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# Topic: Classical logic

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 [No title]   (Site not responding. Last check: 2007-10-07) Logic is traditionally divided into deductive reasoning, concerned with what follows logically from given premises, and inductive reasoning, concerned with how we can go from some number of observed events to a reliable generalization. Some have considered classical logic to be just like a mathematical theory, and in particular the laws of non-contradiction and the excluded middle to be simply axioms of the theory, which have to be assumed without proof. Logics such as fuzzy logic have since been devised with an infinite number of "degrees of truth", e.g., represented by a real number between 0 and 1. www.informationgenius.com /encyclopedia/l/lo/logic_1.html   (1758 words)

 On the Formulae-as-Types Correspondence for Classical Logic   (Site not responding. Last check: 2007-10-07) The possibility of a similar formulae-as-types correspondence for classical logic looks to be a seminal development in this area, but whilst promising results have been achieved, there does not appear to be much agreement of what is at stake in claiming that such a correspondence exists. By regarding the rules which determine the deductive strength of classical logic as structural rules, as opposed to the logical rules associated with specific logical connectives, we extend Prawitz's inversion principle to classical propositional logic, formulated in a theory of Parigot's lambda-mu calculus with eta expansions. Our treatment is the first treatment of induction in classical arithmetic that truly falls under the aegis of the formulae-as-types correspondence, as it is the first that is consistent with the intensional reading of propositional equality. www.linearity.org /cas/thesis   (314 words)

 Classical Free: LOGIC Often the definition of Logic includes the word "correct", as in "Logic, the art of correct reasoning," or "necessary," as in "Logic, the science of necessary inference." While these definitions are useful in their own right, they weight the study of Logic too heavily in the direction of only one kind of human reasoning--the deductive. To define Logic as the art of necessary inference is to rule out the inductive reasoning that leads us to many of the general statements we deduce from in deductive thought. Logic and reason are a gift from God, a gift by which we interact with the world and His Word. www.classicalfree.org /tgc_intro_logic.asp   (1819 words)

 COMPUTABILITY LOGIC: a theory of interactive computation HOMEPAGE Technically CL is a game logic: it understands interactive computational problems as games played by a machine against the environment, their computability as existence of a machine that always wins the game, logical operators as operations on computational problems, and validity of a logical formula as being a scheme of "always computable" problems. In particular, the classical notion of truth turns out to be nothing but computability restricted to the formulas of the classical fragment of the universal language, which makes classical logic a natural syntactic fragment of the universal logic. Computability logic is a formal theory of (interactive) computability in the same sense as classical logic is a formal theory of truth. www.cis.upenn.edu /~giorgi/cl.html   (3951 words)

 Bryn Mawr Classical Review 97.7.27 Barnes correctly infers from Seneca's often extravagant hostility to logic and from the discouragement he offers to Lucilius that logic was an extremely popular philosophical theme in the early empire, a point which emerges even more clearly from his analysis of Epictetus. Logical theory for its own sake, like the commentary-based syllabus of the contemporary philosophical schools, does nothing to establish what is true or to improve the moral quality of our lives. Hypothetical arguments, Barnes argues, are not the province of a separate kind of logic (as Peripatetic hypothetical syllogisms might be thought to be), but rather the dialectical application of perfectly normal Stoic logic, where the second premiss of a modus ponens argument is asserted hypothetically within the context of a dialectical encounter. ccat.sas.upenn.edu /bmcr/1997/97.07.27.html   (1721 words)

 Classical Logic Classical logic provides the basis by which we assess the external world in our daily lives. This proved to be impossible by any logical process but later, the theologian Kierkegaard proposed that logic could be abandoned altogether when dealing with propositions involving God or attributes of the human soul. In contrast, historic Christianity stands on the beliefs that classical logic represents the “truth of God” and sound reason is the foundation of the Christian faith. www.christianapologetic.org /classical_logic.htm   (472 words)

