Abstract
We investigate a hitherto under-considered avenue of response for the logical pluralist to collapse worries. In particular, we note that standard forms of the collapse arguments seem to require significant order-theoretic assumptions, namely that the collection of admissible logics for the pluralist should be closed under meets and joins. We consider some reasons for rejecting this assumption, noting some prima facie plausible constraints on the class of admissible logics which would lead a pluralist admitting those logics to resist such closure conditions.
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1 Introduction
Various forms of logical pluralism have been objected to on grounds of what have been called collapse arguments, which aim to show that the pluralist is, tacitly or unintentionally, actually committed to some kind of logical monism.Footnote 1 We’ll consider two such, namely upward collapse arguments which seek to show that the pluralist is committed to a single deductively strong logic, and downwardFootnote 2 collapse arguments, which seek to show that the pluralist is committed to a single deductively weak logic.Footnote 3 There has been a fairly substantial literature concerning the collapse arguments, to what extent they have force against various forms of pluralism, and how the pluralist may avoid them. In this paper, we shall argue that such arguments build in certain substantial, and questionable, assumptions about the collections of admissible logics, and their closure conditions. We then argue that the pluralist has ways out of these arguments if they admit collections of logics which satisfy (or, more aptly, fail to satisfy) certain order-theoretic properties.
That is, if we consider the collection of admissible logics as ordered in terms of deductive strength, then the properties of this ordering are salient to assessing the effectiveness of collapse arguments. In particular, it seems to be assumed in various collapse arguments that the collection of admissible logics will include logics which are lattice joins and lattice meets, to which the pluralist is then argued to be really committed. We note that if one does not require that the collection of admissible logics is closed under meets and joins, then the strength of the collapse arguments seems to dissipate.Footnote 4 This leaves the question of why one might not want the collection of admissible logics to be so closed, to which we’ll propose a handful of potential answers. Having made these proposals, we shall consider some lingering questions about the normativity of logic, and about the requirement that all admissible logics be globally applicable.
2 Technical preliminaries
We shall assume that logics are Tarskian consequence relations over a fixed language.Footnote 5 That is, let us fix a set Fm of formulas (constructed, as usual, out of some atomic formulas, connectives, and punctuation), and take a logic L to be a relation \(\vdash _{{\textbf {L}}}\;\subseteq \wp (Fm)\times Fm\) satisfying the following constraints:
-
If \(A\in \Gamma \) then \(\Gamma \vdash _{{\textbf {L}}}A\). Reflexivity
-
If \(\Gamma '\supseteq \Gamma \vdash _{{\textbf {L}}}A\) then \(\Gamma '\vdash _{{\textbf {L}}}A\). Monotonicity
-
If \(\Gamma \vdash _{{\textbf {L}}}A\) and for each \(B\in \Gamma \), \(\Delta \vdash _{{\textbf {L}}}B\), then \(\Delta \vdash _{{\textbf {L}}}A\). Cut
-
If \(\Gamma \vdash _{{\textbf {L}}}A\) then for any \(\sigma :Fm\overset{hom.}{\longrightarrow }Fm\), \(\{\sigma B\mid B\in \Gamma \}\vdash _{{\textbf {L}}}\sigma A\).Footnote 6Structurality
We’ll sometimes use the notation \(\Gamma \vdash _{{\textbf {L}}}\Delta \) to mean that \(\Gamma \vdash _{{\textbf {L}}}A\) holds for every \(A\in \Delta \).
This definition of “logic” does wind up committing us to some presuppositions which are contentious in the literature. For instance, by fixing a language Fm, and assuming that the connectives of Fm have a univocal meaning in all the logics considered, we are skating past difficult issues concerning logical pluralism and meaning change, as have been discussed in Kouri Kissel (2018a), for instance.Footnote 7 Furthermore, by restricting our attention to Tarskian consequence relations, we avoid questions concerning whether substructural logics, presented in a certain way, are legitimate.Footnote 8 We focus on this setup in order to facilitate our argument, though recognise that doing so involves making substantial assumptions that may be brought into question.
The major advantage of this definition of logic is that there are a number of results about Tarskian consequence relations, of a kind not yet proved for alternatives. In particular, it is known that the family of all such logics form a complete lattice.Footnote 9 Logics are ordered by subset:
In which case we’ll say that \({\textbf {L}}_2\) is stronger than L\(_1\) (and L\(_1\) is weaker than L\(_2\)).Footnote 10 This order, as has been discussed in Wójcicki (1988), is a complete lattice in that each set of logics \(\{{\textbf {L}}_i\}_{i\in I}\) has a join \(\underset{i\in I}{\bigvee }{} {\textbf {L}}_i\) and a meet \(\underset{i\in I}{\bigwedge }{} {\textbf {L}}_i\). The meet is defined straightforwardly as set intersection:
but the join is not set union, as this may fail to be a logic (for reasons of transitivity), so we require a more involved definition:
So we take not the join, but rather the least logic including the join — that way, we can be sure that what we have is a logic, while still preserving the inferential resources of all the logics to be joined.
