A Classical Prejudice?

Know Techn Pol (2010) 23:25–40DOI 10.1007/s12130-010-9098-4

SPECIAL ISSUE

A Classical Prejudice?

Patrick Allo

Received: 28 April 2010 / Accepted: 18 May 2010 / Published online: 24 July 2010© Springer Science+Business Media B.V. 2010

Abstract In this paper, I reassess Floridi’s solution to the Bar-Hillel–Carnapparadox (the information yield of inconsistent propositions is maximal) byquestioning the orthodox view that contradictions cannot be true. The mainpart of the paper is devoted to showing that the veridicality thesis (semanticinformation has to be true) is compatible with dialetheism (there are truecontradictions) and that, unless we accept the additional non-falsity thesis(information cannot be false), there is no reason to presuppose that there isno such thing like contradictory information.

Keywords Contradiction · Dialetheism · Logic · Method of abstraction ·Paradox · Philosophy of information · Semantic information ·Veridicality thesis

1 Introduction

The classical theory of semantic information, as formulated in Carnap and Bar-Hillel (1952), is a close relative of the usual model theoretic characterisation1

1The original version is based on state descriptions instead of models, but as may be seen fromKemeny (1953), it can be so reformulated.

Postdoctoral Fellow of the Science Foundation (FWO).

P. Allo (B)Centre for Logic and Philosophy of Science, Vrije Universiteit Brussel,Pleinlaan 2, 1050 Brussel, Belgiume-mail: [email protected]

P. AlloIEG, Oxford University, Oxford, UK

Author's personal copy

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of the classical consequence relation. It intendedly complies with the followingplatitude about logical consequence:

A conclusion A follows from premises � iff the content of A does notexceed the combined content of all the premises in �.

Moreover, it does so in a way which is co-extensive with the classical senseof follows from and thus provides what could be called a classical account ofinformational content.

It is therefore not surprising that they share many features. The one featureI am interested here is known as Ex Contradictione Quodlibet or Explosionand is one of the so-called paradoxes of material and strict implication: Froma contradiction, one can derive any old proposition. In a theory of semanticcontent, this corresponds to the fact that the most informative proposition is a(or rather, the) contradictory one. It is this last feature which Floridi (2004a)names the Bar-Hillel–Carnap paradox (BC paradox).

How these paradoxes arise in the respective theories is largely irrelevant forthe point I want to make. The usual approach (often summarised in terms ofthe inverse relationship principle) individuates informational contents as pro-portions of a space of possibilities (e.g. models). When it is assumed that con-tradictions cannot be true at any point in the space of possibilities, they are allassigned the null proportion of the space of possibilities. Since the null propor-tion has the total space as its complement, the corresponding informationalcontent is maximal. In other words, the standard story is just that since contra-dictions exclude all possibilities, they have maximal informational content. Butthis does not mean that no other story can be told. A purely structural account,according to which a consequence relation or individuation of semantic con-tent is characterised by the structural rules of (a) identity, (b) weakening,(c) contraction and (d) cut, leads to exactly the same result.

What matters more is the attitude one adopts in the face of these presumedparadoxes, and this readily reduces to one’s attitude towards contradictions.Either one opts for a traditional Tarskian approach according to which con-tradictions pose a problem because they are false and not (or at least not inthe first place) because they lead to triviality (Tarski 1944) or one believes thatexplosion is the real problem. Floridi’s option, which leads to the developmentof a theory of strongly semantic information, is to follow Tarski’s lead.

The diagnosis proposed in Floridi (2004a) is that a measure or individuationof informational content that is solely based on the individuation of contentas a proportion of a space of possibilities implements a semantic principlethat is simply too weak. The underlying problem is the assumption that truthsupervenes on semantic information. Once a stronger semantic principle—oneaccording to which semantic information encapsulates truth—is implemented,the BC paradox no longer arises (again, the actual implementation is beyondthe scope of this discussion). This is, in rough outline, the philosophical basisof the theory of strongly semantic information (henceforth, TSSI).

I have never been entirely happy with this solution, but for reasons that arediametrically opposed to the usual objections voiced by, e.g. Fetzer (2004a, b).


A Classical Prejudice? 27

My own position is unusual or at least atypical in the following sense: AlthoughI endorse the stronger semantic principle,2 I do not favour the solution to theBC paradox which initially led to the formulation of that principle. This isprimarily because of its structural similarity with the remark that even thoughExplosion is valid, it will never be sound in the sense of having both truepremises and a true conclusion.3 Even if according to logical orthodoxy itcannot be sound, it should still puzzle us that it is valid. Similarly, even thougha strong semantic principle blocks the BC paradox, there is still a problemwith pure individuations of semantic content that are based on the inverserelationship principle.

