Misplaced Pages

#221778

42-523: [REDACTED] Look up LP in Wiktionary, the free dictionary. LP or lp may stand for: Businesses and organizations [ edit ] LP, Limited partnership in corporate law or a Limited Partner in a venture capital fund In politics [ edit ] Labour Party (disambiguation) , in several countries Liberal Party , in several countries Libertarian Party (United States) , or

84-539: A theory contains a single inconsistency, the theory is trivial – that is, it has every sentence as a theorem. The characteristic or defining feature of a paraconsistent logic is that it rejects the principle of explosion. As a result, paraconsistent logics, unlike classical and other logics, can be used to formalize inconsistent but non-trivial theories. The entailment relations of paraconsistent logics are propositionally weaker than classical logic ; that is, they deem fewer propositional inferences valid. The point

126-413: A contradiction, anything follows") can be expressed formally as Which means: if P and its negation ¬ P are both assumed to be true, then of the two claims P and (some arbitrary) A , at least one is true. Therefore, P or A is true. However, if we know that either P or A is true, and also that P is false (that ¬ P is true) we can conclude that A , which could be anything, is true. Thus if

168-430: A contradiction. However, if the rule double negation elimination ( ¬ ¬ A ⊢ A {\displaystyle \neg \neg A\vdash A} ) is added as well, then every proposition can be proved from a contradiction. Double negation elimination does not hold for intuitionistic logic . One example of paraconsistent logic is the system known as LP (" Logic of Paradox "), first proposed by

210-689: A diagnostic and, at times, therapeutic procedure, that is performed to collect a sample of cerebrospinal fluid Other uses in science and technology [ edit ] Liquefied petroleum gas , a hydrocarbon gas mixture Liquid propane Low power electronics Low-power broadcasting , in radio and TV broadcasting Low precipitation supercell , a subtype of highly organized thunderstorm Sound pressure level , Lp Luyten-Palomar, astronomical survey catalog of high proper motion stars (LP numbers). L in zones -45 to -89 deg.; LP in zones +89 to -44 deg. See Star catalogue#Proper motion catalogues Philosophy [ edit ] Logic of Paradox ,

252-422: A dialetheist rationally commits one to some form of paraconsistent logic, on pain of otherwise embracing trivialism , i.e. accepting that all contradictions (and equivalently all statements) are true. However, the study of paraconsistent logics does not necessarily entail a dialetheist viewpoint. For example, one need not commit to either the existence of true theories or true contradictions, but would rather prefer

294-403: A manufacturer of building materials lowercase people , an organization founded by rock band Switchfoot Ladakh Police , police agency of Ladakh, India Lonely Planet , travel publisher Liberapay Science, technology, philosophy [ edit ] Computing and mathematics [ edit ] lp (Unix) , command for printing documents L space ℓ space LPMud ,

336-595: A member of the boy band One Direction. Lil Peep (died 2017), an American rapper and singer-songwriter. Albums [ edit ] LP (Ambulance LTD album) , 2004 LP (Discovery album) , 2009 LP (Holy Fuck album) , 2007 LP (Insomniac Folklore album) , 2010 LP! , by JPEGMafia, 2021 LP1 (Liam Payne album) 2019 LP (Landon Pigg album) , 2006 The LP , by Large Professor, 1996 L.P. (The Rembrandts album) , 1995 LP (Soviettes album) , 2003 Other uses [ edit ] Lateral pass in gridiron football Lesson plan ,

378-576: A member thereof Liberty Party (disambiguation) , in several countries Schools [ edit ] Lycée professionnel , French vocational high schools Lower Primary school, a subdivision of primary schools in certain places Lorne Park Secondary School , a high school in Mississauga, Ontario, Canada Other businesses and organizations [ edit ] LAN Perú , an airline based in Lima, Peru (IATA code LP) Louisiana-Pacific ,

420-654: A paraconsistent logic Music [ edit ] LP record , a long-playing 12- or 10-inch (30 or 25 cm) vinyl record that spins at 33⅓ rpm LP (singer) , American indie pop singer El-P (born 1975), American rapper Latin Percussion , a brand of percussion instruments Laxmikant–Pyarelal (1940–1998), Indian music director duo Gibson Les Paul , electric guitar Linkin Park , an American rock band from Agoura Hills, California Liam Payne (born 29 August 1993) an English singer. He rose to fame as

