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Know Techn Pol (2010) 23:25–40 DOI 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–Carnap paradox (the information yield of inconsistent propositions is maximal) by questioning the orthodox view that contradictions cannot be true. The main part of the paper is devoted to showing that the veridicality thesis (semantic information has to be true) is compatible with dialetheism (there are true contradictions) and that, unless we accept the additional non-falsity thesis (information cannot be false), there is no reason to presuppose that there is no 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 BarHillel (1952), is a close relative of the usual model theoretic characterisation1
1 The
original version is based on state descriptions instead of models, but as may be seen from Kemeny (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, Belgium e-mail:
[email protected] P. Allo IEG, Oxford University, Oxford, UK
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of the classical consequence relation. It intendedly complies with the following platitude about logical consequence: A conclusion A follows from premises iff the content of A does not exceed the combined content of all the premises in . Moreover, it does so in a way which is co-extensive with the classical sense of follows from and thus provides what could be called a classical account of informational content. It is therefore not surprising that they share many features. The one feature I am interested here is known as Ex Contradictione Quodlibet or Explosion and is one of the so-called paradoxes of material and strict implication: From a contradiction, one can derive any old proposition. In a theory of semantic content, 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 for the point I want to make. The usual approach (often summarised in terms of the inverse relationship principle) individuates informational contents as proportions of a space of possibilities (e.g. models). When it is assumed that contradictions cannot be true at any point in the space of possibilities, they are all assigned the null proportion of the space of possibilities. Since the null proportion has the total space as its complement, the corresponding informational content is maximal. In other words, the standard story is just that since contradictions exclude all possibilities, they have maximal informational content. But this does not mean that no other story can be told. A purely structural account, according to which a consequence relation or individuation of semantic content 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 presumed paradoxes, and this readily reduces to one’s attitude towards contradictions. Either one opts for a traditional Tarskian approach according to which contradictions pose a problem because they are false and not (or at least not in the first place) because they lead to triviality (Tarski 1944) or one believes that explosion is the real problem. Floridi’s option, which leads to the development of a theory of strongly semantic information, is to follow Tarski’s lead. The diagnosis proposed in Floridi (2004a) is that a measure or individuation of informational content that is solely based on the individuation of content as a proportion of a space of possibilities implements a semantic principle that is simply too weak. The underlying problem is the assumption that truth supervenes on semantic information. Once a stronger semantic principle—one according to which semantic information encapsulates truth—is implemented, the BC paradox no longer arises (again, the actual implementation is beyond the scope of this discussion). This is, in rough outline, the philosophical basis of the theory of strongly semantic information (henceforth, TSSI). I have never been entirely happy with this solution, but for reasons that are diametrically opposed to the usual objections voiced by, e.g. Fetzer (2004a, b).
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My own position is unusual or at least atypical in the following sense: Although I endorse the stronger semantic principle,2 I do not favour the solution to the BC paradox which initially led to the formulation of that principle. This is primarily because of its structural similarity with the remark that even though Explosion is valid, it will never be sound in the sense of having both true premises and a true conclusion.3 Even if according to logical orthodoxy it cannot be sound, it should still puzzle us that it is valid. Similarly, even though a strong semantic principle blocks the BC paradox, there is still a problem with pure individuations of semantic content that are based on the inverse relationship principle. The aim of this paper is twofold. First, I want to give a more thoughtful articulation of the above worry. Second, I want to take a closer look at how the methodology of the philosophy of information and in particular the method of abstraction (Floridi 2008) influences or determines our logical options and the attitude we should adopt towards contradictions.
2 Dialetheism and Paraconsistency My quibbles with Floridi’s way of dealing with the BC paradox are to a certain extent based on considerations about finding uniform solutions to structurally similar problems. Since I believe that even if unsound, the validity of explosion still poses an independent problem, I have a default commitment to a similar response to Floridi’s appeal to a stronger semantic principle.4 The demand for a uniform solution fires in more than one direction, and three separate cases therefore need to be considered. First, one could point out that the validity of explosion poses a problem because even if contradictions cannot be true, we often need to reason from inconsistent theories. Since I endorse the stronger semantic principle proposed by Floridi, I need not be concerned with the practical need for paraconsistent inference. Of course, I agree that a paraconsistent solution to the pure individuations of semantic content (i.e. the individuation of what Floridi calls weakly semantic information) makes sense. But this is largely uncontroversial. The interesting question is whether this paraconsistent solution should carry over to the strongly semantic case. This question cannot be answered by purely practical considerations.