The aim of this paper is twofold. First, I want to give a more thoughtfularticulation of the above worry. Second, I want to take a closer look at how themethodology of the philosophy of information and in particular the method ofabstraction (Floridi 2008) influences or determines our logical options and theattitude we should adopt towards contradictions.

2 Dialetheism and Paraconsistency

My quibbles with Floridi’s way of dealing with the BC paradox are to a certainextent based on considerations about finding uniform solutions to structurallysimilar problems. Since I believe that even if unsound, the validity of explosionstill poses an independent problem, I have a default commitment to a similarresponse to Floridi’s appeal to a stronger semantic principle.4 The demand fora uniform solution fires in more than one direction, and three separate casestherefore need to be considered.

First, one could point out that the validity of explosion poses a problembecause even if contradictions cannot be true, we often need to reason frominconsistent theories. Since I endorse the stronger semantic principle proposedby Floridi, I need not be concerned with the practical need for paraconsis-tent inference. Of course, I agree that a paraconsistent solution to the pureindividuations of semantic content (i.e. the individuation of what Floridi callsweakly semantic information) makes sense. But this is largely uncontroversial.The interesting question is whether this paraconsistent solution should carryover to the strongly semantic case. This question cannot be answered by purelypractical considerations.

2Basically, my reasons for endorsing the stronger semantic principle is based on the epistemicvalue of information; see, e.g. Dretske (2009).3Sainsbury (1995, 136), but see also the discussion of Philosophy’s Most Dif f icult Problem inWoods (2003, 14–6).4A possible reply is that there is a good reason to assume that the solution should be different,namely the fact that information has to be true whereas mere semantic content does not haveto. This reply is flawed for two reasons: first, because premises have to be true as well; second,because the reply already presupposes that no contradiction can be true and that we only need toavoid explosion because sometimes we have to reason from a mix of true and false premises.


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Second, the BC paradox also has a conditional version—trivial contentdoes not exceed contradictory content—and it is all but clear how a strongersemantic principle can handle this problem. More exactly, I do not think thata solution which takes the conditional content of A, given B to be undefined(or, alternatively, null) if both A and B are false makes any sense (nor do Ithink that Floridi would endorse such a proposal).5 As I explained elsewhere(Allo 2007), this is because conditional content is not in general a factual kindof content. Again, this is not the main focus for this paper. My guess is thatFloridi would favour a solution where the conditional content of A, given B,is only defined if both are contingent, jointly compatible and (but this morecontroversial) the conditional B > A is true.

The third case we should consider bears directly on the interplay betweenexplosion (and the BC paradox) and the assumption that all contradictions arefalse. When it comes to classical logic (and a fortiori in Carnap and Bar-Hillel’stheory of semantic information), the interplay is straightforward: Both featuresare two sides of the same coin. Non-classical logicians are in general morereluctant to adhere to this close connection. Non-dialetheic paraconsistentlogicians will in general claim that even though there are no true contradic-tions, the argument from contradiction to triviality is invalid. Dialetheicparaconsistent logicians, by contrast, precisely reject explosion because theythink that at least some contradictions are (or may be) true (Priest 2006a).The pragmatically inspired argument for paraconsistency described above isdefinitively of the non-dialetheic kind, but does not exhaust the category;there are many other reasons to endorse such a position. Still, given thefocus on strongly semantic information, only a dialetheic approach can make adifference here.

From the above, one might hastily conclude that if dialetheism is true,Floridi’s solution to the BC paradox does not go through. We should be carefulin jumping to this conclusion. Before we can embrace a dialetheic modificationof strongly semantic information—or even just criticise it from a dialetheicperspective—we first need to ensure that they are compatible. This requiresa closer look at the sources of paradox in relation to the methodology of thephilosophy of information and in particular to the method of abstraction andits emphasis on consistency.

