
	Where results make sense

Topic: Double negation elimination

	Propositional Logic Applet
		The negation of a proposition: !p (traditionally a '¬' symbol)
		Since (a+!a) is T regardless of the state of a, the CNF becomes C1C2T*T. Conjunction of T with a true statement does not add anything to the statement.
		This means that the negation of that statement is always true.
	www.oursland.net /aima/propositionApplet.html (906 words)

	Double negative elimination - Wikipedia, the free encyclopedia
		In logic and the propositional logic, the inference rules double negative elimination (also called double negation elimination) and double negative introduction (also called double negation introduction) allow deriving the double negative equivalent by adding (for double negative introduction) or removing (for double negative elimination) a pair of negation signs.
		The rule of double negative introduction states the converse, that double negatives can be added without changing the meaning of a proposition.
		Double negative elimination is a theorem of classical logic, but not intuitionistic logic.
	en.wikipedia.org /wiki/Double_negative_elimination (315 words)

	Intuitionistic logic - Wikipedia, the free encyclopedia
		In intuitionistic logic, only the first is a theorem: Double negation can be introduced, but it cannot be eliminated.
		In classical propositional logic, it is possible to take one of conjunction, disjunction, or implication as primitive, and define the other two in terms of it together with negation, such as in Łukasiewicz's three axioms of propositional logic.
		Negation is as usual defined as ¬A = A→ø, so the value of ¬A reduces to A
	en.wikipedia.org /wiki/Intuitionistic_logic (1091 words)

	Double negative elimination - Encyclopedia, History, Geography and Biography
		ory href="/encyclopedia/Propositional_calculus">propositional calculus, the inference rules double negative elimination (also called double negation elimination) and double negative introduction (also called double negation introduction) allow deriving the double negative equivalent by adding (for double negative introduction) or removing (for double negative elimination) a pair of negation signs.
		Since biconditionality is an equivalence relation, any instance of ¬ ¬ A in a well-formed formula can be replaced by A, leaving unchanged the truth-value of the wff.
		(This distinction also arises in natural language in the form of litotes.) Double negation introduction is a theorem of intuitionistic logic, as is $\neg \neg \neg A \vdash \neg A$ .
	www.arikah.net /encyclopedia/Double_negation_elimination (346 words)

	Official List of Rules and Theorems
		Notice that an ``introduction'' rule is a rule for proving a goal with the appropriate top-level connective or quantifier, while an ``elimination'' rule is a rule for using a premise or previous conclusion with that the appropriate top-level connective.
		Negation basic rules P P and ~P ~P ---------- negation elimination -------- negation elimination Q Q Either form is OK. G, P - Q and ~Q ---------------- negation introduction G - ~P If you don't like the - notation, here's the box format: G (all your earlier knowledge) Goal: ~P
		Q and ~Q (however you prove this) ~P negation introduction, previous box Double Negation These are derived rules, but they are really part of our basic toolbox.
	math.boisestate.edu /~holmes/M387syllabus/node46.html (1231 words)

	Double Negation Introduction and Elimination ~~I and ~~E (Site not responding. Last check: 2007-10-28)
		Double negation elimination (~~E) enables you to move from a sentence ~~p to p.
		2 is not as it stands the negation of the consequent of 1.
		However 3 is the negation of the consequent of 1.
	www.thelogiccourse.com /bluestorm/dnset.html (112 words)

	Andrews. To Truth through Proof. (Site not responding. Last check: 2007-10-28)
		Absolute consistency (there is an unprovable wff) and consistency with respect to negation (there is no wff such that it and its negation are provable) are defined for general logical systems and a condition is given under which the two become equivalent (the provability of "A implies.
		Negation normal form is defined, and the two-dimensional representation is introduced (conjuncts=vertical paths and disjuncts=horizontal paths).
		System F of first-order logic is introduced in the usual way (with disjunction, negation, and universal quantification as primitive)--except he allows propositional, function and relation variables over which we cannot quantify.
	www.andrew.cmu.edu /user/cebrown/notes/tttp.html (6301 words)

