What does it mean for the laws of logic to fail? My task in this paper is to answer this question. I use the resources that Routley/Sylvan developed with his collaborators for the semantics of relevant logics to explain a world where the laws of logic fail. I claim that the non-normal worlds that Routley/Sylvan introduced are exactly such worlds. To disambiguate different kinds of impossible worlds, I call such worlds logically impossible worlds. At a logically impossible world, the laws of logic fail. In this paper, I provide a definition of logically impossible worlds. I then show that there is nothing strange about admitting such worlds

Tanaka, Koji

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What does it mean for the laws of logic to fail? My task in this paper is to answer this question. I use the resources that Routley/Sylvan developed with his collaborators for the semantics of relevant logics to explain a world where the laws of logic fail. I claim that the non-normal worlds that Routley/Sylvan (with his collaborators) introduced are exactly such worlds. To disambiguate different kinds of impossible worlds, I call such worlds logically impossible worlds. At a logically impossible world, the laws of logic fail. In this paper, I provide a definition of logically impossible worlds. I then show that there is nothing strange about admitting such worlds

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Logically Impossible Worlds

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Forthcoming in the Australasian Journal of LogicLogically Impossible WorldsKoji TanakaSchool of PhilosophyResearch School of Social SciencesAustralian National UniversityKoji.Tanaka@anu.edu.au1 Impossible WorldsIf we think of a possible world as representing a way things can be, an impossibleworld represents a way in which things cannot be. The importance of impossibleworlds has been discussed in recent literature.1 One reason for accepting impossi-ble worlds comes from counterfactual reasoning (Nolan (1997), Routley (1989)).Consider, for instance, the following counterfactual: if someone were to squarea circle, she or he would be famous. This counterfactual has an impossible an-tecedent. One way to analyse such a counterfactual is to invoke an impossibleworld where someone does square a circle and evaluate the consequent at theworld or a set of worlds that is or are the closest to that impossible world.But what exactly are impossible worlds? What does it mean to talk about waysin which things cannot be? There are several formulations of impossible worlds inthe literature.2 Some characterise the impossibility of impossible worlds in termsof logic. There have been mainly three ways in which impossible worlds arecharacterised in logical terms. First, an impossible world is a world where A∧¬Ais true for some A (Lycan (1994)). According to this view, an impossible world isa world where some contradiction obtains. Second, an impossible world is a worldthat is governed by some non-classical logic (Cresswell (1973)). According to thissecond formulation, it is assumed that classical logic holds at the actual world, anda world is impossible if the laws of classical logic fail. Because of the assumptionthat the actual world is governed by classical logic, this means that an impossible1For a survey, see Berto (2013) §1.2See Berto (2013), Jago (2015), Nolan (2013).world is a world where the laws of logic are different. Third, an impossible worldis a world where the laws of logic fail (Priest (2008): 172).3The first and the second formulations of impossible worlds in terms of logicgive us an insight into what an impossible world might be like. If impossibility ischaracterised in terms of contradictions, an impossible world is contradictory. Itmight be a world where Sylvan’s Box, a box which is both empty and nonempty,exists (Priest (1997)). An impossible world in this sense might be a world whichis governed by some paraconsistent logic, or classical logic if it is also a trivialworld where everything is true. If an impossible world is a world