Abstract
The paper concerns transparent theories of truth, i.e. theories treating ‘ ‘ϕ’ is true’ as fully intersubstitutable with ϕ, and examines what the prospects are of maintaining a suitably refined version of transparency in view of the problem posed by the semantic paradoxes. In particular, three kinds of transparent theories—theories denying the law of excluded middle, theories denying the law of non-contradiction and theories denying the metarule of contraction—are compared with respect to the two most prominent semantic paradoxes: the Liar and Curry’s. It is argued that there are versions of the Liar paradox that do not rely on the law of excluded middle or the law of non-contradiction, and that such versions are blocked by the first two kinds of theories only by (implausibly) severing important connections between logical consequence and negation. Similarly, it is argued that Curry’s paradox does not rely on the law of excluded middle or the law of non-contradiction, and that it is blocked by the first two kinds of theories only by (implausibly) severing important connections between logical consequence and the conditional. All the paradoxes discussed are shown however to rely on the metarule of contraction, and so the third kind of theory is revealed to have the advantage of offering a unified solution to such paradoxes.
Earlier versions of the material in this paper have been presented in 2012 at the MCMP Conference on Paradox and Logical Revision (Ludwig Maximilian University), at the 5th SPFA Meeting in Braga (University of Minho), at the 10th SIFA Conference in Alghero (University of Sassari), at the LOGOS Semantic Paradoxes and Vagueness Seminar (University of Barcelona) and at an NIP workshop in honour of Crispin Wright (University of Aberdeen). I’d like to thank all these audiences for very stimulating comments and discussions. Special thanks go to Massimiliano Carrara, Roy Cook, Philip Ebert, Hartry Field, Branden Fitelson, Luke Fraser, Patrick Greenough, Michiel van Lambalgen, José Martínez, Sebastiano Moruzzi, Julien Murzi, Sergi Oms, Marco Panza, Bryan Pickel, Graham Priest, Stephen Read, David Ripley, Sven Rosenkranz, Gonçalo Santos, Ricardo Santos, Daniele Sgaravatti, Vladimir Stepanov, Alan Weir, Robert Williams, Timothy Williamson, Crispin Wright and an anonymous referee. I’m also very grateful to the editors Theodora Achourioti, Kentaro Fujimoto, Henri Galinon and José Martínez for inviting me to contribute to this volume and for their support and patience throughout the process. In writing the paper, I’ve benefitted, at different stages, from an AHRC Postdoctoral Research Fellowship and the FP7 Marie Curie Intra-European Fellowship 301493 with project on A Non-Contractive Theory of Naive Semantic Properties: Logical Developments and Metaphysical Foundations (NTNSP), as well as from partial funds from the project FFI2008-06153 of the Spanish Ministry of Science and Innovation on Vagueness and Physics, Metaphysics, and Metametaphysics, from the project FFI2011-25626 of the Spanish Ministry of Science and Innovation on Reference, Self-Reference and Empirical Data, from the project CONSOLIDER-INGENIO 2010 CSD2009-00056 of the Spanish Ministry of Science and Innovation on Philosophy of Perspectival Thoughts and Facts (PERSP) and from the FP7 Marie Curie Initial Training Network 238128 on Perspectival Thoughts and Facts (PETAF).
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- 1.
Against an influential tendency in much of the recent literature, I should make clear that by ‘reject’ and its likes I naturally mean something simply along the lines of refusal to accept the relevant proposition (in particular, I mean something that is compatible with thinking that the relevant proposition might be true). I’ll use ‘deny’ and its likes to mean acceptance of the negation of the relevant proposition.
- 2.
Throughout, I use ‘follow from’ and its relatives to denote the relation of logical consequence (broadly understood so as to encompass also the “logic of truth”) while I use ‘entail’ and its relatives to denote the converse relation. I use ‘equivalence’ and its relatives to denote two-way entailment. Moreover, I use ‘implication’ and its relatives to denote the status of the semantic values of two sentences that is necessary and sufficient for the conditional from the one sentence to the other sentence to be true (just like falsehood is the semantic value of a sentence that is necessary and sufficient for the negation of the sentence to be true).
- 3.
