Logical pluralism
In this essay, the first one that I post in this blog which is in English, I consider one controversial issue in the field of Philosophical Logic. Some of recent discussions in this field are focused in the possibility of a logical pluralism, a philosophical thesis that concerns, as opposed to logical monism, the coexistence of several logical theories. So, this essay is a reconstruction of Field’s picture and argumentation in relation to Logical Pluralism.
1. Nature and types of pluralism
Field’s analysis of pluralism starts noting those general aspects that can be associated to it. Thus, he asserts that “they [pluralists] hold (i) that there is no uniquely correct logic, and (ii) that because of this, some or all debates about logics are illusory, or need to be somehow reconceived as not straightforwardly factual” [1]. Once these common aspects are established, differences between different proposals of pluralism arise by the argumentation offered against monism. And these differences, as it will be seen, will allow a dual evaluation of each considered account, that is, by its correction or its interest (the fact that a particular pluralism could be a controversial and substantive proposal) as a theory.
So, the accession to pluralism implies, apart from the acceptance of several valid logics as a fundamental thesis, the dissolution of a genuine rivalry between them. Not in vain, this rivalry leads to a defense of monism, because the inconsistency between content of two theories, at least with respect to theorems, can cause that a position which accepts both as valid doesn’t be sustainable. It is interesting to note that this is one motivation that leads Field to consider that, in fact, the possible horizon to develop pluralism is very reduced, almost in this limited context of debate. Nevertheless, its necessary to localize and develop previously different concretions of the two theses shown above.
1.1. Naïve Pluralism
The first form of pluralism that Field localizes [2] is the approach that defenses the arbitrariness in the demarcation of what can be considered logic. In other words, what is the matter is to reject by principle the distinction between proper logic concepts and nonlogic concepts and, accordingly, that leads to dissolve differences between logic truths and the rest of them. The discussion that this approach raises can drive to debate, between several formal proposals, which of them is really logic; the acceptance of all those proposals as logics, without further distinction will be that particular pluralism, called here naïve pluralism for clarity. This form of pluralism has been stated, by Beall and Restall, as a non interesting pluralism [3].
Although Field proposes the possibility of developing a controversial aspect about epistemological relevance, privileged or not, that logic can have regarding all other content subject to knowledge, he does not raise this possibility of discussion. His veredict of naïve pluralism coincides with Beall and Restall’s: “In general, then, I’m inclined to agree with pluralism based on the arbitrariness of the demarcation between logic and nonlogic. That kind of pluralism doesn’t strike me as altogether exciting” [4].
1.2. Radical pluralism
The way of eliminating the debate between logics as avoiding to fix the demarcation of their field is not interesting. So, as a task for pluralism, it is necessary to consider another option, that is, to dissolve genuine rivalry. Field considers radical pluralism; its field is applied logic and, precisely the main thesis of that pluralism is that there is no substantive debate between logics:
“[I]n debating the interesting form of pluralism that started this section [radical pluralism], one needs to focus on nonstandard logicians who take their preferred logic as an all-purpose logic. The radical form of pluralism (…) is that there is no genuine debate between advocates of different all-purpose logics” [5]
The justification of radical pluralism is based upon differences of meaning that are established between the content of different theories. Once an application is fixed, if every logic develops a different meaning to the objects that they consider, then if there is chance to rivalry, that rivalry is no properly substantive. Two forms of difference in meaning can be distinguished, and Field has a reply for both [6]. The fact is that Field recognizes that radical pluralism is an interesting approach, and he therefore develops an argumentation to refute it; that is the reason that leads him to conclude that radical pluralism is false.
One difference of meaning has to be previously localized; it is established when two theories have alternative explanations about one formal device, as can be a logical connective, in the sense that the result is not a difference in meaning concerned with one single object, but with two different objects, and each of them is part of each theory that studies it. In that case, pluralism is not interesting, because it simply confirms that two different logics consider different languages, each with their respective connectives; the lack of rivalry here is trivial. Field considers the example of a pluralism that recognizes that there are several senses by which one can understand validity. The result is that different theories do not dispute between them, simply, because they do not consider the same notion of implication. This particular pluralism is not rival of any monism and, therefore, it can not be substantive.
