Abstract
At the core of JC Beall and Greg Restall’s brand of logical pluralism is Generalised Tarski’s Thesis, according to which an argument is valid iff, in every relevant case where every premise is true, so is the conclusion. I argue that the thesis implies that many philosophically interesting substructural logics are not legitimate relations of logical consequence. I then diagnose the clash as due to the fact that, in important ways, the thesis is not sensitive to intensional connections and to plurality in occurrences, values and models. Next, I extend the argument to the effect that the more general conception of logical consequence as guaranteed truth preservation clashes with substructurality. I conclude with a proposal as to how, for the substructural logics in question, we can still uphold a broadly semantic conception of logical consequence: given any such logic L, we can reinterpret truth-preservation conditionals with the notions of conjunction and implication available in L, and say that the fact that, in L, φ0, φ1, φ2 …, φi entail ψ is grounded in the fact that, in L, the conditional ‘If “φ0” is true and “φ1” is true and “φ2” is true … and “φi” is true, “ψ” is true’ is a logical truth. On this proposal, contrary to the contemporary Tarskian vulgate, it is logical consequence that is grounded in logical truth rather than vice versa.
Earlier versions of the material in this chapter have been presented in 2015 at the Veritas Pluralism, Language and Logic Workshop (Yonsei University); in 2018, at the LanCog Seminar (University of Lisbon), at the LOGOS Workshop Pluralism and Substructural Logics (University of Barcelona), at the fifth SBFA Conference (Federal University of Bahia) and at the Workshop Disagreement within Philosophy (Rhine Friedrich-Wilhelm University of Bonn). I’d like to thank all these audiences for the very stimulating comments and discussions. Special thanks go to Agustín Rayo, Colin Caret, Bogdan Dicher, Catarina Dutilh Novaes, Luís Estevinha, Filippo Ferrari, Ole Hjortland, Luca Incurvati, José Martínez, Ricardo Miguel, Sergi Oms, Nikolaj J. L. L. Pedersen, Hili Razinsky, Lucas Rosenblatt, Sven Rosenkranz, Diogo Santos, Ricardo Santos, Erik Stei, Célia Teixeira, Pilar Terrés, Zach Weber, Jack Woods and Jeremy Wyatt. I’m also grateful to the editors Nathan Kellen, Nikolaj J. L. L. Pedersen and Jeremy Wyatt for inviting me to contribute to this volume and for their support and patience throughout the process. The study has been funded by the FCT Research Fellowship IF/01202/2013 Tolerance and Instability: The Substructure of Cognitions, Transitions and Collections. Additionally, the study has been funded by the Russian Academic Excellence Project 5-100. I’ve also benefited from support from the Project FFI2012-35026 of the Spanish Ministry of Economy and Competition The Makings of Truth: Nature, Extent, and Applications of Truthmaking, from the Project FFI2015-70707-P of the Spanish Ministry of Economy, Industry and Competitiveness Localism and Globalism in Logic and Semantics and from the FCT Project PTDC/FER-FIL/28442/2017 Companion to Analytic Philosophy 2.
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Notes
- 1.
Throughout, page references are to BR unless otherwise stated.
- 2.
Throughout, I follow BR in assuming, merely for simplicity, a multiple-premise, single-conclusion framework. The extension of my points to the (superior but) more complex multiple-premise, multiple-conclusion framework is straightforward.
- 3.
BR claim that necessity is important for logical consequence at least partly because it guarantees that logic applies unrestrictedly in hypothetical reasoning (pp. 15–16). But it arguably doesn’t in the first place, since, say, the law of excluded middle (henceforth , ‘LEM’) is a logical truth but it is arguably wrong to reason that, if Brouwer were right, it would be the case that either Goldbach’s conjecture holds or it doesn’t. Notice that the hypothesis is perfectly possible (Brouwer might have held correct views), so it is not even the case that logic applies unrestrictedly to possible hypotheses (which would anyways raise the issue of why it should be more important unrestrictedly to apply to possible hypotheses rather than to, say, interesting hypotheses).
- 4.
Throughout, I work with a standard framework for context-dependent languages where the truth of a sentence is relative both to a context of utterance and to a circumstance of evaluation, and I make such relativities explicit with the relevant parts of the construction ‘φ is true-as-uttered-in-C-as-evaluated-at-E’.
- 5.
