Inconsistent mathematics is the study of the mathematical theoriesthat result when classical mathematical axioms are asserted within theframework of a (non-classical) logic which can tolerate the presenceof a contradiction without turning every sentence into a theorem.

• 1. Foundations of Mathematics
• 2. Arithmetic
• 3. Analysis
• 4. Geometrical Inconsistency
• 5. Chunk and Permeate
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Foundations of Mathematics

Inconsistent Mathematics began historically with foundationalconsiderations. Frege and Russell proposed to found their mathematicson the naive principle of set theory: to every predicate is a set. Butthe naive principle leads rapidly to a proof of the existence of theRussell set, the set of all sets not members of themselves, which bothis and is not a member of itself. This and other set-theoreticparadoxes noted by Russell and others led to attempts to produceconsistent set theories as a foundation for mathematics. Perhaps thebest known of these was Zermelo-Fraenkel set theory ZF. But ZF andothers such as NBG and the like were in various ways ad hoc, having toinclude multiple independent principles instead of a single simplecomprehension axiom. Hence, a number of people including da Costa(1974), Brady (1971, 1989), Priest, Routley, & Norman (1989, pp.152, 498), considered it preferable to retain the full power of thenatural comprehension principle, and tolerate a degree ofinconsistency in set theory. Brady, in particular, has extended,streamlined and simplified these results on naive set theory in hisbook (2006). For a clear account see also Restall’s review(2007).

These constructions require, of course, that one dispenseatleast with that principle of Boolean logicex contradictionequodlibet (ECQ) (from a contradiction every proposition may bededuced, also called explosion). In passing, C.I.Lewis showed that ECQfollows by a simple argument from the principle disjunctive syllogism(DS) (fromA-or-B and not-A deduceB). So DS has to go too. Obviously, ECQ trivialises anyinconsistent theory (triviality = every sentence is provable).Non-triviality should be regarded as a constraint on interestingtheories: a trivial theory is useless for mathematical calculation,since a trivial theory does not distinguish principles in the theoryfrom their negations. Similarly, considerable debate (Burgess 1981,Mortensen 1983), made it clear that dispensing with DS was not socounter-intuitive, especially when a plausible story emerged about thespecial conditions under which DS continues to hold.

It should also be noted that Brady’s construction ofinconsistent naive set theory opens the door to a revival ofFrege-Russell logicism (briefly, that mathematics reduces to logic.)Logicism was widely held, even by Frege himself, to have been badlydamaged by the Russell Paradox. If the Russell Contradiction does notspread, then there is no obvious reason why one cannot hold that naiveset theory provides an adequate foundation for mathematics, and thatnaive set theory is deducible from logic via the naive comprehensionschema. The only change needed is a move to an inconsistency-tolerantlogic. Even more radically, Weber, in related papers (2010), (2012),(2013), has taken the inconsistency to be a positive virtue, since itenables us to settle several questions that were left open by Cantor,namely, that the well-ordering theorem and the axiom of choice areprovable, and that the Continuum Hypothesis is false (2012, 284). Someof these come out provably both true and false; wherein Weber isconcerned to advance proofs of theclassical recapture, whichis the project of showing that traditional results remain provable(2010, 72). This is invigorating new ground. Weber also showedsomething essential to this project, namely, that Cantor’sTheorem continues to hold; that is, it does not depend onoverly-strong logical principles which are contested byparaconsistentists. Retaining Cantor’s Theorem is important inWeber’s view, since different orders of infinity remainavailable in inconsistent set theory.

In addition, mathematics contains machinery for a metalanguage, thatis for talking about mathematics itself. This includes the concepts:(i) names for mathematical statements and other parts of syntax, (ii)self-reference, (iii) proof and (iv) truth. Gödel’scontribution to the philosophy of mathematics was to show that thefirst three of these can be rigorously expressed in arithmeticaltheories, albeit in theories which are either inconsistent orincomplete. The possibility of a well-structured example of the formerof these two alternatives, namely inconsistency, was not takenseriously, again because of belief in ECQ. However, in additionnatural languages seem to have their own truth predicate. Combinedwith self-reference this produces the Liar paradox, “Thissentence is false”, an inconsistency. Priest (1987) and Priest,Routley and Norman (1989, p. 154) argued that the Liar had to beregarded as a statement both true and false, a true contradiction.This represents another argument for studying inconsistent theories,namely the claim that some contradictions are true, also known asdialetheism. Kripke (1975) proposed instead to model a truthpredicate differently, in a consistent incomplete theory. We see belowthat incompleteness and inconsistency are closely related.