To spell out this formalism a bit, what we require for A to follow from \(\Gamma \) in \(\underset{i\in I}{\bigvee }{} {\textbf {L}}_i\) is that A has to follow from \(\Gamma \) given the inferential resources we have in any of the logics \(\{{\textbf {L}}_i\}_{i\in I}\). One way for this to be the case is if \(\Gamma \vdash _{{\textbf {L}}_i}A\) holds for some such logic, but another way for this to happen is if we have multiple logics where we can reason from \(\Gamma \), via a sequence of collections of lemmas (using different logics among \(\{{\textbf {L}}_i\}_{i\in I}\)), to A. The latter option amounts to chaining together inferences in the “joined” logics.
As an example, neither intuitionistic logic J nor the relevant logic R can prove Peirce’s law as a theorem, i.e. for \({\textbf {L}}\in \{{\textbf {J}},{\textbf {R}}\}\):Footnote 11
However, in intuitionistic logic we have the provability of:
and in R we have:
Thus, we have \(\varnothing \vdash _{{\textbf {J}}\,\vee \,{\textbf {R}}}((A\rightarrow B)\rightarrow A)\rightarrow A\) from chaining these together.Footnote 12
3 Collapse arguments
What Caret (2017) calls collapse arguments are among the most venerable points of opposition logical pluralists have faced. The aim of such arguments is to show that the pluralist’s view collapses into logical monism of some form. A version of this objection, the upward collapse argument, has been pressed by Priest (2001) and, following on him, Read (2006), as well as by Keefe (2014).Footnote 13 A further version, the downward collapse argument, has been proposed by Steinberger (2019a) and Bueno and Shalkowski (2009). The upward collapse arguments are aimed at the conclusion that the pluralist is committed to a relatively strong logic, whereas the downward collapse arguments aim to show that the pluralist is committed to a relatively weak logic. Let’s consider these in some more detail.
3.1 The upward collapse argument
Read (2006), developing on Priest (2001), presses the upward collapse argument in the following terms (we’ve altered the notation to match that we’re using here):
Graham Priest challenges Beall and Restall as follows: suppose there really are two equally good accounts of deductive validity, L\(_1\) and L\(_2\), that B follows from A according to L\(_1\) but not L\(_2\), and we know that A is true. Is B true? Does the truth of B follow (deductively) from the information presented?...The answer seems clear: L\(_1\) trumps L\(_2\). After all, L\(_2\) does not tell us that B is false; it simply fails to tell us whether it is true....It follows that in a very real sense, L\(_1\) and L\(_2\) are not equally good. L\(_1\) answers a crucial question that L\(_2\) does not. For Priest’s question is the central question of logic.
(Read 2006, pp. 194–195)
By “the central question of logic”, Read refers to the question “what follows from what”, which Priest, Read, and, it seems, Beall and Restall (2006) all take to be what logic is about. The challenge rests squarely on this notion of trumping,Footnote 14 and it would seem that strength is the appropriate notion here — in Read’s example, if we were comparing CL and J, the argument would have us conclude that the pluralist is really a monist about CL.
We can, it seems, generalise this argument in the following way to consider more than just two logics (which does not seem out of line with how upward collapse arguments have been discussed in the literature).
- (P1):
-
The central role of logic is to characterise the correct patterns of inference.
- (P2):
-
The strongest logic admitted by the pluralist trumps all the other admitted logics.
- (P3):
-
The central role of logic is thus fulfilled by the strongest admitted logic.
The pluralist is committed to monism about their strongest admitted logic.
This just seeks to precisely set out Read’s argument, while generalising it to consider comparisons between classes of logics. As we’ll mention shortly, we’ve omitted a key assumption, namely, that there is a strongest logic under consideration — there certainly is a logic which is the join of all those under consideration, but this argument requires the assumption that it is admissible. We’ll come back to this shortly, but first let’s consider an argument for collapsing in the opposite direction.
3.2 The downward collapse argument
The upward collapse argument sought to undermine versions of pluralism that assume a form of global application of logics — namely that logics seek to do one thing.Footnote 15 The downward collapse argument goes in the opposite direction, seeking to undermine versions of pluralism which do not (seem to) make this assumption, but which, rather opt for some kind of logical relativism.Footnote 16 One such way to go would be to take different admissible logics to apply to different domains of discourse,Footnote 17 such as having intuitionistic logic as admissible and applying to mathematics or ethics (or some other thing supposed to be constructive) and having classical logic as admissible for applications to middle sized dry goods in the middle distance.Footnote 18
Steinberger (2019a) sets out a form of the downward collapse argument, in analogy with upward collapse arguments:Footnote 19
[T]he natural move here is instead to take the intersection of the logics in question. This is in the spirit of logical modesty: When engaging in cross-domain reasoning, we should draw only on principles sanctioned by all the relevant logics.
(Steinberger 2019a, p. 15)
The key idea seems to be that against the relativist-seeming assumption of multiple domains, we should instead recognise the core importance of universal applicability, and require “capital L” Logic to be applicable to every domain.Footnote 20 Assuming that reasoning in different domains will validate different principles of inference, the one true Logic is just that which validates only those principles valid for use in every domain.Footnote 21
Using this as a starting point, we can construct a collapse argument along something like the following lines:
- (P1):
-
The domain-relative pluralist admits multiple correct logics, each of which applies to a particular domain.
- (P2):
-
The intersection of all the domain-relative pluralist’s admitted logics applies to every domain.