When there is a common object belonging to different theories, and each gives a different definition, this difference of meaning is radical, and has to be distinguished from the previous one. In this case, the translation between theories is not homophonic. There can be several pure theories, as intuitionism and classical logic, concerned with, say, what is the proper meaning of negation. But, given that this would be a case of radical difference of meaning in which there are two different connectives, an interesting pluralism does not follow.
An alternative to that radical pluralism occurs when two theories (or more) have to speak about the same, that is, one single negation with one genuine meaning, which is accounted for in different ways. Field contends that, in this case, genuine rivalry actually disappears, and therefore a pluralism can be developed. So, there is an alternative for radical pluralism of being established without dealing with homophonic translations. However, Field considers that “it is hard to see what grounds there could be for a claim of difference of meaning in any such strong sense” [7]. In fact, he marks that, in the case of negation, translation between classical logic, intuitionist logic and a dialetheic logic is not transitive [8]. This would show, therefore, that there are not, at leat in the case of negation, a same intertheoretic meaning to be shared. Hence, even if that substantive form of radical pluralism is interesting, Field’s evaluation of it makes this pluralism false.
1.3. Beall and Restall’s pluralism
The way by which Beall and Restall develop their pluralism is not radical, since their strategy consists in restricting the meaning of validity, that every theory provides, in order to obtain a common meaning. All logics have to be compatible in the sense that they equally grasp this supposed intuitive notion of validity (formulated as a principle, (V)). These are not all-purpose logics and, for this reason, its rivalry is not the same that is stated considering radical pluralism; the rivalry between an intuitionist and a classical logician in respect of the notion of negation. In fact, Beall and Restall’s account reflects a restricted version of every logic, because each of them has to endorse the common notion of validity.
Field’s veredict of this pluralism is surprising: “I’m prepared to concede that Beall-Restall pluralism is correct. But while it is more exciting than the view that ‘implies’ can mean different things, I’m not sure that it’s exciting enough” [9]. Field considers that, accepting Beall and Restall’s movement and, given the restricted form that they offer of intuitionism and relevant logic, there is no difficulty accepting their pluralism. But he proposes, at the same time, that, since it is not substantive to consider that genuine implication has different meanings, it is not interesting to defend that implication has a single meaning, but in a restricted way. This answer seems to concede too much to a proposal, Beall and Restall’s, which is not satisfactory at all for Field. Moreover, Priest’s monist critique [10] to this pluralism accepts Beall and Restall’s notion of validity, so that their dispute contains a substantive argumentation. It seems difficult, so, to hold that Beall and Restall’s pluralism is not, despite correct, interesting. Notwithstanding, Field’s acceptance of this pluralism as a correct proposal can be understood as an acknowledgment of it, as it does not present with so severity the difficulty concerned with differences of meaning, as radical pluralism does.
1.4. The possibility of pluralism
Field’s exposition discards two possible ways of pluralism. The first of them arises when there are several logics that differ in their theorems; the goal for those logics is that all theorems can be proven as necessarily true. But there is a severe difficulty with this pluralism; if there are divergence between logics in terms of their theorems, their disagreement becomes genuine or, just, factual. In this case, one position has to be correct, and the rest have to be incorrect. So pluralism disappears, because its position is not sustainable in this context.
The second alternative to pluralism is localized when different logics share their theorems, but disagree in their valid inferences. At once, Field points out that “the interest of pluralism is vastly reduced when confined to such examples” [11]. In fact, the initial aim of pluralism has been confined to a very particular case; Field has reduced its scope to a non problematical and practically marginal issue. As the author states it, it may be not completely convincing. But this concern can be viewed in more detail.
Field argues that many times such a disagreement is a question of mere verbal disagreement. So to speak, it is a matter of choosing some principles as valid. Thus, in fact, two logics that differ only in their valid inferences can be no genuinely different. I seems that there is a previous decision that leads to have a particular logic with its valid inferences; the foundations of this decision are philosophical and, because of that, they are not strictly concerned with logical pluralism. That seems to be the reason by which Field says that this disagreement is actually mere verbal.
Formally considered, that difference between logics cannot be substantive. However, from a philosophical point of view, it can be interesting to focus the attention in the arguments that support the sides in a verbal disagreement. It is clear that this does not develop a pluralism, but there is no necessity to reject their interest. So, one can explore this way of pluralism, but it should be recognized that it is not a question of formal concern.