To simplify the motivation, mathematics, like many another science,encounters anomalies, puzzles and paradoxes. Paradoxes typicallyappear in the form of contradictions where there are grounds foraccepting incompatible sides of the contradiction. Mathematicsprogresses at least in part by eliminating anomalies in favour ofconsistent reconstructions. This has typically been the method ofaddressing anomalies, and it is facilitated by noticing that thecontradictions emerge in the foundations of mathematics. But it wasnoticed in the later twentieth century that there is another way,namely accept the contradiction and develop mathematical theoriescontaining both sides of the contradiction. This would be impossibleif the contradictory theory was erected on a logical foundationcontaining the Boolean principle Ex Contradictione Quodlibet ECQ, froma contradiction everything follows. So ECQ has to be abandoned, butfortunetely that proves possible, indeed mathematicallystraightforward. What remains is a rich field, of novel mathematicalapplications interesting in their own right, which sidestep the vexingquestions of which foundational principles to adopt, by developingcontradictions in areas of mathematics such as number theory oranalysis which are far from foundations. This is inconsistentmathematics.

2. Arithmetic

But these remarks have been about foundations, and mathematics is notits foundations. Hence there is a further independent motive, to seewhat mathematical structure remainswherever the constraintof consistency is relaxed. But it would be wrong to regard this as inany way a repudiation of the structures studied in classicalmathematics: inconsistent structures represent anaddition toknown structures.

Robert K. Meyer (1976) seems to have been the first to think of aninconsistent arithmetical theory. At this point, he was moreinterested in the fate of a consistent theory, his relevant arithmeticR#. This amounts to the axioms for Peano arithmetic, with a base ofthe quantified relevant logic RQ, and Meyer hoped that the weaker baseof relevant logic would allow more models. He was right. There provedto be a whole class of inconsistent arithmetical theories; see Meyerand Mortensen (1984), for example. In a parallel with the aboveremarks on rehabilitating logicism, Meyer argued that thesearithmetical theories provide the basis for a revived Hilbert Program.Hilbert’s program was the project of rigorously formalisingmathematics and proving its consistency by simple finitary/inductiveprocedures. It was widely held to have been seriously damaged byGödel’s Second Incompleteness Theorem, according to whichthe consistency of arithmetic was unprovable within arithmetic itself.But a consequence of Meyer’s construction was that within hisarithmetic R# it was demonstrable by finitary means that whatevercontradictions there might happen to be, they could not adverselyaffect any numerical calculations. Hence Hilbert’s goal ofconclusively demonstrating that mathematics is trouble-free proveslargely achievable as long as inconsistency-tolerant logics areused.

The arithmetical models used by Meyer and Mortensen later proved toallow inconsistent representation of the truth predicate. They alsopermit representation of structures beyond natural number arithmetic,such as rings and fields, including their order properties.Axiomatisations were also provided. Recently, the finite inconsistentarithmetical collapse models, a strictly larger class than thosestudied by Meyer and Mortensen, have been completely characterised byGraham Priest. Collapse models are obtained from classical models bycollapsing the domain down to congruence classes generated by variouscongruence relations. When members of the same congruence class areidentified, the theories produced are inconsistent. For example,Meyer’s initial construction collapsed the integers under thecongruencemodulo 2. This puts 0 and 2 in the same congruenceclass, so that in a suitable three-valued logic, both 0=2 andnot-(0=2) hold. Priest showed that these models take a certain generalform, see Priest (1997) and (2000). Strictly speaking, Priest went alittle too far in including “clique models”. This wascorrected by Paris and Pathmanathan (2006), and extended into theinfinite by Paris and Sirokfskich (2008). Even more recently, Tedder(2015) obtained axiomatisations for the class of finite collapsemodels with a different background logic, Avron’s A3.

3. Analysis

One could hardly ignore the examples of analysis and its special case,the calculus. For a model-theoretic approach to these see Mortensen(1990, 1995).

Now Meyer’s original approach to the natural numbers, that isR#, was axiomatic rather than model-theoretic. The axiomatic approachhas also been taken inanalysis by McKubre-Jordens and Weber(2012). In axiomatising analysis with a base of paraconsistent logic,their paper pushes Meyer’s approach to arithmetic via R# a longway further. These same authors (2017) rework the theory ofintegration as it was in Archimedes’ hands, which employs themethod of exhaustion, using paraconsistent reasoning. This gives aresult “up to inconsistency”, which means that one is ableto prove “Classical result or contradiction”. Theclassical result can be then seen to be recapturable by the classicalmove disjunctive syllogism applied to the classically-false(inconsistent) second disjunct.