Paradox A also suggests other interesting options that unfortunately there won’t be space to discuss in this paper. Let me however remark that merely denying reasoning by cases and accepting all the other principles employed in paradox A does not suffice to uphold transparency. By LEM, \(\mathsf{t} \vdash T(\ulcorner \lambda \urcorner)\vee \neg T(\ulcorner \lambda \urcorner)\) holds, and so, by transparency, \(\mathsf{t} \vdash T(\ulcorner \lambda \urcorner)\vee T(\ulcorner \lambda \urcorner)\) holds, and hence, by a version of contraction, \(\mathsf{t} \vdash T(\ulcorner \lambda \urcorner)\) holds. By similar reasoning, \(\mathsf{t} \vdash \neg T(\ulcorner \lambda \urcorner)\) holds, and so, by adjunction, \(\mathsf{t} \vdash T(\ulcorner \lambda \urcorner)\hspace{.5ex}\& \hspace{.5ex} \neg T(\ulcorner \lambda \urcorner)\) holds. Since, by LNC, \(T(\ulcorner \lambda \urcorner) \hspace{.5ex}\& \hspace{.5ex} \neg T(\ulcorner \lambda \urcorner) \vdash \mathsf{f}\) holds, it follows, by transitivity, that \(\mathsf{t} \vdash \mathsf{f}\) holds. Denial of reasoning by cases is in fact one of the main features of supervaluationist and revision theories (see e.g. (McGee 1991) and (Gupta and Belnap 1993) respectively), which are not transparent theories. All the theories considered in this paper accept the specific version of reasoning by cases employed in paradox A, although I’ve argued in (Zardini 2011) that theories denying contraction should also deny a more general—and very frequently mentioned—version of that metarule (the issue will briefly crop up in fn 16 and at the end of Sect. 23.3).
- 4.
Transparent theories as a whole can in turn be seen as forming a logical and philosophical natural kind along the correlated dimensions of how deeply one deviates from classical logic and, consequently, of how tightly one can connect truth with reality. Theories that retain full classical logic must give up even the equivalence between \(T(\ulcorner \varphi \urcorner)\) and ϕ, and so must allow for the possibility that truth and reality straightforwardly come apart. Theories that at least are closed under classical laws/rules and structural (but not necessarily operational) metarules must still give up the intersubstitutability of \(T(\ulcorner \varphi \urcorner)\) with ϕ (i.e. transparency), and so must allow for the possibility that truth and reality come apart at least in certain contexts (for example, in the suppositional contexts created by antecedents of conditionals). Transparent theories are characteristic in that, forcing truth and reality to go together in every context, they require a deeper deviation from classical logic consisting in denying some of its laws/rules or structural metarules. Thanks to Gonçalo Santos for raising this issue.
- 5.
Throughout, I assume—plausibly enough in view of our definition of \(\mathsf{f}\)—that [\(\varphi \vdash \mathsf{f}\) holds iff ϕ itself is a logical falsehood] and—plausibly enough in view of our definition of \(\mathsf{t}\)—that [\(\mathsf{t} \vdash \varphi\) holds iff ϕ itself is a logical truth]. (Throughout, I use square brackets to disambiguate constituent structure in English.) I’ll offer some justification for the first assumption in fn 19.
- 6.
As the text already suggests, what I typically have in mind when I talk about “denial of an instance of LEM” is denial because of rejection of its conclusion, rather than denial because the conclusion may fail to have properties over and above truth that are deemed necessary for being a logical truth. Analogous comments apply for my talk below of “denial of an instance of LNC” and “denial of an instance of contraction”.
- 7.
Given that non-LEM and non-LNC theories are already well-known in the literature, a similar presentation of their details would be superfluous. Moreover, given that there are important disagreements of detail among different non-LEM theories and among different non-LNC theories, a single such presentation would be impossible. Finally, given that the arguments of this paper are robust with respect to possible disagreements of detail among different non-LEM theories and among different non-LNC theories, multiple such presentations would be irrelevant.
- 8.
The metarules given for & and ∨ are essentially the metarules for the “multiplicative” operators tensor and par of linear logics and the “intensional” operators fusion and fission of relevant logics. In both these kinds of logics, such operators are opposed to the “additive” or “extensional” operators, which are typically supposed to express our informal notions of conjunction and disjunction (as they occur for example in informal presentations of the semantic paradoxes). Against a likely misunderstanding, I can’t emphasise enough that, with \(\mathbf{IKT}\)’s & and ∨, I intend to give a theory of precisely our informal notions of conjunction and disjunction (hence my use of the standard symbolism), and that I take this interpretation to be warranted by the fact the last four rules just mentioned in the text are valid: together, those rules amount to saying that a conjunction is true iff both of its conjuncts are true, and that a disjunction is true iff either of its disjuncts is, which I take to capture the core of our informal notions of conjunction and disjunction. The divergence of my interpretation with the interpretation typically given for linear and relevant logics is explainable by the fact that those logics lack K-L and K-R, which determines that at least some of the rules just mentioned in the text are not valid for their “multiplicative” or “intensional” operators (hence these logics’ search for other operators that can represent more adequately our informal notions of conjunction and disjunction).
- 9.
To save on brackets, I’ll assume the usual scope hierarchy among the operators (with ∨ binding more strongly than & and ∨, and with these in turn binding more strongly than →) and right associativity for each 2ary operator (so that \(\varphi_{0}\star \varphi_{1}\star \varphi_{2}\ldots \star \varphi_{i}\) reads \(\varphi_{0}\star (\varphi_{1}\star (\varphi_{2}\ldots \star \varphi_{i})))\ldots)\), with ☆ being a 2ary operator).