2 Basically,
my reasons for endorsing the stronger semantic principle is based on the epistemic value of information; see, e.g. Dretske (2009). 3 Sainsbury (1995, 136), but see also the discussion of Philosophy’s Most Dif f icult Problem in Woods (2003, 14–6). 4 A 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 have to. 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 to avoid 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 content does not exceed contradictory content—and it is all but clear how a stronger semantic principle can handle this problem. More exactly, I do not think that a 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 I think 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 kind of content. Again, this is not the main focus for this paper. My guess is that Floridi would favour a solution where the conditional content of A, given B, is only defined if both are contingent, jointly compatible and (but this more controversial) the conditional B > A is true. The third case we should consider bears directly on the interplay between explosion (and the BC paradox) and the assumption that all contradictions are false. When it comes to classical logic (and a fortiori in Carnap and Bar-Hillel’s theory of semantic information), the interplay is straightforward: Both features are two sides of the same coin. Non-classical logicians are in general more reluctant to adhere to this close connection. Non-dialetheic paraconsistent logicians will in general claim that even though there are no true contradictions, the argument from contradiction to triviality is invalid. Dialetheic paraconsistent logicians, by contrast, precisely reject explosion because they think that at least some contradictions are (or may be) true (Priest 2006a). The pragmatically inspired argument for paraconsistency described above is definitively of the non-dialetheic kind, but does not exhaust the category; there are many other reasons to endorse such a position. Still, given the focus on strongly semantic information, only a dialetheic approach can make a difference 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 careful in jumping to this conclusion. Before we can embrace a dialetheic modification of strongly semantic information—or even just criticise it from a dialetheic perspective—we first need to ensure that they are compatible. This requires a closer look at the sources of paradox in relation to the methodology of the philosophy of information and in particular to the method of abstraction and its emphasis on consistency.
3 Consistency and Logical Orthodoxy Orthodoxy has it that contradictions cannot be true, but how much of the tradition is mere prejudice? And more importantly, how much of this orthodoxy should be upheld in the context of Floridi’s solution to the BC paradox? This
5 Alternatively and in analogy with the standard approach to conditional probability, one could say
that the content of A, given B, is only defined if B is not necessarily false. I do not further explore this alternative.
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question cannot be answered in a void. My suggestion is to start from the handful of methodological insights on which the philosophy of information is founded and to see to what extent the veridicality thesis can be used to dismiss the existence of contradictory information. Two types of considerations can be used in defence of the orthodoxy: the alleged consistency of the world and the theoretical virtues of consistency. Each of these should be evaluated on its own merits, that is, without appeal to the disastrous consequences of explosion.6 Apart from sheer prejudice, the most prominent reason for our reluctance to even consider the possibility of true contradictions can be retraced to the correspondence intuition. Put crudely, for contradictions to be true, ultimate reality 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 of the world. As Priest remarks “if one supposes reality to be constituted solely by (non-propositional) objects, like tables and chairs, it makes no sense to suppose that reality is inconsistent or consistent. This is simply a category mistake” (Priest 2006b, 51). Floridi could not agree more. In his view, ultimate reality can at best impose certain limitations on how we perceive the world. Whatever non-propositional objects are, they only function as constraining affordances: “they limit the possible models” (Floridi 2008, 325). To speak of constraining affordances in terms of truth and consistency is again a category mistake. But Floridi makes an even stronger point. Ultimate reality lies beyond the limits of cognition (I am deliberately using the terminology of Priest (2002) to describe Floridi’s broadly Kantian outlook). To describe ultimate reality in terms that can be semantically characterised (i.e. be referred to as true, or consistent) is to construct a model of reality. As far as I can see, there is no reason to assume that the constraints imposed by the world are such that only consistent models are possible, or, at least, no argument to that effect which refers to the properties of reality itself is available to Floridi. A second defence of logical orthodoxy can nevertheless be mounted, one that directly appeals to desirable features of models. In Floridi’s view, what we know are the models we construe of reality, and seeing data as constraining affordances means that there cannot be a single privileged model of reality. To prevent these assumptions from reducing to a full-blown relativism, we need to be able to compare different models, to assess their respective virtues and to exclude or dismiss models with a poor track record. Clearly, consistency is such a virtue—even the dialetheist concedes as much and will try to minimise inconsistency as much as possible (but not more!)—but is it also sufficient to dismiss all inconsistent models? If we assume that, as a virtue of models, consistency is on a par with other theoretical virtues like elegance, explanatory power, simplicity, parsimony and informativeness, there is no reason to dismiss inconsistent models (Priest