I’d like to forestall two likely reactions to this point. Don’t say that what is true is ‘For every context C, if “Snow is white” is true-as-uttered-in-C-as-evaluated-at-C, “Actually, snow is white” is true-as-uttered-in-C-as-evaluated-at-C’. The proposed Ersatz has the embarrassing problem (emerging in the opening ‘what is true [my emphasis, EZ]’) that neither it nor its embedded conditional is non-trivially necessary (so that the spirit if not the letter of the necessity condition would seem violated). What the proposed Ersatz indeed offers is universality over contexts, but that is a far cry from any interesting necessity: for every context C, ‘I exist’ is true-as-uttered-in-C-as-evaluated-at-C, but there is no interesting sense in which the existence of a first person is necessary [Zardini (2012), pp. 266–268 will be excused if he perhaps got too carried away by his ‘semantic modality’]. Along these lines, the proposed Ersatz is not even of the right logical form: for the necessity condition is presumably to the effect that there is a certain unique truth-theoretic property (truth-as-uttered-in-such-and-such-context) which is preserved at every circumstance from premises to conclusion (cf fn 3), whereas the proposed Ersatz is to the effect that every circumstance is such that at it some truth-theoretic property or other (truth-as-uttered-in-a-context-containing-that-circumstance) is preserved from premises to conclusion, with different such properties being preserved at different circumstances. That’s like proposing to understand the idea of universal access to healthcare not to the effect that that there is a certain unique health system that is accessed by everyone, but to the effect that everyone accesses some health system or other, with different such systems being accessed by different people. Also, don’t say that, while, for no context C, ‘If “Snow is white” is true-as-uttered-in-C, “Actually, snow is white” is true-as-uttered-in-C’ is metaphysically necessary, for every context C it is logically necessary. For, without further ado (which, as far as I know, has never been made), the claim that logical consequence is logically necessary is totally vacuous, as it is tantamount to the claim that logical consequence holds with the very special force of … logical consequence (in exactly the same way, also defeasible arguments could be claimed to be ‘defeasibly necessary’!).
- 6.
Such failure of normativity spells disaster for normative constraints on logical consequence of the kind ‘If φ0, φ1, φ2 … ⊢ ψ holds, one should not accept φ0, φ1, φ2 … and reject ψ’, which are endorsed , for example, by Restall (2005); Beall (2015) [see Zardini (2016), pp. 313–314, fn 9 for some further discussion]. Notice that, if one proposed (what BR don’t) that, actually, ‘one should not accept φ0 & φ1 & φ2 … and reject ψ’, even setting aside various issues related to the possible infinity of φ0, φ1, φ2 … it would be hard to see how such proposal does not in effect amount to a wholesale rejection of the normativity of multiple-premise arguments (since the source of the normative fact that one should not accept φ0 & φ1 & φ2 … and reject ψ is presumably the fact that the single-premise argument φ0 & φ1 & φ2 … ⊢ ψ holds), which would be quite sad since the preface paradox can do nothing to undermine the incontrovertible fact that quite a few times multiple-premise arguments do have normative force (in particular, are such that one should not accept each of their premises and reject their conclusion, independently of whether one de facto accepts the conjunction of their premises). Indeed, while, as far as the issues raised by the preface paradox are concerned, in classical logic multiple -premise arguments can be taken to have normative force (when and) only when one should [accept the conjunction of their premises if one accepts each of them] (so that, if a multiple-premise argument has normative force and one accepts each of its premises, since one should then accept the conjunction of its premises, the proposal in question—at least when crucially supplemented by some sort of account of when one should [accept the conjunction of the premises of an argument if one accepts each of them]—by in effect appealing to the normativity of the corresponding single-premise argument can vindicate the claim that one should not reject its conclusion), that is no longer so in some logics with non-standard behaviour of conjunction such as the subvaluationist logic Sb [Jaśkowski (1948)]. In Sb, it is possible, for example, that φ, ψ entail χ with normative force, that one accepts each of φ and ψ but that φ & ψ is inconsistent (let alone entailed by φ, ψ), so that, for non-epistemic reasons (and so for reasons not concerned by the issues raised by the preface paradox), it is not the case that one should accept it. In such a situation, since the argument from φ, ψ to χ has normative force and one accepts each of φ and ψ, one should not reject χ, but, since it is not the case that one should accept φ & ψ, the proposal in question can no longer in effect appeal to the normativity of any φ & ψ-[single-premised] argument to vindicate the claim that one should not reject χ.
- 7.