- 10.
A fascinating earlier reference envisaging failure of contraction (but not in relation to the paradoxes) may be (Pastore 1936), a work that however requires further historical and exegetical investigation. Thanks to Luke Fraser for alerting me to the existence of Pastore’s work.
- 11.
Although I can’t go into the details in this paper, the latter difference leads to Grišin’s theory remaining finitary, while mine goes infinitary, since it introduces quantifiers by means of metarules akin to the ω-rule; the latter difference also leads to major divergences in the deductive systems codifying the respective theories and in the consistency proofs based on those (see (Zardini 2011, pp. 511–512, 524–532)).
- 12.
Ultimately, I don’t think that Grišin’s result forecloses a non-contractive naive theory of sets, but the issue lies beyond the scope of this paper.
- 13.
Contraction on \(\mathsf{t}\) is admissible even in IKT and so I won’t henceforth bother to make it explicit.
- 14.
While paradox A corresponds to the kind of informal presentation of the Liar paradox that proves that the Liar sentence cannot be true and proves that the Liar sentence cannot be untrue (while observing that it must be either true or untrue), paradox B corresponds to the kind of informal presentation of the Liar paradox that first proves that the Liar sentence is untrue and then on that basis proves that it is also true (while observing that it cannot be both true and untrue).
- 15.
Having officially recorded the versions with side-premises and side-conclusions, for simplicity I’ll ignore these in my treatment of reductio.
- 16.
Even adding the further disjunctive assumption that either the ungrounded sentence or its negation holds (an instance of LEM) does not substantially improve the appeal of the inference in the theoretic context of transparent theories. For, in that context, to suppose that the sentence holds is to suppose something intuitively equivalent with its failing to hold (since the sentence actually denies of itself that it holds), and so the fact that the suppositions represented by the two disjuncts can both be so developed as to reach the supposition that the sentence holds is after all not great evidence in favour of the sentence holding (since the supposition so reached is an unstable one). Thanks to Sven Rosenkranz for pressing me on this issue.
- 17.
Classical reductio is traditionally known as ‘consequentia mirabilis’ (or ‘Clavius’ law’, from the (Latin) name of the Counter-Reformation Jesuit priest who, actually in the steps of Girolamo Cardano, did much for promoting this method of proof in mathematics). The two names betray two completely different understandings of the principle. The modern name signals an understanding of the principle under which it implicitly involves deriving the contradiction \(\varphi \hspace{.5ex}\& \hspace{.5ex} \neg \varphi\) from \(\neg \varphi\), and from this fact inferring ϕ (presumably with the implicit thought that the first derivation shows that \(\neg \varphi\) is false, which, in a bivalent spirit, is then taken to suffice for ϕ’s being true). The traditional name signals an understanding of the principle under which it only involves deriving ϕ from \(\neg \varphi\), and from this fact inferring ϕ (presumably with the implicit thought that the first derivation shows that, even if \(\neg \varphi\) is true, ϕ still is, which, in a bivalent spirit, is then taken to suffice for ϕ’s being true), without any derivation of a contradiction justifying the reasoning (one can see the two understandings nicely opposed in an epistolary debate between Christiaan Huygens and André Tacquet; see (Nuchelmans 1992) for a historical reconstruction of the dispute). Neither understanding helps in mitigating the objections against the principle levelled in the text. I’ll discuss how both understandings covertly assume contraction at the end of this section.
- 18.
Because of this, in (Zardini 2011, p. 514, fn 38) I’ve argued also for the terminological point that the traditional name ‘reductio ad absurdum’ for what I’m in this paper calling ‘reductio’ is an egregious misnomer. For a metarule properly called ‘reductio ad absurdum’ should have an input saying that certain premises lead to an absurdity, which is clearly the case for the single-premise reduction theorem and clearly not the case for either classical or intuitionist reductio, which would more properly be called ‘reductio ad ipsius contradictoriam’.
- 19.
