Geometry.net Online Store

1. Propositional Logic [Internet Encyclopedia Of Philosophy] Classical truthfunctional propositional logic is by far the most widely studied These are, of course, cornerstones of Classical propositional logic. http://www.iep.utm.edu/p/prop-log.htm

Propositional Logic Propositional logic , also known as sentential logic Table of Contents (Clicking on the links below will take you to those parts of this article) 1. Introduction 2. History 3. The Language of Propositional Logic a. Syntax and Formation Rules of PL ... 9. Suggestions for Further Reading 1. Introduction A statement can be defined as a declarative sentence, or part of a sentence, that is capable of having a truth-value, such as being true or false. So, for example, the following are statements: George W. Bush is the 43rd President of the United States. Paris is the capital of France. Everyone born on Monday has purple hair. Sometimes, a statement can contain one or more other statements as parts. Consider for example, the following statement: Either Ganymede is a moon of Jupiter or Ganymede is a moon of Saturn. While the above compound sentence is itself a statement, because it is true, the two parts, "Ganymede is a moon of Jupiter" and "Ganymede is a moon of Saturn", are themselves statements, because the first is true and the second is false. The term proposition is sometimes used synonymously with statement . However, it is sometimes used to name something abstract that two different statements with the same meaning are both said to "express". In this usage, the English sentence, "It is raining", and the French sentence "Il pleut", would be considered to express the same proposition; similarly, the two English sentences, "Callisto orbits Jupiter" and "Jupiter is orbitted by Callisto" would also be considered to express the same proposition. However, the nature or existence of propositions as abstract meanings is still a matter of philosophical controversy, and for the purposes of this article, the phrases "statement" and "proposition" are used interchangeably.

2. Propositional Calculus - Wikipedia, The Free Encyclopedia There is no third truthvalue, at least not in Classical logic. Classical propositional calculus as described above is equivalent to Boolean algebra, http://en.wikipedia.org/wiki/Propositional_calculus

var wgNotice = ""; var wgNoticeLocal = ""; var wgNoticeLang = "en"; var wgNoticeProject = "wikipedia"; Propositional calculus From Wikipedia, the free encyclopedia Jump to: navigation search This article may be too long Please discuss this issue on the talk page and help summarize or split the content into subarticles of an article series In logic and mathematics , a propositional calculus (or a sentential calculus ) is a formal system in which formulae representing propositions can be formed by combining atomic propositions using logical connectives , and a system of formal proof rules allows certain formul¦ to be established as "theorems". In general terms, a calculus is a formal system that consists of a set of syntactic expressions ( well-formed formul¦ or wffs ), a distinguished subset of these expressions (axioms), plus a set of formal rules that define a specific binary relation , intended to be interpreted as logical equivalence , on the space of expressions. When the formal system is intended to be a logical system , the expressions are meant to be interpreted as mathematical statements, and the rules, known as inference rules , are typically intended to be truth-preserving. In this setting, the rules (which may include

3. Classical Propositional Logic: Overview simple C++ software for manipulating Classical propositional logic expressions. I will be adding rough content and reworking it as I go along. http://cpl.wikidot.com/

@import url(/commonmodules/css/monetize/textlinkads/MonetizeTextLinkAdsModule.css); @import url(http://www.wikidot.com/commontheme/base/css/style.css?0); @import url(http://www.wikidot.com/commontheme/co/css/style.css?0); Classical Propositional Logic Expression Representation and Manipulation in C++ create account or login Overview Classical Propositional Logic Basic Expression Types ... Truth Function Representation Software Requirements ... Terminology ... Expression Construction ... Other C++ Source Code ... Class CPL_Expression Overview Purpose This is currently an experimental site, which I am constructing while trying to develop some (relatively!) simple C++ software for manipulating classical propositional logic expressions. I will be adding rough content and reworking it as I go along. Progress will be slow as this is just a part-time hobby for me. I will try to add new content at least once a week. My long-term aim is to publish my C++ source code and provide a baseline specification, interface and test environment for whoever may want to have a go at a more complex solution. Silvestro Fantacci Content Classical Propositional Logic Classical propositional logic is a truth-functional and two-valued propositional logic. Propositional logic deals with how propositions (or statements) are constructed/connected with each other.

4. Classical Propositional Logic Classical propositional logic. In this subsection, we shall demonstrate a simple way to prove a tautology in the Classical propositional logic using . http://unit.aist.go.jp/cvs/Agda/tutorial/node151.html

Next: Remark. The law of Up: Examples Previous: Exercises. Contents Index Classical propositional logic In this subsection, we shall demonstrate a simple way to prove a tautology in the classical propositional logic using . The law of excluded middle is a formula of the form , where is a some formula. In intuitionistic logic, the law of excluded middle is not generally provable. In general, there are two ways to prove a classical tautology in : One is to use Glivenko's theorem in propositional logic: is provable in classical logic if and only if is provable in Intuitionistic logic. The other is to define a package and to use a postulate representing the classical principle such as the law of excluded middle. In this subsection, we shall take the latter approach. Now we shall prove the proposition In , we express the proposition as follows: Since we like to use the law of excluded middle, let us open a new file and define the package named Classical as follows: The above postulate declares a new primitive constant em of the given type. Assume

5. What Is Classical Propositional Logic? J-Y. B`eziau, R.P. De The aim of this paper is to try to characterize Classical propositional logic (CPL) with the notion of mathematical structure. http://logic.ru/en/node/354