3 Consistency and Logical Orthodoxy

Orthodoxy has it that contradictions cannot be true, but how much of the tra-dition is mere prejudice? And more importantly, how much of this orthodoxyshould be upheld in the context of Floridi’s solution to the BC paradox? This

5Alternatively and in analogy with the standard approach to conditional probability, one could saythat the content of A, given B, is only defined if B is not necessarily false. I do not further explorethis alternative.



question cannot be answered in a void. My suggestion is to start from thehandful of methodological insights on which the philosophy of information isfounded and to see to what extent the veridicality thesis can be used to dismissthe existence of contradictory information. Two types of considerations can beused in defence of the orthodoxy: the alleged consistency of the world and thetheoretical virtues of consistency. Each of these should be evaluated on its ownmerits, that is, without appeal to the disastrous consequences of explosion.6

Apart from sheer prejudice, the most prominent reason for our reluctanceto even consider the possibility of true contradictions can be retraced to thecorrespondence intuition. Put crudely, for contradictions to be true, ultimatereality would have to be inconsistent as well, and this is quite hard to imagine.Yet, it is harder to turn it into a fool-proof argument for the consistency ofthe world. As Priest remarks “if one supposes reality to be constituted solelyby (non-propositional) objects, like tables and chairs, it makes no sense tosuppose that reality is inconsistent or consistent. This is simply a categorymistake” (Priest 2006b, 51).

Floridi could not agree more. In his view, ultimate reality can at best imposecertain limitations on how we perceive the world. Whatever non-propositionalobjects are, they only function as constraining affordances: “they limit thepossible models” (Floridi 2008, 325). To speak of constraining affordances interms of truth and consistency is again a category mistake. But Floridi makesan even stronger point. Ultimate reality lies beyond the limits of cognition(I am deliberately using the terminology of Priest (2002) to describe Floridi’sbroadly Kantian outlook). To describe ultimate reality in terms that can besemantically characterised (i.e. be referred to as true, or consistent) is to con-struct a model of reality. As far as I can see, there is no reason to assume thatthe constraints imposed by the world are such that only consistent models arepossible, or, at least, no argument to that effect which refers to the propertiesof reality itself is available to Floridi.

A second defence of logical orthodoxy can nevertheless be mounted, onethat directly appeals to desirable features of models. In Floridi’s view, what weknow are the models we construe of reality, and seeing data as constrainingaffordances means that there cannot be a single privileged model of reality. Toprevent these assumptions from reducing to a full-blown relativism, we needto be able to compare different models, to assess their respective virtues andto exclude or dismiss models with a poor track record. Clearly, consistency issuch a virtue—even the dialetheist concedes as much and will try to minimiseinconsistency as much as possible (but not more!)—but is it also sufficient todismiss all inconsistent models?

If we assume that, as a virtue of models, consistency is on a par with othertheoretical virtues like elegance, explanatory power, simplicity, parsimonyand informativeness, there is no reason to dismiss inconsistent models (Priest

6Remember: Explosion and the falsity of contradictions come apart in several non-classical logics.Even the law of non-contradiction is not sufficient to validate explosion.


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2006b, 123–5). If we disqualify inconsistent models, we will have to reject manymore models which, for instance, are ad hoc or do not fully explain all the data.Unless we assume there is a single best model (or just a handful such models),a mass rejection of inconsistent models is clearly undesirable. On this account,inconsistent information makes perfect sense. Since it can be true, it is genuineinformation and not a form of misinformation.

Of course, tradition has it that consistency has a special status. It is not justa theoretical virtue among many others. One only needs to look at the manyfailed or inconclusive attempts to define coherence, and note that indepen-dently of how it is explained, the status of consistency as a necessary conditionremains largely unchallenged. This only makes sense if we have good reasonsto treat consistency differently. Most reasons to that effect fall in either ofthe following categories. On the one hand, one may advance that unlike, say,simplicity, inconsistency is a fully reliable indicator of falsity. On the otherhand, one may point to the fact that unlike the other theoretical virtues, con-sistency is an all-or-nothing affair; a model is either consistent or it is not.With regard to the former, all we have is an argument that is not availableto Floridi. With regard to the latter, it only takes a moment of reflection to seethat the argument only goes through in the presence of explosion. One benefitof using a paraconsistent logic is precisely that degrees of inconsistency can bemeasured, and this is all we need to treat consistency as a virtue that needs tobe balanced with several others.

Perhaps, one may object,7 inconsistencies do indeed carry genuine infor-mation, but only in an uninteresting residual sense of informing us that ourmodel is inconsistent. It is, therefore, primarily a form of meta-informationrather than a form of factual information. The force of this objection dependson the prior assumption that an inconsistency only reports a property of themodel, in the sense that it only informs us of the lack of a given virtue of thatmodel, namely consistency. This crucial assumption is false. What is true isthat contradictions do convey such meta-information and that they are indeedspecial in that sense. This specialness is, however, only due to the fact thatexplicit contradictions (i.e. individual formulae of the form A ∧ ¬A) informus of a global property of a model, namely its inconsistency, whereas there isno formula which explicitly informs us about other more elusive virtues of amodel like for instance its coherence.