As might have been intimated by the qualification ‘apparently’, I don’t think that the issue is purely terminological. I think that we can quite convincingly argue in a variety of ways that logical falsehood just is inconsistency. A first argument for that conclusion mimics the standard argument against the definition of falsehood of a sentence in terms of truth of its negation, which consists in the simple observation that that definition gets things dramatically wrong for sentences belonging to languages which don’t have negation. In a completely analogous fashion, we can argue against the definition of logical falsehood of a sentence in terms of logical truth of its negation, by simply observing that that definition gets things dramatically wrong again for sentences belonging to languages which don’t have negation. And, if the logical falsehood of a sentence is not to be identified with the logical truth of its negation, it’s hard to see what else is left for it to be other than the inconsistency of the sentence. A second argument for the conclusion starts from the observation that inconsistent sentences behave dually with respect to logical truths, in the sense that, just like logical truths correspond to valid no-premise, single-conclusion arguments, inconsistent sentences correspond to valid single-premise, no-conclusion arguments. Assuming very plausibly that logical falsehood behaves dually with respect to logical truth, that forces the identification of logical falsehood with inconsistency. Or, similarly, just as the best judgement that logic can pass on a sentence in itself (i.e. not qua component of more complex sentences) is that it is the conclusion of a valid no-premise, single-conclusion argument, so the worst judgement that logic can pass on a sentence in itself is that it is the premise of a valid single-premise, no-conclusion argument. Assuming very plausibly that, just as the best judgement that logic can pass on a sentence in itself is equal to a judgement of logical truth, so the worst judgement that logic can pass on a sentence in itself is equal to a judgement of logical falsehood, that again forces the identification of logical falsehood with inconsistency. A third argument for the conclusion (and indeed, as we’ll see, for something even stronger) assumes very plausibly that just as (logical) truth is closed under entailment (at least in the sense that, if ϕ is a (logical) truth and ϕ entails ψ, it follows that ψ is a (logical) truth), so (logical) falsehood is closed under logical consequence (at least in the sense that, if ψ is a (logical) falsehood and ψ follows from ϕ, it follows that ϕ is a (logical) falsehood). The argument then bifurcates. As for non-LEM theories, we only need to establish that inconsistency implies logical falsehood. We do so by observing that, at least in the systems of interest for this paper, being inconsistent implies entailing sentences that are (logical) falsehoods by anyone’s lights (for example, ‘For every P, P’), from which the desired implication follows by closure of (logical) falsehood under logical consequence (and, as I’ll observe in the text, the weaker result just implicitly established that inconsistent sentences are merely false is already incompatible with non-LEM theories). As for non-LNC theories, we only need to establish that logical falsehood implies inconsistency. We do so by reducing to absurdity the claim that (it doesn’t because) some logical falsehoods are also logical truths. In turn, we do so by observing that, at least in the systems of interest for this paper, being a logical truth implies being entailed by sentences that are not (logical) falsehoods by anyone’s lights (for example, ‘For some P, P’), from which the desired absurdity follows by closure of (logical) falsehood under logical consequence (and, as I’ll observe in fn 33, the stronger result just implicitly established that logical truths lack mere falsehood is also incompatible with non-LNC theories). (As a matter of fact, non-LEM and non-LNC theorists are wont to reject closure of falsehood under logical consequence. The high implausibility of rejecting such a fundamental principle connecting logical consequence with falsehood is typically masked by recommending an apparently similar principle of closure of rejectability under logical consequence (if ψ ought to be rejected and ψ follows from ϕ, it follows that ϕ ought to be rejected). The recommendation is both disappointing and wrong. The recommendation is disappointing as it offers a superficial, merely normative Ersatz talking about what people ought to do in substitution for a deep, fully descriptive principle connecting logical consequence with semantics and so with the ways the objective world can be independently of people and of what they ought to do (in fact, even if it were true, the normative principle would cry out for a deeper explanation appealing, among other things, to something along the lines of the descriptive principle). The recommendation is also wrong as virtually every counterexample to closure of knowledge and justification can be turned into a counterexample to closure of acceptability and rejectability—those normative notions, as opposed to the semantic notions of truth and falsehood, are not suitable for the formulations of appropriate closure principles.)
- 20.
Throughout, modals and their likes subscripted with ‘L’ express logical modality.
- 21.
Might she not deny instead the duality of necessity L and possibility L ? In our dialectical context, the move would be extremely problematic in several respects. Firstly, short of transparency failing in contexts of logical modality, the move in effect now accepts the metarule from \(\varphi \vdash \mathsf{f}\) to \(\mathsf{t} \vdash\) ‘It is not possible L that ϕ’. If in addition we still have the extremely plausible rule \(\varphi,\) ‘It is not possible L that ϕ’ \(\vdash \mathsf{f}\), the resulting non-LEM theory is trivial (see the final version of paradox B mentioned in Sect. 23.4). Secondly, the duality of necessity L and possibility L is entailed by the duality of universal quantification and particular quantification plus the standard modality-worlds-linking principles ‘It is necessary L that ϕ iff, for every possible L world w, ‘ϕ’ is true at w’ and ‘It is possible L that ϕ iff, for some possible L world w, ‘ϕ’ is true at w’ (together with an appropriate version of transparency for ‘true at w’ like ‘In w, ‘ϕ’ is true at w iff ϕ’ and with the auxiliary assumption ‘ ‘ϕ’ is true at w 0 iff, for every possible L world w 1, in w 1, ‘ϕ’ is true at w 0’). Thirdly, the contrapositive of ‘If ϕ, then it is possible L that ϕ’ suffices to license the inference from ‘ ‘ϕ’ cannot L be true’ to ‘ ‘ϕ’ is not true’, which, as I’ll observe in the text, is already incompatible with non-LEM theories. (See also the similar argument at the end of fn 33 that does not employ the duality of necessity \(_{L}\) and possibility \(_{L}\).) Thanks to Robert Williams for urging me to consider these issues.