6. 2. A Fallibilistic Justification Of Classical Propositional Logic 2. A Fallibilistic Justification of Classical propositional logic. http://www.hf.uio.no/ifikk/filosofi/njpl/vol4no1/connexive/node5.html

Next: 2.1 Conventions Up: Connexive Logic Previous: 1.2 Adaptable Behaviour 2. A Fallibilistic Justification of Classical Propositional Logic Material inference rules are a constitutive aspect of factual orientation. The sequential scheme a a m b b n articulates this informal insight. Rules of this kind do not hold unless there are formal rules of orientation, too. In the first instance, logical relations between various pieces of information depend on the range of inference rules to which they are jointly subjected. However, the use of logical connectives allows articulation of logical relations not just between , but likewise within , assertions. Their introduction will result in a systematic account of formal, and, in particular, of logical inference rules. The present section is meant to justify this strictly inferential understanding of logic. Above, the concept of an inference rule was discussed in terms of epistemic conditions for its application. In this way reasons for a progressive differentiation of such rules could be proposed. This fallibilistic conception of inference figures will now lead to a justification of classical propositional logic and its connexive variant. For this purpose the subsequent argumentation will rely on the following conventions for the presentation of a logical calculus. In fact they have been applied tacitly in previous sections of this article. Subsections 2.1 Conventions

7. Philosophical Dictionary: CPL: Classical Propositional Logic Classical propositional logic A twovalued logic for statements involving the logical terms not , and and or that infers that p is true if not not http://www.maartensz.org/philosophy/Dictionary/L/Logic-CPL.htm

Help Index Maarten Maartensz L Logic-CPL Classical Propositional Logic A two-valued logic for statements involving the logical terms 'not', 'and' and 'or' that infers that 'p' is true if 'not not p' is true. 1. Introducuction: This article gives a truth-valuational semantics for Classic Propositional Logic , briefly CPL In the present treatment I try to be clear about the assumptions made and presume both English and basic mathematics including ordinary algebra of real numbers known to every reader who knows English and finished secondary education. This allows a very simple presentation of classical propositional logic in terms of basic mathematics and English. Part of the reason for this is that there is a far-going parallel between propositional logic and ordinary algebra that was first noticed by George Boole Ordinary algebra is a formal language with statements about numbers and variable terms like "x", "y" for numbers, that enable one to write statements like "x >= x" i.e. "x is greater or equal than x" etc.. Constant terms in algebra are terms for numbers like "0", "1" etc. If the the term "x" is uniformly by respectively "y" and "0" in the statement "x>=x", the statements "y>=y" and "0>=0" result. It should be noted that variables are terms that do not occur normally in ordinary language, for that contains no distinction between variables and constants.

8. Category:MSC2000 03B05 Classical Propositional Logic - Wikisource + A Reduction in the number of the Primitive Propositions of logic. Pages in category MSC2000 03B05 Classical propositional logic http://en.wikisource.org/wiki/Category:MSC2000_03B05_Classical_propositional_log

var categoryTreeCollapseMsg = "collapse"; var categoryTreeExpandMsg = "expand"; var categoryTreeLoadMsg = "load"; var categoryTreeLoadingMsg = "loading"; var categoryTreeNothingFoundMsg = "nothing found"; var categoryTreeNoSubcategoriesMsg = "no subcategories"; var categoryTreeNoPagesMsg = "no pages or subcategories"; var categoryTreeErrorMsg = "Problem loading data."; var categoryTreeRetryMsg = "Please wait a moment and try again."; var wgNotice = ""; var wgNoticeLocal = ""; var wgNoticeLang = "en"; var wgNoticeProject = "wikisource"; Category:MSC2000 03B05 Classical propositional logic From Wikisource Jump to: navigation search Subcategories There is one subcategory to this category. A A Reduction in the number of the Primitive Propositions of Logic Pages in category "MSC2000 03B05 Classical propositional logic" There is one page in this category. A A Reduction in the number of the Primitive Propositions of Logic Retrieved from " http://en.wikisource.org/wiki/Category:MSC2000_03B05_Classical_propositional_logic Category MSC2000 03Bxx General logic Views Category Discussion Edit History Personal tools Log in / create account Navigation Main Page Community portal Community discussion Recent changes ... Donate Search Toolbox What links here Related changes Upload file Special pages ... Permanent link This page was last modified 14:32, 14 July 2006.

9. Intuitionistic Logic -- From Wolfram MathWorld Similarly, intuitionistic predicate logic is intuitionistic propositional logic combined with Classical firstorder predicate calculus. http://mathworld.wolfram.com/IntuitionisticLogic.html

Search Site Algebra Applied Mathematics Calculus and Analysis ... Sakharov Intuitionistic Logic The proof theories of propositional calculus and first-order logic are often referred to as classical logic Intuitionistic propositional logic can be described as classical propositional calculus in which the axiom schema is replaced by Similarly, intuitionistic predicate logic is intuitionistic propositional logic combined with classical first-order predicate calculus. Intuitionistic logic is a part of classical logic, that is, all formulas provable in intuitionistic logic are also provable in classical logic. Although, even some basic theorems of classical logic do not hold in intuitionistic logic. Of course, the law of the excluded middle does not hold in intuitionistic propositional logic. Here are some examples of propositional formulas that are not provable in intuitionistic propositional logic: Here are some examples of first-order formulas that are not provable in intuitionistic predicate logic: Truth tables for propositional connectives define the interpretation of classical propositional calculus over the domain of two elements: true and false . This interpretation is a model of classical propositional calculus, that is