We should not put too much weight on this distinctive feature of consistency.Basically, all it reveals is that there are reliable indicators for inconsistency.Even if we have a good criterion, finding out whether a model is inconsistentmay still be intractable (in the computational sense), hard to uncover orinvolve a large number of premises (as in the Paradox of the Preface).8 The real

7This line of thought is loosely inspired by views of Floridi. I do not recall whether these wereexpressed in print.8Thus, if an inconsistency involves all premises, the only difference between inconsistency and,say, adhocness is that the former is a syntactic feature whereas the latter is not.



issue, however, is whether we can make sense of the claim that contradictionsonly convey information about the (in)consistency of a model. In short, wecannot! Because the distinction between ‘properties of the model’ and ‘prop-erties of what is being modelled’ is itself not factual, but methodological, thereare limits to how we can appeal to that distinction. We believe that consistencyis a virtue of models (and this is not something I dispute), and contradictionstherefore carry valuable meta-information about the inconsistency of a model.This is all the distinction warrants. If, in addition to that, we also want to upholdthat contradictions only carry such meta-information, we need to appeal tothe falsity of all contradictions. Indeed, we need to claim that because a con-tradiction cannot be true, it cannot convey information about what is beingmodelled but only meta-information about the model itself. We already knowthat this argument is not available in the present context.

The last few paragraphs explain why we cannot reject out of hand that con-tradictions may constitute or carry genuine (strongly semantic) information,i.e. information about what is the case. What this information may look likehas hitherto remained implicit. The next two sections fill this gap by investi-gating the informational potential of two common types of dialetheia or truecontradictions.

4 Paradox and Self-Reference

Paradoxical sentences like the liar are the prototype of true contradictions, theraison d’être of dialetheism itself. If there were only one true contradiction, itwould for sure be a sentence asserting its own falsity. This still leaves open thequestion of whether there is genuine paradoxical information. The quickanswer is that, of course, there is no such thing, or, at least, there is no suchthing if paradoxes do require self-reference.9 This, one may say, is one of thelessons of Kripke’s theory of truth (Kripke 1975): Self-referential sentencesare not grounded, and surely, our information should be grounded to qualifyas factual, or to put it in the terminology of the previous section: to conveyfactual as opposed to mere meta-information.

Such a quick appeal to groundedness only leads to a dismissive argumentagainst the possibility of paradoxical information. It for instance presupposesthat our language is not part of the world. Instead of taking a closer look atthe requirement that truth-apt sentences must be grounded, I shall considera more general argument against paradoxical information. That argument isbased on, on the one hand, a general schema proposed by Priest (2002) as the

9An alternative reply would be that liar sentences do not qualify as information because whatevertheir truth value, they will have that truth value as a matter of necessity. As a result, if onlycontingent truths qualify as semantic information, such sentences are easily dismissed. In view ofthe existence of so-called contingent liars (Field 2008, 24), it is immediately clear that this strategywill not work.


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source of all paradoxes—the Inclosure Schema—and, on the other hand, therestrictions imposed by the method of abstraction. What I wish to illustrate isthat, as a general rule, the method of abstraction steers clear of the sourcesof paradox.

The basis of the inclosure schema is the interplay between closure andtranscendence. Given a predicate ϕ, the procedure of diagonalisation is usedto generate an object which in virtue of that procedure is not ϕ (i.e. it tran-scends ϕ), but which, in view of the end result, happens to be ϕ as well (i.e. itis enclosed in ϕ as well). Consider, once more, a liar sentence asserting itsown untruth. In virtue of what it says of itself (formally, in virtue of theconstruction), it fails to be true (transcendence), but because it fails to be trueand what it says, it happens to be true as well (closure).

The general lesson is that contradictions arise at the limits of many concepts.The self-application of a concept is such a limit. Whenever we scrutinise thoselimits, we run into contradiction because the process of doing so has contradic-tory properties: It goes beyond the limit but also stretches the limit. As a result,we both stay within the boundaries of a concept and exceed its boundaries.The suggestion, then, is that when the procedure used to generate an objectwhich exceeds a given predicate is successful, but also happens to fall withinthe bounds of that predicate, we cannot avoid the resulting contradiction. Theend result does not just appear paradoxical, but really is.

