There is More to Negation than Modality

Abstract

There is a relatively recent trend in treating negation as a modal operator. One such reason is that doing so provides a uniform semantics for the negations of a wide variety of logics and arguably speaks to a longstanding challenge of Quine put to non-classical logics. One might be tempted to draw the conclusion that negation is a modal operator, a claim Francesco Berto (Mind, 124(495), 761–793, 2015) defends at length in a recent paper. According to one such modal account, the negation of a sentence is true at a world x just in case all the worlds at which the sentence is true are incompatible with x. Incompatibility is taken to be the key notion in the account, and what minimal properties a negation has comes down to which minimal conditions incompatibility satisfies. Our aims in this paper are twofold. First, we wish to point out problems for the modal account that make us question its tenability on a fundamental level. Second, in its place we propose an alternative, non-modal, account of negation as a contradictory-forming operator that we argue is superior to, and more natural than, the modal account.

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Notes

  1. 1.

    See [9].

  2. 2.

    As a referee presses, it might be a mistake to take the Quinean challenge seriously if it diverts attention from characterization negation to characterizing disagreement instead. However, we do not think meeting the Quinean challenge amounts to characterizing disagreement rather than negation. Meeting the challenge means giving a characterization of negation that makes room for disagreement and ensures that a given characterization of negation not beg the question against e.g. the deviant.

  3. 3.

    Berto’s language includes an additional connective for necessity for which we have no use. This gives us a slight simplification of the set-up. In particular, frames (hence models) for our purposes have only one accessibility relation for compatibility which we denote by R rather than R N .

  4. 4.

    We’re suppressing the model subscripts on \(\Vdash \); when being explicit we write \(x\Vdash _{\mathcal {M}}A\) to mean that A holds at x in model \(\mathcal {M}\).

  5. 5.

    In explaining incompatibility, using expressions like ‘exclude’, ‘preclude’, and ‘rule out’ instead of overtly negative expressions like ‘not’ (as Berto does in the quoted passage above) does not suggest that incompatibility is primitive. It is after all plausible, for instance, that the prefixes ‘ex-’ and ‘pre-’ here signal the use of negation, as does the ‘out’ in ‘ruling out’, or that the expressions in any case have meanings or truth conditions that depend on negation whether or not those expressions contain subexpressions signaling the use of negation.

  6. 6.

    For Price, negation’s primary function is to express when two—let us call them states of affairs—are incompatible. You say you’re going to talk to Fred in the kitchen, we say he’s in the garden, to which you say thanks and set off for the kitchen. If you don’t see the implicit incompatibility between Fred’s being in the garden and his being in the kitchen, we can make it explicit by uttering ‘But he’s not in the kitchen’. Notice that we are talking about incompatibility between states of affairs here, not worlds, assuming worlds are typically not just single states of affairs (if sometimes they may be). Also notice that this story works only when there is a salient contrast between two (or more) states of affairs. Our uttering ‘There’s no beer left’ does not seem to signal or express any implicit incompatibility between the state of there being beer and some one other state. Which other state? For this and other reasons, we remain unconvinced by Price’s claims concerning the primary function of negation.

  7. 7.

    We can think of worlds being compatible with each other in certain respects and incompatible in certain other respects; that is, (in)compatibility can be viewed as a ternary relation between states and respects. Or, we could think of negated statements being context-sensitive and (in)compatibility remaining two-place, with the quantifier in (S∼) being restricted by which respects are salient. Both of these options seem to give us a response to some of the tension just noted in saying that all the worlds where Sam is a gram heavier than she actually is are incompatible with ours, period. But no matter how we restrict the quantifier, many of the issues raised above still stand. For instance, why at all do worlds need to enter into the truth conditions for negated statements?

  8. 8.

    See also [36].

  9. 9.

    It is worth noting that (Contra) is admissible in BD but not LP, even though the former can be seen as a natural generalization of the latter to a paracomplete or partial logic.

  10. 10.

    Such an example is given by the Routley-star—hence compatibility—semantics for LP on which (Contra) fails. See [3, p. 9] for details.

  11. 11.

    One could rebut the counterexample in other ways. For example, one might think that if one’s son’s playing the sax really prevents one’s reading a paper, then surely one’s reading a paper must prevent one’s son’s playing the sax. For how could one be reading a paper while one’s son is playing the sax if the latter prevents the former? One can’t, so it looks like prevention is symmetric after all, despite initial appearances. As Berto notes, the asymmetry may be explained by causal asymmetry since one’s son’s playing the sax causes one not to read a paper, but not conversely. Thanks to [anonymous] for pressing us on this response to the Dunn-Hartonas counterexample. We think there are better counterexamples in any case, as discussed below.

  12. 12.