- (P3):
-
Logic is universal, and hence can be appropriately applied to every domain.Footnote 22
The domain-relative pluralist is committed (via (P1)) to logical monism concerning the intersection of their admitted logics.
This, similarly to the upward collapse argument, seems to make the tacit assumption that the intersection of the pluralist’s admissible logics (the meet of all such logics) will itself be admissible.
We can see traces of this kind of consideration motivating Beall’s (2017, 2018, 2020) recent turn to logical monism with FDE as the one true Logic, and other inferential principles one might adopt as being extra-logical, and adopted for particular theoretical purposes. For Beall, any admissible logics will make use of distributive lattice properties for conjunction and disjunction, and at least double negation equivalence and the De Morgan laws for negation — hence, FDE is the “basement level” consequence relation applying everywhere.
3.3 How to escape collapse arguments
There have been a number of attempts by pluralists (and fellow travelersFootnote 23) to respond to collapse arguments. For instance, we’ve cited Caret (2017) who proposes a contextualist solution. There are surely many such solutions available.
Our focus, however, is on the tacit order-theoretic assumptions. In the upward collapse argument, this is the assumption that the join of all the admissible logics will itself be admissible; in the downward collapse argument, it is that the meet of all the admissible logics will itself be admissible. There may be some reasons for the pluralist to want their set of admissible logics to be so closed, but we’ll consider some reasons why they might not want that, and propose that if they don’t, indeed, want that, then they also have a clear-cut path for escaping the force of the collapse arguments.
The solution proposed is a schematic one — we don’t provide any universal reason for rejecting the assumption that the class of admissible logics must form a (join/meet semi-)lattice, but rather note that doing so provides an escape hatch. It is up to the individual pluralist to assess whether they have reasons for adopting such a class of admissible logics, and, having done so, thumb their nose at the would-be collapser.
4 Why not have a join-semilattice of admissible logics?
In this section we’ll consider two sorts of reasons one might have for resisting admitting the join of one’s admissible logics. Both rely on the fact that the join of a class of logics will, if not itself admissible, be inferentially stronger than all of one’s admissible logics. Thus, if one is pressed to require their admissible logics not to be too strong, then one has reason to push back against this pressure.
4.1 Avoiding triviality
The first reason hearkens back to the by now well-worn cautionFootnote 24 that too much strength is a bad thing, at least when it results in one being committed to the trivial logic:
Clearly, commitment to this logic has a number of undesirable results. For one thing, as a tool for studying good inference, and distinguishing this from bad inference, Triv is more or less useless — according to it, every inference is good.Footnote 25 Furthermore, if one requires something like closure of commitments under this logic (in a manner we’ll discuss further later, according to which one ought (defeasibly) not to accept all the premises and reject the conclusion of a valid argument), then one is stuck in the tricky position of either having to accept everything, or reject everything. This is not an enviable epistemic position in which to find oneself. Even formally speaking, Triv is little more than a convoluted way of studying the language Fm.
It is not too difficult to find examples of seriously proposed logics in the literature which join up to Triv. Let’s consider two such recipes.
4.1.1 Classical logic and a contraclassical logic
Classical logic CL is famously Post-complete: any proper extension of CL (in the same language) will be trivial. That is, if a logic L is such that \({\textbf {CL}}\lneq {\textbf {L}}\) then \({\textbf {L}}={\textbf {Triv}}\). A logic L is contraclassical in the “superficial” sense of Humberstone (2000) just in case L is not a sublogic of CL (i.e. \({\textbf {L}}\nleq {\textbf {CL}}\)). It follows immediately that for any contraclassical logic L, \({\textbf {L}}\vee {\textbf {CL}}={\textbf {Triv}}\), and so if one’s class of admissible logics includes any such pair, then their join will be Triv.
Classical logic is usually taken for granted as admissible. Indeed, it seems to be the default admissible logic (the one we use when “nothing goes wrong”). So we guess the pluralist who admits classical logic won’t have too much work to do in justifying this (perhaps unfortunate) choice. The partisan of a contraclassical logic will have a much harder time — most forms of logical pluralism seem to consider only subclassical logics. Indeed, Beall and Restall (2006) seem to do so in response to Read’s (2006) objection that, for upward collapse reasons, adopting a contraclassical logic will force them to admit Triv.
There are, however, a few fairly well-developed contraclassical logics (at least by the standard of “well-developedness” appropriate for non-classical (or “deviant”, if you follow Haack’s (1996) nomenclature, to apply to only those logics which invalidate some classically valid inference forms) logics beyond intuitionistic logic). Some salient examples include connexive logic (Wansing , 2022; Francez , 2021) and Abelian logic (Meyer & Slaney , 1989; Meyer & Slaney , 2002). The former, at least, has been put forward philosophical purposes (some of which are discussed in Wansing (2022)), and so it is conceivable that there may be some logician who wants to adopt both classical logic and some connexive logic as admissible. Any such logician (perhaps working in the vicinity of Bochum, Germany) will have the grist of a response to collapse worries.