	PHIL 250: Introduction to Symbolic Logic
		One thereby rests that conclusion A on the same set of assumptions on which its double negation depends.
		Since this is neither an introduction rule, nor a genuine elimination rule (dealing with a single dominant occurrence of the operator concerned), there is no rule 'corresponding' to it!
		The rule of double-negation elimination obliterates any distinction between a sentence and its double negation.
	people.cohums.ohio-state.edu /tennant9/tennant_dne.html (62 words)

	Friday, February 4
		The idea is to look at the topmost connective of the theorem we are trying to prove and think about what the previous step of the proof (which presumably uses the introduction rule for that connective) would look like.
		Of course, another possible way to prove a theorem is to assume its negation and try to prove a contradiction from it.
		Continuing, we think that the natural way to prove ~B is by negation introduction, using a proof with premise B and conclusion a contradiction.
	math.boisestate.edu /~holmes/M387syllabus/node25.html (558 words)

	[cs/0301026] Double-Negation Elimination in Some Propositional Logics
		A proof is free of double negation if none of its deduced steps contains a term of the form n(n(t)) for some term t, where n denotes negation.
		The first question asks for conditions on the hypotheses that, if satisfied, guarantee the existence of a double-negation-free proof when the conclusion is free of double negation.
		The second question asks about the existence of an axiom system for classical propositional calculus whose use, for theorems with a conclusion free of double negation, guarantees the existence of a double-negation-free proof.
	arxiv.org /abs/cs.LO/0301026 (196 words)

	Graham Priest - Doubt Truth to Be a Liar - Reviewed by Hartry Field, New York University - Philosophical Reviews - ...
		It's worth noting that a distinction could be made between accepting the truth of a contradiction and accepting that contradiction; Priest defines dialetheism in terms of the former, but takes it to involve the latter as well.
		Turning to Part II (Negation), Chapter 4 argues that the normal understanding of negation requires the Law of Excluded Middle but allows for the acceptance of contradictions.
		Chapter 5 argues against various attempts to show that a negation operator obeying the classical laws must be coherent; also, against attempts to show that a conditional obeying at least the intuitionist laws must be coherent.
	ndpr.nd.edu /review.cfm?id=6101 (2530 words)

	Optional, Break, Exercise
		Double Negation Elimination isn't on, but Double Negation Introduction is: P implies Not-Not-P. But at least that's as bad as it gets: you don't have to worry about distinguishing septuple negations from octuple and such nonsense: Triple Negation reduces to Single!
		Conjunctions (disjunctions, conditionals, existential quantifications) of doubly negated constituents imply the corresponding doubly negated conjunctions (disjunctions, conditionals, existential quantifications) of the (un-negated) constituents: a pair of negation signs immediately governed by one of these operators can be "pulled out in front": (Not-Not-L & Not-Not-R) implies Not-Not-(L & R), and likewise for the others.
		The worst offenders here are the negative universal quantifier rule (which treats negated universal quantifications as if they were existential, and about which I will say no more), the negative conjunction rule (which treats negated conjunctions as disjunctions) and the positive conditional rule (which treats conditionals as if they were disjunctions).
	www.philosophy.unimelb.edu.au /handouts/161016/ex8.html (913 words)

	Reference.com/Encyclopedia/Classical logic
		Aristotle's Organon introduces his theory of syllogisms, which is a logic with a restricted form of judgments: assertions take one of four forms, All Ps are Q, Some Ps are Q, No Ps are Q, and Some Ps are not Q.
		These judgments find themselves if two pairs of two dual operators, and each operator is the negation of another, relationships that Aristotle summarised with his square of oppositions.
		Aristotle explicitly formulated the law of the excluded middle and law of non-contradiction in justifying his system, although these laws cannot be expressed as judgments within the syllogistic framework.
	www.reference.com /browse/wiki/Classical_logic (379 words)

