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Topic: Propositional calculus

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Natural deduction

Double negative elimination

Intuitionistic logic

Axiom

Johnston diagram

Modal logic

Lambda calculus

Proof theory

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	Propositional calculus - Wikipedia, the free encyclopedia
		In mathematical logic the propositional calculus or sentential calculus is a formal deduction system whose atomic formulas are propositional variables.
		In the propositional calculus the language consists of propositional variables (or placeholders) and sentential operators (or connectives).
		When the "atomic sentences" of propositional logic are broken up into terms, variables, predicates, and quantifiers, they yield first-order logic, or first-order predicate logic, which keeps all the rules of propositional logic and adds some new ones.
	en.wikipedia.org /wiki/Propositional_logic (3097 words)

	Propositional Logic [Internet Encyclopedia of Philosophy]
		Propositional logic, also known as sentential logic, is the branch of logic that studies ways of joining and/or modifying entire propositions, statements or sentences to form more complicated propositions, statements or sentences, as well as the logical relationships and properties that are derived from these methods of combining or altering statements.
		In propositional logic, the simplest statements are considered as indivisible units, and hence, propositional logic does not study those logical properties and relations that depend upon parts of statements that are not themselves statements on their own, such as the subject and predicate of a statement.
		However, there are other forms of propositional logic in which other truth-values are considered, or in which there is consideration of connectives that are used to produce statements whose truth-values depend not simply on the truth-values of the parts, but additional things such as their necessity, possibility or relatedness to one another.
	www.iep.utm.edu /p/prop-log.htm (8796 words)

	Propositional calculus (Site not responding. Last check: 2007-11-06)
		A calculus or proof theory is that part of a logical which determines how to construct arguments : to derive conclusions from premises.
		Ideally the axioms inference rules of a calculus are chosen that if the formulas in a set semantically true then any formulas derivable from are also true.
		When the "atomic sentences" propositional logic are broken up into terms variables predicates and quantifiers they yield first-order logic or first-order predicate logic which keeps the rules of propositional logic and adds new ones.
	www.freeglossary.com /Propositional_logic (2103 words)

	Encyclopedia article: Propositional calculus (Site not responding. Last check: 2007-11-06)
		The propositional calculus is a formal deduction system whose atomic formulas are propositional variables.
		It is a set of axioms (which may be an empty or countably infinite set) or axiom schemata, and inference rule (additional info and facts about inference rule) s for deriving valid inferences.
		In the propositional calculus the language consists of propositional variables (or placeholders) and sentential operators (or connective (An uninflected function word that serves to conjoin words or phrases or clauses or sentences) s).
	www.absoluteastronomy.com /encyclopedia/p/pr/propositional_calculus.htm (2711 words)

	Learn more about Propositional calculus in the online encyclopedia. (Site not responding. Last check: 2007-11-06)
		A logical calculus is a formal, axiomatic system for recursively generating well-formed formulas (wffs).
		The propositional calculus is the foundation of symbolic logic; more complex logical calculi are usually defined by adding new operators and rules of transformation to it.
		With wffs and rules of inference, it's possible to derive wffs; the derivation is a valid argument form, while the derived wff is known as a lemma.
	www.onlineencyclopedia.org /p/pr/propositional_calculus.html (480 words)

	propositional calculus (Site not responding. Last check: 2007-11-06)
		Propositional Calculus -- from MathWorld Propositional Calculus -- from MathWorld The formal basis of logic dealing with the notion and usage of words such as "NOT," "OR," "AND," and "implies." Many systems of propositional calculus have been...
		A propositional calculus is a formal, deduction system, or proof theory for reasoning with propositional formulas as...
		A propositional calculus is a formal, axiomatic system, or proof theory for...
	www.mathscoacher.com /articles/34/propositional-calculus.html (385 words)

	Reasoning-1
		Propositional calculus, or propositional logic, is a subset of predicate logic.
		Intuitively speaking, a proposition is the conceptual correspondence of a declarative sentence (in English).
		There are two types of propositions: atomic and compound, the former is represented by an identifier, while the latter is by a structure of propositions connected together by connectives.
	www.cis.temple.edu /~pwang/203-AI/Lecture/Reasoning-1.htm (1036 words)

	websites.NT.Glossary.propositionalCalculus (Site not responding. Last check: 2007-11-06)
		It's elementary objects are called "atomic propositions", often denoted by letters p, q, r, a, b, etc., and we interpret these to be statements to which we are willing to assign a value of either T (for True) or F (for False).
		The other important element in the propositional calculus is a small collection of operators that take propositions and form from them other propositions (this time no longer atomic).
		The truth table approach to propositional calculus is an effective procedure for dealing with small systems of propositions, but there are systems too complex to handle with truth tables.
	www.naturaltheology.net /Glossary/propositionalCalculus.html (1659 words)