6 Remember:
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 many more 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 genuine information and not a form of misinformation. Of course, tradition has it that consistency has a special status. It is not just a theoretical virtue among many others. One only needs to look at the many failed or inconclusive attempts to define coherence, and note that independently of how it is explained, the status of consistency as a necessary condition remains largely unchallenged. This only makes sense if we have good reasons to treat consistency differently. Most reasons to that effect fall in either of the following categories. On the one hand, one may advance that unlike, say, simplicity, inconsistency is a fully reliable indicator of falsity. On the other hand, one may point to the fact that unlike the other theoretical virtues, consistency 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 available to Floridi. With regard to the latter, it only takes a moment of reflection to see that the argument only goes through in the presence of explosion. One benefit of using a paraconsistent logic is precisely that degrees of inconsistency can be measured, and this is all we need to treat consistency as a virtue that needs to be balanced with several others. Perhaps, one may object,7 inconsistencies do indeed carry genuine information, but only in an uninteresting residual sense of informing us that our model is inconsistent. It is, therefore, primarily a form of meta-information rather than a form of factual information. The force of this objection depends on the prior assumption that an inconsistency only reports a property of the model, in the sense that it only informs us of the lack of a given virtue of that model, namely consistency. This crucial assumption is false. What is true is that contradictions do convey such meta-information and that they are indeed special in that sense. This specialness is, however, only due to the fact that explicit contradictions (i.e. individual formulae of the form A ∧ ¬A) inform us of a global property of a model, namely its inconsistency, whereas there is no formula which explicitly informs us about other more elusive virtues of a model 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 inconsistent may still be intractable (in the computational sense), hard to uncover or involve a large number of premises (as in the Paradox of the Preface).8 The real
7 This
line of thought is loosely inspired by views of Floridi. I do not recall whether these were expressed in print. 8 Thus, 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.
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issue, however, is whether we can make sense of the claim that contradictions only convey information about the (in)consistency of a model. In short, we cannot! Because the distinction between ‘properties of the model’ and ‘properties of what is being modelled’ is itself not factual, but methodological, there are limits to how we can appeal to that distinction. We believe that consistency is a virtue of models (and this is not something I dispute), and contradictions therefore 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 uphold that contradictions only carry such meta-information, we need to appeal to the falsity of all contradictions. Indeed, we need to claim that because a contradiction cannot be true, it cannot convey information about what is being modelled but only meta-information about the model itself. We already know that this argument is not available in the present context. The last few paragraphs explain why we cannot reject out of hand that contradictions may constitute or carry genuine (strongly semantic) information, i.e. information about what is the case. What this information may look like has hitherto remained implicit. The next two sections fill this gap by investigating the informational potential of two common types of dialetheia or true contradictions.
4 Paradox and Self-Reference Paradoxical sentences like the liar are the prototype of true contradictions, the raison d’être of dialetheism itself. If there were only one true contradiction, it would for sure be a sentence asserting its own falsity. This still leaves open the question of whether there is genuine paradoxical information. The quick answer is that, of course, there is no such thing, or, at least, there is no such thing if paradoxes do require self-reference.9 This, one may say, is one of the lessons of Kripke’s theory of truth (Kripke 1975): Self-referential sentences are not grounded, and surely, our information should be grounded to qualify as factual, or to put it in the terminology of the previous section: to convey factual as opposed to mere meta-information. Such a quick appeal to groundedness only leads to a dismissive argument against the possibility of paradoxical information. It for instance presupposes that our language is not part of the world. Instead of taking a closer look at the requirement that truth-apt sentences must be grounded, I shall consider a more general argument against paradoxical information. That argument is based on, on the one hand, a general schema proposed by Priest (2002) as the
9 An
alternative reply would be that liar sentences do not qualify as information because whatever their truth value, they will have that truth value as a matter of necessity. As a result, if only contingent truths qualify as semantic information, such sentences are easily dismissed. In view of the existence of so-called contingent liars (Field 2008, 24), it is immediately clear that this strategy will not work.