	EMail Msg <199208280557.AA14282@venera.isi.edu>
		Furthermore, the first two steps in a proof by P's rules are always the same: (1) draw a double negation around the empty set; (2) insert the antecedent of the implication into the if-context.
		His discovery of 1896 was that by combining the relational graphs with an oval for a negative context, you got a complete system of FOL with some very nice rules of inference.
		However, he preferred the form with conjunction, negation, and existential because it gave simpler representations for most of the kinds of statements he wanted to make.
	www-ksl.stanford.edu /email-archives/interlingua.messages/232.html (1833 words)

	Defining double negation elimination -- Restall 8 (6): 853 -- Logic Journal of IGPL
		Defining double negation elimination -- Restall 8 (6): 853 -- Logic Journal of IGPL
		negations which verify a number of different theses.
		Please note that abstracts for content published before 1996 were created through digital scanning and may therefore not exactly replicate the text of the original print issues.
	jigpal.oxfordjournals.org /cgi/content/abstract/8/6/853 (144 words)

	Wikinfo \| Intuitionistic logic
		A more familiar example of a classical tautology which is invalid in intuitionistic logic concerns the so-called 'double negation elimination'.
		In intuitionistic logic, only the first is a theorem---double negation can be introduced, but not eliminated.
		The interpretation of negation in intuitionistic logic is different from its counterpart in classical logic.
	www.wikinfo.org /wiki.php?title=Intuitionistic_logic (859 words)

	Inference in First Order Logic: Horn Clauses
		The "universal elimination" rule lets us use a universally quantified sentence to reach a specific conclusion, i.e.
		The "existential elimination" rule lets us convert an existentially quantified sentence into a form without the quantifier.
		The "existential elimination" rule is also called the Skolemization rule, after a mathematician named Thoralf Skolem.
	www.sdsc.edu /~tbailey/teaching/cse151/lectures/chap09a.html (1010 words)

	PHIL 250: Introduction to Symbolic Logic
		The classical rule of reductio ad absurdum (CR)'symmetrizes' the operation of negation: it enables one to deduce A from ¬¬A. (The converse implication can be proved using only the introduction and elimination rules for negation.)
		Compare and contrast CR with the introduction rule for negation, and see how the negation sign has changed position.
		; whereas in CR, the negation sign occurs in the (eventually discharged) assumption for reductio.
	people.cohums.ohio-state.edu /tennant9/tennant_cr.html (121 words)

	EMail Msg <9208280153.AA18905@herodotus.cs.uiuc.edu>
		But it is fairly easy to define a less awkward normal form which has negations pushed inwards using DeMorgans laws but does not do this distribution, and redefine resolution on that form.
		But you also imply that it has some operational significance, so that a 'context' somehow restricts the attention of a search process: >Furthermore, they allow graphs to be imported >into a small context where only the information relevant to a proof >needs to be considered.
		Surely, to apply double-negation elimination wherever possible might seem to be a useful heuristic.
	www.ksl.stanford.edu /email-archives/srkb.messages/156.html (714 words)

	Greg Restall * Defining Double Negation Elimination
		Here is a PDF file of the paper for you to download, print and read.
		In his paper “Generalised Ortho Negation” J. Michael Dunn mentions a claim of mine to the effect that there is no condition on `perp frames’ equivalent to the holding of double negation elimination (from ~~A to infer A).
		In this paper I correct my error and analyse the behaviour of conditions on frames for negations which verify a number of different theses.
	consequently.org /writing/defdneg (156 words)

	Propositional Logic Applet
		The negation of a proposition: !p (traditionally a '¬' symbol)
		Since (a+!a) is T regardless of the state of a, the CNF becomes C1C2T*T. Conjunction of T with a true statement does not add anything to the statement.
		This means that the negation of that statement is always true.
	www.wu.ece.ufl.edu /books/CS/TheoreticalCS/logic.html (1382 words)