To be clear, the idea is that the objects are different, but the logical operations on them are the same: for example, there is a single operation of disjunction, which obeys LEM on the real numbers of classical analysis, whereas it does not obey it on the real numbers of intuitionist analysis. An alternative idea would be that the objects are the same, but the logical operations on them are different: for example, classical analysis talks about classical disjunction which obeys LEM on the real numbers, whereas intuitionist analysis talks about intuitionist disjunction which does not obey LEM on the same numbers. Such alternative is not only in itself unnatural, it is also incoherent in view of the well-known fact [Popper (1948)] that, on a shared domain of objects and properties, intuitionist logic would seem to collapse on classical logic: letting ‘orC’, ‘orI’, ‘identicalC’, ‘identicalI’, ‘notC’ and ‘notI’ express classical and intuitionist disjunction, identity and negation respectively, by using the deductive rules appropriate for each logical expression we can derive ‘r is identicalI with 0 orI r is notI identicalI with 0’ from ‘r is identicalC with 0 orC r is notC identicalC with 0’, thereby making a hash of the intuitionist continuum. While the classical-analysis/intuitionist-analysis case is only one example of logical pluralism, I take the overall thrust of this discussion to provide some evidence to the effect that, in those cases where a logic is motivated by the aim of accounting for the specific behaviour of objects and properties in a certain domain (as the specific logics alluded to in sections “Transitivity” and “Contraction” are), plurality of legitimate relations of logical consequence is accommodated by distinguishing domains rather than by multiplying logical operations. In other cases, a logic is motivated instead by the aim of accounting for certain general, domain-unspecific features of logical consequence (as the specific logics alluded to in sections “Reflexivity”, “Monotonicity ” and “Commutativity ” are)—in those cases, plurality of legitimate relations of logical consequence is indeed most plausibly accommodated by multiplying logical operations. In both kinds of cases, contrary to logical pluralism as typically understood in the contemporary debate (as per the next paragraph in the text), plurality of legitimate relations of logical consequence is ultimately accommodated by distinguishing which fully interpreted sentences they apply to.
- 8.
While there is usually a slide in going from its being legitimate to hold that φ does not entail an absurdity to its being legitimate to accept φ (for example, if φ is ‘The number of stars in the universe is odd’), there is no slide here in going from its being legitimate to hold that ‘The Liar sentence is true iff it is not’ does not entail ‘The Earth is flat’ to its being legitimate to accept ‘The Liar sentence is true iff it is not’, since the possibility that doing the former is not legitimate is essentially the only reason for doubting that doing the latter is legitimate.
- 9.
Throughout, by ‘entail’ and its relatives I mean the converse of the relation of logical consequence. By ‘imply’ and its relatives I mean instead an operation expressed by a conditional operator.
- 10.
Given this ultimate aim (and the fact that the abundant literature on GTT has already done a good job in this respect), in this chapter I won’t delve into other very plausible counterexamples, nor into the very obvious fact that, just as GTT is a very natural generalisation of an unduly restrictive notion of logical consequence, there are very natural generalisations of GTT itself in several directions (a fact to be handled with some caution, see fn 26).
- 11.
Although, should a substructural logic prove to be part of the best solution to, for example, the semantic paradoxes, I myself would find it terminologically misguided not to label it ‘legitimate relation of logical consequence’ (surely, whatever logic governs truth deserves to be labelled ‘legitimate relation of logical consequence’!).
- 12.
In the text, I’ll briefly mention what I regard as the best such reasons not to commit to denial of (I)—an issue which obviously lies well beyond the scope of this chapter—but simply to make more vivid how interesting philosophical positions denying (I) would unhelpfully be outlawed by GTT. Analogous comments apply for the other structural properties to be considered below in the text.
- 13.
That is, by relying in a non-deviant way on one’s belief in the premises of the argument and on one’s inference from them to the conclusion: one can perhaps acquire a new justification for believing, say, ‘There are circular arguments whose conclusion I’ve inferred from their premises’ by ‘going through’ the relevant instance of (I), but one would thereby be relying in a clearly deviant way on one’s belief in the premise of the argument and on one’s inference from it to the conclusion. As elsewhere [see e.g. Chisholm (1966), p. 30 for the much discussed case of deviant causation, in whose debate it would ironically seem presupposed that deviance only affects causation and not also rational connection], it is a tricky issue, lying beyond the scope of this chapter, to spell out exactly what such deviance is.
- 14.
Even the crooked simplification argument, say, φ & ψ ⊢ φ passes muster, since one can be told that Al met with Bob and Cate and acquire a new justification for believing that Al met with Bob by inferring it by simplification from what one has been told.
- 15.