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source of all paradoxes—the Inclosure Schema—and, on the other hand, the restrictions imposed by the method of abstraction. What I wish to illustrate is that, as a general rule, the method of abstraction steers clear of the sources of paradox. The basis of the inclosure schema is the interplay between closure and transcendence. Given a predicate ϕ, the procedure of diagonalisation is used to generate an object which in virtue of that procedure is not ϕ (i.e. it transcends ϕ), but which, in view of the end result, happens to be ϕ as well (i.e. it is enclosed in ϕ as well). Consider, once more, a liar sentence asserting its own untruth. In virtue of what it says of itself (formally, in virtue of the construction), it fails to be true (transcendence), but because it fails to be true and 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 those limits, we run into contradiction because the process of doing so has contradictory 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 object which exceeds a given predicate is successful, but also happens to fall within the bounds of that predicate, we cannot avoid the resulting contradiction. The end result does not just appear paradoxical, but really is. Let us, for the sake of argument, grant that at least some instances of the inclosure schema generate genuine paradoxes. Can we deny that the result of diagonalisation qualifies as information? As I read it, it is a presupposition of the method of abstraction that it can indeed disqualify observables (or interpreted typed variables) in virtue of their unexpected behaviour (I shall make this more precise in a moment). Given that (a) information is always assessed at a given level of abstraction and that (b) levels of abstraction are collections of observables (Floridi 2008, 305–9), only the non-paradoxical qualifies as information. This is the hallmark of any restriction strategy which aims at the avoidance of paradoxes: It denies that such limits exist (e.g. there is no set of all sets). The method of abstraction is not any different in that respect. A more revealing insight can nevertheless be extracted from the above considerations. Consider a typed variable which can take any of the following values: ϕ, ¬ϕ and neither. Because it is so defined that ϕ and ¬ϕ are exclusive (though not exhaustive; the neither option is available), it is sufficient for that variable to take both values to be considered ill-typed. If a typed variable fails to behave as intended, the resulting interpreted typed variable cannot be accepted as an observable. Hence, no level of abstraction will depend on ill-typed variables.10 The relevant contrast, here, is that between intended behaviour and actual behaviour. The inclosure schema is the general
10 Whether there are no observables based on ill-typed variables or no levels of abstraction based on ill-typed interpreted variables is just a matter of terminology.
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pattern which characterises the limit cases where intended behaviour and actual behaviour do not agree. Dialetheism embraces these mismatches (i.e. it does not deny the Existence condition, see Priest (2002, 9.2)), but because of its commitment to successful construction, the method of abstraction can only dismiss such mismatches (hence, deny the existence condition). The situation described in the previous paragraph is not just a methodological difference. There is a genuine philosophical disagreement on the following two issues: successful construction and the existence of the limit. With regard to the former, the disagreement concerns the question of how we should react to the unintended behaviour of what we construct. This can be illustrated with a reference to the behaviour of negation. As Priest has often stressed (Priest 2006b, Chapt. 4 & 5), it does not suffice to define negation in such a way that ϕ 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 truth conditions, or is defined in terms of excluded middle and non-contradiction), the truth conditions alone cannot be used to show that truth and falsity are exclusive—that is, unless we already assume that they are exclusive (which is just to say that Explosion is valid). From the standpoint of the method of abstraction, only one reaction is possible. If truth and falsity are intended to be exhaustive and exclusive, then negation should behave accordingly in its intended domain of application. Put differently, intended behaviour and a well-defined domain of application are the distinctive features of successful construction. By tying intended behaviour to a previously defined domain of application, the features of limiting cases on which the inclosure schema trades are well beyond the reach of the method of abstraction. To describe the limit, we need something like the most general level of abstraction or a level of abstraction that can be used to describe itself (again, the analogy with other restriction strategies like Russell’s and Tarski’s is obvious). If no such level of abstraction exists, transcendence and closure remain disconnected and the unintended behaviour of a previously conceived predicate or condition is blocked. This concludes the discussion of paradoxical information. So far, the provisional conclusion seems that Floridi would not have to appeal to the properties of classical logic (mainly explosion) or to the necessary falsity of contradictions (consistency is not assumed to be given) to dismiss contradictory information of the paradoxical kind. All that is required is an emphasis on successful construction.11 Because the latter has independent theoretical virtues as well as practical advantages, the resulting argument is more convincing than what we saw in Section 3.