Even so, there is chance to pluralism: “the best case for significant pluralism in logic lies with examples that (a) don’t seem like verbal variants, and where in addition (b) the logics disagree only in inferences that have at least one premise, not in their theorems” [12]. In order to exemplify this account, it can be supposed that a solution to the liar paradox has been obtained. In that case, two logics can be very similar but inconsistent between them. This can be a case of pluralism. It is clear that these case, even if it is possible to localize them, are very rare.
2. Discussion
It has been seen that Beall and Restall’s pluralism is considered correct but uninteresting by Field. But more things can be said about that. There us a broad discussion in Field’s paper concerning the possibility of raising a pluralism such that Beall and Restall’s proposal, in the sense that its main aim cannot be sustained if the relation between and intuitive account of validity and the necessary truth preservation is observed.
2.1. First proposal of a definition of validity
Beall and Restall express their proposal of an intuitive notion of validity as (V):
“(V) A conclusion, A, follows from premisses,
, if and only if any case in which each premise in
Their account relates this principle with three tenets:
“(1) The pretheoric (or intuitive) notion of consequence is given by (V).
(2) A logic is given by a specification of the cases to appear in (V). Such a specification of cases can be seen as a way of spelling out truth conditions of the claim expressible in the language in question.
(3) There are at least two different specifications of cases which may appear in (V).” [14]
For present purposes the discussion will focus on (1), which is considered as self-explanatory by Beall and Restall; the discussion will show that this is far away to be uncontroversial. However, Priest agrees with it when he defines deductive validity as a concrete case of (V), with which Beall and Restall, in fact, agree, since they favour a model theoretic account of validity:
“
iff every model of
is a model of
” [15]
Actually, that was Field’s movement, because, at once, he points that (V) is not adequate to express necessary truth preservation, so in order to fill that aim it is better to give a model theoretic definition as Priest does.
Anyway, assuming that (V), or some variant, can grasp properly the necessary truth preservation, it is not clear how it expresses, as Beall and Restall pretend, the intuitive notion of validity. This relation is what will be discussed here, with the referent of Field’s argumentation. There are two levels of discussion that will be followed; the first, concerning the intuitive notion of validity, and the second, concerning its extensional scope.
2.2. The intuitive notion of validity
Kreisel’s argument [16] shows that the intuitive notion of validity coincides with the classical notion, so there is no disagreement on the question of which mathematical notion coincides with genuine validity. But, in fact, this argument only shows that
That is, the extensional identity between the intuitive notion of validity and the classical one. What this argyment does not show is the identity of meaning at the intuitive level, that is, that the classical notion of validity defines the intuitive notion. What Kreisel obtains is a mathematically useful explanation in terms of capturing the extensional identity, but he cannot say anymore.
In fact, Field asserts [17] that (V) is clearly insufficient to capture the intuitive meaning of implication. He fives two reasons for that. By the first, the difference between a particular notion of validity adapted to a definition of model and the genuine notion of validity is stated. The key is that while different ways of defining models in order to obtain different particular notions of validity (as intuitionist validity or classical one) can be accepted, that does not lead to accept that one of them, or, moreover, none of them grasp the truly (or genuine) notion of validity. Field opens the door to a pluralism over this genuine notion, which defend that there is no one single acceptable notion; the direct answer is that this pluralism is trivial, in the sense that accepting the variation in the definition of model in order to produce several notions of validity is neither interesting nor fruitful.
This is an interesting point, and shows clearly the distance between the mathematical character of theories obtained from logics and the genuine meaning of the notions that they, as formal constructions with a canonical application, pretend to grasp. It seems that there is, on the one hand, a neutral mathematical construction and, on the other, a philosophical position with a content that has the aim to capture notions which are formulated in that content. So to speak, every logic is concerned, in a way or another, with this philosophical content, but it has been noted that, in a particular application, as a theory, it only gives a mathematical definition, and not an intuitive one.
The second reason has formal character, and it is concerned with the scope of quantification of the model domain. The fact s that this domain is a set, and without a domain expressed in set theory, “truth-in-a-model wouldn’t be set theoretically definable, and so validity wouldn’t be set-theoretically definable, and that would undermine the model-theoretic point of the definition” [18]. But, being a set, that domain cannot be the whole universe, which is too much big to be a model domain. Thus, the notion of validity in a model cannot be the same that works in the factual or natural world.