which is gov-erned, for instance, by intuitionistic logic, it is a world where mathematics is doneintuitionistically. At such a world, a mathematician cannot rely on the Law ofExcluded Middle to prove anything.4 But what does it mean for the laws of logicto fail?My task in this paper is to answer this question. I will use the resources thatRoutley/Sylvan developed with his collaborators for the semantics of relevant log-ics to explain a world where the laws of logic fail. I will claim that the non-normalworlds that Routley/Sylvan (with his collaborators) introduced are exactly suchworlds.5 To disambiguate different kinds of impossible worlds, I will call suchworlds logically impossible worlds. At a logically impossible world, the laws oflogic fail. In this paper, I will provide a definition of logically impossible worlds.I will then show that there is nothing strange about admitting such worlds.2 Logically Impossible WorldsWe can define logically impossible worlds by elaborating on some features of thesemantics for relevant logics that were introduced by Routley/Sylvan (with hiscollaborators). For simplicity, I consider the simplified semantics for the relevantlogic B (Priest and Sylvan (1992)). The semantics for B is a structure 〈W,N,R, I〉where W is a set of worlds, N is a set of normal worlds, R ⊂ W ×W ×W, and Iis a map from propositions to worlds. R at normal worlds satisfies the normalitycondition:〈w,w′,w′′〉 ∈ R iff w′ = w′′ for every w ∈ N and every w′,w′′ ∈ W.In other words, R at normal worlds is binary and at non-normal world is ternary.3As we will see later, Priest (2008) equates a world where the laws of logic fail with a worldwhere the laws of logic are different. In the last section of this paper, I will argue that that is amistake.4See Detlefsen (1990) for gaining an understanding of what that world might be like.5Nolan (2013) makes a brief suggestion along this line (p. 361) but no sustained discussion hasbeen given in the literature as far as I know.2For the sake of illustration, I will consider a so-called relational semantics andassume an evaluation of propositions to be not a function but a relation betweena proposition and a value. So I maps a proposition, p, to an extension Ep (theset of worlds where p is true) and an anti-extension Ap (the set of worlds wherep is false), i.e., I(p) = 〈Ep, Ap〉. A two-valued semantics requires the Routley-star operation to specify the truth condition for negation. As will become clear,however, negation is not what is at issue in defining logically impossible worlds.In order to focus on the main issue, I will examine a relational semantics.6An evaluation of a proposition, ρ, can be given as:〈p, 1〉 ∈ ρw iff w ∈ Ep.〈p, 0〉 ∈ ρw iff w ∈ Ap.Evaluations of complex formulas involving conjunction and disjunction also comein two clauses: one for extension and another for anti-extension.Our main interest is with conditionals as an examination of conditionals cangive us a definition of logically impossible worlds. Conditionals are evaluated interms of ternary relations:〈A→ B, 1〉 ∈ ρw iff for every w′,w′′ ∈ W such that 〈w,w′,w′′〉 ∈ R, if〈A, 1〉 ∈ ρw′ then 〈B, 1〉 ∈ ρw′′ .〈A → B, 0〉 ∈ ρw iff for some w′,w′′ ∈ W such that 〈w,w′,w′′〉 ∈ R,〈A, 1〉 ∈ ρw′ and 〈B, 0〉 ∈ ρw′′ .Because of the normality condition, the evaluations of conditionals at normalworlds can be simplified as:〈A→ B, 1〉 ∈ ρw iff for every w′ ∈ W, if 〈A, 1〉 ∈ ρw′ then 〈B, 1〉 ∈ ρw′ .〈A→ B, 0〉 ∈ ρw iff for some w′ ∈ W, 〈A, 1〉 ∈ ρw′ and 〈B, 0〉 ∈ ρw′ .Validity is then defined in terms of truth-preservation at all normal worlds:Σ |= A iff for every 〈W,N,R, I〉 and every w ∈ N, if 〈B, 1〉 ∈ ρw forevery B ∈ Σ, then 〈A, 1〉 ∈ ρw.Note that because we have a relational semantics, falsity does not exclude thesentence being true as well. In order to show that a conditional, for instance A→B, fails to be a logical truth at a world, we need to demonstrate not that it is false,〈A→ B, 0〉 ∈ ρw, but that it fails to be true at that world, i.e., 〈A→ B, 1〉 < ρw.6The relevant logic based on a relational semantics is, in fact, not the same as B. The