11 By successful construction, I only mean construction according to the specification, i.e. the agreement between intended and actual behaviour. Whether this also involves the successful construction of a (physical) artefact is a separate issue we can refer to as implementability. The latter 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 predicates overlap. More exactly, they overlap when applied at the borders of their intended domain of application. In the case of the liar and other typical paradoxes, the limit is reached by self-application. This is often a reason to consider such uses illegitimate. Presumably, the feeling that something has gone wrong is more closely related to the involved circularity than to the resulting contradiction. If so, we may wonder whether a non-circular overlap of the extension and anti-extension of a predicate could, with the same conviction, be diagnosed as illegitimate. Unless one already presupposes a classical account of negation, I do not think it should. Overlaps between extension and anti-extension are due to overdefined predicates. The cogency of the latter kind of predicates is the crux of what Mares (2004) calls semantic dialetheism, the thesis that inconsistencies result from how our predicates (or, more broadly, natural languages) apply to the world. At least in one respect, this is a brand of dialetheism that nicely combines with the method of abstraction. Semantic dialetheism is defended as an intermediate position: It is stronger than the more popular doxastic motivations for paraconsistency (the position I dismissed in Section 2 for not meeting the demands of strongly semantic information), but not as radical as metaphysical dialetheism. The disagreement between the latter two positions concerns the level at which contradictions (or inconsistencies) should be situated. According to metaphysical dialetheism, the world is (or may be) irreducibly inconsistent; according to semantic dialetheism, some perfectly fine descriptions of the world are inconsistent. At first, the distinction between metaphysical and semantic dialetheism may seem incompatible with the broader methodology sketched in the previous sections. Consistency does not as such qualify as a property of the world itself. Yet, metaphysical dialetheism does not hold that a world made of nonpropositional objects can itself be inconsistent. Rather, the idea is that the facts of the world are such that “any consistent description of the world will [be] misdescribing at least one fact” (Mares 2004, 270). When characterised in this manner, metaphysical dialetheism is clearly intelligible, even if one thinks that all we can know are our models of the world.12 In its most useful form, semantic dialetheism reduces to the following two theses: (a) consistent descriptions (or models) of the world are possible, but (b) such (re)descriptions may on the whole be worse off than inconsistent descriptions. Thus, as soon as we grant that consistency is on a par with other theoretical virtues of models, semantic dialetheism can be used to motivate inconsistent levels of abstraction, here understood as levels of abstraction
12 Still, if one believes that we can only reason about relations between models and not about relations between the world in itself and our models, one has to conclude that metaphysical dialetheism cannot be defended.
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where observables may have inconsistent types. Nothing else is required to speak of strongly semantic inconsistent or contradictory information. So far, this leaves open the question of how overdefined predicates may come into existence. Basically, two aspects of how we construct models play a role (Mares 2004, sect. 8). The first one is the use of predicates outside their originally intended domain of application. The second one is related to the fallibility of conceiving incompatible properties. Even if we think that P and not P are jointly exclusive because the properties required for something to be 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 the domain of application is a sensible thing to do if we want our models to be more inclusive and thus more informative, and by doing so, we often have to give up the total control over the avoidance of overlaps we had in the original domain. Even more, this total control can already be absent in the original or intended domain of application. The above description looks like an easy target for the proponent of the method of abstraction. The purpose of the method of abstraction, as well as of the broader field of formal methods it is based on, is to avoid situations like the one described above. The whole point is to guarantee the consistency of our models and to preserve consistency when we move to new, more encompassing and 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 do not have to deny the purpose of formal methods to make my point. I only need to point out that methods intended to ensure consistency may still fail to deliver. This probably will not convince the proponent of orthodox formal methods: If the method does not deliver, then the outcome does not meet the specifications and this is sufficient to disqualify it. The second criticism is that this method does not sufficiently take the costs of maintaining consistency into account. If semantic dialetheism has one virtue, then it is that it allows us to favour inconsistent descriptions whenever there are good reasons to do so. Since finding successor concepts for our actual overlapping concepts often means that 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 reason to opt for, or at least not to dismiss, inconsistent descriptions. If we think of the method of abstraction as a means to obtain the best or most useful models of reality, then a strong adherence to the consistency ideology is at odds with that purpose. To claim otherwise is to give consistency a special status, and we have seen (Section 3) that the usual arguments which motivate this special status are simply not available.