10. Tutch User's Guide: Proofs In Propositional Logic 3.4 propositional logic IV, Classical Proofs . The remaining constructs of intuitionistic propositional logic are disjunction , truth T and falsehood F. http://www.andrew.cmu.edu/course/15-317/software/tutch-0.52/doc/html/tutch_3.htm

Up Top Contents [Index] 3. Proofs in Propositional Logic We explain how to code natural deduction style proofs in Tutch, how to check validity of the coded proof and how to submit proofs as homework solutions. 3.1 Propositional Logic I Proofs of Conjunctions and Implications 3.2 Propositional Logic II Proofs of Equivalences, Incomplete Proofs 3.3 Propositional Logic III Proofs of Disjunctions and Negations 3.4 Propositional Logic IV Classical Proofs 3.5 Requirements and Submission Submitting your Proofs Up Top Contents [Index] 3.1 Propositional Logic I The proof rules for conjunction and implication in natural deduction are: AndI If A true and B true then AndEL If then A true AndER If then B true ImpI If then ImpE If and A true then B true The following Tutch file ` prop0.tut ' is a proof of the proposition A proof begins with the keyword ` proof ', followed by a identifier (here ` mp '), a colon ` ', then the goal proposition (here ` '), an equal symbol ` ' and then the proof, enclosed between ` begin ' and ` end '. The last proposition of the proof must match the goal.

11. Propositional Logic. Mathematical Logic. Part 2. In Section 2.7 we will use these formulas to prove the elegant and nontrivial Glivenko s theorem a) A is provable in the Classical propositional logic http://www.ltn.lv/~podnieks/mlog/ml2.htm

propositional logic, propositional calculus, intuitionistic, logic, propositional, minimal, constructive, intuitionist, computer, independent, constructive logic, minimal logic, intuitionistic logic, calculus, Glivenko, independence, embedding Back to title page Left Adjust your browser window. In this book, constructive logic is used as a synonym of intuitionistic logic Right 2. Propositional Logic Proving formulas containing implication only Proving formulas containing conjunction Proving formulas containing disjunction Formulas containing negation - minimal logic ... George Boole (1815-1864): "In 1854 he published An Investigation into the Laws of Thought, on Which are founded the Mathematical Theories of Logic and Probabilities. Boole approached logic in a new way reducing it to a simple algebra, incorporating logic into mathematics. He pointed out the analogy between algebraic symbols and those that represent logical forms. It began the algebra of logic called Boolean algebra which now finds application in computer construction, switching circuits etc." (according to MacTutor History of Mathematics archive See also: G.Boole

12. Boolean Propositional Logic The most well known, and probably the simplest of these logics is known as Classical or boolean propositional logic, in which it is assumed that all http://www.rbjones.com/rbjpub/logic/log003.htm

Boolean Propositional Logic See also An introduction to Propositional Logics Boolean Operators semi-formal and formal descriptions of a propositional logic. informal semi-formal and formal descriptions of a first-order predicate logic. Sentential Connectives in Natural Languages It may be observed in the workings of natural languages that there are certain constructions which have the following features. They are sentential operators They operate on one or more complete sentences to give a new sentence. They are truth functional operators The truth of the resulting sentence can be determined knowing only the truth values of the sentences from which it was constructed. An example is the construction known as conjunction . This consists in conjoining two sentences with the connective and For example, the conjunction of the two sentences:: Grass is green Pigs don't fly. Is the sentence: Grass is green and pigs don't fly. The conjunction of two sentence will be true if, and only if, each of the two sentences from which it was formed is true. Other propositional connectives include: p or q known as the disjunction of "p" and "q".

13. A Sequent Proof System For Classical Propositional Logic A Sequent Proof System for Classical propositional logic. http://www.cs.uwyo.edu/~jlc/prop_gloss/node19.html

Next: A strategy for the Up: Decidability Previous: Decidability A Sequent Proof System for Classical Propositional Logic Consider the propositional proof system shown in Figure Figure 1: Proof System for Classical Propositional Logic Recall that a sound rule preserves validity, i.e. the validity of its premises implies the validity of its conclusion. A proof rule is said to be invertible when every assignment satisfying the conclusion also satisfies all the premises. For the rules used here, if any premise of an invertible rule is falsified by a given three-valued assignment, then the conclusion is falsified by the same assignment. Each of the proof rules has been formally shown to be both sound and invertible. Negation on the left *T formula_not_left_sound concl,M,N:Formula list. p:Formula. *T formula_not_left_falsifiable concl,M,N:Formula list. p:Formula. a:Assignment. Conjunction on the left *T formula_and_left_sound concl,M,N:Formula list. q,r:Formula *T formula_and_left_falsifiable concl,M,N:Formula list. q,r:Formula.