	CIS 301 Captain's log
		August 30: Examples of relations and functions; Propositional logic; propositions and propositional connectives; sequents; conjunction introduction and elimination; double negation introduction and elimination; proofs of some simple sequents.
		Taking a cue from the proof rules for conjunction, an elimination rule and an introduction rule for universal quantification was presented.
		An important trick is to decompose the desired postcondition into two components which can then serve as loop invariant and (negation of) loop guard.
	www.cis.ksu.edu /~tamtoft/CIS301/Fall02/log.html (1500 words)

	Citations: Stockholm Studies in Philosophy - Prawitz, Almqvist (ResearchIndex)
		He introduced proof reduction rules that essentially push double negation onto simpler formulae.
		that pushes double negation of an implicational formula into its components.
		Computationally the dual constructs of abstraction ff: Gamma and naming [fi] Gamma) which witness the elimination and introduction of absurdity respectively) give expression to a kind of generic jump operator.
	citeseer.ist.psu.edu /context/347258/0 (677 words)

	2002 Abstracts of MCS Reports and Preprints
		This paper introduces face elimination as the basic technique for accumulating Jacobian matrices by using a minimal number of arithmetic operations.
		A proof is free of double negation if none of its deduced steps contains a term of the form n(n(t)) for some term t, where n denotes negation.
		The first question asks for conditions on the hypotheses that, if satisfied, guarantee the existence of a double-negation-free proof when the conclusion is free of double negation.
	www-fp.mcs.anl.gov /division/publications/abstracts/abstracts02.htm (11977 words)

	File prop_logic.ML (Isabelle2005: October 2005)
		Not, Or, And), and ) ( perform double-negation elimination.
		SAnd (fm1, fm2) = And (fm1, fm2); (* ------------------------------------------------------------------------- ) ( simplify: eliminates True/False below other connectives, and double- ) ( negation ) ( ------------------------------------------------------------------------- ) ( prop_formula -> prop_formula *) fun simplify (Not fm) = SNot (simplify fm)
		only variables may be negated, but not subformulas).
	www.cl.cam.ac.uk /Research/HVG/Isabelle/dist/library/HOL/prop_logic.ML.html (1384 words)

	Fitch's Paradox of Knowability
		Williamson (1982) argues that Fitch's result is not a refutation of anti-realism, but a reason for the anti-realist to accept intuitionistic logic.
		Without double negation elimination one cannot derive Fitch's conclusion ‘all truths are known’ (at line 11) from ‘there is not a truth that is unknown’ (line 10).
		But compare with Wansing (2002), where a paraconsistent constructive relevant modal logic with strong negation is proposed to block the paradox.
	www.seop.leeds.ac.uk /entries/fitch-paradox (6217 words)

	Exercise 3
		The general answer is that the system-- only the rules of negation introduction and double negation elimination are needed to show this-- wouldn't be sound without it: we could deduce any formula we wanted, true, false, or ridiculous.
		Now use our "negation in, negation out" shuffle to get P as an item of the main derivation.
		(f) For every sentence of the language, either it or its negation (but not, by (a), both!) belongs to a truth set.
	www.philosophy.unimelb.edu.au /handouts/161016/ex3.html (1548 words)

	[No title]
		Inference rules: - Inference is to generate new sentences that are necessarily true, given that the old sentences are true.
		o Existential Elimination there_is v a ------------------------ SUBST({v/k}, a) e.g., from there_is x Kill(x, Victime), we can infer Kill(Murderer, Victim), as long as Murderer does not appear elsewhere in the knowledge base.
		^ Pn -> Q where P1, P2,..., Pn and Q are atomic sentences (with no negation) e.g., rainy(x) ^ cold(x) -> snowy(x).
	www.eecis.udel.edu /~lliao/cis670/120302.txt (569 words)

Try your search on: Qwika (all wikis)