6 Negation, Falsity, and Acceptance A last issue we cannot entirely ignore is the relation between negation, falsity and 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-A are logically independent. This account of negation is often called the nontruth-functional account of negation and is found in the works of Da Costa and in some paraconsistent logics discussed in Batens (1980). The hallmark of this account is that falsity does not reduce to truth of negation. On a second account, negation remains truth functional and falsity can be defined as truth of negation. This is Priest’s preferred account and leads to the view that the joint truth of A and not-A also leads to their joint falsity. In other words, contradictions are always false but sometimes also true. This view is often equated with dialetheism and is the main reason why the dialetheist can maintain 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 problems they encounter are of course be different. If, as with a non-truth-functional negation, A and not-A are logically independent, 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 any non-classical negation (Quine 1986; Slater 1995) and are therefore just an expression of logical orthodoxy, but the stronger objection that non-truthfunctional negations even fail to be ‘contradictory forming’ operators (Priest and 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 implies that some formulae can be true as well as false, the following questions come up. 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 on how Sainsbury contrasts dialetheism with rational dialetheism (Sainsbury 1995, Chapt. 6). According to the former, there are true contradictions; according to the latter, it is also rational to believe (or accept) true contradictions. Rational dialetheism (if not vacuous) entails dialetheism simpliciter, but not vice versa. This means it is possible to endorse the latter while denying the former. A similar distinction could perhaps be made with regard to information. Endorsing true contradictions does not mean we have to endorse the possible existence of genuine contradictory information. The main argument for reasoning like this is that a commitment to contradictory information forces us to accept the existence of false information, and, as argued at length in Floridi (2005a), false information is no information at all.13 This type of rejection of contradictory information does not have to be accepted by the dialetheist. Floridi’s non-falsity thesis (false information does not count as information) depends on the veridicality thesis (information must
13 Remark that in this case the move from semantic dialetheism to informational dialetheism does not involve considerations about what we can and/or should believe but only depends on the nature of semantic information itself.
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be true), but the two theses are only equivalent on the assumption that truth and falsity are exclusive and exhaustive. As a result, the rejection of informational dialetheism will have to rely on the non-falsity thesis as an independent premise (in addition to, or as a replacement of the veridicality thesis). I do not want to take a definite position with regard to this issue but will again just make 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 be used to defend that thesis. Apart from that, because for most propositions the non-falsity thesis is already implied by the veridicality thesis, it does not add anything worthwhile to the latter. Second, we could ask whether the non-falsity thesis can even have its intended force. That is, the formulation of the nonfalsity thesis will itself depend on our means to express ‘true only’ and make sure 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 have to be rejected by the dialetheist; see, e.g. Beall (2009, Chapt. 3) on incoherent operators. Whether we can successfully define a kind of information that is true only will therefore unavoidably depend on how we express the consistency condition.
7 Concluding Remarks What I tried to show in the previous sections is that a dialetheic account of strongly semantic information is not easily dismissed. My personal view is that such an account has many virtues but that an approach which systematically enforces consistency remains possible. I have briefly mentioned several ways of conceiving such an account (e.g. by claiming that information should be ‘true only’) and have pointed out why this may be problematic. More importantly, I have often emphasised where arguments for a consistent approach fail and what the costs of an overall consistent approach are. In brief: We cannot appeal to the consistency of the world, and we should be aware of the overall cost of maintaining consistency. In this final section, I want to consider how this affects the broader project of the philosophy of information. First, I believe that a dialetheic account of strongly semantic information calls for a revision of how the veridicality thesis relates to the solution of the BC paradox. Second, I think we should be more attentive to the substantial philosophical import of certain methodological assumptions. The veridicality thesis is of independent value, and does not in itself rely on assumptions about how the BC paradox needs to be handled. The theory of strongly semantic information (Floridi 2004a), by contrast, was formulated
14 When formulated with a dialetheic negation of the kind favoured by Priest, saying that information must be true and may not be false does not prevent information from being false. What is required is something like ‘true only’ or ‘untruth’ rather than truth and falsity.
