativity, cointerpretability , tolerance in logic tolerance , cotolerance and arithmetic complexities. Main contributors to the field are Alessandro Berarducci, Petr H jek, Konstantin Ignatiev, Giorgi Japaridze , Franco Montagna, Vladimir Shavrukov, Rineke Verbrugge, Albert Visser and Domenico Zambella. References http www.csc.villanova.edu japaridz Giorgi Japaridze and Dick de Jongh, The Logic of Provability . In Handbook of Proof Theory , S.Buss, ed. Elsevier, 1998, pp. 475 546. Category Modal logic ... more details
Merge to Bayesian logic discuss Talk Probabilistic logic Merger proposal date March 2011 The aim of a probabilistic logic or probability logic is to combine the capacity of probability theory to handle uncertainty with the capacity of deductive logic to exploit structure. The result is a richer and more expressive formalism with a broad range of possible application areas. Probabilistic logic is a natural extension of traditional logic truth tables the results they define are derived through probabilistic ... proposals for probabilistic logics The term probabilistic logic was first used in a paper by Nils ... ref name N86 Nilsson, N. J., 1986, Probabilistic logic, Artificial Intelligence 28 1 71 ... of provability , which generalizes the concepts of ordinary logical entailment for math V math and classical ... or epistemic probabilities as a general notion for both logical entailment provability and probability . The idea is to augment standard propositional logic by considering an epistemic operator ... proposed by fuzzy logic can be used to obtain a logic in which the models are the probability distributions ... Logic, Artificial Intelligence 70 1 2 33 52. ref . In such a logic the question of the consistency ... logic ref name JO1 J sang, A., 2001, A logic for uncertain probabilities, International Journal ... Logic, Journal of Multiple Valued Logic and Soft Computing , 15 1 , pp.5 38, 2008. ref . Possible ... logic Imprecise probability Logic , Deductive logic , Non monotonic logic Probabilistic database Probability , Probability theory Probabilistic argumentation Reasoning Subjective logic Uncertainty Upper ... Logic . CSLI Publications Univ. of Chicago Press . Bacchus, F., 1990. Representing and reasoning ... . Number 166 in Mathematics Studies. North Holland. H jek, A., 2001, Probability, Logic, and Probability Logic, in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic , Blackwell. Henry E. Kyburg, Jr. Kyburg, H. E. , 1970. Probability and Inductive Logic Macmillan. Kyburg, H. E., 1974. The Logical ... more details
Provabilitylogic Quantification Quantum logic Q.E.D. Quod erat demonstrandum Reductio ad absurdum ...Outline of logic header Logic is the formal science of using reason . It is considered a branch of both philosophy and mathematics . One of the aims of logic is to identify the correct or validity valid ... argument arguments . Logic investigates and classifies the structure of statements and arguments ... language . The scope of logic can therefore be very large, ranging from core topics such as the study ... reasoning, and arguments involving causality . Foundations of logic Main Philosophy of logic Philosophical logic Columns width 270px col1 Analytic synthetic distinction Antinomy A priori and a posteriori ... Probability col4 Quantification Reason Reasoning Reference Semantics Syntax logic Truth Truth value Validity Traditional logic Main Term logic Classical logic Columns width 270px col1 Baralipton Baroco Bivalence Boolean logic Boolean valued function Categorical proposition Commutativity of conjunction ... Polysyllogism Port Royal Logic Premise col4 Prior Analytics Relative term Sorites paradox Square of opposition Sum of Logic Syllogism Term logic Tetralemma Truth function Informal logic and critical thinking Main Informal logic Critical thinking Columns width 270px col1 Argument Accuracy and precision ... thinking Informal logic Inquiry Interpretive discussion Narrative logic Occam s razor Opinion Practical ... of justification Topical logic Vagueness Weak mindedness Fallacies Main List of fallacies Formal Fallacy Informal Fallacy Relevance fallacies Formal and mathematical logic Main Formal logic Mathematical logic Mathematical logic, symbolic logic and formal logic are largely, if not completely synonymous ... logical validity is being studied. Symbols and strings of symbols Logical symbols Main Table of logic ... variable Predicate variable Literal mathematical logic Literal Metavariable col2 Logical constant ... Axiom Atomic sentence Clause logic Contingency philosophy Contingent proposition Contradiction ... more details
saved book title Logic and Metalogic subtitle cover image cover color Logic and Metalogic Main article Logic History History of logic Topics in logic Term logic Aristotelian logic Propositional calculus Predicate logic Modal logic Informal logic Mathematical logic Algebraic logic Multi valued logic Fuzzy logic Metatheory Metalogic Philosophical logicLogic in computer science Controversies in logic Principle of bivalence Paradoxes of material implication Paraconsistent logic Is logic empirical? Category Wikipedia books on logicLogic ... more details