This uncontroversial fact would easily be accounted for if, when transmission occurs, the justification for believing the conclusion were simply identical with the justification for believing the premises, for then it would be obvious that the justification for believing the conclusion of (I) cannot be new with respect to the justification for believing its premise. Pace Moruzzi and Zardini (2007), p. 181, that would seem however a simplistic view of what happens when transmission occurs (a mistake for which I assume the sole responsibility!), since, presumably, also the justification one has for inferring the conclusion from the premises is part of the justification one acquires for believing the conclusion. While the ‘mereology of justification’ is still in its infant days, a natural speculation is that nothing having as a part a justification j for believing φ can be a new justification with respect to j for believing φ, given which it still follows that the justification for believing the conclusion of (I) cannot be new with respect to the justification for believing its premise (assuming throughout, extremely plausibly, that one can only acquire a justification for believing the conclusion if one has a justification for believing the premises, and that the latter is then part of the former). Notice that, if the necessary condition for validity in question is the stronger one to the effect that one can use a valid argument in at least some context to acquire a first justification for believing its conclusion, the relevant uncontroversial fact is instead that in no context can one acquire a first justification for believing φ by inferring it by (I) from φ, which is easily accounted for, since it is obvious that the justification for believing the conclusion of (I) cannot be first with respect to the justification for believing its premise [see e.g. Zardini (2014b) for a recent discussion of transmission].
- 16.
Again, even simplification (fn 14) passes muster (if vacuously so), since, arguably, φ & ψ is always true in virtue of the (sometimes non-logical) fact described by φ, the (sometimes non-logical) fact described by ψ and the (always logical) fact that φ and ψ entail φ & ψ.
- 17.
I’m not saying that x-is-true-in-virtue-of-the-truth-of-y is irreflexive (i.e. such that, for every x, it is not the case that x is true in virtue of the truth of x), since it isn’t [Zardini (2018b)].
- 18.
To belabour the point, φ does not preserve the truth of φ in virtue of the meaning of some expressions occurring in it, even if it is crucial for that preservation that, given the specific meanings involved in φ, the very same meanings are involved in φ. Compare: I don’t have the same age as my mother’s only son in virtue of being 38-year old, even if it is crucial for that sameness that, given my specific age, that very same age is had by my mother’s only son.
- 19.
Read (1981,2003) offers a very different defence of the connection between non-monotonic logics and truth preservation by reinterpreting truth preservation in terms of a relevant implication (‘If every premise is true, so is the conclusion’). Such reinterpretation would seem however miles away from the original idea of truth preservation as presented in section “Reflexivity” (whose intelligibility Read does not contest), which is simply about the conclusion being true except the premises are not: that requires a ‘relevance connection’ between the premises being true and the conclusion being true just as little as the fact that an object weighs 1,171,979 grams except it is not now in my right pocket (which is presumably tantamount to every object that is now in my right pocket weighing 1,171,979 grams) requires a ‘relevance connection’ between an object being now in my right pocket and its weighing 1,171,979 grams. Having noted this, Read’s idea of reinterpreting truth preservation in terms of non-classical notions is in my view insightful in the respects I myself will exploit in sections “A Logic-Relative Version of GTT; an Absolute-Truth Formulation of a Logic-Relative Version of GTT” and “From Logical Consequence Back to Logical Truth”.
- 20.
That follows, for example, under the [logical-consequence]-first doctrine, according to which logical truth is grounded in logical consequence as being merely a limit case of logical consequence with no premises and one conclusion. BR seem actually wary of the doctrine (pp. 12–13, despite p. 3), but they do accept at least that logical truths are co-extensional with the single conclusions that follow from no premises, which amply suffices for the implication at issue in the text.
- 21.
And not simply as a necessary truth, for it is crucial for what is the logic of BX (i.e. what follows from what in BX, what is a logical truth in BX, what is a logical falsity in BX, etc.) that LEM have the same status as the other axioms of BX, which it is again crucial for what is the logic of BX that it be the one of logical truth. Analogous comments apply to the next two occurrences in the text of ‘as a logical truth’.
- 22.