relationalsemantics generates a logic that does not validate contraposition whereas B does. For details, seeRestall (1995).3Now, a conditional is said to be relevant if the antecedent and the consequentshare some propositional variables. For a conditional to be a logical truth of rele-vant logics, it must be relevant. The following conditionals fail to be logical truthsaccording to relevant logics: A→ (B ∨ ¬B), (A ∧ ¬A)→ B, and A→ (B→ B).The first conditional (A → (B ∨ ¬B)) can be invalidated by allowing someworlds to be incomplete: neither B nor ¬B are true. If A is true at such anincomplete world, the antecedent of the conditional (A) is true but the conse-quent (B ∨ ¬B) is not true at that world. So, for some w′ ∈ W, 〈A, 1〉 ∈ ρ′w but〈B∨ ¬B, 1〉 < ρ′w. Hence, for some normal world w ∈ N, 〈A→ (B∨ ¬B), 1〉 < ρw.Once we introduce incomplete worlds to the set of worlds, the conditional can beshown to be invalid.The second conditional ((A ∧ ¬A) → B) can be invalidated in terms of incon-sistent worlds where both A and ¬A are true. If B is not true at such an inconsistentworld, the conditional can be shown to be not true. Suppose that 〈A∧¬A, 1〉 ∈ ρw′for some w′ ∈ W. If 〈B, 1〉 < ρw′ , then 〈A ∧ ¬A, 1〉 ∈ ρw′ and 〈B, 1〉 < ρw′ for somew′ ∈ W. So, for some normal world w ∈ N, 〈(A ∧ ¬A) → B), 1〉 < ρw. So theconditional is invalid.From the perspective of relevant logics, it is the third conditional, A → (B →B), that is the most interesting. In most relevant logics, B → B comes out tobe a logical truth. For every w′ ∈ W, if 〈B, 1〉 ∈ ρw′ then 〈B, 1〉 ∈ ρw′ . So〈B→ B, 1〉 ∈ ρw for any w ∈ N. Hence the conditional is valid. So A→ (B→ B)has a consequent that is a logical truth. This is where the distinction betweennormal and non-normal worlds becomes important.Routley and Routley (1972) developed semantics for First Degree Entailment(FDE). FDE is a conditional-free fragment of (full) relevant logics. When Routleyand Meyer (1972, 1973) extended the Routley-Routley semantics to (full) relevantlogics, they held that the real world must be among the worlds that has a ‘privi-leged status’ (Routley and Meyer (1973): 205). To identify a privileged world,7Routley and Meyer introduced non-normal worlds which do not have such a sta-tus. The introduction of non-normal worlds allowed them to invalidate irrelevantconditionals such as A→ (B→ B).At a non-normal world, the truth value of a conditional, A → B, is not deter-mined by the truth values of A and B at the same world. It is determined by thosevalues at different worlds that are related in a ternary manner. What is importanthere is not the ternary relations but the fact that the antecedent and the consequentare evaluated at different worlds. That allows a non-normal world to make B→ Bcome out not true.7There may be more than one privileged world. But completeness requires there to be only onesuch world.4But, as we saw above, B → B is a logical truth in the system. Non-normalworlds are parts of the structure that validate B → B as a logical truth. Yet it isat these worlds that B → B fails to be true. So, a non-normal world is part of thestructure that defines the logical truths of the system but it is a world where thosevery logical truths fail to hold.It can now be shown that a non-normal world is a world where the laws of logicfail and is, thus, a logically impossible world. Consider a conditional A → Bwhere A and B do not contain any conditionals. The semantics for FDE is asystematic account of the truth-preserving relationship between the antecedent,A, and the consequent, B. It was developed to study the relationship betweenA and B in terms of A |= B. When it is extended to (full) relevant logics, theanalysis of the relation between A and B is used to study the conditional A → Bin terms of |= A → B. This means that if A → B is a logical truth, the conditionalexpresses the validity of the truth-preserving relation between the antecedent andthe consequent. In other words, in relevant logic, a logical truth in the form ofconditional expresses a law of logic.There is then a sense in which logical truths of