4.1.2 An explosive logic and an inconsistent logic
For our purposes, an explosive logic L is one in which (assuming the language includes a negation connective \(\lnot \)) the argument form \(A,\lnot A\vdash _{{\textbf {L}}} B\) is valid.Footnote 26 Futhermore, we’ll call a logic L inconsistentFootnote 27 just in case there is some \(A\in Fm\) such that \(\varnothing \vdash _{{\textbf {L}}}A\) and \(\varnothing \vdash _{{\textbf {L}}}\lnot A\).Footnote 28
It is straightforward that if L\(_1\) is an explosive logic and L\(_2\) is an inconsistent logic then \({\textbf {L}}_1\vee {\textbf {L}}_2={\textbf {Triv}}\), for note the following reasoning:
and Structurality ensures that for any \(\Gamma \cup \{A\}\subseteq Fm\), \(\Gamma \vdash _{{\textbf {L}}_1\,\vee \,{\textbf {L}}_2}A\), and thus \({\textbf {L}}_1\vee {\textbf {L}}_2={\textbf {Triv}}\).
Now, while not all connexive logics are inconsistent, some, like Wansing’s C (Niki & Wansing , n.a.) are. Abelian logic (Meyer & Slaney , 1989) is also an inconsistent logic, as are connexive extensions of relevant logics (Mortensen , 1984). Furthermore, there are interesting subtleties concerning how the inconsistencies of these logics spread (or don’t spread) to premise sets, as has been investigated in Mangraviti & Tedder (n.a.). So it is, again, not inconceivable that one may desire to admit some inconsistent logic. Furthermore, many of the most popular logics on the scene are explosive (including CL, of course, but also intuitionist logic, fuzzy logics, and so on). So, as with CL, the pluralist seeking to admit an explosive logic will probably not run into too much in the way of challenges to their admission.
4.2 Preserving systematic properties
Concern with triviality is, however, just one reason to oppose admitting logics which are too strong. Another reason to be skeptical of such admission would be a desire that all admissible logics satisfy some systematic properties which limits the joint validities of these systems. Let’s consider some examples.
4.2.1 Multiple routes to paraconsistency
In trying to avoid explosion, while preserving as much else of classical logic as possible, there are a few natural avenues one will land on. We can delineate some such options by considering a version of Lewis’ (1932) “proof” of explosion:
The two most popular options are to reject the inference rule (\(\star \)), leading to such non-adjunctive paraconsistent logics as Schotch et al. (2009). Another popular option is to reject disjunctive syllogism \((A\vee B)\wedge \lnot A\vdash B\), obtaining paraconsistent logics such as LP (Priest , 1979), and the usual relevant logics (Anderson & Belnap , 1975).
As the proof shows, any logic including all these inferences will be explosive. So if one is a particularly hardcore paraconsistentist, who holds that every admissible logic should be non-explosive, then if one adopts the non-adjunctive strategy for some purposes (such as reasoning about combining sets of beliefs or commitments) and another for some other purposes (such as reasoning about true contradictions, assuming there are any), then one has good reason to reject join closure.
4.2.2 Preserving relevance
The most common way of formalise the notion of relevance in relevant logics is in terms of the variable sharing property (VSP), namely: a logic L has VSP just in case \(\varnothing \vdash _{{\textbf {L}}}A\rightarrow B\) holds only if there is an atomic formula in common between A and B. While there are other ways of formalising the notion of relevance, this is the standard: it assumes that the implication connective \(\rightarrow \) captures the real notion of entailment, and that Tarskian consequence relations are, at best, an instrument for investigating logics.
Probably the most famous relevant logic (FDE aside) is the logic R, discussed earlier. A proof that R has VSP is one of the earliest results in the field, but there are a number of other systems, including some logics which are not subsystems of R, which also have VSP. One salient example is the logic TM, which comprises the extension of the logic T of ticket entailment, discussed in Anderson and Belnap (1975, Ch. 5) by the so-called mingle axiom:
In the extension of R by this axiom, obtaining the system RM, we can quickly obtain results which injure VSP, such as the following theorem:Footnote 29
So TM is not a subsystem of R, but it has been shown, in Méndez et al. (2012), that TM has VSP. So it is compatible with being a relevantist pluralist to admit both TM and R, but admitting their join is to admit the irrelevant system RM.Footnote 30
4.2.3 Preserving the disjunction property
Another property that many have taken seriously, this time for reasons of constructivity, is the disjunction property, enjoyed by a logic L just in case whenever \(\varnothing \vdash _{{\textbf {L}}}A\vee B\) then either \(\varnothing \vdash _{{\textbf {L}}}A\) or \(\varnothing \vdash _{{\textbf {L}}}B\). This is a feature of intuitionistic logic J (a proof can be found in Chagrov & Zakharyaschev 1997, §2.8), but it also holds of the contraction-free relevant logic RW, as was proved in Slaney (1984). So a pluralist of a certain constructivist stripe may well have grounds to admit both of these logics, while rejecting their join \({\textbf {CL}}={\textbf {J}}\vee {\textbf {RW}}\) which, famously, lacks the disjunction property.
These are some examples of systematic properties close to (at least one of the) authors’ heart(s), but there are likely to be others which similarly militate against admitting the join of a collection of admitted logics. This provides us a couple of avenues for the logic-join resister, so let us briefly turn our attention to meets.