Throughout, I understand the elements of a series 〈x0, x1, x2 …〉 to be the members of the set {x0, x1, x2 …}, and, crucially, I understand the premises 〈φ0, φ1, φ2 …〉 to be the members of the set {φ0, φ1, φ2 …}, so that, for example, the only premise in 〈φ, φ〉 is φ (as a consequence of both understandings, I identify the premises 〈φ0, φ1, φ2 …〉 with the elements of that series). The understanding of premises just flagged as crucial would seem mandated by the traditional idea that premises are sentences [which, in general, I myself have defended in Zardini (2018e); BR, pp. 8–12, also consider judgement types and propositions, which however mandate the same understanding of premises]. In connection with non-contractive logics, sometimes [e.g. Girard (1995), p. 2] a different idea is aired to the effect that premises are ‘tokens’ (of some kind or other). I find extant proposals in this direction deeply problematic [see Zardini (2018b) for details], and I am much more attracted to seeing non-contractive logics as being sensitive to the number of occurrences of the same premise (i.e. sentence), so that, for example, the series 〈φ, φ〉 does not represent two tokens of φ, say φ19 and φ79 (whatever these may be!), but the double occurrence of φ. I’ll henceforth presuppose such broad understanding of non-contractive logics. The understanding of course invites the question how, when testing for logical consequence, there can be sensitivity to the number of occurrences if one only considers whether the premises (i.e. certain sentences) are true. My reply begins with the observation that, when, in the metatheory, one considers whether ‘the premises’ φ0, φ1, φ2 … are true, one is not simply considering whether φ0 is true, whether φ1 is true, whether φ2 is true …; one is combining φ0, φ1, φ2 … (to be sure, in that order) and considering whether they are all true—i.e. whether φ0 is true together with φ1 together with φ2 … It is then the mode of such combination that, if one’s metatheory is in a substructural logic, can be sensitive to the number of occurrences (and, as for the logics in section “Commutativity ”, to their order): for example, if φ occurs twice as a premise as in 〈φ, φ〉, when testing for logical consequence one should not consider whether simply φ is true, but whether φ is true together with φ, and one might thereby be using a mode of combination under which the latter question may receive an answer different from the one received by the former question. I’ll make important use of this reply in the broader context of sections “A Logic-Relative Version of GTT; an Absolute-Truth Formulation of a Logic-Relative Version of GTT” and “From Logical Consequence Back to Logical Truth”.
- 23.
Clearly, on this kind of logic of context change, also all the other structural properties considered in this chapter are not valid, but I think that it is more insightful to place it in the non-commutative camp rather than in any other substructural camp.
- 24.
For our purposes, it is important to distinguish between models and tests. Roughly, models are single evaluations of the sentences of the whole language, whereas tests are partial checks for logical consequence looking whether, if the series of premises have the relevant value in the relevant model(s), the conclusion has the relevant value in the relevant model. Importantly, one test might look at different models for different occurrences of the premises or the conclusion. As a baby example, a language might have only two different models, and logical consequence for that language be determined by the tests checking for the fact that it is not the case that the series of premises are true in one model and the conclusion is not true in the other model [see Zardini (2014a) for a more adult example].
- 25.
I should note that the generalisations required by non-monotonic and non-contractive logics would seem in an importantly different ballpark, since their implementation is much less straightforward, and typically requires the adoption, at some level or other, of the operations of, respectively, intensional implication and multiplicative conjunction characteristic of those logics (I’ll myself follow this strategy in sections “A Logic-Relative Version of GTT; an Absolute-Truth Formulation of a Logic-Relative Version of GTT” and “From Logical Consequence Back to Logical Truth”).
- 26.
All this of course does not mean that we should simply replace GTT with some sort of GGTT that allows for all possible generalisations in these directions (or others), since a few such generalisations will yield relations that are obviously not instances of the kind Legitimate Relation of Logical Consequence (but are not ruled out by BRT2 either). In a familiar oscillation, in the attempt at weakening too strong a condition that undergenerates with respect to a target kind, we end up with too weak a condition that overgenerates with respect to the kind. There is no (non-trivial) essence of Legitimate Relation of Logical Consequence.
- 27.
Throughout, bear in mind that, as per regionalism (section “Generalised Tarski’s Thesis”), in some cases such ‘interpretation’ does not involve different logical operations and is more akin to the way in which, say, playing football and playing chess represent two different ‘interpretations’ of playing even though these do not involve different properties playing0 and playing1 exemplified (on the one hand by people and on the other hand) by football and chess, respectively.
- 28.
That is only a very rough gloss and you shouldn’t read too much into it, not the least because the sense in which ‘the collective truth of the series of premises generates the truth of the conclusion’ varies dramatically from logic to logic (recall the last paragraph in the text). Still, the gloss is evocative at least for the specific substructural logics alluded to in Sect. 2, constituting a programmatic slogan congenial to the foundational role I’ll argue in section “From Logical Consequence Back to Logical Truth” these conditionals have. Thanks to Luís Estevinha for feedback on this matter.
- 29.
If the logic in question has more than one operation of universal quantification, implication or conjunction, I assume that there is a most appropriate one for the role these conditionals are supposed to play, and I focus on that one (the assumption is arguably satisfied for all the specific substructural logics alluded to in Sect. 2).
- 30.