- 22.
I should note that the argument in the text does not fall afoul of the worry I’ve raised above regarding the inference from intensional facts to categorical ones. For that worry concerned the specific case in which the intensional fact is that an ungrounded sentence is strongly equivalent with its own negation, while the different specific case in which the intensional fact is that a sentence entails \(\mathsf{f}\) is a case in which the intensional fact itself already intuitively involves, if not even is constituted by, the categorical fact that the sentence is a logical falsehood (or by the categorical fact that the sentence is inconsistent). Thanks to Sven Rosenkranz and an anonymous referee for raising this issue.
- 23.
Independently of the issue of which formulation of non-LEM theories is most appropriate, the observation in the text shows that a non-LEM theory becomes completely indistinguishable from supervaluationist and revision theories in a language lacking disjunction (and the resources to define it) but expressive enough as to generate semantic paradoxes (the language needed to generate the essence of paradox B is a good example of such a language).
- 24.
Such assumptions start to look less plausible vis-à-vis the alternative assumptions made by supervaluationist and revision theories once it is realised that, keeping fixed the duality of conjunction and disjunction and the idea that some sentences are such that both they and their negation are logical falsehoods, the latter assumptions allow supervaluationist and revision theories to uphold the very plausible claim that the negation of any contradiction is a logical truth, while the former assumptions force non-LEM theories to endorse the very implausible claim that the negations of certain contradictions are logical falsehoods.
- 25.
Contrast with IKT, whose fundamental thought that some sentences are such that they fail to contract is strong enough to characterise it against all the other main alternative theories. Of course, IKT too has its own variations. In (Zardini 2013a), I study in some detail a particularly natural one that replaces the multiplicative operators with the additive ones, and argue that such variation suffers from a lack of connection between logical consequence on the one hand and conjunction and disjunction on the other hand similar to the lack of connection between logical consequence on the one hand and negation on the other hand suffered by non-LEM theories.
- 26.
Thanks to Branden Fitelson, David Ripley, Sven Rosenkranz and an anonymous referee for criticisms of an earlier draft of this paragraph.
- 27.
- 28.
Since we’ve reserved ‘law of excluded middle’ (i.e. ‘LEM’) for the logical claim expressed by ‘\(\mathsf{t}\vdash \varphi \vee \neg \varphi\)’, let’s use ‘excluded middles’ to refer to the typically non-logical claims expressed by instances of \(\varphi \vee \neg \varphi\).
- 29.
I hasten to add that I don’t mean to imply that a theory must solve all the paradoxes just mentioned in the text by deploying its fundamental thought, since, at least for the first two kinds of paradoxes, a theory might also appeal to independently plausible truth-theoretic principles, and, at least in the case of the truth-teller paradox, such appeal might suffice to yield already a solution to the paradox (for example, one might argue that the truth-teller simply lacks truth on the strength of general considerations concerning truth and grounding in reality, see (Priest 2006, p. 66)). Thanks to Patrick Greenough for urging this clarification.
- 30.
I’ve argued that there are paradoxical sentences which are not indeterminate, and that is enough to undermine the modification discussed in the text. It then becomes a secondary question whether the non-LEM theorist should say that LEM fails for whichever is the false sentence in the pair of opposite Epimenides sentences considered in the text. For what it’s worth, to me it would be very plausible to say that it does, and so concede that, in some cases, LEM fails even if a sentence is not indeterminate. But I suppose that one could instead say that it does not. To me, that would be very implausible. Obviously, since the sentence is false, the relevant excluded middle is true. But, since the sentence might have been indeterminate and cannot be known a priori not to be such (for the number of stars in the universe might have had the other parity and cannot be known a priori not to have it), to conclude from the truth of the relevant excluded middle that the relevant instance of LEM is valid is in effect to concede that, in some cases, a sentence is a logical truth even if it is not necessarily such and even if it cannot be known a priori, as well as to concede that, at least given transparency and in the broad sense explained in fn 2, there is a logical proof of the parity of the number of stars in the universe—in fact, a logical proof of every actual truth. Notice that exactly the same points apply to an Epimenides sentence which we do know to be false, such as ‘This sentence is not true and there are no stars’ (given which it would then become extremely natural to give the same treatment also to sentences which are necessarily false and known a priori (but not logically) to be false, such as ‘This sentence is not true and something is not part of itself’). Given this, the non-LEM theorist would have to concede for the Epimenides sentences she knows to be false (not only that they are nevertheless paradoxical but also) that LEM nevertheless fails for them (as she would have to for other paradoxical sentences), even if she believes them to be false, and so even if she does not reject their negation. Thanks to Timothy Williamson for discussion of this question.
- 31.