14. DI & CoS - Classical And Intuitionistic Logic System SKS is a set of rules for Classical propositional logic presented in the calculus of structures. Like sequent systems and unlike natural deduction http://alessio.guglielmi.name/res/cos/CL/index.html

This page is no longer updated, please refer to this page Alessio Guglielmi's Research Deep Inference and the Calculus of Structures / Classical and Intuitionistic Logic Deep Inference and the Calculus of Structures Classical and Intuitionistic Logic So far, for classical logic in the calculus of structures we achieved: the cut rule trivially reduces to atomic form; one can show cut elimination for the propositional fragment by the simplest argument to date; the propositional fragment is fully local, including contraction; first order classical logic can be entirely made finitary; cut elimination and decomposition theorems are proved. We can present intuitionistic logics in the calculus of structures with a fully local, cut-free system. The logic of bunched implications BI can be presented in the calculus of structures. Japaridze's cirquent calculus benefits from a deep-inference presentation, in particular in the case of propositional logic. The basic proof complexity properties of propositional logic are known. Atomic Cut Elimination for Classical Logic System SKS is a set of rules for classical propositional logic presented in the calculus of structures. Like sequent systems and unlike natural deduction systems, it has an explicit cut rule, which is admissible. In contrast to sequent systems, the cut rule can easily be restricted to atoms. This allows for a very simple cut elimination procedure based on plugging in parts of a proof, like normalisation in natural deduction and unlike cut elimination in the sequent calculus. It should thus be a good common starting point for investigations into both proof search as computation and proof normalisation as computation.

15. Pc.ml: A Tableau Prover For Classical Propositional Logic | The Tableau WorkBenc Compilation and Running of the TWB pc.ml A Tableau Prover for Classical propositional logic pcseq.ml a sequent calculus for Classical proposition http://twb.rsise.anu.edu.au/pc_ml

@import "/modules/book/book.css"; @import "/modules/node/node.css"; @import "/modules/system/defaults.css"; @import "/modules/system/system.css"; @import "/modules/user/user.css"; @import "/sites/all/modules/cck/content.css"; @import "/sites/all/modules/geshifilter/geshifilter.css"; @import "/sites/all/modules/cck/fieldgroup.css"; @import "/modules/comment/comment.css"; @import "/themes/garland/style.css"; @import "/themes/garland/print.css"; The Tableau WorkBench (TWB) About Us Demo Download ... Site News Tutorial: The Tableau Work Bench: Theory and Practice Basic Notions Defining Histories and Variables Defining the Strategy Introduction and Examples ... Introduction and Examples pc.ml: A Tableau Prover for Classical Propositional Logic Connectives The keyword CONNECTIVES in the TWB introduces a list of string separated with a simicolon ";". Each string can be used as a connective in the grammar definition (below) and it will become a reserved keyword. Connectives can be of variable length, can be composed by a mix of numbers, symbols and characters. Restrictions Double semicolon ";;" cannot be used.

16. JSTOR Symbolic Logic (Propositional Logic). Classical propositional calculus. The axiomatic systems of Classical propositional logic are given. That is, the chapter begins with the Lukasiewicz axioms http://links.jstor.org/sici?sici=0022-4812(197012)35:4<580:SL(L>2.0.CO;2-D

17. Some Benchmark Formulae For Intuitionistic Propositional Logic propositional logic is of course of less interest than firstorder logic, that the unprovable formulae should all be provable in Classical logic, http://www.dcs.st-and.ac.uk/~rd/logic/marks.html

Some benchmark formulae for intuitionistic propositional logic Roy Dyckhoff, University of St Andrews, 30 June 1997 Abstract We propose a small collection of problem classes, mainly gathered from other sources, for benchmarking theorem provers for intuitionistic propositional logic. Our approach is based on that of Heuerding et al [HS] , in proposing not just certain particular formulae but classes of formulae that will scale up for faster machines and better techniques. Nonprovable formulae are used as well as provable ones, to ensure that provers tuned to spot positive answers have some work to do. Introduction [HS] has proposed a suite of benchmarks for several propositional modal logics, using a method which is intended to allow the comparison of provers even when faster machines and better techniques make old benchmarks obsolete. Our purpose here is to apply the same method to develop a benchmark suite for intuitionistic propositional logic, providing data that will allow a rough comparison of several techniques. One intended use of the benchmarks is for a comparison of submissions of results by others, in association with the Tableaux'98 meeting. For details of this, see

18. PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE (BEOGRAD) (N.S.), Vol. 69(83), Pp. 27-3 Abstract Ra\v skovic 3 introduced a conservative extension of Classical propositional logic with some probability operators and proved corresponding http://www.emis.de/journals/PIMB/083/5.html

Vol. 69(83), pp. 27-33 (2001) Previous Article Next Article Contents of this Issue Other Issues ... EMIS Home CRAIG INTERPOLATION THEOREM FOR CLASSICAL PROPOSITIONAL LOGIC WITH SOME PROBABILITY OPERATORS Abstract: Classification (MSC2000): Full text of the article: Compressed PostScript file (60 kilobytes) PDF file (156 kilobytes) Electronic fulltext finalized on: 5 Feb 2002. This page was last modified: 5 Feb 2002. Mathematical Institute of the Serbian Academy of Science and Arts ELibM for the EMIS Electronic Edition

19. Phase Semantics And Sequent Calculus For Pure Noncommutative Classical Linear Pr Phase semantics and sequent calculus for pure noncommutative Classical linear propositional logic. Source, Journal of Symbolic logic archive http://portal.acm.org/citation.cfm?coll=GUIDE&dl=GUIDE&id=133251

20. PHIL 2340: Alternative Systems Of Propositional Logic (In fuzzy propositional logic, as in Classical propositional logic, an interpretation is just an assignment of truth values to atomic sentences although http://www.trinity.edu/cbrown/logic/alternatives.html