462-422: A teacher's detailed description of the course of instruction for a class Let's Play , a style of video series documenting the playthrough of a video game Liquidity provider in finance Listening post , a facility established to monitor radio and microwave signals Little person, someone affected by dwarfism Lower Peninsula of Michigan United Nations laissez-passer , a travel document issued by

SECTION 10

#1732766280222

504-521: A theory entails all possible consequences) are assumed inseparable, granted that negation is available. These views may be philosophically challenged, precisely on the grounds that they fail to distinguish between contradictoriness and other forms of inconsistency. On the other hand, it is possible to derive triviality from the 'conflict' between consistency and contradictions, once these notions have been properly distinguished. The very notions of consistency and inconsistency may be furthermore internalized at

546-424: A true hypothesis and a true implication lead to a true conclusion, we must have that a not-true ( f ) conclusion and a true ( t or b ) hypothesis yield a not-true implication. If all the propositional variables in Γ are assigned the value b , then Γ itself will have the value b . If we give X the value f , then So Γ→ X will not be a tautology. Limitations: (1) There must not be constants for

588-510: A type of virtual world server software created in 1989 LP or lp, the device name for a printer in some computer operating systems A legacy abbreviation derived from line printer but now used for other types of printer Larch Prover , in automated theorem proving system Linear programming , in applied mathematics LivePerson , software company producing AI chatbots Logic programming Medicine and psychology [ edit ] Licensed Psychologist Lumbar puncture ,

630-451: A weaker standard like empirical adequacy , as proposed by Bas van Fraassen . In classical logic Aristotle's three laws, namely, the excluded middle ( p or ¬ p ), non-contradiction ¬ ( p ∧ ¬ p ) and identity ( p iff p ), are regarded as the same, due to the inter-definition of the connectives. Moreover, traditionally contradictoriness (the presence of contradictions in a theory or in a body of knowledge) and triviality (the fact that such

672-403: Is also a tautology of paraconsistent logic (by merging b into t ). This logic is sometimes referred to as "Pac" or "LFI1". Some tautologies of paraconsistent logic are: Some tautologies of classical logic which are not tautologies of paraconsistent logic are: Suppose we are faced with a contradictory set of premises Γ and wish to avoid being reduced to triviality. In classical logic,

714-505: Is both paraconsistent and paracomplete (a subsystem of intuitionistic logic). The other three simply do not allow one to express a contradiction to begin with since they lack the ability to form negations. Here is an example of a three-valued logic which is paraconsistent and ideal as defined in "Ideal Paraconsistent Logics" by O. Arieli, A. Avron, and A. Zamansky, especially pages 22–23. The three truth-values are: t (true only), b (both true and false), and f (false only). A formula

756-478: Is different from Wikidata All article disambiguation pages All disambiguation pages LP">LP The requested page title contains unsupported characters : ">". Return to Main Page . Logic of Paradox Inconsistency-tolerant logics have been discussed since at least 1910 (and arguably much earlier, for example in the writings of Aristotle ); however, the term paraconsistent ("beside

798-513: Is easy to check that this valuation constitutes a counterexample to both explosion and disjunctive syllogism. However, it is also a counterexample to modus ponens for the material conditional of LP. For this reason, proponents of LP usually advocate expanding the system to include a stronger conditional connective that is not definable in terms of negation and disjunction. As one can verify, LP preserves most other inference patterns that one would expect to be valid, such as De Morgan's laws and

840-777: Is not truth-functional as one might expect a 'but not' operator to be; similarly, the intuitionistic implication operator cannot be treated like " ¬ ( A ∧ ¬ B ) ". Dual-intuitionistic logic also features a basic connective ⊤ which is the dual of intuitionistic ⊥: negation may be defined as ¬ A = (⊤ # A ) A full account of the duality between paraconsistent and intuitionistic logic, including an explanation on why dual-intuitionistic and paraconsistent logics do not coincide, can be found in Brunner and Carnielli (2005). These other logics avoid explosion: implicational propositional calculus , positive propositional calculus , equivalential calculus and minimal logic . The latter, minimal logic,