In view of this dialectic, one may try to take a rather different, more concrete approach focussing for example on a specific model-theoretic construction and saying that it is from the fundamental thought behind the construction that both denial of LEM and denial of the single-premise reduction theorem flow (although it should be mentioned that a non-LEM theorist in the spirit of (Field 2008) would be reluctant to assign such a fundamental explanatory role to the model theory). A natural candidate for such a proposal is the strong-Kleene construction of (Kripke 1975). There is no doubt of course about the technical fact that the overall construction invalidates both LEM and the single-premise reduction theorem. What is open to doubt, however, is whether there is a single fundamental thought behind the overall construction. Let me explain. The basic thought behind the construction seems to be the “gaps-and-grounds” picture supporting the package constituted by the strong-Kleene valuation scheme together with the definition of truth at a later stage in terms of how things are at the earlier stage(s). Such basic thought thus includes the thought that sentences might have a gappy status (not to be identified with lack of truth and falsehood) which is a fixed point for negation and that is inherited by a disjunction from both of its disjuncts, a thought which, given plausible additional assumptions, is indeed sufficient to invalidate LEM. But such basic thought remains silent about the single-premise reduction theorem. That issue is only addressed by the additional theoretic decision of defining (single-premise, single-conclusion) logical consequence as downwards preservation of the truth-entailing non-gappy status (at the relevant fixed point(s)). But, precisely in the context of the basic thought in which downwards preservation of the truth-entailing non-gappy status does no longer coincide with upwards preservation of the untruth-entailing non-gappy status, that decision is arbitrary and indeed questionable in that it gives more importance to the truth-entailing non-gappy status than to the untruth-entailing one. The arbitrariness and indeed questionability of the decision may not be immediately apparent because one can still infer from the truth of the premise of a “valid” argument the truth of the conclusion, but it does emerge once it is noticed that one can no longer infer from the untruth of the conclusion of a “valid” argument the untruth of the premise (see also the principle of closure of (logical) falsehood under logical consequence discussed at the end of fn 19). Once the alternative but more natural definition requiring both downwards preservation of the truth-entailing non-gappy status and upwards preservation of the untruth-entailing non-gappy status is adopted, the single-premise reduction theorem is validated. Moreover, the basic thought by itself already seems strongly to suggest the single-premise reduction theorem, and seems in any case to have consequences incompatible with non-LEM theories. For that thought also involves the thought that a sentence is true in virtue of its positive grounding in reality, from which it seems to follow that sentences that are not so grounded are not true (since they lack that in virtue of which a sentence is true). But, in the Kripke construction, if a sentence is a logical falsehood, it is not positively grounded in reality, and so it is not true. This strongly suggests the single-premise reduction theorem, and is in any case incompatible with non-LEM theories. Thanks to José Martínez, Sebastiano Moruzzi and Bryan Pickel for urging me to consider this alternative proposal.
- 32.
Mutatis mutandis, all the three points made in fn 21 apply if she denies the duality of necessity L and possibility L (see also the similar argument in fn 33 that does not employ the duality of necessity L and possibility L ).
- 33.
Similarly, if a sentence is a logical truth, certainly it must L lack falsehood and so its negation must L lack truth and hence is inconsistent. I suppose that the non-LNC theorist will have to deny the inference from ‘ϕ is a logical truth’ to ‘ϕ must L lack falsehood’. The non-LNC theorist has thus to deny that even the best judgement that logic can pass on a sentence in itself suffices for the necessary L lack of falsehood of that sentence—indeed, it’s easy to see that the non-LNC theorist even has to deny that it suffices for the mere lack of falsehood of that sentence (in this sense, although logic would still be powerful enough to establish many truths, somehow it would no longer be powerful enough to establish any lack of falsehood). Such a lack of connection between logical consequence on the one hand and lack of falsehood and logical modality (and even mere lack of falsehood) on the other hand strikes me as a great cost of these theories.
- 34.
Another aspect of this lack of connection (strictly related to the principle of closure of (logical) falsehood under logical consequence discussed at the end of fn 19) concerns the traditional idea that logical consequence consists in the impossibility L that [the premise is true and the conclusion is not true]. Non-LEM theories need to reject that a premise entails a conclusion only if it is impossible L that [the premise is true and the conclusion is not true], since they accept that \(\lambda\) entails \(\mathsf{f}\), and so would have to accept that it is impossible L (i.e. it is not possible L ) that [\(\lambda\) is true and \(\mathsf{f}\) is not true]. But, by the duality of necessity L and possibility L , that implies that it is necessary L that it is not the case that [\(\lambda\) is true and \(\mathsf{f}\) is not true], and so, by the relevant De Morgan rule and closure of necessity L under entailment, it would be necessary L that either \(\lambda\) is not true or \(\mathsf{f}\) is true, and hence, reasoning by cases, by the properties of \(\mathsf{f}\) and closure of necessity L under entailment, it would be necessary L that \(\lambda\) is not true, which is however unacceptable for non-LEM theories. Non-LNC theories need to reject that a premise entails a conclusion if it is impossible L that [the premise is true and the conclusion is not true], since they accept that it is necessary L that λ is not true. By the contrapositive of simplification and closure of necessity L under entailment, that implies that it is necessary L that it is not the case that [λ is true and \(\mathsf{f}\) is not true], and so, by the duality of necessity L and possibility L , that it is not possible L (i.e. it is impossible L ) that [λ is true and \(\mathsf{f}\) is not true], and hence non-LNC theories would have to accept that λ entails \(\mathsf{f}\), which is however unacceptable for them.