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with the specific intent of solving the paradox in question (Sequoiah-Grayson 2007). Because TSSI is itself a further elaboration of the veridicality thesis, the fact that it solves the BC paradox is easily considered as evidence in favour of the 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 space or defined by means of ⊥→ p for arbitrary p), then (a) a theory of semantic content that is based on a too weak semantic principle will unavoidably (i.e. even if the underlying logic is paraconsistent) lead to the conclusion that the semantic content of ⊥ is maximal and (b) a stronger semantic principle which stipulates that only the contingently true can be informative indeed solves this problem. This is nevertheless independent from (c) the logical status of contradictions (contradictions do not have to imply ⊥) and (d) the status of ⊥ itself. The dialetheist will deny that contradictions entail ⊥ since this would validate explosion, but will not necessarily deny that some contradictions imply ⊥ and are therefore unacceptable.15 In other words, TSSI does indeed solve the ⊥-version of the BC paradox, but the solution no longer ranges over all contradictions. When it comes to paradox, the received view is that the paradoxes are of utmost philosophical relevance. They are not mere puzzles. By giving a solution to the liar paradox, we explain how we can have, say, a truth predicate and an underlying logic which enjoy certain properties. This tells us something about our language and about the nature of truth. Floridi, in discussion, is sceptical about the philosophical relevance of paradoxes like the liar. In a first sense, this scepticism is warranted. From the standpoint of the method of abstraction, self-referential expressions are problematic. They are mistakes, uninteresting consequences of failing to keep an eye on what counts as an observable. In a second sense, however, the importance of paradoxes is underestimated. I shall just give two examples. The paradox of the liar shows that we cannot have both a classical logic and a fully transparent truth predicate (i.e. a predicate T such that A and TA are inter-substitutional in all non-opaque contexts). By arguing that a selfreferential sentence does not count as a genuine observable, we acknowledge this fact and start to give a solution of a certain type. What the actual solution precisely is, is a non-trivial issue. This open problem is material for another paper. 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 path in the study of truth constrain each other. What paradoxes also illustrate is that defining concepts in a certain way does not guarantee their intended behaviour. Even more, if we force our concepts to behave as intended across the board, then this is not just a source of paradox,
15 Remark that this restricted ability to reject some contradictions is—at least if we deny trivialism and have the means to avoid Curry’s paradox (Beall 2009, Chapt. 2)—all we need to recapture what Floridi (2007) refers to as syntactically erasing information.
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but it can even lead to incoherent concepts. In other words, the successful construction of concepts can lead to triviality unless we limit their scope of application. The price we have to pay is twofold. We cannot unrestrictedly apply our concepts, and we cannot express these limitations because the latter would have to apply unrestrictedly. As pointed out in Priest (2002), even a Kantian position cannot consistently be expressed. There is no reason to assume 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 a big issue. As I pointed out elsewhere (Allo 2010), the method largely defines the philosophical basis of PI. Yet, two built-in assumptions may need closer scrutiny: (a) the emphasis on consistency in model construction and (b) the role of successful construction. With regard to consistency, I have argued (Section 5) that informational dialetheism is compatible with a stronger semantic principle based on the veridicality thesis and that a view where consistency is balanced with other theoretical virtues of models may in fact better serve the purpose of the method of abstraction. With regard to successful construction, the situation is more delicate. The main dilemma is that we have to give up either consistency or expressive completeness (i.e. semantic closure). In my view, the default preference for a loss in expressive completeness is a methodological bias. By accepting that the overlap between the extension and the antiextension of certain predicates does not always lead to ill-typed variables, we have already given up the non-falsity thesis. This means that there must be a separate reason to reject expressive completeness or the existence (or perhaps just 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 Philosophy Conference in Trondheim, Norway. The topic came up in discussions with Luciano Floridi regularly. This is the first published version of ideas from the original presentation and the outcome of the ensuing discussions.
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