Let us, for the sake of argument, grant that at least some instances of theinclosure schema generate genuine paradoxes. Can we deny that the result ofdiagonalisation qualifies as information? As I read it, it is a presupposition ofthe method of abstraction that it can indeed disqualify observables (or inter-preted typed variables) in virtue of their unexpected behaviour (I shall makethis more precise in a moment). Given that (a) information is always assessedat a given level of abstraction and that (b) levels of abstraction are collectionsof observables (Floridi 2008, 305–9), only the non-paradoxical qualifies asinformation. This is the hallmark of any restriction strategy which aims at theavoidance of paradoxes: It denies that such limits exist (e.g. there is no set ofall sets). The method of abstraction is not any different in that respect.

A more revealing insight can nevertheless be extracted from the aboveconsiderations. Consider a typed variable which can take any of the followingvalues: ϕ, ¬ϕ and neither. Because it is so defined that ϕ and ¬ϕ are exclu-sive (though not exhaustive; the neither option is available), it is sufficientfor that variable to take both values to be considered ill-typed. If a typedvariable fails to behave as intended, the resulting interpreted typed variablecannot be accepted as an observable. Hence, no level of abstraction willdepend on ill-typed variables.10 The relevant contrast, here, is that betweenintended behaviour and actual behaviour. The inclosure schema is the general

10Whether there are no observables based on ill-typed variables or no levels of abstraction basedon ill-typed interpreted variables is just a matter of terminology.



pattern which characterises the limit cases where intended behaviour andactual behaviour do not agree. Dialetheism embraces these mismatches (i.e. itdoes not deny the Existence condition, see Priest (2002, 9.2)), but because ofits commitment to successful construction, the method of abstraction can onlydismiss such mismatches (hence, deny the existence condition).

The situation described in the previous paragraph is not just a methodologi-cal difference. There is a genuine philosophical disagreement on the followingtwo issues: successful construction and the existence of the limit. With regardto the former, the disagreement concerns the question of how we should reactto the unintended behaviour of what we construct. This can be illustrated witha reference to the behaviour of negation. As Priest has often stressed (Priest2006b, Chapt. 4 & 5), it does not suffice to define negation in such a way thatϕ and ¬ϕ are (for all ϕ) exclusive and exhaustive for it to behave that way.Even if our negation is assumed to be Boolean (i.e. is given its usual truthconditions, or is defined in terms of excluded middle and non-contradiction),the truth conditions alone cannot be used to show that truth and falsity areexclusive—that is, unless we already assume that they are exclusive (whichis just to say that Explosion is valid). From the standpoint of the method ofabstraction, only one reaction is possible. If truth and falsity are intended tobe exhaustive and exclusive, then negation should behave accordingly in itsintended domain of application.

Put differently, intended behaviour and a well-defined domain of applica-tion are the distinctive features of successful construction. By tying intendedbehaviour to a previously defined domain of application, the features oflimiting cases on which the inclosure schema trades are well beyond the reachof the method of abstraction. To describe the limit, we need something like themost general level of abstraction or a level of abstraction that can be used todescribe itself (again, the analogy with other restriction strategies like Russell’sand Tarski’s is obvious). If no such level of abstraction exists, transcendenceand closure remain disconnected and the unintended behaviour of a previouslyconceived predicate or condition is blocked.

This concludes the discussion of paradoxical information. So far, the provi-sional conclusion seems that Floridi would not have to appeal to the propertiesof classical logic (mainly explosion) or to the necessary falsity of contradictions(consistency is not assumed to be given) to dismiss contradictory informationof the paradoxical kind. All that is required is an emphasis on successfulconstruction.11 Because the latter has independent theoretical virtues as wellas practical advantages, the resulting argument is more convincing than whatwe saw in Section 3.

11By successful construction, I only mean construction according to the specification, i.e. theagreement between intended and actual behaviour. Whether this also involves the successfulconstruction of a (physical) artefact is a separate issue we can refer to as implementability. Thelatter is undoubtedly relevant but beyond the scope of the present discussion.


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5 Semantic Dialetheism

Paradoxes arise because the extension and anti-extension of some predi-cates overlap. More exactly, they overlap when applied at the borders oftheir intended domain of application. In the case of the liar and other typicalparadoxes, the limit is reached by self-application. This is often a reason to con-sider such uses illegitimate. Presumably, the feeling that something has gonewrong is more closely related to the involved circularity than to the resultingcontradiction. If so, we may wonder whether a non-circular overlap of theextension and anti-extension of a predicate could, with the same conviction, bediagnosed as illegitimate. Unless one already presupposes a classical accountof negation, I do not think it should.