Symbolic Logic: Alternative Systems of Propositional Logic 1. Equivalent Systems of Propositional Logic Two formal systems are "equivalent" if all and only the derivations made possible by one are made possible by the other. That is, a set of rules P for propositional logic is equivalent to a different set P* if and only if, for any collection of premises P , P , . . . P n and conclusion C, if you can derive C from P , P , . . . P n using P then you can derive C from P , P , . . . P n using P* , and if you can derive C from P , P , . . . P n using P* , you can derive it using P How to prove that two formal systems are equivalent. Roughly, for each system, figure out what derivations are made possible by any rules it has that the other doesn't have and show that you can get the same results using only the rules the other has. For example: contradiction elimination allows you to derive Q (no matter what Q is!) from . We can show that this can be done using only the remaining rules of our system: assume Q; get

21. IngentaConnect Search Results 15 articles with title/keywords/abstract containing nonClassical propositional logic. Key. Free Content - Free Content. New Content - New Content http://www.ingentaconnect.com/search;jsessionid=aw09mfoabvhw.alexandra?database=

22. Constructive Propositional Logic Next Classical propositional logic Up Examples from Introductory Previous Most such courses begin with rules in which the propositional connectives http://www.cs.cornell.edu/info/Projects/Nuprl/book/node81.html

Next: Classical Propositional Logic Up: Examples from Introductory Previous: Examples from Introductory Constructive Propositional Logic Many people have seen formal proofs only during courses in logic. Most such courses begin with rules in which the propositional connectives and correspond to the natural language connectives ``and'',``or'',``not'' and ``implies'', respectively. If the nature of propositions is left unanalyzed except for their propositional structure, and if the unanalyzed parts are denoted by variables such as P Q and R , then the resulting forms are called propositional formulas . For example, is an instance of a propositional formula. Let us take a propositional formula which we recognize to be true and analyze why we believe it. We will then translate the argument into Nuprl . Consider the formula . We argue for its truth in the following fashion. If we assume then we need to show . Supposing is also true, we need to show ; that is, we must show that P is true if Q is. Therefore assume

23. Atlas: Algebraic Analysis Of Visser's Formal Propositional Logic By Majid Alizad In 1981 Albert Visser characterized a propositional logic that is embedded into IPL, and Boolean algebras play for Classical propositional logic, CPL. http://atlas-conferences.com/cgi-bin/abstract/caug-82

Atlas home Conferences Abstracts about Atlas ALGEBRAIC AND TOPOLOGICAL METHODS IN NON-CLASSICAL LOGICS III (TANCL'07) August 5-9, 2007 St Anne's College, University of Oxford Oxford, England Organizers Mai Gehrke and Hilary Priestley View Abstracts Conference Homepage Algebraic analysis of Visser's formal propositional logic by Majid Alizadeh Institute for Studies in Theoretical Physics and Mathematics. University of Tehran In 1981 Albert Visser characterized a propositional logic that is embedded into GL formal propositional logic, FPL . In this talk we introduce the variety of IPL , and Boolean algebras play for classical propositional logic, CPL L -algebras. Date received: July 22, 2007 Atlas Conferences Inc. Document # caug-82.

24. Broadview Press: Logical Options Trees for Classical propositional logic. 1.3.1. Tree Rules for Classical propositional logic. 1.3.2. Trees as a Test for Validity http://www.broadviewpress.com/bvbooksprintable.asp?BookID=237

25. Logic For Computer Scientists/Propositional Logic - Wikibooks, Collection Of Ope This section introduces propositional logic. We will study syntax and model theoretic semantic of a language of Classical propositional logic and we http://en.wikibooks.org/wiki/Logic_for_Computer_Scientists/Propositional_Logic

var wgNotice = ""; var wgNoticeLocal = ""; var wgNoticeLang = "en"; var wgNoticeProject = "wikibooks"; Logic for Computer Scientists/Propositional Logic From Wikibooks, the open-content textbooks collection Logic for Computer Scientists Jump to: navigation search edit Propositional Logic This section introduces propositional logic. We will study syntax and model theoretic semantic of a language of classical propositional logic and we investigate various calculi for deciding certain properties of sentences in this language. 4.1 Preliminaries 4.2 Syntax 4.3 Semantics 4.4 Equivalence and Normal Forms ... 4.7 Analytic Tableaux Retrieved from " http://en.wikibooks.org/wiki/Logic_for_Computer_Scientists/Propositional_Logic Views Module Discussion Edit this page History Personal tools Log in / create account Navigation Main Page Help Cookbook Wikijunior ... Donations Community Bulletin Board Community portal Reading room Help cleanup ... Contact us Search Toolbox What links here Related changes Upload file Special pages ... Permanent link This page was last modified 13:41, 29 May 2007. All text is available under the terms of the GNU Free Documentation License (see for details).