- 35.
To the best of my knowledge, such theories have not been investigated in relation to the semantic paradoxes. I won’t go into their details in this paper—suffice it to say that they naturally arise by dualising, respectively, the familiar Kripke construction based on the supervaluationist evaluation scheme and the familiar revision-sequence construction. Such theories are very similar to one another in the respects that are relevant for our discussion: in particular, they accept both a sentence and its negation without accepting any contradiction (more strongly, they hold that the contradiction is a logical falsehood and accept LNC in its full generality). The underlying idea, to put it very roughly, is that conjunction is sensitive to compatibility relationships between the conjuncts.
- 36.
A comment analogous to that in fn 23 applies concerning the complete indistinguishability of all these theories in expressively impoverished paradoxical languages.
- 37.
Such assumptions start to look less plausible vis-à-vis the alternative assumptions made by subvaluationist and non-standard revision theories once it is realised that, keeping fixed the duality of conjunction and disjunction and the idea that some sentences are such that both they and their negation are logical truths, the latter assumptions allow subvaluationist and non-standard revision theories to uphold the very plausible claim that the negation of any excluded middle is a logical falsehood, while the former assumptions force non-LNC theories to endorse the very implausible claim that the negations of certain excluded middles are logical truths.
- 38.
Comments analogous to those in fn 30 apply concerning the question whether the non-LNC theorist should say that LNC fails for whichever is the false-only sentence in the pair of opposite Epimenides sentences considered in the text.
- 39.
Comments analogous to those in fn 31 apply concerning a rather different, more concrete approach focussing for example on a specificmodel-theoretic construction and saying that it is from the fundamental thought behind the construction that both denial of LNC and denial of the single-conclusion demonstration theorem flow. In particular, the first comment in fn 31 has an analogue to the effect that the decision of defining (single-premise, single-conclusion) logical consequence as downwards preservation of truth-entailing non-glutty or glutty status (at the relevant fixed point(s)), without requiring upwards preservation of untruth-entailing non-glutty or glutty status, is arbitrary and indeed questionable. The second comment in fn 31 has an analogue to the effect that the basic thought behind the relevant construction involves the thought that a sentence is untrue in virtue of its negative grounding in reality, from which it seems to follow that sentences that are not so grounded lack untruth (since they lack that in virtue of which a sentence is untrue).
- 40.
Compare the similar argument consisting only in the inference from \(T(\ulcorner \lambda \urcorner)\vdash \neg T(\ulcorner \lambda \urcorner)\) and \(\neg T(\ulcorner \lambda \urcorner)\vdash T(\ulcorner \lambda \urcorner)\) to \(\mathsf{t}\vdash\) ‘Classical mathematics is inconsistent’. Although that argument is classically valid, there is no obligation for a unified transparent solution to deploy its fundamental thought in order to deny the only inference employed in the argument—in the theoretic context of transparent theories, that inference is, without further justification, very dubious at best. This is not to deny of course that the inference might be further justified by appeal to more fundamental and, even in the theoretic context of transparent theories, apparently compelling principles, so as to produce a genuine paradox for those theories that should ideally be blocked by deploying their fundamental thought. That is in fact what I’ve done for the single-premise reduction theorem and the single-conclusion demonstration theorem (with the main aim of showing that non-LEM and non-LNC theories need to appeal to some rather unobvious considerations concerning the lack of connection between logical consequence and negation that are quite foreign to the issue as to whether LEM or LNC are valid), and what I’ll proceed to do for reductio (with the main aim of showing that the justifying arguments do involve contraction). Thanks to Sven Rosenkranz and an anonymous referee for criticisms of an earlier draft of this paragraph.
- 41.
Having officially recorded the version with side-premises and side-conclusions, for simplicity I’ll ignore these in my treatment of the metarule of absorption.
- 42.
Comments analogous to the first two in fn 19 apply concerning the only apparent terminological character of the point. In particular, the second comment in fn 19 has an analogue to the effect that, just as the best judgement that logic can pass on a sentence in itself—that it is the conclusion of a no-premise, single-conclusion argument—is equal to a judgement of validity for the categorical statement consisting in that sentence, so the best judgement that logic can pass on a sentence in itself with respect to a sentence in itself—i.e. that it is the conclusion of a single-premise, single-conclusion argument whose premise is the latter sentence—should be equal to a judgement of validity for the hypothetical statement from the latter sentence to the former sentence—i.e. to a logical implication from the latter to the former. A third argument for the conclusion that logical implication just is entailment assumes very plausibly that entailment, usually presented as a metalinguistic relation, is ultimately just a kind of object-linguistic implication (lying at one extreme of a spectrum at whose other extreme lies material implication), which should then be identified with logical implication.