5 Why not have a meet-semilattice of admissible logics?
In a sense, the considerations in favour of resisting closing the set of admissible logics under meet are dual to those for joins. In this case, the limiting logic is not Triv but rather the logic of conclusion:
In the Tarskian setting, this is the minimum logic, admitting as valid only those inferences which literally just recapitulate a premise in the conclusion. While not as obviously undesirable as Triv, there are reasons to be skeptical of Incl. For one thing, under almost any way of reading the phrase “normative guidance” (a range of such options are surveyed in MacFarlane (n.a.)), Incl provides no normative guidance. What guidance it provides boils down, more or less, to the rule don’t accept and reject the same proposition, which is likely to be, at best, otiose (depending on your views about commitment and acceptance/rejection).
There are, however, some salient differences. In particular, it seems to require more radical differences among one’s set of admissible logics in order for them to meet to Incl than for them to join to Triv. In order for this to happen, one must already admit some logics which have no substantive validities in common, so to speak. Given that there is fairly widespread agreement on the inferential behaviour of at least some connectives (most notably conjunction), this seems like an unlikely scenario.
Having said this, however, it seems that there are some reasons one might not want to admit the meet of one’s admissible logics. One such reason is that the meet might be too weak to fulfill one’s goals; another is that one might take the various applications of one’s logics to be incongruent to such an extent that one should not expect there to be any overlap. Let’s consider some variations on these themes arising in the literature, and how they put pressure on meet closure.
5.1 Not meeting the goals of logic
If one admits logics on the grounds of their ability to facilitate some goal, then it may well be that rather different logics can do so successfully, while their meet may not. There is a considerable number of logical pluralists putting forward such a goal-oriented view of logic, each of which possibly lends itself to endorse a certain collection of logics but not admit their meet, as shown by the following examples.Footnote 31
5.1.1 Logical eclecticism
Consider Shapiro’s (2014) logical eclecticism, according to which logics are to be admitted on the basis of their capacity to provide for formal frameworks for fruitful mathematical theories. On the grounds of this, Shapiro admits CL and J, but leaves the door open that many other logics may also work for similar aims.
On one hand, it seems that one could build logics in a way quite different from CL or J, so long as one adapted one’s mathematical axioms to suit, and still obtain interesting mathematical theories, where the logic allows one to draw significant conclusions beyond those assumed in the axiomatic basis. On the other hand, however, it seems that if one’s logic is too weak, then one will be unable to draw much of anything that isn’t already assumed (Incl is the limit case of this tendency for weak logics). For example, a pluralism that admitted sufficiently different logics would find a rather weak meet: for instance, the meet of quantum logic (for details, see the classic Birkhoff & von Neumann (1936) or Goldblatt (1974)), intuitionistic logic, and some weak paraconsistent or relevant logic such as FDE, would lead one to a rather minimal extension of lattice logic, by a negation and conditional which obeys very few properties. While not quite the minimal Tarskian consequence relation, this would still be extremely weak.Footnote 32
This provides at least an indication that taking meets may not provide the basis for interesting mathematical theorising. For instance, such considerations have caused some consternation in the area of relevant and inconsistent formal mathematics, where it is a common refrain that very weak logics provide little to no inferential power to draw significant consequences not already apparent in the axioms.Footnote 33
Another broader form of elective pluralism was developed by Teresa Kouri Kissel (2018b, 2019). Her approach generalises Shapiro’s pluralism to account for deductive goals other than formalising mathematics. This generalising move makes it, if anything, even less plausible that the meet of a collection of logics will itself be suitable for some goal. If we broaden the kinds of goals we can use logics for, it makes it, prima facie, less likely that there will be substantial overlap between the logics used to satisfy those different goals.
5.1.2 Telic pluralism
Leon Commandeur (2022) has suggested that logic might not only meaningfully serve one (exceptional) goal, but is adequately described as fulfilling a plurality of goals resulting in his “Telic Pluralism”. This view on logic straightforwardly results in a logical pluralism when different admissible goals are attained by using different logics. The examples Commandeur uses to motivate his position suggest that this is the case. Some such goals are: providing a formalisation of logical consequence in natural languages, capturing the structure of mind- and language-independent reality, or model information-flow.Footnote 34 It is reasonable to assume that these goals are attainable only by using different logics. However, taking the meet of all so admitted logics will most likely not be of instrumental use to attain any of said goals. This provides the telic pluralist with a reason to admit a plurality of different logics while rejecting to endorse their meet.
5.2 Pespectival pluralism
The second possibility alluded to above is that the things to which we apply different logics might be so different that one would expect, prima facie, that there is no significant overlap.
Recently, Roy Cook (2023) aimed to fill a lacuna in the landscape of logical pluralism by promoting the idea of different perspectives being the source of plurality in logics. The general idea is that, accepting insights from standpoint epistemology, epistemic subjects might reason in different, but in the context of their respective perspectives correct, ways. Standpoints are usually taken to be influenced by certain social and political circumstances, but the general idea of a plurality of epistemic perspectives is sufficient for our purpose and does not depend on any one way of accounting for standpoints. If the difference in perspectives can be taken to also affect the logical parts of reasoning, then pluralism about logic is correct, assuming a close tie between logic and reasoning. It is, however, easy to see that the meet of all so admitted logics will only by chance correspond to the logic used by a group of epistemic subjects from their specific standpoint. Thus, also perspectival pluralism provides reasons for rejecting the meet of all admitted logics.Footnote 35
5.3 The difference between rejecting joins and meets
So far the provided reasons for resisting meet closure are, in a sense, less concrete than the kinds of considerations we bring to bear in the case of rejecting join closure. Nonetheless it seems that there is the grist here for a case to be made that meet closure need not be required of the admissible logics of every kind of logical pluralist.