26. Classical Algebra Supplement Supplement to the Classical Algebra text propositional logic is that part of logic that deals with combining statements using connectives such as AND, http://www.math.uwaterloo.ca/~wgilbert/Books/Supplement/Sect01PropLogic.html

Supplement to the Classical Algebra text Chapter 0: Logic and Proofs Mathematics makes precise use of language in stating and proving its results. We use the English language to express our ideas and arguments, though we give some common English words a more precise meaning so as to make them unambiguous. Good mathematical writing consists of complete sentences, allowing for the fact that symbols stand for words. For example, `A = B.' is a sentence in which the subject is A, the verb is `equals' and the object is B. In mathematics, we tend to use more complicated and compound expressions than we do in everyday language, so this chapter explains some methods for dealing with these expressions. We also introduce the more common types of mathematical reasoning we use in proofs. 0.1 PROPOSITIONAL LOGIC Logic is the study of correct reasoning. The rules of logic give precise meaning to mathematical statements and allow us to make correct arguments about these statements. 0.1.1 Definition In mathematics, a statement

27. PlanetMath: Logical Axiom For example, in Classical propositional logic, or sometimes known as Boolean propositional logic, the following collection constitutes a set of axioms http://planetmath.org/encyclopedia/LogicalAxiom.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... RSS Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About logical axiom (Definition) There are in general two ways of arriving at tautologies in any logical system. One is done semantically , via a truth function on the set of propositional variables , and extending the function to the larger set consisting of all (well-formed) formulas , and then declaring a tautology as any formula that is always mapped to The second way is done mechanically, or in logical jargon, syntactically , via a special set of well-formed formulas , called logical axioms , together witha ( finite ) set of rules of inference . Any formula axioms by applying the rules of inferences is called a tautology in question can be obtained by a finite sequence of formulas, such that is an axiom

28. Pkhakadze The ideas of Herbrand based on FregeHilbert’s Classical formalistic comprehensions which in their turn are based on Boolean propositional logic (Classical http://www.viam.science.tsu.ge/curi/logic/pkhakadze.htm

Curriculum Vitae Konstantine Pkhakadze Name : Konstantine Pkhakadze Date and place of birth : January 19, 1960, Tbilisi, Georgia Nationality : Georgian Address: I. Vekua Institute of Applied Mathematics Tbilisi State University, 2 University St. 380043,Tbilisi, Georgia E-mail: pkhakadz@viam.hepi.edu.ge Tel : (+995) (32) - 32 59 29 (h); (+995) (32) - 30 35 81 (w). Education May 1994 Thesis: ,, Propositional i-algebra and some of its applications”(in Georgian). Tbilisi state University. Advisor-Dr. Kh. Rukhaia. 1989-1992 Post-graduate study : Tbilisi State University; specialization- Algebra, Mathematical Logic and Number Theory. Advisor-Dr. Kh. Rukhaia. Completed thesis in September 1992. 1976-1981 Undergraduate study: Tbilisi State University, Faculty of Mechanics and Mathematics 1976 School graduation: Graduated from a school in Tbilisi, specialized in mathematics and physics. Employment Senior Researcher of the Department of Methodology and Mathematical Logic at the I. Vekua Institute of Applied Mathematics; Tbilisi State University. Head of the Department - Dr. Kh. Rukhaia Research assistant of the Department of System Programming at the I. Vekua Institute of Applied Mathematics; Tbilisi State University. Head of the Department - Dr. J. Antidze.

29. The Questia Online Library After a brief recapitulation of Classical logicto establish notation and set Nevertheless, the restriction to propositional logic has its downside. http://www.questia.com/PM.qst?a=o&se=gglsc&d=5001902120

30. Classical Logic = Fibred MLL, , March 25, 2005Accepted For A Short Theorem 1 (Soundness and Completeness)A formula of Classical propositional logic is true (valid) iff it has a combinatorial proof. http://hermes.aei.mpg.de/arxiv/05/04/028/article.xhtml

Classical logic = Fibred MLL Dominic Hughes Stanford University March 25, 2005Accepted for a short presentation at Logic in Computer Science '05. Syntactically, classical logic decomposes thus Classical logic = MLL + Superposition where MLL is Multiplicative Linear Logic, and superposition means contraction (binary case) and weakening (nullary case). Proof nets for classical logic have been proposed , containing explicit contraction and weakening nodes. Just as the logical sequent rules of conjuction (tensor) and disjunction (par) are represented explicitly in the net, so are the structural rules, contraction and weakening. Unfortunately, this explicit representation of structural rules retains some of the redundancy (or `syntactic bureaucracy') of sequent calculus. For example, in the net presentation of classical categories in one has to quotient by equations on nets, involving contraction and weakening. Thus, proof nets for classical logic fail to achieve the same elegance as the proof nets for MLL. We present a representation of a proof which is more abstract than a proof net, which we call a combinatorial proof : superposition is modelled mathematically , as a lax form of fibration, rather than syntactically (as in proof nets, which involve contraction and weakening nodes). This draws a nice boundary between logical rules and structural rules: the former are modeled with explicit nodes, the latter as actual superposition, in an abstract mathematical sense, without nodes. We can summarise the situation thus:

31. School Of Computer Science Syllabus Pages (School Of Computer Science - The Univ It is assumed that students will be familiar with Classical propositional logic (Boolean logic). Further knowledge of the subject is not assumed, http://www.cs.manchester.ac.uk/undergraduate/programmes/courseunits/syllabus.php

32. Berti, Massimiliano Classical logic. Appendix The fundamental metatheorem for the Classical propositional logic.- A proof system for the Classical logic http://www.yurinsha.com/410/p6.htm

33. CST LECTURES: Lecture Classical propositional logic also can be given a semantics in any Boolean algebra, each formula A being assigned a value in the Boolean algebra so that http://www.cs.man.ac.uk/~petera/Padua_Lectures/lect6.html

Lectures on Constructive Set Theory (Padua, Spring 1998) Lecture 6: 20th May (16.30 - 18.00) See Lecture 5 The Semantics of Intuitionistic Logic In lecture 5 we were concerned with the formal rules of deduction for intuitionistic logic (and classical logic). This was a matter of syntax; i.e. the rules are explained purely in terms of the syntactic forms of the logical formulae and not in terms of their meaning. We introduce some notation. Formal Deduction if B can be deduced from zero, one or more of A1,...,An as assumptions, using the rules of inference of intuitionistic logic (or sometimes classical logic, depending on the context). Note the special case when n=0. We now want to consider the question: In what sense are the rules of deduction correct/sound? This question is a matter of semantics; i.e. it concerns the possible meaning of the logical formulae. We will first briefly consider the standard classical answer which is in terms of truth values The classical standard semantics What do we mean by an interpretation here?