logic syntactic component , which among other things determines the notion of formal proof provability ...In logic , the term decidable refers to the existence of an effective method for determining membership in a set of formulas. Logical system s such as propositional logic are decidable if membership in their set ... mathematical logic theory set of formulas closed under logical consequence in a fixed logical ... in the context of first order logic where G del s completeness theorem establishes the equivalence of semantic and syntactic consequence. In other settings, such as linear logic , the syntactic consequence provability relation may be used to define the theorems of a system. A logical system is decidable ... system. For example, propositional logic is decidable, because the truth table truth table ... order logic is not decidable in general in particular, the set of logical validities in any signature logic signature that includes equality and at least one other predicate with two or more arguments ... logic, such as second order logic and type theory , are also undecidable. The validities of monadic predicate calculus with identity are decidable, however. This system is first order logic restricted ... alone. For example, ternary logic Kleene s logic has no theorems at all. In such cases, alternative ... s, or the logical consequence consequence relation , A A of the logic. Decidability of a theory A theory mathematical logic theory is a set of formulas, which here is assumed to be closed ... logic, although the set of validities the smallest theory is decidable. A consistent theory which has ... of first order logic is semi decidable, but not decidable. In this case, it is because there is no effective ... used as a synonym for independence mathematical logic independent statement . See also Portal Logic ... Introduction to first order logic title Handbook of Mathematical Logic publisher North Holland location Amsterdam series Studies in Logic and the Foundations of Mathematics isbn 978 0 444 86388 ... more details
Dynamic logic may mean In modal logic, dynamic logic modal logic is a modal logic for reasoning about dynamic behaviour in digital electronics, dynamic logic digital logic is used for circuit design disambig ... more details
Symbolic logic may refer to First order logic , a system of formal logic Mathematical logic , a field of mathematics mathdab Category Logic ... more details
Possible worlds Provabilitylogic Regular modal logic Two dimensionalism Modal verb Col 4 of 4 Portal ... entries logicprovabilityProvabilityLogic by Rineke Verbrugge. Edward N. Zalta , 1995, http mally.stanford.edu ...gallery File DavidLewis2.jpg thumb David Lewis gallery Refimprove date August 2008 Modal logic is a type of mathematical logic Formal logic formal logic that extends the standards of formal logic to include ... modes or moods or modalities represented in modal logic, namely, Logical possibility possibility ... modal logic represents modalities using modal operator s. For example, It might rain today and It is possible that rain will fall today both contain the notion of possibility. In a modal logic this is represented ... Diamond math for Possibly . In a classical modal logic , each can be expressed by the other with negation ... that it will not rain today. Development of modal logic Although Aristotle s logic is almost entirely ... seen as anticipations of modal logic and its connection with potentiality and time. Modal logic as a self ... essence and accident philosophy accident . C. I. Lewis founded modern modal logic in his 1910 Harvard ... Symbolic Logic with C. H. Langford , which introduced the five systems S1 through S5 . Ruth C. Barcan later Ruth Barcan Marcus developed the first axiomatic systems of quantified modal logic &mdash ... modern temporal logic , closely related to modal logic, in 1957 by adding modal operators F and P meaning henceforth and hitherto . Vaughan Pratt introduced dynamic logic modal logic dynamic logic in 1976. In 1977, Amir Pnueli proposed using temporal logic to formalise the behaviour of continually operating concurrent programs. Flavors of temporal logic include propositional dynamic logic PDL , propositional linear temporal logic PLTL , linear temporal logic LTL , computational tree logic CTL , Hennessy Milner logic Hennessy&ndash Milner logic , and T . The mathematical structure of modal logic, namely Boolean algebra structure Boolean algebra s augmented with unary operation s often called ... more details