To set aside distracting issues concerning the opacity of absolute truth [e.g. Zardini (2015b)], in this discussion I assume that it makes sense to extend the target logics with a quoting singular term ⌜φ⌝ for each sentence φ of their original language and with a truth predicate T such that, for every sentence φ of their original language, T⌜φ⌝ is intersubstitutable with φ. Such extensions are straightforward (indeed, on my view, just as logical as, say, the extension of the conjunction-free fragment of intuitionist logic with conjunction), contrary to those that would be needed to develop a theory of truth-in-a-case (as per the third last paragraph in the text). (Such extensions are also harmless, since they only licence the intersubstitutability of T ⌜φ⌝ with φ if φ is T-free.) Moreover, focusing on languages for which such extensions make sense is justified since, in this discussion, my aim is to defend anti-universalist claims rather than universalist ones. Thanks to José Martínez and Ricardo Santos for feedback on some of these issues.
- 31.
I should really be a bit more precise about what it is for a logic to satisfy the absolute-truth formulation of the logic-relative version of GTT. For our purposes, anticipating a bit, a natural way of making that notion precise is to say that a logic L satisfies the absolute-truth formulation of the logic-relative version of GTT iff [φ0, φ1, φ2 …, φi ⊢L ψ holds iff T ⌜φ0⌝ & T ⌜φ1⌝ & T ⌜φ2⌝ … & T ⌜φi⌝ → T ⌜ψ⌝, as interpreted by L, is a logical truth in L] (where, by fn 30, the last claim is equivalent with the claim that φ0 & φ1 & φ2 … & φi → ψ, as interpreted by L, is a logical truth in L, which makes satisfaction of the absolute-truth formulation of the logic-relative version of GTT by L a matter of L’s conjunction and implication correlating in the familiar ways to premise combination and entailment in L, respectively). Thanks to Ricardo Santos for urging this clarification.
- 32.
Yes, including non-contractive logics, even given what I’ve said in section “A Logic-Relative Version of GTT; an Absolute-Truth Formulation of a Logic-Relative Version of GTT”, since we’re considering a formulation of the logic-relative version of GTT in terms of plain [collective-truth]-generation conditionals, without cases or designated values: while it is the case that, if φ gets a designated value, so does φ ⊗ φ, we do not have that, if φ is true, so is φ ⊗ φ [Zardini (2011)].
- 33.
Throughout, I understand ‘ground’ and its relatives in a suitably broad fashion, so as to include also the case of reduction (as is particularly plausible in the case of the [logical-consequence]-first doctrine).
- 34.
The second main conjunct in its full strength is not guaranteed by the first one, and so, if at all desired, it must be added separately. Still, it represents the arguably most plausible grounding of logical truth in logical consequence, and that’s why, throughout, we’re focussing on it. Notice that ‘most plausible’ doesn’t imply ‘plausible’: following, but from no premises, would seem to make just as little sense as arriving, but at no places! Sometimes, the problem is fudged by invoking the empty set and saying that logical truths are those sentences that follow from the empty set; however, while it is useful formally to model logical consequence as a relation between sets (or, in view of substructurality, series), only an empty-set mystic would think that logical truths are characterised by a distinctive logical relation to the empty set (as opposed to many other more relevant objects). Some other times, the problem is fudged instead by invoking the 0ary operator t and saying that logical truths are those sentences that follow from t; however, while it is useful to introduce t in the formal study of logical consequence (especially in view of substructurality), its informal understanding is as of the ‘conjunction’ of all logical truths, which arguably prevents grounding a sentence being a logical truth in its following from t. Notice that an analogous problem affects the related idea that logical falsity is grounded in logical consequence as being merely a limit case of logical consequence with one premise and no conclusions (whereas, FWIW, the more general notion of inconsistency of a series of premises can still be grounded in logical consequence and logical falsity in terms of the series entailing the ‘disjunction’ of all logical falsities). Thanks to Ricardo Miguel for pressing me on the formulation of the [logical-consequence]-first doctrine.
- 35.
Having noted all this, I’ll (pick up from fn 31 and) henceforth focus on the finite case myself.
- 36.
Even setting aside the problem raised in the text, the extension of these logics with the desired operations typically requires complications going so far beyond the basic, natural framework of the logics as to make the [logical-truth]-first doctrine hardly credible for them. Moreover, the resulting operations are typically so tailor-made to fit the target relation of logical consequence that, given on the one hand the valid arguments of one of these logics and on the other hand its putatively grounding logical truths, the by far most plausible account is that the putatively grounding logical truths are what they are because the valid arguments are what they are rather than vice versa.
- 37.