34. Chapter 4: Groundwork - Propositional Logic And Tableaux GROUNDWORK propositional logic AND TABLEAUX. THE last chapter developed certain powerful systems corresponding both to the Classical and to the http://www.clas.ufl.edu/users/jzeman/modallogic/chapter04.htm

GROUNDWORK - PROPOSITIONAL LOGIC AND TABLEAUX T HE last chapter developed certain powerful systems corresponding both to the classical and to the intuitionist propositional calculi. These systems enable us to find a proof for a given formula of IC or PC iff that formula is provable; in effect, they provide us with a decision procedure for each of these systems. One might initially look upon the present chapter as an attempt to refine the methods of the last chapter by making these decision procedures easier to apply. We shall, indeed, begin our presentation as if this were our motivation. It will turn out, however, that the present subject-matter is considerably more than a mere aid to calculation, ancillary to the sequent logics of the last chapter. It provides us with an introit to the semantics of the classical PC and, as it will turn out, to the semantics of many modal systems as well. Proof tableaux Powerful though the sequent-logics are, they have certain mechanical disadvantages for use as decision procedures. Not the least of these is one determined by a feature of these logics very important in making them powerful; this is the subformula property. Once a formula appears in a cut-free proof in a sequent-logic, it does not later disappear, but must be written at each following step of the proof, either as a subformula of a larger formula or itself as a parametric formula. If a hundred steps follow the introduction of a , then a is written out a hundred times. It might be well to seek to find a method in which some of the work involved in writing and rewriting such formulas is avoided. Such a method was provided by

35. Author-index.html Inferencesearching algorithms in Classical propositional logic (with . Embedding of Classical propositional logic into its implicative fragment and into http://www.iph.ras.ru/~logic/author-index.en.html

BIBLIOGRAFICAL INDEX of Proceedings of the Research Logical Seminar of Institute of Philosophy Russian Academy of Sciences (1982-2000) I. Modal and Relevant Logics (Proceedings of Scientific Seminar in Logic of the Institute of Philosophy). M., 1982. II. Logical Investigations (Proceedings of Scientific Seminar in Logic of the Institute of Philosophy). M., 1983. III. Many-valued, Relevant and paraconsistent (Proceedings of Scientific Seminar in Logic of the Institute of Philosophy). M., 1984. IV. Non-classical Logics (Proceedings of Scientific Seminar in Logic of the Institute of Philosophy). M., 1985. V. Non-standard semantics of Non-classical Logics (Proceedings of Scientific Seminar in Logic of the Institute of Philosophy). M., 1986. VI. Non-classical Logics and propositional attitudes (Proceedings of Scientific Seminar in Logic of the Institute of Philosophy). M., 1987. VII. Non-classical Logics and its Applications (Proceedings of Scientific Seminar in Logic of the Institute of Philosophy). M., 1989. VIII. Philosophical Foundations of Non-classical Logics (Proceedings of Scientific Seminar in Logic of the Institute of Philosophy). M., 1990.

36. MATHEMATICAL LOGIC FOR COMPUTER SCIENCE Prerequisites Sets; Inductive Definitions and Proofs; Notations; Classical propositional logic Propositions and Connectives; propositional Language http://www.worldscibooks.com/compsci/3434.html

Home Browse by Subject Bestsellers New Titles ... Browse all Subjects Search Bookshop New Titles Editor's Choice Bestsellers Book Series ... World Scientific Series in Computer Science - Vol. 47 MATHEMATICAL LOGIC FOR COMPUTER SCIENCE 2nd Edition by Lu Zhongwan (Chinese Academy of Science, Beijing) Mathematical logic is essentially related to computer science. This book describes the aspects of mathematical logic that are closely related to each other, including classical logic, constructive logic, and modal logic. This book is intended to attend to both the peculiarities of logical systems and the requirements of computer science. In this edition, the revisions essentially involve rewriting the proofs, increasing the explanations, and adopting new terms and notations. Contents: Prerequisites: Sets Inductive Definitions and Proofs Notations Classical Propositional Logic: Propositions and Connectives Propositional Language Structure of Formulas Semantics Tautological Consequence Formal Deduction Disjunctive and Conjunctive Normal Forms Adequate Sets of Connectives Classical First-Order Logic: Proposition Functions and Quantifiers First-Order Language Semantics Logical Consequence Formal Deduction Prenex Normal Form Axiomatic Deduction System: Axiomatic Deduction System Relation between the Two Deduction Systems Soundness and Completeness: Satisfiability and Validity Soundness Completeness of Propositional Logic Completeness of First-Order Logic Completeness of First-Order Logic with Equality Independence