Robert M. author link Robert M. Solovay title Provability Interpretations of Modal Logic journal ...Mathematical logic also known as symbolic logic is a subfield of mathematics with close connections to foundations of mathematics , theoretical computer science and philosophical logic . ref Undergraduate ... study of logic and the applications of formal logic to other areas of mathematics. The unifying themes in mathematical logic include the study of the expressive power of formal system s and the deductive power of formal mathematical proof proof systems. Mathematical logic is often divided ... share basic results on logic, particularly first order logic , and definable set definability . In computer ... logic encompasses additional topics not detailed in this article see logic in computer science for those. Since its inception, mathematical logic has contributed to, and has been motivated by, the study ... Mathematical logic emerged in the mid 19th century as a subfield of mathematics independent of the traditional study of logic CITEREFFerreir.C3.B3s2001 Ferreir s 2001 , p. 443 . Before this emergence, logic was studied with rhetoric , through the syllogism , and with philosophy . The first ... the foundations of mathematics. Early history See History of logic Sophisticated theories of logic were developed in many cultures, including Logic in China China , Logic in India India , Logic in Greece Greece and the Logic in Islamic philosophy Islamic world . In the 18th century, attempts to treat the operations of formal logic in a symbolic or algebraic way had been made by philosophical mathematicians ... and little known. 19th century symbolic logic In the middle of the nineteenth century, George Boole and then Augustus De Morgan presented systematic mathematical treatments of logic. Their work, building ... of logic into a sufficient framework for the study of foundations of mathematics CITEREFKatz1998 ... Frege presented an independent development of logic with quantifiers in his Begriffsschrift , published ... more details
In mathematics, logic can refer to consistent theory logiclogic , an infinitary extension of first order logiclogic , a deductive system in set theory developed by Hugh Woodin mathdab ... more details
Unreferenced date December 2006 For the subject in computer programming dynamic logic modal logic In integrated circuit design, dynamic logic or sometimes clocked logic is a design methodology logic family in Digital circuit digital logic that was popular in the 1970s and has seen a recent resurgence ... . Dynamic logic is distinguished from so called static logic in that it uses a clock signal in its implementation of combinational logic circuits. The usual use of a clock signal is to synchronize transitions in sequential logic circuits, and for most implementations of combinational logic a clock signal is not even needed. Terminology In the context of logic design, the term dynamic logic is more commonly used as compared to clocked logic , as it makes clear the distinction between this type of design and static logic . To additionally confuse the matter, clocked logic is sometimes used as a synonym for sequential logic . This usage is nonstandard and should be avoided. Static versus dynamic logic Advert section date October 2010 The largest difference between static and dynamic logic is that in dynamic logic, a clock signal is used to evaluate combinational logic . However, to truly comprehend the importance of this distinction, the reader will need some background on static logic. In most types of logic design, termed static logic , there is at all times some mechanism to drive the output either high or low. In many of the popular logic styles, such as Transistor transistor logic ... not qualify as distinct from static logic. In contrast, in dynamic logic , there is not always a mechanism ... high or low during distinct parts of the clock cycle. Dynamic logic requires a minimum clock rate .... Static logic has no minimum clock rate the clock can be paused indefinitely. While it may seem ... CPUs use dynamic logic ref http www.anandtech.com show 1647 11 ref , only CPUs designed with fully ... logic, when properly designed, can be over twice as fast as static logic. It uses only the faster ... more details
Philosophy sidebar The history of logic is the study of the development of the science of valid inference logic . While many cultures have employed intricate systems of reasoning, and logical methods are evident ... in three traditions those of Logic in China China , Indian logic India , and Greek philosophy Greece . Of these, only the treatment of logic descending from the Greek tradition, particularly Aristotelian logic , found wide application and acceptance in science and mathematics. Logic was known as dialectic or analytic in Ancient Greece. Aristotle s logic was further developed by Logic in Islamic ... as barren by historians of logic. ref name ReferenceA Oxford Companion p. 498 Bochenski, Part I Introduction, passim ref Logic was revived in the mid nineteenth century, at the beginning of a revolutionary ... or mathematical logic during this period is the most significant in the two thousand year history of