	Stupid Question. \| Lambda the Ultimate
		It's a really nice introduction to design issues in structural proof theory and the problems that calculi moulded on the sequent calculus have with modal logic, but she was uncomfortable with introducing talk of design patterns into a work on proof theory, so the definition of calculus was left implicit.
		A propositional calculus can be defined as a formal systems in various ways; variations in the syntax, proof theory and model theory are common.
		A proof calculus can be a sort of pattern, then it is like the differential calculus before Cauchy introduced his underpinnings; or it can be fully rigorous, in which case it is like the integral calculus after Lebesgue.
	lambda-the-ultimate.org /node/view/533 (2846 words)

	Logic Notes
		In the propositional calculus, the basic unit of inference is a proposition, which is just a statement about the world that is either true or false.
		A propositional symbol, such as P or Q, represents a sentence about the world, such as "It is raining".
		Another form of proof in the propositional calculus is to derive one expression from another by replacing expressions with equivalent expressions.
	starbase.trincoll.edu /~ram/cpsc352/notes/logic/propcalc.html (551 words)

	Functional satisfaction (Site not responding. Last check: 2007-11-06)
		By “simple” we mean a technique inspired by the semantics of the propositional calculus, not a sophisticated, resource aware, technique such as binary decision diagrams.
		The trick is to consider a continuation-based semantics of the propositional calculus.
		and of the functional connectors as a denotational, continuation-based, semantics of the propositional calculus.
	pauillac.inria.fr /~maranget/enum (2489 words)

	Comparing the expressive power of the propositional logic with the calculus of classes (Site not responding. Last check: 2007-11-06)
		Using this we have expressed the lengthy argument of Lewis Carroll in the propositional calculus since all the statements are universal in Example 2.7.11 of LMCS.
		The simplest ``upgrade'' of the propositional calculus which is adequate to handle the I,O statements is the monadic predicate calculus which deals with quantified first-order statements about unary predicates.
		In summary we have a translation of arguments in the propositional calculus into arguments in equations in the calculus of classes, and conversely.
	www.math.uwaterloo.ca /~snburris/htdocs/scav/compar/compar.html (240 words)

	Propositional Calculus (Site not responding. Last check: 2007-11-06)
		The propositional calculus is based on statements which have truth values (true or false).
		The calculus provides a means of determining the truth values associated with statements formed from ``atomic'' statements.
		If propositional logic is to provide us with the means to assess the truth value of compound statements from the truth values of the `building blocks' then we need some rules for how to do this.
	www.oopweb.com /Prolog/Documents/prologbook/Volume/node16.html (150 words)

	Parameterizing Models of Propositional Calculus Formulas (Site not responding. Last check: 2007-11-06)
		Extending the idea to predicate calculus is likely to be reasonable only under some restrictions.
		Thus parameterizing the models of the axioms for a group is the problem of group classification with its hundred year history.
		There may be straightforward ways of going from a parameterization of a theory to parameterizations of some kinds of elaborations of the theory.
	www-formal.stanford.edu /jmc/parameterize/parameterize.html (403 words)

	Philosophy and Computers --- Links - Propositional Calculus (Site not responding. Last check: 2007-11-06)
		Overview of The Propositional Calculus - Well organized web page with definitions of symbols, sentences, and semantics.
		Intuitionist Propositional Calculus - A Constructive Completeness Proof for this.
		Propositional Calculus - This long page talks about propositional calculus in detail.
	www.ic.sunysb.edu /Class/phi365/prop_calc.html (174 words)

	Amazon.com: Introduction to Mathematical Logic: Books (Site not responding. Last check: 2007-11-06)
		The name propositional calculus is given to any one of various logistic systems-which, however, are all equivalent to one another in a sense which will be made clear later.
		The discussion of the propositional calculus is continued in the next chapter where a new system of propositional calculus is obtained by dropping the constants from the first one and adding another symbol (negation).
		He discusses the highly interesting and thought-provoking intuitionistic propositional calculus, due to A. Heyting, which is a formalization of the famous mathematical intuitionism of L.E.J. Brouwer.
	www.amazon.com /exec/obidos/tg/detail/-/0691029067?v=glance (2267 words)