37. Intuitionistic Logic (Stanford Encyclopedia Of Philosophy/Winter 2002 Edition) In Classical propositional logic, if (A is provable) implies (B is provable), then (A B) is provable; thus every Classically admissible rule is Classically http://www.science.uva.nl/~seop/archives/win2002/entries/logic-intuitionistic/

This is a file in the archives of the Stanford Encyclopedia of Philosophy how to cite this entry CITATION INFO Stanford Encyclopedia of Philosophy A B C D ... Z content revised DEC Intuitionistic Logic Intuitionistic logic encompasses the principles of logical reasoning which were used by L. E. J. Brouwer in developing his intuitionistic mathematics, beginning in [1907]. Because these principles also underly Russian recursive analysis and the constructive analysis of E. Bishop and his followers, intuitionistic logic may be considered the logical basis of constructive mathematics. Philosophically, intuitionism differs from logicism by treating logic as a part of mathematics rather than as the foundation of mathematics; from finitism by allowing (constructive) reasoning about infinite collections; and from platonism by viewing mathematical objects as mental constructs with no independent ideal existence. Hilbert's formalist program, to justify classical mathematics by reducing it to a formal system whose consistency should be established by finitistic (hence constructive) means, was the most powerful contemporary rival to Brouwer's developing intuitionism. In his 1912 essay Intuitionism and Formalism Brouwer correctly predicted that any attempt to prove the consistency of complete induction on the natural numbers would lead to a vicious circle.

38. TabVis: Tableau Calculus Proof System TabVis is a proof system for Classical propositional logic. It uses the Tableau calculus. The Tableau can be constructed interactively and is visualized. http://stud4.tuwien.ac.at/~e0225493/TabVis/index_eng.html

TabVis: An Interactive Proof System For The Tableau Calculus Updated August, 2005. Diese Seite gibts auch auf Deutsch Contents Introduction The Tableau Calculus Basic Properties Rules ... Technical Resources Introduction TabVis is a proof system for classical propositional logic. It uses the Tableau calculus. The Tableau can be constructed interactively and is visualized. Furthermore TabVis can be used to search for a proof automatically. Historically, semantical tableau calculi where first introduced independently by Beth (1955), Hintekka (1955) and Schütte (1956). The calculus TabVis uses, was proposed by Smullyan (1968). He used signed formulas in his tableaux and introduced the unifying notation. TabVis uses the Tableau calculus of Smullyan to draw and interactively construct proofs in this system. Its main purpose is to assist students in getting comfortable with the Tableau calculus for classical propositional logic. The Tableau Calculus Basic Properties Rules A Proof in the Tableau Calculus Soundness and Completeness of the calculus For simplicity, I will use the same logic symbols as they are used in TabVis. So "-" stands for the logical not and , "v" for the or implication Basic Properties The Tableau calculus is based on eight properties that are valid under any interpretation in classic propositional logic (from here on I will stop mentioning that we talk about classical propositional logic; all future statements refer to this logic unless stated otherwise). These are:

39. NcDP: A Non-Clausal Davis-Putnam Prover ncDP is a nonclausal theorem prover for Classical propositional logic. It is a generalization of the well-known Davis - Putnam - Logemann - Loveland http://www.leancop.de/ncdp/

ncDP: A Non-Clausal Davis-Putnam Prover What is ncDP ? Documentation Source Code Related Links ... Theoretical CS What is ncDP ? ncDP is a non-clausal theorem prover for classical propositional logic. It is a generalization of the well-known Davis - Putnam - Logemann - Loveland decision procedure for propositional formulas in clausal form. By working entirely on the original (non-clausal) formula, any translation steps to clausal form are avoided. This yields a compact code and a strong performance. Features of ncDP Theorem prover for classical propositional logic. Based on a generalized non-clausal Davis-Putnam calculus. Sound and complete. Decision procedure for propositional logic. Source code available for popular Prolog systems, including ECLiPSe Prolog, SWI-Prolog and SICStus Prolog. Strong performance. Simple input format. Documentation ncDP is described within a chapter of a book and in a technical report. The book chapter contains a description of the non-clausal Davis-Putnam calculus and the source code. Some performance results are presented as well.

40. The Language Of Science / Logic (Hartley Slater) which were subsequently extended to form ‘propositional logic’. being available alternatives to what is then called ‘Classical propositional logic’. http://www.polimetrica.eu/site/?p=25

41. HKUST Institutional Repository: Item 1783.1/785 Title, Reducing strong equivalence of logic programs to entailment in Classical propositional logic. Authors, Lin, Fangzhen. Keywords, Strong equivalence http://repository.ust.hk/dspace/handle/1783.1/785

Home Institutional Repository Library Home Quick Links ... How Do I.. Search Advanced Search Full-Text Search Home About the ... Top 20 Browse Communities Titles Authors By Date ... CSE Conference Papers Please use this identifier to cite or link to this item: http://hdl.handle.net/1783.1/785 Title: Reducing strong equivalence of logic programs to entailment in classical propositional logic Authors: Lin, Fangzhen Keywords: Strong equivalence Classical propositional logic Logic programs Issue Date: Citation: Proceedings of KR 2002 Abstract: URI: http://hdl.handle.net/1783.1/785 Appears in Collections: CSE Conference Papers Files in This Item: File Description Size Format gkipkr02.pdf Adobe PDF View/Open Recommend / E-mail this item Powered by HP/MIT's DSpace software, Version 1.3.2 OAI-PMH and SRW/U compliant