Overlaps between extension and anti-extension are due to overdefinedpredicates. The cogency of the latter kind of predicates is the crux of whatMares (2004) calls semantic dialetheism, the thesis that inconsistencies resultfrom how our predicates (or, more broadly, natural languages) apply to theworld. At least in one respect, this is a brand of dialetheism that nicely com-bines with the method of abstraction. Semantic dialetheism is defended asan intermediate position: It is stronger than the more popular doxastic moti-vations for paraconsistency (the position I dismissed in Section 2 for notmeeting the demands of strongly semantic information), but not as radicalas metaphysical dialetheism. The disagreement between the latter two posi-tions concerns the level at which contradictions (or inconsistencies) shouldbe situated. According to metaphysical dialetheism, the world is (or may be)irreducibly inconsistent; according to semantic dialetheism, some perfectly finedescriptions of the world are inconsistent.

At first, the distinction between metaphysical and semantic dialetheism mayseem incompatible with the broader methodology sketched in the previoussections. Consistency does not as such qualify as a property of the world itself.Yet, metaphysical dialetheism does not hold that a world made of non-propositional objects can itself be inconsistent. Rather, the idea is that thefacts of the world are such that “any consistent description of the world will[be] misdescribing at least one fact” (Mares 2004, 270). When characterised inthis manner, metaphysical dialetheism is clearly intelligible, even if one thinksthat all we can know are our models of the world.12

In its most useful form, semantic dialetheism reduces to the following twotheses: (a) consistent descriptions (or models) of the world are possible, but(b) such (re)descriptions may on the whole be worse off than inconsistentdescriptions. Thus, as soon as we grant that consistency is on a par with othertheoretical virtues of models, semantic dialetheism can be used to motivateinconsistent levels of abstraction, here understood as levels of abstraction

12Still, if one believes that we can only reason about relations between models and not aboutrelations between the world in itself and our models, one has to conclude that metaphysicaldialetheism cannot be defended.



where observables may have inconsistent types. Nothing else is required tospeak of strongly semantic inconsistent or contradictory information.

So far, this leaves open the question of how overdefined predicates maycome into existence. Basically, two aspects of how we construct models play arole (Mares 2004, sect. 8). The first one is the use of predicates outside theiroriginally intended domain of application. The second one is related to thefallibility of conceiving incompatible properties. Even if we think that P andnot P are jointly exclusive because the properties required for something tobe P are incompatible with the properties required for something to be non-P,there is no guarantee that this is so. Both are quite common. Extending thedomain of application is a sensible thing to do if we want our models to bemore inclusive and thus more informative, and by doing so, we often have togive up the total control over the avoidance of overlaps we had in the originaldomain. Even more, this total control can already be absent in the original orintended domain of application.

The above description looks like an easy target for the proponent of themethod of abstraction. The purpose of the method of abstraction, as well as ofthe broader field of formal methods it is based on, is to avoid situations like theone described above. The whole point is to guarantee the consistency of ourmodels and to preserve consistency when we move to new, more encompassingand more informative models (Floridi 2008, 325). I do not want to deny that,but would at least like to make a few critical comments. The first is that I donot have to deny the purpose of formal methods to make my point. I onlyneed to point out that methods intended to ensure consistency may still failto deliver. This probably will not convince the proponent of orthodox formalmethods: If the method does not deliver, then the outcome does not meetthe specifications and this is sufficient to disqualify it. The second criticism isthat this method does not sufficiently take the costs of maintaining consistencyinto account.

If semantic dialetheism has one virtue, then it is that it allows us to favourinconsistent descriptions whenever there are good reasons to do so. Sincefinding successor concepts for our actual overlapping concepts often meansthat we have to sacrifice some virtues of our original descriptions of the world,it is at least conceivable that the cost of restoring consistency can be a reasonto opt for, or at least not to dismiss, inconsistent descriptions. If we think of themethod of abstraction as a means to obtain the best or most useful models ofreality, then a strong adherence to the consistency ideology is at odds with thatpurpose. To claim otherwise is to give consistency a special status, and we haveseen (Section 3) that the usual arguments which motivate this special status aresimply not available.

6 Negation, Falsity, and Acceptance

A last issue we cannot entirely ignore is the relation between negation, falsityand acceptance. Let us start with the connection between negation and falsity.


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On one account, a paraconsistent negation has to be such that A and not-Aare logically independent. This account of negation is often called the non-truth-functional account of negation and is found in the works of Da Costaand in some paraconsistent logics discussed in Batens (1980). The hallmarkof this account is that falsity does not reduce to truth of negation. On asecond account, negation remains truth functional and falsity can be definedas truth of negation. This is Priest’s preferred account and leads to the viewthat the joint truth of A and not-A also leads to their joint falsity. In otherwords, contradictions are always false but sometimes also true. This view isoften equated with dialetheism and is the main reason why the dialetheist canmaintain that the law of non-contradiction is true. Both types of negation can,however, be used to spell out the details of semantic dialetheism. The problemsthey encounter are of course be different.