	FAQ on Project 1 and Propositional Calculus
		An extension to propositional calculus is “predicate calculus”.
		A predicate is not a function, it is a proposition when used with variables.
		In this project, what we require is the design of the syntax of a “new” programming language based on extended propositional calculus.
	www.cs.bilkent.edu.tr /~inazli/FAQ.htm (414 words)

	Week 11 Responses
		The predicate calculus seems to be a straightforward enough means for the purpose of simulated thought and problem solving.
		The only thing that I think is necessary to make a prediacte calculus system have the came cognitive capabilities as a human is some way to give the system some sense of the "real world" which its model corresponds to.
		Predicate Calculus (at least what we have discussed so far) seems pretty straightforward to me. I am interested in the sentence "Every farmer who has a donkey beats it" and why it is so impossible to 'translate' into the predicate calculus without all sorts of unintended implications.
	mainline.brynmawr.edu /Courses/cs372/fall98/Responses/Week11.html (3429 words)

	[No title]
		Proposition Atom B_on_top_of_A A B The assignment of a proposition to a particular atom is called a denotation.
		An interpretation that satisfies a sentence (by causing it to be evaluated to the value of true) is called a model of that sentence.
		COMP-4640: Intelligent & Interactive Systems The Propositional Calculus Validity A sentence that is true under all interpretations is said to be valid.
	www.eng.auburn.edu /~gvdozier/chapter6.doc (885 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Propositional Calculus is a formal system with no axioms.
		It is a system with a set of rules used to produce statements which would be true in all conceivable worlds.
		Propositional Calculus is a solid, consistent formal system, but its simple rules make it somewhat limiting.
	www.pha.jhu.edu /~leslieh/IntSem/proposit.html (221 words)

	Digital Logic and Boolean Algebra (Site not responding. Last check: 2007-11-06)
		Predicate calculus, which is at the basis of the Prolog programming language.
		Propositional logic deals with statements (propositions) which make claims which can be either true or false.
		In predicate calculus we restrict the meaning of predicate to possession of a property or attribute, e.g.
	www.engr.udayton.edu /faculty/jloomis/ece314/notes/carch/node4.html (3242 words)

	lec3
		Propositional calculus is something of a toy calculus.
		Propositional calculus (also called "sentential calculus" or "sentential logic") deals with whole, unanalysed, sentences or propositions.
		A more sophisticated logic, such as predicate calculus, is needed to exhibit the full structure of the argument.
	www.cit.gu.edu.au /~terryd/subjects/logic/lecture3.html (2161 words)

	Chapter 3: Groundwork - Propositional Sequent-Logic
		This theory of deduction is informal, for it is stated in the unformalized language of mathematical English (this is in contrast to the objects of study, the propositional calculi themselves, which are indeed formalized, and formalized within the metalanguage of mathematical English).
		We shall therefore define new systems, systems which will contain the notation of the propositional calculus and more besides, systems which may be thought of as formalizing part of the process of deducing propositional calculus statements from other propositional calculus statements.
		As we have remarked, the systems of the present chapter will include all the symbols of the propositional calculus and the rules of formation for those symbols; a set of symbols will be called a wff here iff it is a wff in the standard propositional calculi.
	www.clas.ufl.edu /users/jzeman/modallogic/chapter03.htm (5162 words)

	Table of Contents
		Lucid, non-intimidating presentation of propositional logic, propositional calculus and predicate logic by Russian scholar.
		Axiomatic construction of mathematical theories in the language of predicate logic with equality Appendix I. A proof of the duality principle for propositional logic Appendix II.
		A proof of the deduction theorem for the propositional calculus Appendix III.
	www.doverpublications.com /cgi-bin/toc.pl/0486645614 (169 words)

	Software Build and Fix: Tips (Site not responding. Last check: 2007-11-06)
		Even so, if you are conscientious about handling errors and if you don't know about the propositional calculus, then I'll bet you've had a similar experience.
		What the the propositional calculus gives you is ways of recognizing equivalent boolean expressions without resorting to truth tables.
		Actually, this interpretation of the propositional calculus is inherent in the name.
	www.mapfree.com /sbf/tips/logic.html (967 words)

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