It is certainly important and worthy to pursue this direction, but amild caution is entered here: the axiomatic project is a bit differentfrom inconsistent mathematics. As noted earlier, Meyer in this phasewas consistentist – he sought a consistent theory with aninconsistency-tolerant logic. With similar motivation, he was alsoconcerned to try to settle what he called “the gammaproblem”, which was essentially the question of whether theaxiomatic theory R# could be shown to contain classical Peanoarithmetic as a sub-theory. If this were so, then his proof ofnontriviality for R# would immediately yield a new proof of thenegation consistency of classical Peano arithmetic! Note that thiswould not be contrary to Godel’s Second Theorem, sincepresumably the proof of the gamma result would not be confined tofinitary techniques. (In the case of Meyer’s theory, it turnedout to be not so.)

There have proved to be many places throughout analysis where thereare distinctive inconsistent insights. The examples in the remainderof this section are drawn from Mortensen (1995). For example: (1)Robinson’s (1974) non-standard analysis was based oninfinitesimals, quantities smaller than any real number, as well astheir reciprocals, the infinite numbers. This has an inconsistentversion, which has some advantages for calculation in being able todiscard higher-order infinitesimals. Interestingly, the theory ofdifferentiation turned out to have these advantages, while the theoryof integration did not. A similar result, using a different backgroundlogic, was obtained by Da Costa (2000). (2) Another place to findapplications of inconsistency in analysis is topology, where onereadily observes the practice of cutting and pasting spaces beingdescribed as “identification” of one boundary withanother. One can show that this can be described in an inconsistenttheory in which the two boundaries are both identical and notidentical, and it can be further argued that this is the most naturaldescription of the practice. (3) Yet another application is the classof inconsistent continuous functions. Not all functions which areclassically discontinuous are amenable of inconsistent treatment; butsome are, for examplef(x)=0 for allx<0andf(x)=1 for allx≥0. The inconsistentextension replaces the first < by ≤, and has distinctivestructural properties. These inconsistent functions may well have someapplication in dynamic systems in which there are discontinuous jumps,such as quantum measurement systems. Differentiating such functionsleads to the delta functions, applied by Dirac to the study of quantummeasurement. (4) Next, there is the well-known case of inconsistentsystems of linear equations, such as the system (i)x+y=1, plus (ii)x+y=2. Suchsystems can potentially arise within the context of automated control.Little work has been done classically to solve such systems, but itcan be shown that there are well-behaved solutions within inconsistentvector spaces. (5) Finally, one can note a further application intopology and dynamics. Given a supposition which seems to beconceivable, namely that whatever happens or is true, happens or istrue on an open set of (spacetime) points, one has that the logic ofdynamically possible paths is open set logic, that is to sayintuitionist logic, which supports incomplete theories par excellence.This is because the natural account of the negation of a propositionin such a space says that it holds on the largest open set containedin the Boolean complement of the set of points on which the originalproposition held, which is in general smaller than the Booleancomplement. However, specifying a topological space by its closed setsis every bit as reasonable as specifying it by its open sets. Yet thelogic of closed sets is known to be paraconsistent, ie. supportsinconsistent nontrivial theories; see Goodman (1981), for example.Thus given the (alternative) supposition which also seems to beconceivable, namely that whatever is true is true on a closed set ofpoints, one has that inconsistent theories may well hold. This isbecause the natural account of the negation of a proposition, namelythat it holds on the smallest closed set containing the Booleannegation of the proposition, means that on the overlapping boundaryboth the proposition and its negation hold. Thus dynamical theoriesdetermine their own logic of possible propositions, and correspondingtheories which may be inconsistent, and are certainly as natural astheir incomplete counterparts.

On closed set logic and boundaries as a natural setting forcontradictory theories, see Mortensen (2003, 2010). Weber and Cotnoir(2015) also explore the inconsistency of boundaries, arising from theincompatibility of the three principles (i) there are boundaries, (ii)space is topologically connected, and (iii) discrete entities can bein contact (i.e., no space between them). This is a very interestingproblem, as all three are plausible; in particular there do seem to beboundaries in our world. An initially surprising feature of thisaccount is that boundaries come out as “empty”; after all,null entities are contrary to the spirit of mereology. But this is notso shocking as it turns out that they are only empty in the sense thatthey have members inconsistently.