 Intuitionistic Logic Intuitionistic logic encompasses the principles of logical reasoning which were used by L. Brouwer in developing his intuitionistic mathematics, beginning in [1907]. Classical logic is finitistically interpretable in the negative fragment of intuitionistic logic. While identity can of course be added to intuitionistic logic, for applications (e.g., to arithmetic) the equality symbol is generally treated as a distinguished predicate constant satisfying nonlogical axioms (e.g., the primitive recursive definitions of addition and multiplication) in addition to reflexivity, symmetry and transitivity. plato.stanford.edu /entries/logic-intuitionistic   (6042 words)

 Logical Pluralism | Log This situation has the absurd consequence that one might concede that the conclusion of an argument was true (since the argument had true premises and was truth-preserving); yet should refuse to infer the conclusion from the premises, in the absence of demonstration of the relevance of the premises to the conclusion. His paper on the role of logic in AI is very clear, and contains an excellent discussion of the role of logic as a determiner of valdi inference, as opposed to logic as a tool for constructing or analysing derivations. Different logics are given by choosing different syntactic units (is identity one of them? is necessity?) or by choosing different fixed interpretations of the syntactic units so chosen. pluralism.pitas.com   (9922 words)

 Scholars' Online Academy Logic: Christian classical education for homeschool and home education.   (Site not responding. Last check: 2007-10-07) (Logic) is the art which directs the very act of reason, that is, through which one may advance with order, ease, and correctness in the act of reason itself. Formal Logic, sometimes called "Minor Logic", deals with the structure of reasoning and thus is concerned chiefly with the method of deriving one truth from another. Logic is concerned merely with the fidelity and accuracy with which a certain process is performed, a process which can be performed with any materials, with any assumption. www.islas.org /classes/logic.html   (445 words)

 Liar Paradox [Internet Encyclopedia of Philosophy] It lacks a classical truth value as does the odd sentence "The present king of France is bald." Kripke trades infinite syntactic complexity for infinite semantic complexity. Some of the solutions to the Liar Paradox require a revision in classical logic, the formal logic in which sentences of a formal language have exactly two possible truth values (TRUE, FALSE), and in which the usual rules of inference allow one to deduce anything from an inconsistent set of assumptions. Some logicians argue that classical logic is not the incumbent which must remain in office unless an opponent can dislodge it, although this is gospel for other philosophers of logic (probably because of the remarkable success of two-valued logic in expressing most of modern mathematical inference). www.iep.utm.edu /p/par-liar.htm   (3427 words)

 HYLE 5-1 (1999): Atomism and the Reasoning by a Non-Classical Logic Moreover, this discovery proves that non-classical logic is not a side effect of just linguistic relevance in scientific theories, but it pertains to the core of the most advanced physical theories of the present century. It is clear why the discovery of non-classical logic in a physical theory by Birkhoff and von Neumann was not bypassed by physicists following classical logic: they indeed recognized the relevance of non-classical logic, since for the first time a DNS concerns the definition of a state of the system at issue. Here, on the basis of an analysis of original texts, we suggested that classical chemistry — as well as some of the theories belonging to physics and mathematics — has been shaped by authoritative scientists through a kind of logic which is at radical variance with that of the tradition of Newtonian mechanics. www.hyle.org /journal/issues/5/drago.htm   (4421 words)

 Topics in Logic: Extensions and Alternatives to Classical Logic   (Site not responding. Last check: 2007-10-07) Modal Logic (the logic of possibility and necessity; add sentential operators for 'it is necessary that' and 'it is possible that', where the latter can be defined as 'it is not necessary that it is not the case that'. Deontic Logic (logic of obligation: add sentential operators for 'it is obligatory that' and 'it is permissible that', where permissibility can be defined as 'not obligatory that not:' Again the main interest may be philosophical, in ethics. Paraconsistent Logic (logics that allow contradictions, sentences of the form P and ~P, to be true. www.trinity.edu /cbrown/topics_in_logic/modifications.html   (596 words)