Having said this, we can also find some systematic properties not preserved under meets, and thus obtain some concrete examples as in the join case. In general, this will turn not on what becomes provable, as in the case of taking a join, but rather with what becomes unprovable when we take the meet. This is because, dual to the join case, if an inference form is invalid in any of the meet-ed logics, then it will be invalid in the meet, which is where we can run into the sorts of problems with weakness mentioned above. This can, however, break some desired properties. We’ll just go into one example in detail, concerning our old friend the disjunction property.Footnote 36
5.3.1 Example: The disjunction property again
We saw above how taking the join of logics satisfying the disjunction property might result in a logic which doesn’t enjoy this property, and taking the meet of some such logics can also have this result. For example, take \({\textbf {RW}}\wedge {\textbf {J}}\), as above. Note then if \({\textbf {L}}\in \{{\textbf {RW}},{\textbf {J}}\}\):
because we have:
and in both of these systems:
Yet, clearly:
This is an eminently generalisable failure, as it will hold whenever we take the meet of a collection of independent logics all of which validate the inference rule of disjunction-introduction. So if one is a pluralist who insists on the disjunction property, then one also has as much reason to be suspicious of meet closure as of join-closure.Footnote 37
6 A lingering thought about normativity
We’ve set out our main point — that there is a, to our knowledge, unexplored avenue of response for pluralists against collapse objections, and furthermore that this avenue may well be desirable for pluralists of a certain stripe. Admittedly, this stripe may well be uncommon, and perhaps uninhabited, nonetheless it seems to provide a live option. Furthermore, it points to a lacuna in discussions surrounding this issue. Before concluding let’s consider one remaining point, and a potential problem, to which we were led by the preceding discussion.
In our treatment of the upward collapse argument, we’ve noted a straightforward formal way around the problem, but this may seem quite unsatisfactory to the would-be collapse-pusher. Indeed, there is something to the intuitive appeal of the argument which we have not addressed. In particular, it seems that there is a case to be made that if one admits a collection of logics, then in endorsing all of them one is granting a salient claim about normativity: namely, that one will not go wrong in reasoning in accordance with each of these logics. If one further admits that all the logics ought to apply globally, then it seems one is committed to the claim that one can’t ever go wrong in any context by reasoning in accordance with any of the admitted logics.
It seems to us that this intuition is trading on a plausible account of the normativity of logic: namely, if one admits the logic L and \(\Gamma \vdash _{{\textbf {L}}}A\), then one ought (defeasibly) not to accept all of \(\Gamma \) while (simultaneously) rejecting A. If we take this as the salient kind of normativity of logic, then there is a sense in which one can never go wrong by reasoning in the join of all one’s admitted logics. To do so just requires that one chain together inferences in one’s admitted logics. Furthermore, it seems that if one cannot go wrong with any individual step, then one cannot go wrong in chaining the steps together.Footnote 38 So on this kind of picture of the normativity of logic, one cannot go wrong by using the join of some admitted logics (as doing so either involves using one of the admitted logics or using just such a chaining together of inferences from admitted logics) and so one has little ground to reject its admission. This seems to put pressure on the pluralist who wants to avail themselves of our suggestion of resisting the join-closure of the collection of admissible logics.
When discussing the normativity of logic, there are some salient distinctions to draw, which we’ll draw following (loosely) on the heels of Steinberger (2019b). The above charactersation of normativity seems to best fit the third-person appraisal kind discussed in Steinberger, but there is another which seems to be salient, especially when considering the kinds of proposals given in Section 4.2. The constraints considered there, especially as concerns relevance, can be seen as involving directives about how (not) to reason. This sort of directive normativity has a more first-person flavour, embodying rules salient for how an agent goes about drawing inferences, not about how their inferences, or indeed their total set of commitments, are assessed from the outside.
According to this distinction, there seems to be a way to explain the salient intuition. One reason why we might want to rule out such chaining, even while admitting something like global applicability of all the admissable logics, is that while one can’t get into trouble of the commitment-position-incoherence sort, nonetheless one ought not reason that way. According to the relevantist pluralist, while irrelevant reasoning may not wind up with you doing some bad accepting and rejecting, nonetheless the reasoning itself is bad qua being irrelevant.
This solution (or at least explanation) may share some affinity with some recent proposals to solve the collapse problem such as proposed by Blake-Turner (2021) or Tajer (n.a.). The general idea behind both of theirs and our solution is to attribute different normative roles to different logics. This avoids the “trumping”-mechanism described in Section 3.1, since the different kinds of normativity can still be thought of as trumping each other in a practical sense, but no longer render any of the logics normatively idle. Following Tajer, we could assign different bridge-principles to the logics endorsed by the relevantist pluralist and their non-relevant join. In the proposed scenario, the explicitly endorsed logics encompass a normativity close to MacFarlane’s (n.a.) principle Wo+, while their join only exhibits something close to Wp+.Footnote 39
So while the intuition that reasoning in the join of a collection of logics each of which one admits won’t get one into trouble, nonetheless there can be a principled reason to be skeptical of such reasoning.