conditional form express thelaws of logic. And non-normal worlds introduced to the semantics of relevantlogic are exactly the worlds where the laws of logic, understood as above, fail.They are logically impossible worlds. Thus, once we define logically impossibleworlds in terms of non-normal worlds that Routley/Sylvan (with his collaborators)introduced to invalidate irrelevant conditionals such as A→ (B→ B), we can gaina clear understanding of what makes a world logically impossible.3 Non-Normal WorldsThere is nothing strange or mysterious about logically impossible worlds as Icharacterised them above. Such worlds are part of the semantic resources thatKripke (1963, 1965) introduced for the systems of modal logic developed byC.I. Lewis (especially for S 2 and S 3) and E.J. Lemmon (especially E2 and E3).This means that logically impossible worlds have been with us since the beginningof possible worlds semantics.8Among the systems of modal logic are S 2 and S 3 as well as E2 and E3, theso-called non-normal systems. The main characteristic of non-normal systems isthe failure of necessitation: it is not the case that if |= A then |= A. In order todevelop semantics for these systems, Kripke introduced non-normal worlds whereA fails to hold for any A. Once we have these worlds, even if |= (A ∨ ¬A) (asA ∨ ¬A is true at every world), 6|= (A ∨ ¬A).8Material contained in this section draws on Tanaka (2013).5To make things more precise, let 〈W,N,R, ν〉 be a structure where W is a setof worlds, N is a set of normal worlds, R ⊂ W × W and ν is an evaluation ofpropositions at a world. For simplicity, we assume R to be universal. For any Aand any w ∈ W − N, νw(A) = 0.Validity can be defined in two ways. First, it can be defined in terms of allnormal worlds. Let’s call such validity weak validity and represent it as |=w. Thisgives rise to a Lewis non-normal system. Second, validity can be defined in termsof all worlds. Let’s call it strong validity and represent it as |=s. This gives rise toa Lemmon system of modal logic.9Logical truths are special cases of validity. So A is a weak logical truth, |=w,if A is true at every normal world in every model. A strong logical truth, |=s, isdefined in terms of all worlds. So |=s A if A is true at every world in every model.Let’s define the generality of logic in terms of the claim that a logical truthexpresses a truth no matter what the situation turns out to be. So to say that thenecessity of A is general is to say that A is a logical truth. Let |= A representthe generality of logical truth A. Then |=s A expresses that the necessity of A isgeneral, i.e., |=s A⇔ |= A.If a law of logic is understood to be expressed by a logical truth, then |= Ameans that A is a law of logic. Thus, there is a sense in which laws of logic areexpressed by logical truths of the form A. If laws of logic are expressed by A,then the failure of A implies the failure of the laws of logic. So, the fact that Afails to be true for any A invokes the idea that there are no laws of logic. Thus, anon-normal world introduced by Kripke can be characterised as a world where thelaws of logic fail. Just like a non-normal world of relevant logic, it is a logicallyimpossible world.4 Logically Impossible Worlds vs. Worlds Where the Laws ofLogic are DifferentBefore we end our discussion about logically impossible worlds, let’s consideran alternative characterisation that might be thought to define logically impossi-ble worlds. Priest (2008) characterises a world where the laws of logic fail assynonymous with a world where the laws of logic are different. For instance, hewrites:non-normal worlds ... are worlds where the laws of logic are different.Let us call these ‘logically impossible worlds’. (p. 171)A little bit later he talks about a logically impossible world as a world ‘at whichlogical laws fail’ (p. 172).9Lemmon (1957). For a discussion, see Hughes and Cresswell (1996).6Priest elaborates on his characterisation of logically impossible worlds, worldswhere the laws of logic are different, in analogous to physically impossible worlds.At a world