logic .... ref name Oxford Companion p. 500 Oxford Companion p. 500 ref Progress in mathematical logic in the first ... and Alfred Tarski Tarski , had a significant impact on analytic philosophy and philosophical logic , particularly from the 1950s onwards, in subjects such as modal logic , temporal logic , deontic logic , and relevance logic . Prehistory of logic File All Gizah Pyramids.jpg alt The four great pyramids ... has been employed in all periods of human history. However, logic studies the principles of valid reasoning ... Babylonian astronomers in the 8th and 7th centuries BC employed an internal logic within their predictive ... Logic in Greek philosophy Before Plato While the ancient Egyptians empirically discovered some truths ... and falsity. ref Kneale, p. 16 ref Plato s logic File Academia mosaic.jpg alt Mosaic seven men standing ... Plato 428 347 include any formal logic, ref Kneale p. 17 ref but they include important contributions to the field of philosophical logic . Plato raises three questions What is it that can ... plato.stanford.edu entries aristotle logic Def Aristotle s Logic . Stanford University , 18 March ... more details
dablink Not to be confused with intensional logic with an s rather than a t in the initial word . Intentional Logic A Logic Based on Philosophical Realism is a book by Henry Babcock Veatch published in 1952. book stub Category Philosophy books Category Logic literature ... more details
Unreferenced stub auto yes date December 2009 Strict logic is essentially synonymous with relevant logic , though it can be characterized proof theory proof theoretically as ordinary logic without weakening , or linear logic with Idempotency of entailment contraction . See also Substructural logic DEFAULTSORT Strict Logic Category Substructural logicLogic stub ... more details
In mathematical logic and rewriting system s, terms are expressions which can be obtained from variable logic variables and function symbol logic function symbols . Terms serve to denote objects. See also Term first order logic Mathlogic stub Category Mathematical logic Category Rewriting systems ... more details
Wiktionarypar logicLogic may refer to Science and technology Logic , the study of the principles and criteria of valid inference and demonstration Mathematical logic , a branch of mathematics that grew out of symbolic logic Philosophical logic Digital logic , a class of digital circuits characterized by the technology underlying its logic gates Software Logic Studio , a music production suite by Apple Inc. Logic Pro , a MIDI sequencer and Digital Audio Workstation application, part of Logic Studio Dolby Pro Logic , also known as Pro Logic, a surround sound processing technology See also Logarithm disambig el he ko nl Logica doorverwijspagina ja ... more details
Binary logic could refer to any two valued logic , especially in social sciences classical propositional logic propositional two valued logic, also called boolean logic in engineering, which is the logical foundation of digital electronics circuits implementing boolean logic see logic gate s Should not to be confused with binary numeral system . dab ... more details
Logic system may refer to A type of Formal system Logic System, a musical project of Japanese composer and programmer Hideki Matsutake disambig ... more details
In mathematical logic , a logical system has the erasure property if and only if no subset of the propositions can be added to another subset of the propositions to refute a consequence. For instance, if proposition A means the store is open from 8 00 to 22 00 and proposition B means except Tuesdays , the system AB does NOT have erasure. See also Monotonic logic in mathematical logic http plato.stanford.edu entries peirce logic Peirce s Logic at the Stanford Encyclopedia of Philosophy mathlogic stub Category Mathematical logic ... more details
For the subject in digital circuit s also known as clocked logic dynamic logic digital logic Dynamic logic is an extension of modal logic originally intended for reasoning about computer programs and later .... Language Modal logic is characterized by the modal operator s math Box p math box p asserting that math ... is possibly the case. Dynamic logic extends this by associating to every action math a , math the modal operators math a , math and math langle a rangle , math , thereby making it a multimodal logic ... exists , math quantifiers. Dynamic logic permits compound actions built up from smaller actions. While ... s regular expression operators are a good match to modal logic. Given actions math a , math and math ... but does terminate. Axioms These operators can be axiomatized in dynamic logic as follows, taking as already given a suitable axiomatization of modal logic including such axioms for modal operators ... n generalized to arbitrary actions math a , math . Derivations The modal logic axiom math a p equiv ... between implication and inference is the same in dynamic logic as in any other logic whereas the implication ... the dynamic nature of dynamic logic moves this distinction out of the realm of abstract axiomatics ... 