7 Concluding remarks
This paper has been aimed at problematising key order-theoretic assumptions which are (often tacitly) employed in collapse arguments. Our goal is not to argue directly against the monist, or even for a specific form of pluralism, but rather to intervene in the debate, arguing that these assumptions need an explicit defense if the collapse arguments are to carry weight. Even if there were no actual pluralists who adopted a class of logics appropriate to avoid both directions of the collapse argument, according to our lights, our points will still, hopefully, be helpful as providing some important points to note in adjudicating collapse debates in the future.
There is a great deal more to be said here, including, perhaps, some reasons for adopting the kinds of pluralism we have suggested here which go beyond avoiding collapse problems. Our lingering last comments on normativity also beg for more spelling out. We’ll not attempt to do either of these things here, but note these as potentially fruitful avenues of research.
Notes
The phrase “collapse argument” has been used by Caret (2017) to unify a broad collection of objections. We’ll focus on a couple of variants, but won’t try to be exhaustive in covering the variations available.
We adopt the terminology of collapsing “upward” and “downward” from Steinberger (2019a).
Words like “strong” and “weak” as applied to logic are used in various ways, and there are often unhelpful ambiguities. The way we use the terms has it that a logic L\(_1\) is stronger than L\(_2\) just in case the former validates all the arguments of the latter (and, perhaps, then some).
A similar critical point has been made by Timo Meier though his reaction to it is very different from ours Meier (n.a.).
For the sake of simplicity, we concern ourselves just with propositional logics.
The notation “\(\sigma :Fm\overset{hom.}{\longrightarrow }\ Fm\) indicates that \(\sigma \) is a homomorphism: that is, for each n-ary logical connective \(\otimes \), \(\sigma \otimes (A_1,\dots , A_n)=\otimes (\sigma A_1,\dots , \sigma A_n)\). That is to say, \(\sigma \) is any substitution of formulas for formulas which preserves logical form. So what this constraint enforces is that logical consequence is a matter of logical form, and doesn’t depend on the particular formulas mentioned. Further details on algebraic logic can be found in Font (2016) and Cintula & Noguera (2021), and we’ll assume a familiarity with some such basic notions. We’ll employ the standard convention of writing \(A_1,\dots , A_n\vdash B\) in place of \(\{A_1,\dots , A_n\}\vdash B\).
Issues surrounding whether the same connective means different things in different logics may arise in our setting (though cross-language comparisons will not), but we set those issues aside as tangential for our purposes.
In order to accommodate our style of argument beyond the Tarskian setting, we’d need a bit more investigation into other frameworks for consequence relations, which are starting to come up for investigation, for instance in Cintula & Paoli (2021) and Badia et al. (n.a.). We encourage the development of such projects, and expect that a straightforward variation on our work here will transfer there.
We’ll proceed assuming a basic familiarity with lattice and order theory. All of our results can be found in Davey & Priestley (2002).
From here on in, we’ll use expressions like \({\textbf {L}}\) and \(\vdash _{{\textbf {L}}}\) more or less interchangeably.
Hence, note, that \({\textbf {J}}\,\vee \,{\textbf {R}}={\textbf {CL}}\), where CL is classical logic, as both \({\textbf {J}},{\textbf {R}}\lneq {\textbf {CL}}\) and the axiomatic extension of J by \(((A\rightarrow B)\rightarrow A)\rightarrow A\) (or \(\lnot \lnot A\rightarrow A\)) is CL. Note that this example serves to illustrate the reason why the join of a set of logics is not just the union.
A formulation of the general problem can already be found in Williamson (1988).
The idea that a stronger logic trumps a weaker logic in cases of disagreement depends on the assumptions that both logics exhibit the same kind of normativity for thought and that invalidity-claims don’t provide normative guidance at all. Both assumptions have recently been called into question in Blake-Turner (2021) and Evershed (2021) respectively. We will set those issues for now, but we will discuss a related idea in Section 6.
The global scope of logical consequence is thus also mentioned as a premise in the generalised version of the argument found in Stei Rivalry (2020).
For a general formulation of logical relativism see Shapiro (2014, pp. 7–8).
Haack calls this “local pluralism” (Haack 1978, p. 223).
A related point has been made by Priest (2006, §10.14) in response to the question of whether logic is global or local. He, however, does not push this as an argument against the pluralist, but rather as reason to think that the dispute between the logical globalist and the logical “localist” is not significant. Questions about whether disagreements over logic are meaningful are rather common, but getting into that topic would take us too far afield. A version of a downward collapse argument that is developed in analogy to the problem of mixed inferences for alethic pluralists can be found in Keefe (2018, p. 441).
Bueno and Shalkowski develop a version of the downward collapse argument. They argue that Beall and Restall’s conceptualisation of necessity as enjoining satisfaction preservation in all cases naturally leads to the universality of logical consequence. They describe the downward collapse of Beall and Restall’s pluralism in the following way: “Only a very weak consequence relation survives this scrutiny, according to their accounting of the necessity constraint as quantification over all cases. This relation is just the intersection of the inferences treated as valid by classical, constructive, and paraconsistent logics. Some fragments of positive logic and some rules of identity will survive.” (Bueno and Shalkowski 2009, p. 299).