Since, in arguing against the [logical-truth]-first doctrine, we’ve also touched on some dramatic cases (B3 and K3) which, at least on their standard sentential fragment, have no logical truths whatsoever but still have a wealth of valid rules, it’s just fair to mention that similar dramatic cases of logics [like TS of Cobreros et al. (2012)] which, at least on their standard sentential fragment, have no valid rules whatsoever but still have a wealth of valid metarules argue against the [logical-consequence]-first doctrine [as does the fact that the valid rules of ST of Cobreros et al. (2012) coincide with those of classical logic in spite of ST quite clearly being different from classical logic!]. The point iterates at higher orders. A thoroughly diverse picture of logics thus emerges, on which, for some logics, what is most fundamental are their logical truths (as I’ll argue starting from the next paragraph in the text); for some other logics, their valid rules; for yet some other logics, their valid metarules … Thanks to Bogdan Dicher, Sergi Oms and Lucas Rosenblatt for insisting on these points.
- 38.
A further reinforcement: the condition that the conclusion is true except the premises are not would seem to be all about getting things right—no matter how inelegantly—in going from the premises to the conclusion; that is, just not getting them wrong; that is, not going from premises that are true to a conclusion that is not.
- 39.
Notice that both the truth-preservation account of logical consequence and GTT constitute a very specific version of reliabilism, where the relevant property a valid argument is reliable about is truth (rather than knowledge, unfalsity, truth together with unfalsity, etc.).
- 40.
That is, φ and φ, φ agree that their only premise, φ, is true. The additional strength of φ, φ over φ consists in representing φ as being true also as occurring twice. While that entails φ’s truth, it is not entailed by it. That’s not enough to make the additional strength of φ, φ relevant for the truth-preservation account of logical consequence: for example, the property of being known by me is also such that, while my knowledge of φ entails φ’s truth, it is not entailed by it, but that property is clearly irrelevant for the account. Having made this point, there is indeed an important question as to how to understand the idea that a premise is true ‘as occurring twice’, which I’ve addressed in fn 22.
- 41.
It’s worth noting that, even if preservation is dropped in favour of a relation relating possibly distinct values (say, connection), the immediately resulting account (‘guaranteed connection between the truth-theoretic values of the premises and of the conclusion’) is still all about ‘truth-theoretic’ values such as truth, falsity, unfalsity, etc., whereas the distinction among designated values in the model theory of some non-reflexive and non-transitive logics does not sustain any such interpretation [for example, think of the non-[truth-theoretic] interpretation offered by Zardini (2008), pp. 347–349 of the distinction between two different kinds of designated values relevant for a wide family of non-transitive logics].
- 42.
We can further argue not only that substructural logical consequence is not grounded in truth preservation, and not only that substructural logical consequence is not co-extensional with truth preservation, but also that substructural logical consequence does not even require truth preservation. The best example for this is arguably offered by the non-contractive approach to the semantic paradoxes mentioned in section “Contraction” [especially as further developed in Zardini (2018f)], on which one can warrantedly accept, say, that the Liar sentence λ is true while holding that the argument from λ, λ to ‘Snow is black’ is valid, and so while holding that the premise (which occurs twice, see fn 22) of the argument is true and its conclusion is not. Substructural logical consequence does not require truth preservation. Thanks to Sven Rosenkranz for prompting this expansion of the argument.
- 43.
To be clear, the claim is not that the intended class is not co-extensional with any class defined in purely non-logical terms—not the least because, under very plausible mathematical assumptions, such co-extensionality does obtain in the case of many logics! Especially for such logics, the claim is not that logical consequence is not correlated with individual-truth preservation over a class of tests defined in purely non-logical terms; it is rather that logical consequence is not grounded in such truth preservation, as it is only correlated with it because the class of tests defined in purely non-logical terms just so happens to be co-extensional with the intended class, which is in turn only characterisable as such partly in logical terms [for example, as the class of all and only those tests involving logically possible models, cf Etchemendy (1990), pp. 107–124].
- 44.
This is not to deny that something of heuristic value can be gained by working with the notion of a sentence being true in every logically possible model rather than with the notion of a sentence being logically necessary, just like something of heuristic value can be gained by working with the notion of a sentence being true at every metaphysically possible world rather than with the notion of a sentence being metaphysically necessary: in both cases, quantificational reasoning, based on the well-understood notion of truth-in-a-model or truth-at-a-world, can be easier than modal reasoning. But in neither case is the heuristics plausibly taken as evidence of the grounding of modal facts in quantificational ones concerning models or worlds.
- 45.
Notice that this proposal does not require us to use L (in particular, its tricky conjunction and implication) in our own theory: it’s sentences that are logical truths, and so, on this proposal, we only need to mention—rather than use—the relevant sentences and hence mention, rather than use, the relevant operators and the logic governing them. Instead of employing ourselves the straightjacket of [individual-truth] preservation, we refer to L’s own interpretation and evaluation of [collective-truth] generation. (Don’t complain that this might give us less of a clue as to which arguments are valid in L: the proposal is about what grounds the fact that an argument is valid in L, not how we get to know it.) Having noted this, as will emerge in particular in the next paragraph in the text, the proposal does rely on a canonical specification of the relevant sentences (grounding is sensitive to how the sentences are presented!), so that its understanding does require appreciation of some general semantic features of such sentences.