If, as with a non-truth-functional negation, A and not-A are logically inde-pendent, it is a common objection that ‘not’ fails to be a genuine negation.The actual force of this objection lies beyond the scope of the present paper,so I will only remark that (a) similar objections can be raised against anynon-classical negation (Quine 1986; Slater 1995) and are therefore just anexpression of logical orthodoxy, but the stronger objection that non-truth-functional negations even fail to be ‘contradictory forming’ operators (Priestand Routley 1989; Priest 2006b, Chapt. 4) is not so easily put aside and(b) the reasons for treating negation as primitive may indeed be prudential(preventing inconsistencies to spread) rather than semantic.

If, as with a truth-functional negation, the truth of contradictions impliesthat some formulae can be true as well as false, the following questions comeup. What should we accept? Should we favour the truth, or avoid falsity?Clearly, if there are true contradictions, we cannot do both. This issue bears onhow Sainsbury contrasts dialetheism with rational dialetheism (Sainsbury 1995,Chapt. 6). According to the former, there are true contradictions; according tothe latter, it is also rational to believe (or accept) true contradictions. Rationaldialetheism (if not vacuous) entails dialetheism simpliciter, but not vice versa.This means it is possible to endorse the latter while denying the former. A simi-lar distinction could perhaps be made with regard to information. Endorsingtrue contradictions does not mean we have to endorse the possible existenceof genuine contradictory information. The main argument for reasoning likethis is that a commitment to contradictory information forces us to accept theexistence of false information, and, as argued at length in Floridi (2005a), falseinformation is no information at all.13

This type of rejection of contradictory information does not have to beaccepted by the dialetheist. Floridi’s non-falsity thesis (false information doesnot count as information) depends on the veridicality thesis (information must

13Remark that in this case the move from semantic dialetheism to informational dialetheism doesnot involve considerations about what we can and/or should believe but only depends on thenature of semantic information itself.



be true), but the two theses are only equivalent on the assumption that truthand falsity are exclusive and exhaustive. As a result, the rejection of infor-mational dialetheism will have to rely on the non-falsity thesis as an indepen-dent premise (in addition to, or as a replacement of the veridicality thesis). I donot want to take a definite position with regard to this issue but will again justmake two comments. First, we could ask for evidence for the non-falsity thesis.As far as I can see, only a strong preference for keeping things consistent can beused to defend that thesis. Apart from that, because for most propositions thenon-falsity thesis is already implied by the veridicality thesis, it does not addanything worthwhile to the latter. Second, we could ask whether the non-falsitythesis can even have its intended force. That is, the formulation of the non-falsity thesis will itself depend on our means to express ‘true only’ and makesure that it has an extension that effectively does not overlap with falsity.14

Such expressions are a well-known source of extended paradox and even haveto be rejected by the dialetheist; see, e.g. Beall (2009, Chapt. 3) on incoherentoperators. Whether we can successfully define a kind of information that istrue only will therefore unavoidably depend on how we express the consistencycondition.

7 Concluding Remarks

What I tried to show in the previous sections is that a dialetheic account ofstrongly semantic information is not easily dismissed. My personal view is thatsuch an account has many virtues but that an approach which systematicallyenforces consistency remains possible. I have briefly mentioned several waysof conceiving such an account (e.g. by claiming that information should be ‘trueonly’) and have pointed out why this may be problematic. More importantly,I have often emphasised where arguments for a consistent approach fail andwhat the costs of an overall consistent approach are. In brief: We cannot appealto the consistency of the world, and we should be aware of the overall cost ofmaintaining consistency.

In this final section, I want to consider how this affects the broader projectof the philosophy of information. First, I believe that a dialetheic account ofstrongly semantic information calls for a revision of how the veridicality thesisrelates to the solution of the BC paradox. Second, I think we should be moreattentive to the substantial philosophical import of certain methodologicalassumptions.

The veridicality thesis is of independent value, and does not in itself relyon assumptions about how the BC paradox needs to be handled. The theoryof strongly semantic information (Floridi 2004a), by contrast, was formulated

14When formulated with a dialetheic negation of the kind favoured by Priest, saying that infor-mation must be true and may not be false does not prevent information from being false. What isrequired is something like ‘true only’ or ‘untruth’ rather than truth and falsity.