That logic is general/universal/topic-neutral/etc. is oftentimes seen as part of the traditional and predominant view of logic. So, e.g., Martin and Hjortland (citing Frege and Kant) claim this to be one of “Logic’s special properties” that the Anti-exceptionalists reject (Martin & Hjortland 2022, pp. 8–9). Apart from such more traditional views on logic, the universality of logic is also of importance in Russell’s argument for nihlism in Russell (2018).
For logical pluralists who consider not differing domains but rather something else, like differing subjects, please feel free to make the necessary substitutions and assess the argument having done so. Strictly speaking, it seems that the logical relativist must (in our understanding and following Shapiro (2014)) always take logic to be relative to some domain. This should then also count for the universal logic. This means that the universal logic is also relative, but to a special domain: the absolute domain or the domain of all domains. Related discussion can be found in Rayo & Uzquiano (2006). It’s not clear what difference, if any, this consideration makes for our point, so we won’t pursue it further.
We are included at least among these, though whether we are actually pluralists or not, and if so what kind we are, is a question whose answer may differ according to the context of the asking.
One upshot of this is that it doesn’t allow us to distinguish logically valid from other good forms of argument, such as strong inductive or abductive arguments. Thanks to Anna-Sara Malmgren for noting this point.
One might prefer to widen this definition to call a logic explosive if it validates arguments like the following, making use of some different vocabulary:
$$\begin{aligned} \lnot A&\vdash A\rightarrow B\\ \varnothing&\vdash (\lnot A\wedge A)\rightarrow B\\ \varnothing&\vdash \lnot A\rightarrow (A\rightarrow B) \end{aligned}$$but we’ll stick to the simple presentation for now.
Note that we use the phrase “inconsistent logic” differently than some in the literature — such as Font (2016) — who use this phrase to pick out what we have called the trivial logic Triv. Our terminology follows Wansing (n.a.), Niki & Wansing (n.a.), and Mangraviti & Tedder (n.a.), who study logics with inconsistent sets of theorems.
As with “explosive logic” there are a handful of different options for cashing out this notion — such as requiring that there be some A such that \(\varnothing \vdash _{{\textbf {L}}}A\wedge \lnot A\) — but we stick with this simple presentation.
Salient information of RM can be found in Anderson and Belnap (Anderson & Belnap 1975, §29.5).
If one decides that the global scope of logic is in fact a necessary criterion for something to be a capital “L” Logic, then our argument will lead to nihilism rather than pluralism. Starting from the point of logical relativism to begin with, a pluralism that doesn’t admit the meet of its accepted logics to itself be admissible seems (at least to us) like the more natural solution. Thanks to Teresa Kouri Kissel for pushing us on that point.
A related point has been made by Bueno and Shalkowski (2009, pp. 299–300), related to the passage we cited earlier.
For salient discussion, see Weber (2021), where a relevant implication connective is combined with an irrelevant, but still non-contractive, implication in order to draw significant consequences from a naïve theory of sets.
Note that only the case of eclectic pluralism really depends on the meet of all admitted logics being deductively weaker, while the other two examples provide reasons independent of logical strength. Thanks to Franci Mangraviti for pointing this out and suggesting perspectival pluralism as a reason for rejecting meets.
An example which involves a bit more technical complexity concerns whether a logic has finite characteristic matrix or is tabular — information on matrix semantics can be found, e.g., in Font (2016, Ch. 4). The meet of a collection of tabular logics need not be tabular. As an example, consider the class of logics induced by some finite Heyting algebra. Because J has the finite model property (see Ono 2019, §7.4), it is the meet of all such logics, and yet, famously, J does not have a finite characteristic matrix.
Thanks to Shay Logan for suggesting this point.
There is a related discussion surrounding the rule of Cut in non-transitive logics (for instance in Ripley (2013)), but we set this issue aside for now.
Tajer in his Tajer (n.a.) avoids the upward collapse by pairing up stronger logics with weaker bridge-principles and vice versa. It is interesting to note the affinity to our proposal here for the kind of normativity one might contravene by inferring “in the join”, so to speak, and suggests a potential avenue for further comparison which we’ll not pursue here.
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Acknowledgements
We’d like to thank Jack Himelright, Teresa Kouri Kissel, Shay Logan, Franci Mangraviti, Blane Worley, and Joseph Zalewski for comments on drafts, as well as two anonymous referees for the journal, and audiences in Bochum, Bergen, Lillehammer and Teplá for helpful discussion. Tedder-specific acknowledgements: As my doktorvater, Jc Beall, and his work, have had a big influence on me, and so I’m happy that this paper can appear in this special issue.
Funding
Tedder received fellowship funding from the Alexander von Humboldt foundation. Open Access funding enabled and organized by Projekt DEAL.
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Gooßens, D., Tedder, A. No cause for collapse. AJPH 2, 37 (2023). https://doi.org/10.1007/s44204-023-00083-1
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DOI: https://doi.org/10.1007/s44204-023-00083-1