    Here Berto also cites the actual world as being self-incompatible for a dialetheist like Graham Priest. But why would the actual world be incompatible with itself? What does that mean? We guess the idea is that the actual world makes certain contradictions true, pace Priest, so it must be incompatible with itself, assuming compatibility semantics. Aside from question-beggingness, here we are explaining self-incompatibility in terms of negation—specifically, in terms of whether or not a sentence and its negation are true. Yet the modalist needs the explanation to go the other way round.

  13. 13.

    In passing, Berto refers to (LINK) in fn. 27, but he makes no further mention of it, including the fact that it entails reflexivity of compatibility.

  14. 14.

    Berto never says so but we assume so given his notation, and we can anyway define such a relation in terms of the proper inclusion relation and identity and substitute that in for \(\sqsubseteq \) in (LINK).

  15. 15.

    For an up-to-date survey on various accounts of negation, see [19].

  16. 16.

    See [35] for other syntactic accounts of contraries, subcontraries and contradictories. For some reasons for preferring a semantic over syntactic characterization of negation, see [11, Chap. 2].

  17. 17.

    Note here that we may take truth and negation as primitive, and define falsity in terms of truth of negation. The problem with doing so, however, is that it settles too many questions concerning the relation between the two truth values, and whether they are both sui generis notions or whether one is definable from the other. There will be many cases where we will want to distinguish falsity from untruth (e.g. when there are truth-value gaps), and taking each as primitive allows us to do so.

  18. 18.

    For a rejection of gaps, see e.g. [29, §4.7], and for a rejection of gluts, see e.g. [21].

  19. 19.

    E.g. if {A : ⊩ A} ∩ {A : >⫥ A} = ∅ (there are no gluts) and {A : ⊩ A} ∩ {A : >⫥ A} = S for S is the set of all sentences of the language (there are no gaps), then we need only one of either ⊩ or ⫥.

  20. 20.

    20 As with compatibility semantics, semantic consequence is defined in terms of the preservation of truth over \(\Vdash \).

  21. 21.

    See [28, Chapter 5] for a sustained attack on classical negation.

  22. 22.

    See [25] concerning these matters.

  23. 23.

    We are not here discounting the school of negationless constructivism.

  24. 24.

    The truth and falsity conditions for negation will obviously have to be relativized to worlds, assuming a world semantics for N3.

  25. 25.

    It is here interesting to note that we cannot assume truth and falsity to be exhaustive without the conditional collapsing into the classical material conditional, thereby forgoing constructiveness. Failure of excluded middle remains a key feature of constructive logic even when negation is strong.

  26. 26.

    Such naysayers include e.g. David Lewis [21] and Hartley Slater [34]. Note that Slater rejects the paraconsistent negation of LP as being genuine on the grounds that is not contrary-forming, but in so doing he presupposes a classicist understanding of contrariety. In response to Slater, Jean-Yves Béziau [6] agrees with Slater in that no paraconsistent negation can be a contradictory-forming operator, but disagrees with the claim that negation must be a contradictory-forming operator. The reason for this is that according to Béziau’s definition of a contradictory-forming operator, only classical negation is contradictory-forming. This illustrates a critical point of departure from our view, according to which some paraconsistent negations are contradictory-forming.

  27. 27.

    The Routley star semantics for K 3 is obtainable by adding one further condition to the Routley star semantics for BD. See [30, Chapter 8] for details.

  28. 28.

    One could deny that the many-valued semantics is relevant here when negation is given a Routley star reading, and that the star semantics takes precedence over the many-valued one. The compatibility account is then not sensitive to which values are taken to be designated, because the correct semantics does not employ three or more values and the notion of a designated value apart from truth. As we see it, however, compatibility semantics should tell us when an operator, inferentially construed, is a negation or not: it is iff it satisfies the core principles given by the compatibility account. The problem is that this conflicts with the intuitive idea that if such an operator has an intuitive semantics relative to which it corresponds to some operation o, then o too should count as a negation no matter which values are taken as designated (equivalently, no matter whether we think of truth and falsity as being exclusive or exhaustive). This conflict simply does not arise on our account.

  29. 29.

    For more details on this, see [14].

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Acknowledgments

We would like to thank Brendan Balcerak Jackson, Francesco Berto, Thomas Müller, Heinrich Wansing, and an anonymous referee for helpful discussion. We would also like to thank the participants of the Munich Centre for Mathematical Philosophy (MCMP) Colloquium.

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Correspondence to Hitoshi Omori.

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This research was funded by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement nr 263227. Hitoshi Omori is a Postdoctoral Research Fellow of the Japan Society for the Promotion of Science (JSPS).

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De, M., Omori, H. There is More to Negation than Modality. J Philos Logic 47, 281–299 (2018). https://doi.org/10.1007/s10992-017-9427-0

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Keywords

  • Negation
  • Compatibility
  • Modality
  • Contradictory