882-422: Is only one of many paraconsistent logics that have been proposed. It is presented here merely as an illustration of how a paraconsistent logic can work. One important type of paraconsistent logic is relevance logic . A logic is relevant if it satisfies the following condition: It follows that a relevance logic cannot have ( p ∧ ¬ p ) → q as a theorem, and thus (on reasonable assumptions) cannot validate

SECTION 20

#1732766280222

924-639: Is that a paraconsistent logic can never be a propositional extension of classical logic, that is, propositionally validate every entailment that classical logic does. In some sense, then, paraconsistent logic is more conservative or cautious than classical logic. It is due to such conservativeness that paraconsistent languages can be more expressive than their classical counterparts including the hierarchy of metalanguages due to Alfred Tarski and others. According to Solomon Feferman : "natural language abounds with directly or indirectly self-referential yet apparently harmless expressions—all of which are excluded from

966-429: Is that, if ¬ A , then A is excluded and B can be inferred from A ∨ B . However, if A may hold as well as ¬A , then the argument for the inference is weakened. Yet another approach is to do both simultaneously. In many systems of relevant logic , as well as linear logic , there are two separate disjunctive connectives. One allows disjunction introduction, and one allows disjunctive syllogism. Of course, this has

1008-453: Is true if its truth-value is either t or b for the valuation being used. A formula is a tautology of paraconsistent logic if it is true in every valuation which maps atomic propositions to { t , b , f }. Every tautology of paraconsistent logic is also a tautology of classical logic. For a valuation, the set of true formulas is closed under modus ponens and the deduction theorem . Any tautology of classical logic which contains no negations

1050-474: Is true, and V ( A , 0 ) {\displaystyle V(A,0)\,} means that A {\displaystyle A\,} is false. A formula must be assigned at least one truth value, but there is no requirement that it be assigned at most one truth value. The semantic clauses for negation and disjunction are given as follows: (The other logical connectives are defined in terms of negation and disjunction as usual.) Or to put

1092-530: The Argentinian logician Florencio González Asenjo in 1966 and later popularized by Priest and others. One way of presenting the semantics for LP is to replace the usual functional valuation with a relational one. The binary relation V {\displaystyle V\,} relates a formula to a truth value : V ( A , 1 ) {\displaystyle V(A,1)\,} means that A {\displaystyle A\,}

1134-464: The Tarskian framework." This expressive limitation can be overcome in paraconsistent logic. A primary motivation for paraconsistent logic is the conviction that it ought to be possible to reason with inconsistent information in a controlled and discriminating way. The principle of explosion precludes this, and so must be abandoned. In non-paraconsistent logics, there is only one inconsistent theory:

1176-521: The United Nations to its staff Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title LP . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=LP&oldid=1250423031 " Category : Disambiguation pages Hidden categories: Short description

1218-512: The consistent") was first coined in 1976, by the Peruvian philosopher Francisco Miró Quesada Cantuarias . The study of paraconsistent logic has been dubbed paraconsistency , which encompasses the school of dialetheism . In classical logic (as well as intuitionistic logic and most other logics), contradictions entail everything. This feature, known as the principle of explosion or ex contradictione sequitur quodlibet ( Latin , "from

1260-410: The contrapositive is added. Intuitionistic logic allows A ∨ ¬ A not to be equivalent to true, while paraconsistent logic allows A ∧ ¬ A not to be equivalent to false. Thus it seems natural to regard paraconsistent logic as the " dual " of intuitionistic logic. However, intuitionistic logic is a specific logical system whereas paraconsistent logic encompasses a large class of systems. Accordingly,

1302-506: The deduction theorem as well as the axioms which are the introduction and elimination rules for the logical connectives (where possible). To this end, we add a third truth-value b which will be employed within the compartment containing the contradiction. We make b a fixed point of all the logical connectives. We must make b a kind of truth (in addition to t ) because otherwise there would be no tautologies at all. To ensure that modus ponens works, we must have that is, to ensure that