Category theory throws light on many mathematical structures. It hascertainly been proposed as an alternative foundation for mathematics.Such generality inevitably runs into problems similar to those ofcomprehension in set theory; see, e.g., Hatcher 1982 (pp.255–260). Hence there is the same possible application ofinconsistent solutions. There is also an important collection ofcategorial structures, the toposes, which support open set logic inexact parallel to the way sets support Boolean logic. This has beentaken by many to be a vindication of the foundational point of view ofmathematical intuitionism. However, it can be proved that that toposessupport closed set logic as readily as they support open set logic, todate the only category-theoretic semantics for a paraconsistent logic.This should not be viewed as an objection to intuitionism, however, somuch as an argument that inconsistent theories are equally reasonableas items of mathematical study. See Mortensen (1995 Chap 11, co-authorLavers). This position has now been taken up, extended and ablydefended by Estrada-González (2010, 2015a, 2015b). The same author(2016) undertakes to provide a category-theoretic description oftrivial theories, with the aim of showing that triviality is not suchan uninteresting feature for mathematical theories to have. Thepresent author remains unconvinced, since a trivial theory is surelyuseless for mathematical calculation; but the ingenuity of thearguments must be conceded.

Duality between incompleteness/intuitionism andinconsistency/paraconsistency has at least two aspects. First there isthe above topological (open/closed) duality. Second there is Routley *duality. The Routley Star * of a set of sentencesS, isdefined asS* =_df {A:~A is not inS}. Discovered by the Routleys (1972)as a semantical tool for relevant logics, the * operation dualisesbetween inconsistent and incomplete theories of the large naturalclass of de Morgan logics. For example, starring Peano arithmeticgives a deductively-closed inconsistent complete non-trivial classicalarithmetical theory containing all of PA, which presents aninteresting challenge for the Godel incompleteness results, seeMortensen (2013). Both kinds of duality interact as well, where the *gives distinctive duality and invariance theorems for open set andclosed set arithmetical theories. On the basis of these results, it isfair to argue that both kinds of mathematics, intuitionist andparaconsistent, are equally reasonable.

4. Geometrical Inconsistency

A geometrical development is the application to explaining thephenomenon of inconsistent pictures. The best known of these areperhaps M. C. Escher’s masterpiecesBelvedere,Waterfall andAscending and Descending. In fact thetradition goes back millennia to Pompeii. Escher seems to have derivedmany of his intuitions from the Swedish artist Oscar Reutersvärd,who began his inconsistent work in 1934. Escher also activelycollaborated with the English mathematician Roger Penrose. There havebeen several attempts to describe the mathematical structure ofinconsistent pictures using classical consistent mathematics, bytheorists such as Cowan, Francis and Penrose. As argued in Mortensen(1997), however, no consistent mathematical theory can capture thesense that one is seeing an impossible thing. Only an inconsistenttheory can capture the content of that perception. This amounts to anappeal to a cognitive justification of paraconsistency. One can thenproceed to display inconsistent theories which are candidates for suchinconsistent contents. There is an analogy with classical mathematicson this point: projective geometry is a classical consistentmathematical theory which is interesting because we are creatures withan eye, since it explains why it is that things look the way they doin perspective.

Inconsistent geometrical studies are further developed in Mortensen(2002a), where category theory is applied to give a generaldescription of the relationships between the various theories andtheir consistent cut-downs and incomplete duals. For an informalaccount which highlights the difference between visual“paradoxes” and the philosophically more common paradoxesof language, such as the Liar, see Mortensen (2002b).

More recently, inconsistent mathematical descriptions have beenobtained for several classes of inconsistent figures, exemplified byEscher’s Cube (found in his printBelvedere), theReutersvärd-Penrose triangle, and others. See Mortensen(2010).

This raises another interesting issue. Ever since Euclid, the methodof reductio ad absurdum has been thoroughly exploited in geometricalproofs (assume he opposite of what you want to prove and deduce acontradiction). But here we have a different technique (employ acontradiction to describe how inconsistent figures look, todemonstrate that the content is genuinely contradictory). Ifsuccessful, this is a demonstration that human conceptualisationexceeds the merely possible (consistent). For discussion see Mortensen(2019, 2022).

One more point: do these geometrical paradoxes lend support to fulldialetheism? The answer is no. If you can draw a contradictory picturethensomethingexists that is inconsistent. And, distinctinconsistent figures imply distinctions between types ofcontradictions which ECQ does not permit. But this should not bepushed too far. Nothing says that because a picture has a content,that content must somehow betrue: many pictures arefictional in content. For more on this theme see Mortensen(2019a).