- 46.
That this philosophical approach to grounding logical consequence for the specific substructural logics alluded to in Sect. 2 is viable is confirmed by technical facts concerning the formal definitions of some of these logics. To give one glaring example (whose obvious import for the foundational issues we’re investigating seems to me to have hitherto been grossly overlooked), as I’ve already in effect remarked in section “Monotonicity”, relevant logics are traditionally presented basically in terms of their logical truths, and, when they are taken to be non-monotonic, their relation of logical consequence is typically defined in terms of φ0, φ1, φ2 …, φi ⊢L ψ holding iff φ0 → (φ1 → (φ2 … → (φi → ψ))) …) is a logical truth in L. To give another glaring example (whose obvious import …), typical non-contractive logics have a semantic presentation employing a certain family of lattices (section “A Logic-Relative Version of GTT; an Absolute-Truth Formulation of a Logic-Relative Version of GTT”): in such presentation, their relation of logical consequence is not defined, as is usual in broadly algebraic treatments of logics, in terms of preservation of designated value from φ0, φ1, φ2 … and φi to ψ in every model (for, as per the argument in section “A Logic-Relative Version of GTT; an Absolute-Truth Formulation of a Logic-Relative Version of GTT”, that definition would give the wrong results), but in terms of φ0 ⊗ φ1 ⊗ φ2 … ⊗ φi → ψ getting designated value in every model (that is, being a logical truth).
- 47.
While this deft movement from use to mention is illuminating, the light it sheds is admittedly somewhat faint. But then there is only so much light to be had at these depths.
- 48.
Don’t say that the fact that φ0, φ1, φ2 …, φi ⊢L ψ holds is grounded instead in the fact that the object-language, conjunction-free conditional φ0 → (φ1 → (φ2 … → (φi → ψ))) …) is a logical truth in L (cf fn 46), so that, in particular, the fact that, in LW, ‘Snow is white’, ‘Grass is green’ entail ‘Snow is white and⊗ grass is green’ is grounded in (the logical necessity of) the fact that, if→ snow is white, then→, if→ grass is green, then→ snow is white and⊗ grass is green. Setting aside the adhocness of such deviation given the route we’ve followed starting from GTT, the deviation stumbles on exactly the same problem raised in the text when grounding the fact that, in LW, by modus ponens, ‘If→ snow is white, then→ grass is green’, ‘Snow is white’ entail ‘Grass is green’, since it grounds it in (the logical necessity of) the tautologous fact that, if→, if→ snow is white, then→ grass is green, then→, if→ snow is white, then→ grass is green. Thanks to José Martínez for comments on some of these issues.
- 49.
Semantic ascent could probably not fulfil this function if alethic deflationism held. Ergo, by modus tollens …
- 50.
In turn, it is plausible that logical truth, even in those logics, is not primitive, and we may expect that its account will appeal to properties of the logical operations used or mentioned in a logical truth (the latter disjunct being particularly relevant when the logical truth involves [collective-truth]-generation conditionals). If so, whether structural properties are valid is grounded in whether certain sentences involving [collective-truth]-generation conditionals are logical truths (see fn 51 for a specific proposal), which is in turn grounded in the properties of logical operations. Substructurality is a logically interesting but philosophically shallow phenomenon caused by logically hidden but philosophically active underlying logical operations. Structure is grounded in operations. Thanks to Ricardo Miguel for pushing me on this.
- 51.
Although a full treatment of the status of metarules (and metametarules, and metametametarules …) lies beyond the scope of this chapter, a natural way of extending the approach we’ve been pursuing to metarules for the specific substructural logics alluded to in Sect. 2 is to say that the fact that a metarule is valid (over and above its being admissible) in L is grounded in the fact that the conditional having as antecedent the conjunction of the series of [collective-truth]-generation conditionals corresponding to the ‘premise rules’ of the metarule and as consequent the [collective-truth]-generation conditional corresponding to the ‘conclusion rule’ of the metarule is a logical truth in L (and then iterate this strategy for metametarules, metametametarules, metametametametarules …). Thanks to Bogdan Dicher and Lucas Rosenblatt for their questions about the status of metarules.
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Zardini, E. (2018). Generalised Tarski’s Thesis Hits Substructure. In: Wyatt, J., Pedersen, N., Kellen, N. (eds) Pluralisms in Truth and Logic. Palgrave Innovations in Philosophy. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-319-98346-2_11
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