where the laws of physics are different, objects might ‘accelerate pastthe speed of light’ (p. 172). According to Priest, at such a world, the laws ofphysics might permit physical objects to move differently from the physical lawsof ‘our’ world. Nevertheless, at that world, there are laws of physics. It is just thatthey are different.Now, Priest (1992) distinguishes the worlds where the laws of logic are dif-ferent from the worlds where the laws of logic fail. However, he claims that thenon-normal worlds which are instrumental for invalidating A → (B → B) are‘worlds where the laws of logic are different, not worlds where the logically im-possible happens’ (p. 296). Logically impossible worlds defined in this paper are,however, not the worlds where the laws of logic are different. What characterisesthem is the failure of the laws of logic. At a non-normal world, logical truths ofthe characteristic form (conditional in the case of relevant logics and necessarystatement in the case of modal logics) fail to hold. However, it is not that differentlogical truths hold at a non-normal world. If A → B expresses a law of logic, itis the failure of the conditional that defines a world being logically impossible. Itdoes not entail, however, that the non-normal world is part of the structure thatprovides a different logical truth and the logical truth holds at that non-normalworld. No laws of logic hold at a non-normal world. So a logically impossibleworld is a world where the laws of logic fail, but it is not a world where the lawsof logic are different.ReferencesBerto, Francesco (2013) ‘Impossible Worlds’, The Stanford Encyclopedia of Philosophy(Winter 2013 Edition), Edward N. Zalta (ed.), Stanford University.Cresswell, M. (1973) Logics and Languages, London: Methuen.Detlefsen, Michael (1990) ‘Brouwerian Intuitionism’, Mind 99: 501-534.Hughes, G.E., and M.J. Cresswell (1996) A New Introduction to Modal Logic, London:Routledge.Jago, Mark (2015) ‘Impossible Worlds’, Noûs 49: 713-728.Kripke, S.A. (1963) ‘Semantical Analysis of Modal Logic I. Normal Modal Proposi-tional Calculi’, Zeitschrifft fur mathematische Logik und Grundlangen der Mathe-matik 9: 67-96.——– (1965) ‘Semantical Analysis of Modal Logic II. Non-Normal Modal Proposi-tional Calculi’, The Theory of Models: Proceedings of the 1963 International Sym-posium at Berkeley, J.W. Addison, L. Henkin and A. Tarski (eds.), Amsterdam:North-Holland Publishing Company: 206-20.Lemmon, E.J. (1957) ‘New Foundations for Lewis Modal Systems’, The Journal of7Symbolic Logic 22: 176-186.Lycan, W. (1994) Modality and Meaning, Dordrecht: Kluwer.Nolan, Daniel (1997) ‘Impossible Worlds: A Modest Approach’, Notre Dame Journalof Formal Logic 38: 535-572.——– (2013) ‘Impossible Worlds’, Philosophy Compass 8: 360-372.Priest, Graham (1992) ‘What is a Non-Normal World?’, Logique et Analyse 35: 219-302.——– (1997) ‘Sylvan’s Box’, Notre Dame Journal of Formal Logic 38: 573-582.——– (2008) An Introduction to Non-Classical Logic (Second Edition), Cambridge:Cambridge University Press.——– and Sylvan, Richard (1992) ‘Simplified Semantics for Basic Relevant Logics’,Journal of Philosophical Logic 21: 217-232.Restall, Greg (1995) ‘Four-Valued Semantics for Relevant Logics (and Some of TheirRivals)’, Journal of Philosophical Logic 24: 139-160.Routley, R. (1989) ‘Philosophical and Linguistic Inroads: Multiply Intensional RelevantLogics’, J. Norman and R. Routley (eds.), Directions in Relevant Logic, Dordrecht:Kluwer: 269-304.Routley, R. and Meyer, R.K. (1972) ‘The semantics of entailment – II’, Journal of Philo-sophical Logic 1: 53-73.——– and Meyer, R.K. (1973) ‘The semantics of entailment – I’ Truth, Syntax andModality, H. Leblanc (ed.), Amsterdam: North-Holland: 199-243.——– and Routley, V. (1972) ‘The Semantics of First Degree Entailment’, Noûs 6: 335-359.Tanaka, Koji (2013) ‘Making Sense of Paraconsistent Logic: The Nature of Logic, Clas-sical Logic and Paraconsistent Logic’, Paraconsistency: Logic and Applications,K. Tanaka et al. (eds.), Dordrecht: Springer: 15-25.8

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