1, and therefore is not valid. Derived rules of inference As for modal logic, the inference rules modus ponens and necessitation suffice also for dynamic logic as the only primitive rules it needs, as noted above. However, as usual in logic, many more rules can be derived from these with the help of the axioms. An example instance of such a derived rule in dynamic logic is that if kicking a broken ... is broken, dynamic logic expresses this inference as math b to k b vdash b to k b , math , having as premise ... x , math is 8 to begin with, or 6.5, whence this proposition is not a theorem of dynamic logic ... referential opacity of modal logic in the case when a modality can interfere with a substitution. When ... manner of first order logic to obtain Peano s celebrated axiom math Phi 0 land forall i Phi ... more details
Hybrid logic refers to a number of extensions to propositional logic propositional modal logic with more expressive power, though still less than first order logic . In formal logic , there is a trade off between expressiveness and computational tractability how easy it is to computer compute automated reasoning reason with logical languages . The history of hybrid logic began with Arthur Prior s work in tense logic. ref cite web url http plato.stanford.edu entries logic hybrid title Hybrid Logic author Torben Bra ner date 2008 work Stanford Encyclopedia of Philosophy accessdate 1 February 2011 ref Unlike ordinary modal logic, hybrid logic makes it possible to refer to states possible worlds in formulas. This is achieved by a class of formulas called nominals , which are true in exactly one state, and by the use of the operator, which is defined as follows sub i sub p is true iff if and only if p is true in the unique state named by the nominal i i.e., the state where i is true . Hybrid logics with extra or other operators exist, but is more or less standard. Hybrid logics have many features in common with temporal logic s which use nominal like constructs to denote specific points in time , and they are a rich source of ideas for researchers in modern modal logic. They also have applications in the areas of feature logic , model theory , proof theory , and the logical analysis of natural language . It is also deeply connected to description logic because the use of nominals allows one to perform assertional ABox reasoning, as well as the more standard terminological TBox reasoning. References reflist Further reading P. Blackburn. 2000. Representation, reasoning and relational structures a hybrid logic manifesto. Logic Journal of the IGPL , 8 3 339 365. External links http hylo.loria.fr Hybrid Logics Home Page http plato.stanford.edu entries logic hybrid Stanford Encyclopedia of Philosophy entry on Hybrid Logic Category Modal logiclogic stub ... more details
Affine logic is a substructural logic whose proof theory rejects the structural rule of Idempotency of entailment contraction . It can also be characterized as linear logic with weakening . The name affine logic is associated with linear logic , to which is differs by allowing the weakening rule. Jean Yves Girard introduced the name as part of the geometry of interaction semantics of linear logic, which characterises linear logic in terms of linear algebra here he alludes to affine transformation s on vector spaces. ref Jean Yves Girard , 1997. http www.seas.upenn.edu sweirich types archive 1997 98 msg00134.html Affine . Message to the TYPES mailing list. ref The logic predated linear logic. V. N. Grishin used this logic in 1974, ref Grishin, 1974, and later, Grishin, 1981. ref after observing that Russell s paradox cannot be derived in a set theory without contraction, even with an unbounded comprehension axiom . ref Cf. Frederic Fitch s demonstrably consistent set theory ref Likewise, the logic formed the basis of a decidable subtheory of predicate logic , called Direct logic Ketonen & Wehrauch, 1984 Ketonen & Bellin, 1989 . Affine logic can be embedded into linear logic by rewriting the affine arrow math A rightarrow B math as the linear arrow math A circ B otimes top math . Whereas full linear logic ie. propositional linear logic with multiplicatives, additives and exponentials is undecidable, full affine logic is decidable. Affine logic forms the foundation of ludics . Notes references References V.N. Grishin, 1974. A nonstandard logic and its application to set theory, Russian . Studies in Formalized Languages and Nonclassical Logics Russian , 135 171. Izdat, Nauka, Moskow. . V.N. Grishin, 1981. Predicate and set theoretic calculi based on logic without contraction ... on Direct Logic. In Linear Logic and its Implementation . See also Strict logic and relevant logic Category Substructural logiclogic stub ... more details
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