- 43.
The point is in effect conceded by a prominent non-LNC theorist who has often emphasised the importance of offering a unified solution to the semantic paradoxes: “[…] the curried versions of the paradoxes belong to a quite different family. Such paradoxes do not involve negation and, a fortiori, contradiction”, “They are paradoxes that involve essentially conditionality […] Genuine Curry paradoxes are therefore ones that depend on a mistaken theory of the conditional, and are perhaps best thought of as more like the ‘paradoxes of material implication' ” (Priest 2003, pp. 169, 278). In fact, the claim could be made that the fundamental version of Curry’s paradox does not satisfy the inclosure schema that (Priest 2003) has argued to be at the root of the semantic paradoxes (where these are understood as that kind of paradox that is instantiated by the Liar paradox). Although a proper treatment of this issue lies beyond the scope of this paper (see (Zardini 2013b)), I should record that, if that claim were correct, it would seem to me more plausible to take it to reflect badly on the inclosure schema as a diagnosis of the semantic paradoxes rather than on Curry’s paradox as a semantic paradox (also given the fact that the rationale behind the claim would equally well establish that Epimenides’ paradox and related paradoxes using material implication are not semantic paradoxes!). It may also be worth mentioning that a non-LEM or non-LNC theory might be such that, if the relevant instances of LEM or LNC are added for a certain fragment of the language, that fragment “behaves classically”, and so in particular obeys the single-premise deduction theorem (the theory in (Field 2008) is an example of such a system). Even this (possible) technical fact would however be very far from indicating that, in some reasonable sense, the single-premise deduction theorem fails in such a theory because LEM or LNC fail in the theory. Compare: the technical fact about intuitionist logic that, if the relevant instances of Peirce’s law (\(\mathsf{t}\vdash ((\varphi \rightarrow \psi)\rightarrow \varphi) \rightarrow \varphi\)) are added for a certain fragment of the language, that fragment “behaves classically”, and so in particular obeys LEM, is very far from indicating that, in some reasonable sense, LEM fails in intuitionist logic because Peirce’s law fails in the logic. Thanks to Graham Priest for putting forth to me the claim about the inclosure schema mentioned in this fn.
- 44.
As usual, states-of-affairs are abstract entities that can either obtain or fail to obtain. A locution like ‘State-of-affairs \(s_{0}\) leads to state-of-affairs \(s_{1}\)’ must be understood as ‘The obtaining of state-of-affairs \(s_{0}\) leads to the obtaining of state-of-affairs \(s_{1}\)’.
- 45.
On this view, the state-of-affairs t 0 expressed by the Curry sentence τ identical to \(T(\ulcorner \tau \urcorner)\rightarrow \top\) is unstable even if \(\varphi \vdash \psi \rightarrow \varphi\) does hold in IKT (by I, K-L and →-R), and so even if t 0 unproblematically obtains. For it is still the case that t 0 leads to the state-of-affairs t 1 expressed by \(T(\ulcorner \tau \urcorner)\) without thereby co-obtaining with it: t 0 does obtain, and does obtain even if t 1 does, and so leads to t 1 and co-obtains with it, but it does all this only because it is a consequence of the state-of-affairs expressed by ⊤—the source of t 0 co-obtaining with t 1 relies in the state-of-affairs expressed by ⊤ rather than in t 0 itself. In this (intended) sense, t 0 leads to t 1 without thereby co-obtaining with it (contrast, for example, with the state-of-affairs that snow is white leading to the state-of-affairs that ‘Snow is white’ is true and thereby co-obtaining with it). Thus, on this view, t 0 is unstable, contraction on τ cannot be built into the relevant system (although, since \(\mathsf{t}\vdash \top\) holds in the system, contraction on τ can be derived in it) and Curry’s paradox cannot be used to prove ⊤ (since the contraction step is only justified by an antecedent proof of ⊤). Instability only defeasibly brings about failure of contraction. Notice that, while non-LEM and non-LNC theories can uphold claims analogous to the second and third one in the second last sentence, it is utterly unclear whether they can identify an analogous defeasible cause that would support those claims (for one thing, indeterminacy or overdeterminacy won’t do). Analogous points apply of course to the relevant versions of Epimenides’ paradox. Thanks to Daniele Sgaravatti for discussion of this issue.
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Zardini, E. (2015). Getting One for Two, or the Contractors’ Bad Deal. Towards a Unified Solution to the Semantic Paradoxes. In: Achourioti, T., Galinon, H., Martínez Fernández, J., Fujimoto, K. (eds) Unifying the Philosophy of Truth. Logic, Epistemology, and the Unity of Science, vol 36. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9673-6_23
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