LP - Misplaced Pages Continue

1344-449: The disadvantages entailed by separate disjunctive connectives including confusion between them and complexity in relating them. Furthermore, the rule of proof of negation (below) just by itself is inconsistency non-robust in the sense that the negation of every proposition can be proved from a contradiction. Strictly speaking, having just the rule above is paraconsistent because it is not the case that every proposition can be proved from

1386-421: The dual notion to paraconsistency is called paracompleteness , and the "dual" of intuitionistic logic (a specific paracomplete logic) is a specific paraconsistent system called anti-intuitionistic or dual-intuitionistic logic (sometimes referred to as Brazilian logic , for historical reasons). The duality between the two systems is best seen within a sequent calculus framework. While in intuitionistic logic

1428-480: The following usual Boolean properties hold: double negation as well as associativity , commutativity , distributivity , De Morgan , and idempotence inferences (for conjunction and disjunction). Furthermore, inconsistency-robust proof of negation holds for entailment: (A⇒(B∧¬B))⊢¬A. Another approach is to reject disjunctive syllogism. From the perspective of dialetheism , it makes perfect sense that disjunctive syllogism should fail. The idea behind this syllogism

1470-413: The inference from { p , ¬ p } to q . Paraconsistent logic has significant overlap with many-valued logic ; however, not all paraconsistent logics are many-valued (and, of course, not all many-valued logics are paraconsistent). Dialetheic logics , which are also many-valued, are paraconsistent, but the converse does not hold. The ideal 3-valued paraconsistent logic given below becomes the logic RM3 when

1512-527: The object language level. Paraconsistency involves tradeoffs. In particular, abandoning the principle of explosion requires one to abandon at least one of the following two principles: Both of these principles have been challenged. One approach is to reject disjunction introduction but keep disjunctive syllogism and transitivity. In this approach, rules of natural deduction hold, except for disjunction introduction and excluded middle ; moreover, inference A⊢B does not necessarily mean entailment A⇒B. Also,

1554-418: The only method one can use is to reject one or more of the premises in Γ. In paraconsistent logic, we may try to compartmentalize the contradiction. That is, weaken the logic so that Γ→ X is no longer a tautology provided the propositional variable X does not appear in Γ. However, we do not want to weaken the logic any more than is necessary for that purpose. So we wish to retain modus ponens and

1596-470: The same point less symbolically: (Semantic) logical consequence is then defined as truth-preservation: Now consider a valuation V {\displaystyle V\,} such that V ( A , 1 ) {\displaystyle V(A,1)\,} and V ( A , 0 ) {\displaystyle V(A,0)\,} but it is not the case that V ( B , 1 ) {\displaystyle V(B,1)\,} . It

1638-402: The sequent is not derivable, in dual-intuitionistic logic is not derivable . Similarly, in intuitionistic logic the sequent is not derivable, while in dual-intuitionistic logic is not derivable. Dual-intuitionistic logic contains a connective # known as pseudo-difference which is the dual of intuitionistic implication. Very loosely, A # B can be read as " A but not B ". However, #

1680-467: The trivial theory that has every sentence as a theorem. Paraconsistent logic makes it possible to distinguish between inconsistent theories and to reason with them. Research into paraconsistent logic has also led to the establishment of the philosophical school of dialetheism (most notably advocated by Graham Priest ), which asserts that true contradictions exist in reality, for example groups of people holding opposing views on various moral issues. Being

1722-563: The truth values because that would defeat the purpose of paraconsistent logic. Having b would change the language from that of classical logic. Having t or f would allow the explosion again because would be tautologies. Note that b is not a fixed point of those constants since b ≠ t and b ≠ f . (2) This logic's ability to contain contradictions applies only to contradictions among particularized premises, not to contradictions among axiom schemas. (3) The loss of disjunctive syllogism may result in insufficient commitment to developing

LP - Misplaced Pages Continue

1764-491: The usual introduction and elimination rules for negation, conjunction , and disjunction. Surprisingly, the logical truths (or tautologies ) of LP are precisely those of classical propositional logic. (LP and classical logic differ only in the inferences they deem valid.) Relaxing the requirement that every formula be either true or false yields the weaker paraconsistent logic commonly known as first-degree entailment (FDE). Unlike LP, FDE contains no logical truths. LP

#221778