 Fuzzy Logic For example, considering a set of tall people in the classical logic, one has to decide where is the border between the tall people and people that are not tall. Fuzzy logic seems to be a general case for the classical logic and as such it does not present any better solutions for problems that might be easily solved using the "crisp" sets. Fuzzy logic seems to be a general case for the classical logic. www-pub.cise.ufl.edu /~ddd/cap6635/Fall-97/Short-papers/24.htm   (1613 words)

 Classical Logic One desideratum of the enterprise is that the logical structures in the regimented language should be transparent. Logic books aimed at mathematicians are likely to contain function letters, probably due to the centrality of functions to mathematical discourse. The rules in D are chosen to match logical relations concerning the English analogues of the logical terminology in the language. plato.stanford.edu /entries/logic-classical   (11934 words)

 Prof. David Harel - Books In Dynamic Logic, such programs are first-class objects on a par with formulas, complete with a collection of operators for forming compound programs inductively from a basis of primitive programs. Apart from the obvious heavy reliance on classical logic, computability theory and programming, the subject has its roots in the work of Thiele [198] and Engeler [42] in the late 1960's, who were the first to advance the idea of formulating and investigating formal systems dealing with properties of programs in an abstract setting. Dynamic Logic, which emphasizes the modal nature of the program/assertion interaction, was introduced by Pratt in 1976 [162]. www.wisdom.weizmann.ac.il /~dharel/dynamic_logic.html#preface   (848 words)

 translating classical to linear logic (338 lines) The second one (you called it Girard's Second Translation) is a translation of cut-free classical logic, based on the idea of translating positive and negative occurrences of the same classical connective into different linear connectives. They have the property of translating faithfully proofs in a two-sided version of sequent- calculus for classical logic into proofs in two sided sequent- calculus for classical linear logic (faithful in the sense that the 'skeleton' of the derivation remains unchanged); moreover cut-free proofs remain cut-free after translation. It is true that I could normalize classical proofs before translating them, but this procedure makes difficult to <> which program you obtain out of a classical proof. www.cis.upenn.edu /~bcpierce/types/archives/1992/msg00085.html   (1127 words)

 G. Aldo Antonelli --- Curriculum Vitae   (Site not responding. Last check: 2007-10-07) Logicism without Logic (joint work with Robert May), 2002 annual meeting of the Association for Symbolic Logic, Las Vegas, June 1-4, 2002 (special session on the philosophy of mathematics). General Extensions for Default Logic, Seminar in Applications of Logic, CUNY Graduate Center, March 19, 1996; and Logic Colloquium, Group in Logic and the Methodology of Science, University of California, Berkeley, November 8, 1996. Topics in Logic: (1) the logic of agency; (2) the metatheory of PA; Winter Quarter 2002. www.hamclub.uci.edu /~aldo/vita.html   (1966 words)

 Wadler: Linear Logic This paper introduces a new way of attaching proof terms to proof trees for classical linear logic, which bears a close resemblance to the way that pattern matching is used in programming languages. The presentation of linear logic is simplified by basing it on the Logic of Unity. Past attempts to apply Girard's linear logic have either had a clear relation to the theory (Lafont, Holmström, Abramsky) or a clear practical value (Guzmán and Hudak, Wadler), but not both. homepages.inf.ed.ac.uk /wadler/topics/linear-logic.html   (1065 words)

 Introduction to Schola Classical Tutorials   (Site not responding. Last check: 2007-10-07) Schola Classical Tutorials offers live group tutorials over the internet in the subjects of a classical liberal arts curriculum: the classical languages, the great books of literature and history, and rhetoric. Schola is dedicated not only to the idea of classical Christian education but also to furthering it through the use of low-cost internet technologies available to most families. Schola's tutorials are offered for students who are willing to devote themselves to a course of serious liberal arts study before they enter college or the world of employment and family. www.schola-tutorials.com   (437 words)

 Classical Two-valued Logic   (Site not responding. Last check: 2007-10-07) A sentence S of L is a logical consequence of a set of sentences Ss of L (Ss = S), if S is true in every structure in which all of the members of Ss are true. The purpose of the completeness theorem is to show that every logical consequence of a set of nonlogical axioms can be proved from these nonlogical axioms by means of the logical axioms and rules. Construct an alternate semantics for propositional logic based on the idea that the valuation function maps atomic formulas to elements in the model. cs.wwc.edu /~aabyan/Logic/Classical.html   (1759 words)