2.3. Necessary truth preservation versus intuitiveness
Therefore, Field argues against the capability of a model theoretic notion of validity to grasp properly, as a matter of natural meaning, the intuitive notion of validity. But he defends more; (V) does not capture even extensionally the genuine notion of validity, unless a normative aspect, which is in that notion, is ignored.
That aspect is concerned with the epistemological account of implication; in Field’s words, “our views about implication constrain our views about how we ought to reason, or (perhaps better) about the proper interrelations among our beliefs” [19]. It is not clear which is exactly Field’s point, but it seems very reasonable that the notion of validity is contained in the particular structure that rationality imposes to all reasonings. If that is a question in which beliefs play a role is not so interesting than the fact that a particular set (and set is not taken as a formal notion here) of relations, given by pure reason, constrain the natural way of making implications and, therefore, constrain the genuine notion of validity.
Once this point has been stated, Field uses Curry’s paradox to hold his thesis. His main point is to show that the aim to derive that implication coincides with necessary truth preservation is a fallacy. That derivation will use inference principles such as elimination and introduction of truth predicate and introduction and elimination of the conditional. What Curry’s paradox shows is that those principles are mutually inconsistent. It does not matter which of them is rejected or accepted; the fact is that one cannot make a derivation using those four principles, and if the identity between the notion of implication and a necessary truth preservation principle as (V) is pretended to hold, the four rules have to be used.
This argument has not to be seen as in itself definitive. It is only an illustration, and the Curry’s paradox is only a resource to show how the thesis is articulated. The fact, even if one rejects this particular paradox, is that there is no way to give an account of truth predicate that, at the same time, preserves truth and allows to accept inference rules (that are intuitively valid) as modus ponens.
So, even if one does not feel satisfied with truth predicate account that is available, this does not open the door to give an alternative account that solves this difficulty. It is a more profound matter and it requires to consider the structure of logic and reasoning as itself. Not in vain, the issues that arise with this problem are such as the universality of the language or the self reference.
2.4. Concluding remarks
As a recapitulation, Field manifests his conclusion:
“Indeed, for nearly every ways of dealing with the truth-theoretic paradoxes, it is in consistent to hold that the logic one accepts actually preserves truth. By ‘the logic one accepts’ I mean the logic that one thinks should normatively govern one (e.g. govern one’s inferential practice, or the interrelations among one’s beliefs (…)). And I mean to include in this logic not just the logic of the usual sentential connectives and quantifiers, but also the logic of the truth predicate. At least in this broad sense of logic, virtually no way of dealing with the paradoxes can consistently take its own logic to be truth preserving.” [20]
Even if Field’s answer goes far beyond the rejection of the possibility of a pluralism such that Beall and Restall rise [21], it is very informative of the way that they hold the foundations of their account. To construct a philosophical thesis over a logical matter requires that this logical question would be really clear. But if it is not possible to identify whatever account of necessary truth preservation that is given fits with the genuine notion of validity, one cannot rise a pluralism based in a pretended unified notion of validity.
3. Appendix A: Kreisel’s argument
Kreisel’s argument shows the extensional identity between an intuitive notion of consequence (expressed by 
(*) Completeness theorem: 
(1) 
(2) 
Therefore,
(C)
Notes
[1] Field, H., Pluralism in Logic, in The Review of Symbolic Logic, 2, 2 (2009), p. 342.
[2] Ibid., pp. 342-343.
[3] Beall, J.C.; Restall, G., Logical Pluralism, in Australian Journal of Philosophy, 78, 4 (2000), p. 480.
[4] Field, H., Op. cit., p. 343.
[5] Ibid., p. 344.
[6] Idem.
[7] Field, H., Op. cit., p. 345.
[8] Ibid., pp. 346-347.
[9] Ibid., p. 346.
[10] Priest, G., Doubt truth to be a liar; Oxford, Oxford University Press, 2006, pp. 200-206.
[11] Field, H., Op. cit., p. 357.
[12] Ibid., p. 258.
[13] Beall, J.C.; Restall, G., Op. cit., p. 476.
[14] Ibid., pp. 476-477.
[15] Priest, G., Op. cit., p. 180.
[16] See appendix A, below.
[17] Field, H., Op. cit., p. 348.
[18] Idem.
[19] Ibid., p. 349.
[20] Ibid., p. 351.
[21] In fact, it serves as an argumentation even against Priest, as it has been seen.