38 P. Allo

with the specific intent of solving the paradox in question (Sequoiah-Grayson2007). Because TSSI is itself a further elaboration of the veridicality thesis, thefact that it solves the BC paradox is easily considered as evidence in favour ofthe veridicality thesis. I think it is better not to use it that way.

What remains of the original solution to the BC paradox is the following:If ⊥ is such that it is necessarily false (i.e. true at no point in the logical spaceor defined by means of ⊥→ p for arbitrary p), then (a) a theory of semanticcontent that is based on a too weak semantic principle will unavoidably(i.e. even if the underlying logic is paraconsistent) lead to the conclusion thatthe semantic content of ⊥ is maximal and (b) a stronger semantic principlewhich stipulates that only the contingently true can be informative indeedsolves this problem. This is nevertheless independent from (c) the logical statusof contradictions (contradictions do not have to imply ⊥) and (d) the status of⊥ itself. The dialetheist will deny that contradictions entail ⊥ since this wouldvalidate explosion, but will not necessarily deny that some contradictions imply⊥ and are therefore unacceptable.15 In other words, TSSI does indeed solvethe ⊥-version of the BC paradox, but the solution no longer ranges over allcontradictions.

When it comes to paradox, the received view is that the paradoxes are ofutmost philosophical relevance. They are not mere puzzles. By giving a solu-tion to the liar paradox, we explain how we can have, say, a truth predicateand an underlying logic which enjoy certain properties. This tells us somethingabout our language and about the nature of truth. Floridi, in discussion, issceptical about the philosophical relevance of paradoxes like the liar. In afirst sense, this scepticism is warranted. From the standpoint of the methodof abstraction, self-referential expressions are problematic. They are mistakes,uninteresting consequences of failing to keep an eye on what counts asan observable. In a second sense, however, the importance of paradoxes isunderestimated. I shall just give two examples.

The paradox of the liar shows that we cannot have both a classical logic anda fully transparent truth predicate (i.e. a predicate T such that A and T〈A〉are inter-substitutional in all non-opaque contexts). By arguing that a self-referential sentence does not count as a genuine observable, we acknowledgethis fact and start to give a solution of a certain type. What the actual solutionprecisely is, is a non-trivial issue. This open problem is material for anotherpaper. Here, the point is only that we learn something from the liar paradox.As illustrated in Beall and Glanzberg (2008), the nature path and the logic pathin the study of truth constrain each other.

What paradoxes also illustrate is that defining concepts in a certain way doesnot guarantee their intended behaviour. Even more, if we force our concepts tobehave as intended across the board, then this is not just a source of paradox,

15Remark that this restricted ability to reject some contradictions is—at least if we deny trivialismand have the means to avoid Curry’s paradox (Beall 2009, Chapt. 2)—all we need to recapturewhat Floridi (2007) refers to as syntactically erasing information.



but it can even lead to incoherent concepts. In other words, the successfulconstruction of concepts can lead to triviality unless we limit their scope ofapplication. The price we have to pay is twofold. We cannot unrestrictedlyapply our concepts, and we cannot express these limitations because the latterwould have to apply unrestrictedly. As pointed out in Priest (2002), evena Kantian position cannot consistently be expressed. There is no reason toassume that Floridi is not in a similar situation.

The methodology of the philosophy of information is, in view of the above,clearly not as neutral as one might have thought. This is not necessarily abig issue. As I pointed out elsewhere (Allo 2010), the method largely definesthe philosophical basis of PI. Yet, two built-in assumptions may need closerscrutiny: (a) the emphasis on consistency in model construction and (b) therole of successful construction. With regard to consistency, I have argued(Section 5) that informational dialetheism is compatible with a stronger seman-tic principle based on the veridicality thesis and that a view where consistencyis balanced with other theoretical virtues of models may in fact better serve thepurpose of the method of abstraction. With regard to successful construction,the situation is more delicate. The main dilemma is that we have to give upeither consistency or expressive completeness (i.e. semantic closure). In myview, the default preference for a loss in expressive completeness is a method-ological bias. By accepting that the overlap between the extension and the anti-extension of certain predicates does not always lead to ill-typed variables, wehave already given up the non-falsity thesis. This means that there must be aseparate reason to reject expressive completeness or the existence (or perhapsjust the usefulness) of a universal level of abstraction.

Acknowledgements A very early version of the material in this paper was presented as“Semantic Information and Logical Orthodoxy” at the 2006 European Computing and PhilosophyConference in Trondheim, Norway. The topic came up in discussions with Luciano Floridi regu-larly. This is the first published version of ideas from the original presentation and the outcome ofthe ensuing discussions.

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