 Re: Classical Logic and Coq   (Site not responding. Last check: 2007-10-07) At 15:02 27/04/01 +0200, Herman Geuvers wrote: > I have just completed a short proof of the fact that a strong form of > classical logic is inconsistent in Coq. I would personally not call it an inconsistency of a strong form of classical logic, but rather an internal proof of undecidability of the underlying logic. But it ought definitely be pointed out to users who may be tempted to mix classical reasoning with constructive operators in higher order logic, with catastrophic effects indeed. www.seas.upenn.edu /~sweirich/types/archive/1999-2003/msg00665.html   (110 words)

 Intuitionistic Completeness and Classical Logic, D. C. McCarty We show that, if a suitable intuitionistic metatheory proves that consistency implies satisfiability for subfinite sets of propositional formulas relative either to standard structures or to Kripke models, then that metatheory also proves every negative instance of every classical propositional tautology. Since reasonable intuitionistic set theories such as HAS or IZF do not demonstrate all such negative instances, these theories cannot prove completeness for intuitionistic propositional logic in the present sense. 121 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1988. projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.ndjfl/1074396309   (279 words)

 DI & CoS - Classical and Intuitionistic Logic This freedom in applying inference rules allows to express logical systems that are difficult or impossible to express in the cut-free sequent calculus and it also allows for a more fine-grained analysis of derivations than the sequent calculus. This paper presents a system for intuitionistic logic in which all the rules are local, in the sense that, in applying the rules of the system, one needs only a fixed amount of information about the logical expressions involved. Logic Journal of the Interest Group in Pure and Applied Logics, Vol. alessio.guglielmi.name /res/cos/CL   (1075 words)

 The Calculus of Structures - Classical Logic We present an inference system for classical first order logic in which each inference rule, including the cut, only has a finite set of premises to choose from. The main conceptual contribution of this paper is the possibility of separating different sources of infinite choice, which happen to be entangled in the traditional cut rule. The calculus of structures is a framework for specifying logical systems, which is similar to the one-sided sequent calculus but more general. www.wv.inf.tu-dresden.de /%7Eguglielm/Research/CL/CL.html   (669 words)

 christian classical homeschool? The classical book list entitled the "Great Books of the Western World" contains the writings of Plato, Sophocles, Aristotle, and other men who declare that the answers to life's mysteries and problems are found in men and not in God. The focus of the traditional classical approach is on Greek mythology, philosophy, logic, and Latin. Proponents of classical education defend the study of mythology (which is really the study of false gods, idols, and/or demons) by saying that the myths are an integral part of our Western literary heritage. homeschoolinformation.com /Approaches/classical.htm   (1789 words)

 [Coq-Club] intuitionistic vs. classical logic in computer science   (Site not responding. Last check: 2007-10-07) The idea was (I outrageously simplify) that stating things like "It is possible to (say) solve this or that kind of equations" is very difficult in a classical setting. One can show the algorithm and say: "this algorithm solves the problem." But the fact that this statement actualy gives the algorithm is implicit. Henk then argued that even going into calculability, the (classical) definition of Turing machines, etc, raised difficulties and was not totaly satisfactory. pauillac.inria.fr /pipermail/coq-club/2002/000609.html   (239 words)

 DI & CoS - FAQ-URR Take, say, sequent system LK for classical logic: it is faithfully simulated by system KS in the calculus of structures, without additional cost in complexity. The only exception is Schütte's presentation of logic, which uses deep inference, but doesn't drop the distinction between formula and sequent and does not exploit a premiss-conclusion symmetry (for example, its cut rule has two premisses). For systems without involutive negation, like intuitionistic logic, it is possible to use the calculus of structures in its present form by using polarities, or asymmetric logical relations like implication. alessio.guglielmi.name /res/cos/faq.html   (2118 words)

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