5. Chunk and Permeate

Recently, an alternative technique for dealing generally withcontradictions has emerged. Brown and Priest (2004) have proposed atechnique they call “Chunk and Permeate”, in whichreasoning from inconsistent premisses proceeds by separating theassumptions into consistent theories (chunks), deriving appropriateconsequences, then passing (permeating) those consequences to adifferent chunk for further consequences to be derived. They suggestthat Newton’s original reasoning in taking derivates in thecalculus, was of this form. This is an interesting and novel approach,though it must meet the objection that to believe a conclusionobtained on this basis, one should believe all the premisses equally;and so an argument of the more common form, appealing to all thepremisses without fragmenting them, should be eventually forthcoming.The objection is thus that Chunk and Permeate is part of the contextof discovery rather than the context of justification.

Later, Benhamet. al. (2014) have extended these methods tothe Dirac delta function. This broadens the class of applications, andso strengthens the technique. However, it also becomes clear there,that there is a close parallel between (one large class of) Chunk andPermeate applications, and (consistent) non-standard analysis:wherever Chunk and Permeate takes a derivative by shifting chunks toone where infinitesimals are zero, non-standard analysis takes aderivative by defining derivatives to be “standard partsonly”. Of course, equivalence between these two techniques doesnot show which is explanatorily deeper. Developments are to be awaitedwith interest.

6. Conclusion

To conclude: there has been appearing lately quite a bit ofphilosophical material, that is sympathetic to the cause ofinconsistent mathematics. Colyvan (2000) addresses the issue thatinconsistent mathematical theories imply inconsistent mathematicalobjects as their subject-matters. He also takes up the important taskof providing an account of how inconsistent mathematics can have abranch which isapplied mathematics. Priest (2013), likeColyvan, notes that inconsistent mathematics adds to the platonistmix. Berto (2007) usefully surveys paradoxes and foundational issues,and sets out some of the arithmetical results that bear on importantphilosophical issues like the Incompleteness Theorems. Van Bendegem(2014) pursues the interesting motivation that change is always astate of anomaly, so that always changing implies always anomalous.Examples include infinitesimals, complex numbers and infinity. Cautionshould be adopted over thinking that inconsistency is alwaysanomalous, however, if only because it is simply more material formathematical study.

It should be emphasised again that these structures do not in any waychallenge or repudiate existing mathematics, but rather extend ourconception of what is mathematically possible. This, in turn, sharpensthe issue of Mathematical Pluralism; see e.g., Davies (2005), Hellmanand Bell (2006), or Priest (2013). Various authors have differentversions of mathematical pluralism, but it is something along thelines that incompatible mathematical theories can be equally true. Thecase for mathematical pluralism rests on the observation that thereare different mathematical “universes” in which different,indeed incompatible, mathematical theorems or laws hold. Well-knownexamples are the incompatibility between classical mathematics andintuitionist mathematics, and the incompatibility between ZF-likeuniverses of sets respectively with, and without, the Axiom of Choice.It seems absurd to say that ZF with Choice is true mathematics and ZFwithout Choice is false mathematics, if they are both legitimateexamples of mathematically well-behaved theories.

The primary question for the philosophy of mathematics is surelywhat is mathematics. Duality operations like topologicalduality or Routley* reinforce the point that incomplete/inconsistentduals are equally reasonable as examples of mathematics. From thispoint of view, disputes about which of intuitionist or classical orinconsistent mathematics to accept seem pointless; they are all partof the subject matter of mathematics. This point is made effectivelyby Shapiro (2014, in contrast see his 2002). Shapiro’sdistinctive position has other ingredients: mathematics as the scienceof structure, and mathematical pluralism implying logical pluralism(on logical pluralism see also Beall and Restall 2006); but we do nottake these up here.

For what it is worth, the present writer thinks that some version ofmathematical pluralism is obviously true, if one takes mathematics tobe firstly about mathematical theories allowing for inconsistency ,and only secondly about the objects internal to those theories. Thereis, of course, no problem about incompatible theories co-existing, ifconsidered as structures of propositions. The primacy of theoriesfits, too, with the natural observation that the epistemology ofmathematics is deductive proof. It is only if one takes as a startingpoint the primacy of the mathematical object as the truth-maker oftheories, that one has to worry about how their objects manage toco-exist.

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contradiction |logic: paraconsistent |mathematics, philosophy of |self-reference